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SSC/Stability/BiPolytropes/HeadScratching

2021-08-19T23:28:43Z

174.64.8.247: /* Summary of (Inadequate) Detailed Analyses */

__FORCETOC__

=Broader Examination of Bipolytrope Stability=



{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |<font size="-1">[[H_BookTiledMenu#MoreStabilityAnalyses|<b>Our<br />Broader<br />Analysis</b>]]</font>
|}
==Overview==

<table border="0" cellpadding="8" align="right">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1:  Equilibrium Sequences<br />of Pressure-Truncated Polytropes
</th>
</tr>
<tr>
<td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
</td>
</tr>
</table>
We expect the content of this chapter — which examines the relative stability of bipolytropes — to parallel in many ways the content of an [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|accompanying chapter in which we have successfully analyzed the relative stability of pressure-truncated polytopes]]. Figure 1, shown here on the right, has been copied from [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|a closely related discussion]]. The curves show the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>. ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence.) On each sequence for which <math>~n \ge 3</math>, the green filled circle identifies the model with the largest mass. We have shown ''analytically'' that the oscillation frequency of the fundamental-mode of radial oscillation is precisely zero<sup>&dagger;</sup> for each one of these maximum-mass models. As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.

----

<sup>&dagger;</sup>In each case, the fundamental-mode oscillation frequency is precisely zero if, and only if, the adiabatic index governing expansions/contractions is related to the underlying ''structural'' polytropic index via the relation, <math>~\gamma_g = (n + 1)/n</math>, and if a constant surface-pressure boundary condition is imposed.

----

In another [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|accompanying chapter]], we have used purely analytic techniques to construct equilibrium sequences of spherically symmetric bipolytropes that have, <math>~(n_c,n_e) = (5,1)</math>. For a given choice of <math>~\mu_e/\mu_c</math> — the ratio of the mean-molecular weight of envelope material to the mean-molecular weight of material in the core — a physically relevant sequence of models can be constructed by steadily increasing the value of the dimensionless radius at the core/envelope interface, <math>~\xi_i</math>, from zero to infinity. Figure 2, whose content is essentially the same as [[SSC/Structure/BiPolytropes/Analytic51#Model_Sequences|Figure 1 of this separate chapter]], shows how the fractional core mass, <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, varies with the fractional core radius, <math>q \equiv r_\mathrm{core}/R</math>, along sequences having seven different values of <math>~\mu_e/\mu_c</math>, as labeled: 1 (black), ½ (dark blue), 0.345 (brown), ⅓ (dark green), 0.316943 (purple), 0.309 (orange), and ¼ (light blue).

When modeling bipolytropes, the default expectation is that an increase in <math>~\xi_i</math> along a given sequence will correspond to an increase in the relative size — both the radius and the mass — of the core. This expectation is realized along the Figure 2 sequences that have the largest mean-molecular weight ratios: <math>~\mu_e/\mu_c</math> = 1 and ½. But the behavior is different along the other five illustrated sequences. For sufficiently large <math>~\xi_i</math>, the relative radius of the core begins to decrease; along each sequence, a solid purple circular marker identifies the location of this ''turning point'' in radius. Furthermore, along sequences for which <math>~\mu_e/\mu_c < \tfrac{1}{3}</math>, eventually the fractional mass of the core reaches a maximum and, thereafter, decreases even as the value of <math>~\xi_i</math> continues to increase; a solid green circular marker identifies the location of this ''maximum mass turning point'' along each of these sequences. (Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence|yet another accompanying chapter]].)

<font color="red">'''The principal question is:'''</font> ''Along bipolytropic sequences, are maximum-mass models associated with the onset of dynamical instabilities?''

==Planned Approach==

<table border="0" cellpadding="5" width="30%" align="right">
<tr>
<th colspan="2" align="center">Figure 2: Equilibrium Sequences of Bipolytropes <br />with <math>~(n_c,n_e) = (5,1)</math> and Various <math>~\mu_e/\mu_c</math></th>
</tr>
<tr>
<td align="center" colspan="2" bgcolor="white">
[[Image:TurningPoints51Bipolytropes.png|300px|center]]
</td>
</tr>
</table>
Ideally we would like to answer the just-stated "principal question" using purely analytic techniques. But, to date, we have been unable to fully address the relevant issues analytically, even in what would be expected to be the simplest case:   [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|bipolytropic models that have <math>~(n_c,n_e) = (0, 0)</math>]]. Instead, we will streamline the investigation a bit and proceed — at least initially — using a blend of techniques. We will investigate the relative stability of bipolytropic models having <math>~(n_c,n_e) = (5,1) </math> whose ''equilibrium structures'' are completely defined analytically; then the eigenvectors describing radial modes of oscillation will be determined, one at a time, by solving the relevant LAWE(s) numerically. We are optimistic that this can be successfully accomplished because we have had experience numerically integrating the LAWE that governs the oscillation of:
* [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Isolated n = 3 polytropes]] — including a quantitative comparison against [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild's (1941)] published work;
* [[SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres|Pressure-truncated isothermal spheres]] — including a quantitative comparison against the published analysis of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)]; and
* [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Pressure-truncated n = 5 polytropes]].

A key reference throughout this investigation will be the paper by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)]. They studied ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models.'' Specifically, their underlying equilibrium models were bipolytropes that have <math>~(n_c,n_e) = (1, 5)</math>. In an [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|accompanying chapter]], we describe in detail how Murphy & Fiedler obtained these equilibrium bipolytropic structures and detail some of their equilibrium properties.

Here are the steps we initially plan to take:

* Governing LAWEs:
** Identify the relevant LAWEs that govern the behavior of radial oscillations in the <math>~n_c = 5</math> core and, separately, in the <math>~n_e = 1</math> envelope. Check these LAWE specifications against the published work of [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)].
** Determine the matching conditions that must be satisfied across the core/envelope interface. Be sure to take into account the critical interface ''jump'' conditions spelled out by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)], as we have already discussed in the context of an [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|analysis of radial oscillations in zero-zero bipolytropes]].
* Determine what surface boundary condition should be imposed on physically relevant LAWE solutions, i.e., on the physically relevant radial-oscillation eigenvectors.
* Initial Analysis:
** Choose a maximum-mass model along the bipolytropic sequence that has, for example, <math>~\mu_e/\mu_c = 1/4</math>. Hopefully, we will be able to identify precisely (analytically) where this maximum-mass model lies along the sequence. <font color="red">'''Yes!'''</font> Our [[SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass|earlier analysis]] does provide an analytic prescription of the model that sits at the maximum-mass location along the chosen sequence.
** Solve the relevant eigenvalue problem for this specific model, initially for <math>~(\gamma_c, \gamma_e) = (6/5, 2)</math> and initially for the fundamental mode of oscillation.

=Summary of (Inadequate) Detailed Analyses=
<table border="1" align="center" width="75%" cellpadding="8">
<tr>
<th align="center" colspan="7">Key Models Along the <math>\mu_e/\mu_c = 1/4</math> Equilibrium Sequence</th>
</tr>
<tr>
<td align="center"><math>\xi_i</math></td>
<td align="center"><math>q</math></td>
<td align="center"><math>\nu</math></td>
<td align="center"><math>R^*</math></td>
<td align="center"><math>M^*_\mathrm{tot}</math></td>
<td align="center">Significance</td>
<td align="center">Figure Marker</td>
</tr>
<tr>
<td align="center">4.9379</td>
<td align="center">0.0482</td>
<td align="center">0.1394</td>
<td align="center">40.945</td>
<td align="center">43.437</td>
<td align="center"><math>\nu_\mathrm{max}</math></td>
<td align="left"><font size="-1">Dark-green circular marker in Fig. 2</font></td>
</tr>
<tr>
<td align="center">2.2805</td>
<td align="center">0.1246</td>
<td align="center">0.0931</td>
<td align="center">12.648</td>
<td align="center">38.970</td>
<td align="center"><math>M^*_\mathrm{tot}\biggr|_\mathrm{min}</math></td>
<td align="left" cellpadding="5"><font size="-1">Light-blue diamond marker in Fig. 4</font></td>
</tr>
</table>

==Attempts to Identify Marginally Unstable (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Bipolytropes==

<table border="0" cellpadding="5" align="right">
<tr>
<th align="center">Figure 3:  Conflicting Instability Regions</th>
</tr>
<tr>
<td align="center" colspan="10">[[File:EigenvectorStability.png|300px|Conflicting Instability Regions]]</td>
</tr>
</table>
<ul>
<li>[[SSC/Stability/BiPolytropes#Virial_Analysis|Virial Analysis]]</li>
<li>Solving the Relevant LAWE:
<ul>
<li>[[SSC/Stability/BiPolytropes#Review_of_the_Analysis_by_Murphy_.26_Fiedler_.281985b.29|Review of the Analysis by Murphy & Fiedler (1985b)]]</li>
<li>[[SSC/Stability/BiPolytropes#Radial_Oscillations_of_.28nc.2C_ne.29_.3D_.285.2C_1.29_Models|Radial Oscillations of 51 Models]]   <math>\Leftarrow</math>   [[SSC/Stability/BiPolytropes#Interface|Interface]] probably not handled correctly here.</li>
<li>[[SSC/Stability/BiPolytropes#Variational_Principle|Variational Principle]]</li>
</ul>
</li>
<li>[[SSC/StabilityConjecture/Bipolytrope51|K-BK74 Conjecture]]<br />
Through this analysis, we were quite successful at generating a reasonably shaped radial-oscillation eigenfunction for the equilibrium model that lies along the sequence at the maximum mass fraction, <math>\nu_\mathrm{max}</math>. We obtained the eigenfunction by subtracting the structural profile of one model (immediately to the left of this maximum) from the structural profile of a separate model (immediately to the right of this maximum). But, after doing so, we realized that the result could not be physically justified. Although we had been careful to ensure that the core mass fraction <math>(\nu)</math> was identical in the two separate models, we had not paid attention to the ''total'' mass; it had not been held fixed. Hence, we cannot claim to have performed a valid dynamical perturbation of models in the vicinity of <math>\nu_\mathrm{max}</math>.
<br /> <br />

The eigenfunction that was constructed in this manner was nevertheless eye-opening! It appears to contain a sizable step function at the interface; that is to say, it appears as though the radial-displacement function at the surface of the core is offset (discontinuously) from the radial-displacement function at the base of the envelope. We had not allowed this to happen in our separate investigation of [[SSC/Stability/BiPolytropes#Radial_Oscillations_of_.28nc.2C_ne.29_.3D_.285.2C_1.29_Models|Radial Oscillations of 51 Models]].
</li>
</ul>

We will attempt to incorporate these new insights into our analyses that follow.
 <br />

=Rethink Evolution and Stability=
<table border="0" align="left" cellpadding="3"><tr>
<th align="center">Figure 4:   Mass vs. Radius</th></tr>
<tr><td align="center">
[[File:MassVradius0.25.png|center|320px|Mass versus Radius m_e = 0.93]]
</td></tr>
</table>
Consider a system that has <math>m_3 = (3\mu_e/\mu_c) = 0.930</math> and that slowly evolves along the appropriate (orange), <math>m_3 =</math> constant equilibrium sequence. It begins its evolution with a very small core — that is, with <math>\xi_i</math> nearly zero, and with <math>q</math> and <math>\nu</math> both very small. The location of the core-envelope interface, <math>\xi_i</math>, moves slowly outward (in Lagrangian mass space) as the ashes left from hydrogen burning build up the mass of the core at the expense of the envelope. This means that (slow) evolution proceeds along the <math>q - \nu</math> equilibrium sequence in a counter-clockwise direction.

[[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Schwarzschild and his collaborators]] noticed that, as an evolution proceeds along an equilibrium sequence for which (in our example case) <math>m_3 \le 1</math>, the fractional core mass increases only up to a limiting value, <math>\nu_\mathrm{max}</math>; in our case, <math>\nu_\mathrm{max} \approx 0.33</math>. As the core-envelope interface location, <math>\xi_i</math>, attempts to increase to a value larger than the value associated with the model at <math>\nu_\mathrm{max}</math>, something rather drastic must happen — at least on a secular time scale associated with nuclear burning. We have wondered whether a ''dynamical'' instability is also encountered at this "turning point" along the equilibrium sequence. Up to now, all of our (inadequate) detailed analyses have been focused on securing an answer to this question.

As the figure on the left illustrates, along this same sequence, the normalized radius <math>(R^*)</math> starts off small and it steadily grows as the evolution proceeds, but the normalized total mass <math>(M^*_\mathrm{tot})</math> starts off large and initially decreases. In reality, we expect the system to conserve its total mass throughout the evolution. Given that the mass has been normalized via the expression,
<div align="center">
<math>M^*_\mathrm{tot} = M_\mathrm{tot} \biggl[ \frac{G^{3/2} \rho_c^{1 / 5}}{K_c^{3/2}} \biggr] \, ,</math>
</div>
we appreciate that a decrease in the dimensionless mass (as depicted in the figure) can quite naturally be attributed to a steady increase in the specific entropy of the core material, <math>K_c</math>. This evolution along the equilibrium sequence will happen on a secular, rather than dynamical, time scale that is set by the rate at which <math>K_c</math> increases — that is, at a rate set by nuclear burning. But at each point along the sequence, we can check to see whether the equilibrium configuration is dynamically stable. We expect that the turning point along the <math>M^*(R^*)</math> sequence is an indication of transition from a (dynamically) stable to (dynamically) unstable state. We should be able to apply the B-KB74 conjecture to get a good idea of what the unstable eigenfunction looks like at this turning point.

==Differentiate M<sup>*</sup> With Respect to &ell;<sub>i</sub>==

In an accompanying discussion titled, [[SSC/StabilityConjecture/Bipolytrope51#New_Derivation|''New Derivation'']], we examined how the core mass-fraction <math>(\nu)</math> varies with <math>\ell_i \equiv \xi_i/\sqrt{3}</math>. Here, we want to examine how the total mass <math>(M^*_\mathrm{tot})</math> varies with <math>\ell_i</math>. In what follows, we borrow heavily from various analytic expressions that have been obtained via this separate ''New Derivation''; and, as in this earlier analysis, numerical evaluations (in parentheses) come from '''Example #1''' for which, <math>\mu_e/\mu_c = 0.25</math> and <math>\xi_i = 0.5</math>, which implies that <math>m_3 = 0.75</math> and <math>\ell_i = (12)^{-1 / 2}</math>.

<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>M^*_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl[- \frac{\eta_s^2}{\theta_i} \cdot \biggl(\frac{d\phi}{d\eta}\biggr)_s \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl[\frac{A\eta_s}{\theta_i} \biggr]
= 40.0934</math>,
</td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\theta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>(1 + \ell_i^2)^{-1 / 2} = 0.960768923</math>,
</td>
</tr>

<tr>
<td align="right">
<math>\eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{m_3 \ell_i}{(1+\ell_i^2)} = 0.199852016</math>,
</td>
</tr>

<tr>
<td align="right">
<math>\Lambda_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{m_3\ell_i}\biggl[1 + (1 - m_3)\ell_i^2 \biggr] = 4.715027199\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\eta_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr) + \frac{m_3\ell_i }{(1 + \ell_i^2)}
= 3.132453649\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\eta_i(1+\Lambda_i^2)^{1 / 2}
=
\frac{m_3\ell_i}{(1+\ell_i^2)}\biggl\{
1 + \frac{1}{m_3^2 \ell_i^2}\biggl[1 + (1 - m_3)\ell_i^2 \biggr]^2
\biggr\}^{1 / 2}
</math>
</td>
</tr>
</table>
Now, the differentiation:
<table border="0" align="center" cellpadding="5">

<tr>
<td align="right">
<math>\frac{d\theta_i}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-\ell_i (1 + \ell_i^2)^{-3 / 2} \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{d\Lambda_i}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{2(1-m_3)\ell_i}{m_3\ell_i} - \frac{1}{m_3\ell_i^2}\biggl[1 + (1 - m_3)\ell_i^2 \biggr]
=
\frac{1}{m_3\ell_i^2}\biggl\{2(1-m_3)\ell_i^2 - [1 + (1 - m_3)\ell_i^2 ] \biggr\} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{d\eta_s}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{d}{d\ell_i} \biggl[
\frac{m_3\ell_i }{(1 + \ell_i^2)}
\biggr]
+
\frac{d}{d\ell_i}
\biggl(\tan^{-1}\Lambda_i\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
\frac{m_3}{(1 + \ell_i^2)}
\biggr]
-
\biggl[
\frac{2m_3\ell_i^2 }{(1 + \ell_i^2)^2}
\biggr]
+
\biggl\{ [1 + (1 - m_3)\ell_i^2 ]^2 + m_3^2 \ell_i^2 \biggr\}^{-1} \biggl\{
2m_3(1-m_3)\ell_i^2
- m_3 [1 + (1 - m_3)\ell_i^2 ]
\biggr\} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{dA}{d\ell_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1+\Lambda_i^2)^{1 / 2} \frac{d}{d\ell_i}\biggl[ \frac{m_3 \ell_i}{(1+\ell_i^2)} \biggr]
+
\biggl[ \frac{m_3 \ell_i}{(1+\ell_i^2)} \biggr] \frac{d}{d\ell_i}(1+\Lambda_i^2)^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1+\Lambda_i^2)^{1 / 2} \biggl[ \frac{m_3 }{(1+\ell_i^2)} - \frac{2m_3 \ell_i^2}{(1+\ell_i^2)^2} \biggr]
+
\biggl[ \frac{m_3 \ell_i}{(1+\ell_i^2)} \biggr]\frac{\Lambda_i }{(1+\Lambda_i^2)^{1 / 2}} \frac{d\Lambda_i}{d\ell_i}
</math>
</td>
</tr>
</table>

=See Also=

* [[SSC/Structure/BiPolytropes/Analytic51|Equilibrium Structure of <math>(n_c, n_e) = (5, 1)</math> Bipolytropes]]
* [[SSC/Stability/BiPolytropes|SSC/Stability/BiPolytropes]]   <math>\Leftarrow</math>  "Our Broader Analysis" tile used to point to this chapter.
* [[SSC/StabilityConjecture/Bipolytrope51|B-KB74 Conjecture RE: Bipolytrope <math>(n_c, n_e) = (5, 1)</math>]]

{{ SGFfooter }}

SSC/Stability/BiPolytropes/HeadScratching

2021-08-16T22:22:33Z

174.64.8.247: /* Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes */

SSC/Stability/BiPolytropes/HeadScratching

2021-08-16T22:19:09Z

174.64.8.247: /* Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes */

SSC/Stability/BiPolytropes/HeadScratching

2021-08-16T22:05:48Z

174.64.8.247: /* Summary of (Inadequate) Detailed Analyses */

SSC/Stability/n5PolytropeLAWE

2021-08-01T21:03:09Z

174.64.8.247: Created page with "__FORCETOC__  =Radial Oscillations of n = 5 Polytropic Spheres= ==Background== ===General Form of the LAWE for Spherical Polytropes===..."

__FORCETOC__

=Radial Oscillations of n = 5 Polytropic Spheres=

==Background==

===General Form of the LAWE for Spherical Polytropes===
In an [[User:Tohline/SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{User:Tohline/Math/EQ_RadialPulsation01}}
</div>

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Because this widely used form of the radial pulsation equation is not dimensionless but, rather, has units of inverse length-squared, we have found it useful to also recast it in the following [[User:Tohline/SSC/Perturbations#Dimensionless_Expression|dimensionless form]]:
<div align="center">
<math>
\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 ,
</math><br />
</div>
where,
<div align="center">
<math>~g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c} \, ,</math>       and       <math>~\tau_\mathrm{SSC} \equiv \biggl[\frac{R^2 \rho_c}{P_c}\biggr]^{1/2} \, .</math>
</div>

In a [[User:Tohline/SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|separate discussion]], we showed that specifically for isolated, ''polytropic'' configurations, this linear adiabatic wave equation (LAWE) can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} +
\biggl[\frac{\omega^2}{\gamma_g \theta} \biggl(\frac{n+1 }{4\pi G \rho_c} \biggr) -
\biggl(3-\frac{4}{\gamma_g}\biggr) \cdot \frac{(n+1)V(x)}{\xi^2} \biggr] x </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{(n+1)}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} +
\frac{(n+1)}{\theta}\biggl[\frac{\sigma_c^2}{6\gamma_g } -
\frac{\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] x \, ,</math>
</td>
</tr>
</table>
</div>
where we have adopted the dimensionless frequency notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sigma_c^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{3\omega^2}{2\pi G \rho_c} \, .</math>
</td>
</tr>
</table>
</div>

===Specifically for n=5 Configurations===

Here we focus on an analysis of the ''specific'' case of isolated, <math>~n=5</math> polytropic configurations, whose unperturbed equilibrium structure can be prescribed in terms of analytic functions. Our hope — as yet unfulfilled — is that we can discover an analytically prescribed eigenvector solution to the governing LAWE.

From our discussion of the [[User:Tohline/SSC/Structure/Polytropes#Primary_E-Type_Solution_2|equilibrium structure of isolated, <math>~n=5</math> polytropes]], we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( 1 + \frac{\xi^2}{3} \biggr)^{-1/2} =3^{1/2} ( 3 + \xi^2 )^{-1/2}\, .</math>
</td>
</tr>
</table>
</div>
Hence, we know as well that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\theta}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{\xi}{3}\biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3/2} = - 3^{1/2}\xi ( 3 + \xi^2 )^{-3/2} \, .</math>
</td>
</tr>
</table>
</div>

The LAWE therefore becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{(n+1)}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} +
\frac{(n+1)}{\theta}\biggl[\frac{\sigma_c^2}{6\gamma_g } -
\frac{\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] x </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{6}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} +
\biggl[\frac{\sigma_c^2}{\gamma_g } \cdot \frac{1}{\theta} -
\frac{6\alpha}{\xi} \cdot \frac{1}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] x</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[4 - \frac{6\xi^2}{(3+\xi^2)} \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} +
\biggl[\frac{\sigma_c^2}{3^{1/2} \gamma_g } \cdot (3+\xi^2)^{1/2} -
\frac{6\alpha}{(3+\xi^2)}\biggr)\biggr] x \, .</math>
</td>
</tr>
</table>
</div>

Or,
<div align="center" id="n5LAWE">
<table border="1" cellpadding="8" align="center">
<tr><th align="center">LAWE for <math>~n=5</math> Polytropes</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3+\xi^2) \frac{d^2x}{d\xi^2} + \biggl[12 - 2\xi^2 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} +
\biggl[\frac{\sigma_c^2}{3^{1/2} \gamma_g } \cdot (3+\xi^2)^{3/2} -
6\alpha \biggr] x </math>
</td>
</tr>
</table>
</td></tr></table>
</div>

==Numerical Integration of LAWE==

By numerically integrating the above LAWE using the algorithm outlined in a [[User:Tohline/SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|separate chapter]], we have examined the properties of the displacement function that describes radial modes of oscillation in pressure-truncated, n = 5, polytropic configurations. Our brief description, here, of these modes parallels the more detailed description of radial oscillation modes in truncated isothermal spheres that has been [[User:Tohline/SSC/Stability/Isothermal#Previously_Published_Eigenvalues_and_Eigenfunctions|presented in a separate chapter]].

The animation sequence that appears in the right panel of Composite Display 1 shows how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the n=5 polytropic sphere, out to <math>~\xi = 10</math> — for sixteen different values of the square of the eigenfrequency, <math>~\sigma_c^2</math>, as denoted at the top of each animation frame. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies: (a) the radius at which the logarithmic derivative of the displacement function, <math>~d\ln x/d\ln\xi </math>, is negative three; and (b) the radius, <math>~\tilde\xi</math>, at which the n = 5 polytropic configuration has been truncated. As displayed here, in every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.

The left panel of Composite Display 1 is also animated and has been provided in support of the animation on the right. Specifically, the number written at the top of each left-panel frame quantitatively identifies the radial location, <math>~\tilde\xi</math>, of the surface of the relevant truncated polytropic configuration; and, on each frame, "×" marks the location of that truncated configuration on the mass-radius equilibrium sequence.

<div align="center" id="n5TruncatedMovie">
<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="2">Composite Display 1:   Numerically Generated Fundamental-Mode Eigenvectors</th>
</tr>
<tr>
<td align="center">
[[File:N5Truncated2.gif|600px|n5 Truncated movie]]
</td>
<td align="center">
Excel File:<br />
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n5Eigenvectors/n5TruncatedSphere.xlsx --- worksheet = OursPt1]]
Movie File:<br />
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/n5movie/ --- worksheet = n5Truncated2.gif]]
</td>
</tr>
</table>
</div>

Each full loop through the left-panel animation sequence can be viewed as evolution along the equilibrium sequence from <math>~\tilde\xi = 0.75</math> to <math>~\tilde\xi = 5</math>, then back again. During this evolution, the "×" marker moves through both turning points along the sequence: the maximum radius configuration — at <math>~\tilde\xi= \sqrt{3}</math> — and the maximum mass configuration — at <math>~\tilde\xi= 3</math>. Notice that <math>~\sigma_c^2</math> is positive for all models having <math>~\tilde\xi < 3</math> while it is negative for all models having <math>~\tilde\xi > 3</math>. Hence, models having <math>~\tilde\xi > 3</math> are dynamically unstable and, as best we have been able to determine via these numerical integrations, the transition from stable to unstable models — that is, the ''marginally'' unstable model — occurs at <math>~\tilde\xi = 3</math>. (Via an ''analytic'' analysis, we prove, below, that this association is precise.) For emphasis, the "×" marker (left panel) and the numerical value recorded for <math>~\sigma_c^2</math> (right panel) have been colored red for models that are not stable.

=Search for Analytic Solutions to the LAWE=

==Eureka Moment==

<font color="red">Note from J. E. Tohline on 3/6/2017:</font>  Yesterday evening, after I finished putting together the [[#n5TruncatedMovie|above animation sequence]] using an Excel workbook, I noticed that the eigenfunction of the fundamental mode for the marginally unstable model <math>~(\sigma_c^2 = 0)</math> resembles a parabola. In an effort to see how well a parabola fits at least the central portion of this eigenfunction, I returned to my Excel spreedsheet and, in a brute-force manner, began to search for the pair of coefficients that would provide a best fit. What I discovered was that a parabola with the following formula fits perfectly!
<div align="center" i="AnalyticSoln">
<table border="1" cellpadding="10" align="center">
<tr>
<th align="center" colspan="1">Fundamental Mode Eigenfunction <br />
when <math>~\sigma_c^2 = 0</math> and <math>~\gamma = 6/5 ~\Rightarrow~\alpha=- 1/3</math></th>
</tr>

<tr>
<td align="center">
<math>~x = x_0 \biggl[ 1 - \frac{\xi^2}{15} \biggr]</math>
</td>
</tr>
</table>
</div>
For the ''specific'' normalization used in the above animation sequence, <math>~x_0 = \tfrac{5}{2}</math>. Let's demonstrate that this eigenvector provides a solution to the LAWE for <math>~n=5</math> polytropes; for simplicity, we will set <math>~x_0 = 1</math>:
<div align="center" id="Proof">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dx}{d\xi} = -\frac{2\xi}{15} \, ;</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~\frac{d^2x}{d\xi^2} = -\frac{2}{15} \, .</math>
</td>
</tr>
</table>
</div>

<div align="center" id="Proof">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Rightarrow ~~~
(3+\xi^2) \frac{d^2x}{d\xi^2} + \biggl[12 - 2\xi^2 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} +
\biggl[ \cancelto{0}{\frac{\sigma_c^2}{3^{1/2} \gamma_g }} \cdot (3+\xi^2)^{3/2} + 2\biggr] x
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2}{15}(3+\xi^2) -\frac{2}{15} \biggl[12 - 2\xi^2 \biggr] + 2\biggl[1 - \frac{\xi^2}{15}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( - \frac{6}{15} - \frac{24}{15} + 2\biggr)
+\xi^2 \biggl( -\frac{2}{15} + \frac{4}{15} - \frac{2}{15} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
0 \, .
</math>
</td>
</tr>
</table>
</div>

Q. E. D.  I don't think that anyone has previously appreciated that the LAWE in this case admits to an analytic eigenvector solution.

Now, let's see how the boundary condition comes into play. We see that the logarithmic derivative of the parabolic eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\ln x}{d\ln \xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2\xi^2}{15} \biggl[ 1 - \frac{\xi^2}{15}\biggr]^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2\xi^2}{(15-\xi^2)} \, .</math>
</td>
</tr>
</table>
</div>
We desire a surface boundary condition that gives, <math>~d\ln x/d\ln\xi = -3</math>. This will only happen when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- \frac{2\xi^2}{(15-\xi^2)}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-3</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~2\xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3(15 - \xi^2)</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3 \, .</math>
</td>
</tr>
</table>
</div>
Hence, although the parabolic eigenfunction provides an accurate solution to the <math>~n=5</math> LAWE throughout the entire configuration — that is, for all <math>~\xi</math> — the desired surface boundary condition will only be satisfied if the polytrope is truncated at <math>~\xi_\mathrm{surf} = 3</math>. The parabolic eigenfunction is therefore only physically relevant to the model that sits at the point along the equilibrium sequence that is associated with the <math>~P_\mathrm{max}</math> turning point.



Let's express the parabolic displacement function, <math>~x</math>, as a function of the Lagrangian mass coordinate, instead of as a function of <math>~\xi</math>. Drawing upon our [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation#Exploration|accompanying discussion]] where we have used <math>~\tilde\xi</math> to denote the truncation edge, we know that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_\xi(\xi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi \biggl\{
\biggl[ \frac{4\pi}{2^5\cdot 3}\biggr]^{1/2} \tilde\xi^{-6}
\biggl( 1+\frac{\tilde\xi^2}{3} \biggr)^{3}\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
and that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
r_\xi (m_\xi)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tilde{r}_\mathrm{edge}
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2} \, ,
</math>
</td>
</tr>
</table>
</div>
<span id="DefineTildeC">where,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\tilde{C}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{3^2}{\tilde\xi^2}\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr) \, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\tilde{r}_\mathrm{edge}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} {\tilde\xi}^{-6} \biggl(1+\frac{\tilde\xi^2}{3}\biggr)^3
\, .
</math>
</td>
</tr>
</table>
</div>

By equating <math>~r_\xi(\xi)</math> with <math>~r_\xi(m_\xi)</math>, we find,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi \biggl\{
\biggl[ \frac{4\pi}{2^5\cdot 3}\biggr]^{1/2} \tilde\xi^{-6}
\biggl( 1+\frac{\tilde\xi^2}{3} \biggr)^{3}\biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} {\tilde\xi}^{-6} \biggl(1+\frac{\tilde\xi^2}{3}\biggr)^3
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2} \, .
</math>
</td>
</tr>
</table>
</div>
This means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x_0 \biggl\{1 -
\frac{1}{15} \biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]
\biggr\} \, ;
</math>
</td>
</tr>
</table>
</div>
and, specifically for the critical case of <math>~\tilde\xi = 3</math>, in which case, <math>~\tilde{C} = 4</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x_0 \biggl\{1 -
\frac{1}{15} \biggl[\frac{3^2m_\xi^{2/3}}{4 - 3 m_\xi^{2/3}}\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

{{ SGFworkInProgress }}

==Setup Using Lagrangian Radial Coordinate==

===Individual Terms===
From our [[User:Tohline/SSC/FreeEnergy/PowerPoint#Case_M_Equilibrium_Conditions|accompanying discussion]], we have, for pressure-truncated, <math>~n=5</math> polytropic spheres

<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n}\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4\pi}{2^5\cdot 3^5}\biggr]^{1/2}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-2} \, ,
</math>
</td>
</tr>
</table>
</div>
which matches the expression derived in an [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_Equation|ASIDE box found with our introduction of the Lane-Emden equation]], and
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{P_\mathrm{e}}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{2^3\cdot 3^3}{4\pi}\biggr]^{3}
\tilde\theta^{6}( -\tilde\xi^2 \tilde\theta' )^{6} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>~R_\mathrm{norm}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} = \biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P_\mathrm{norm}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} = \frac{K^{10}}{G^{9} M_\mathrm{tot}^{6} } \, ,</math>
</td>
</tr>
</table>
</div>

and, from [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|our more detailed analysis]],

<table border="0" cellpadding="3" align="center">
<tr>
<td align="right">
<math>
~{\tilde\theta}_5 = 3^{1 / 2} \biggl( 3 + {\tilde\xi}^2\biggr)^{-1/2}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
~\biggl(- {\tilde\xi}^2 {\tilde\theta}^'_5\biggr) = 3^{1 / 2} {\tilde\xi}^3 \biggl( 3 + {\tilde\xi}^2\biggr)^{-3/2} \, .
</math>
</td>
</tr>
</table>

Hence,

<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4\pi}{2^5\cdot 3^5}\biggr]^{1/2}
\tilde\xi \biggl[ 3^{1 / 2} {\tilde\xi}^3 \biggl( 3 + {\tilde\xi}^2\biggr)^{-3/2} \biggr]^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4\pi}{2^5\cdot 3^5}\biggr]^{1/2}
\tilde\xi \biggl[ 3^{-1} {\tilde\xi}^{-6} \biggl( 3 + {\tilde\xi}^2\biggr)^{3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{4\pi}{2^5\cdot 3^7}\biggr]^{1/2}
{\tilde\xi}^{-5} \biggl( 3 + {\tilde\xi}^2\biggr)^{3} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_\mathrm{e}}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{2^3\cdot 3^3}{4\pi}\biggr]^{3}
\biggl[ 3^{1 / 2} \biggl( 3 + {\tilde\xi}^2\biggr)^{-1/2} \biggr]^{6} \biggl[ 3^{1 / 2} {\tilde\xi}^3 \biggl( 3 + {\tilde\xi}^2\biggr)^{-3/2} \biggr]^{6}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{2^3\cdot 3^3}{4\pi}\biggr]^{3}
\biggl[ 3^{3} \biggl( 3 + {\tilde\xi}^2\biggr)^{-3} \biggr] \biggl[ 3^{3} {\tilde\xi}^{18} \biggl( 3 + {\tilde\xi}^2\biggr)^{-9} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~\biggl[ \frac{2^3\cdot 3^5}{4\pi}\biggr]^{3}
{\tilde\xi}^{18} \biggl( 3 + {\tilde\xi}^2\biggr)^{-12} \, .
</math>
</td>
</tr>
</table>
</div>

Now, given that the [[User:Tohline/SSC/Virial/FormFactors#Summary_.28n.3D5.29|structural form-factors for <math>~n=5</math> configurations]] are,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
( 1 + \ell^2 )^{-3/2} = 3^{3 / 2} (3 + {\tilde\xi}^2)^{-3 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, ,
</math>
</td>
</tr>
</table>
we understand that the central density is,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho_c = \frac{\bar\rho}{ {\tilde\mathfrak{f}}_M }</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[3^{3 / 2} (3 + {\tilde\xi}^2)^{-3 / 2} \biggr]^{-1} \biggl[ \frac{3 M_\mathrm{tot}}{4 \pi R_\mathrm{eq}^3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( \frac{3}{4\pi}\biggr)
\biggl[ \frac{2^5\cdot 3^6}{4\pi}\biggr]^{ 3 / 2} (3 + {\tilde\xi}^2)^{3 / 2} M_\mathrm{tot} \biggl[ R_\mathrm{norm}
{\tilde\xi}^{-5} \biggl( 3 + {\tilde\xi}^2\biggr)^{3} \biggr]^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2^{5}\cdot 3^{20}}{\pi^5}\biggr]^{ 1 / 2} {\tilde\xi}^{15} (3 + {\tilde\xi}^2)^{-15 / 2} M_\mathrm{tot} R^{-3}_\mathrm{norm}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2^{5}\cdot 3^{20}}{\pi^5}\biggr]^{ 1 / 2} \biggl[{\tilde\xi} (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{15} M_\mathrm{tot}^{-5} \biggl( \frac{G}{K} \biggr)^{-15/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2\cdot 3^{4}}{\pi}\biggr]^{ 5 / 2} \biggl[{\tilde\xi} (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{15} \biggl( \frac{K^3}{G^3M_\mathrm{tot}^2} \biggr)^{5/2} \, .
</math>
</td>
</tr>
</table>

<span id="r0">Now let's derive the prescription for the Lagrangian radial coordinate in the context of pressure-truncated,</span> <math>~n=5</math> polytropes.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_0 \equiv a_5 \xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\frac{3K}{2\pi G} \biggr]^{1 / 2} \rho_c^{-2/5} \xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\frac{3K}{2\pi G} \biggr]^{1 / 2} \xi \biggl\{
\biggl[ \frac{2\cdot 3^{4}}{\pi}\biggr]^{ 5 / 2} \biggl[{\tilde\xi} (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{15} \biggl( \frac{K^3}{G^3M_\mathrm{tot}^2} \biggr)^{5/2}
\biggr\}^{-2/5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\frac{3K}{2\pi G} \biggr]^{1 / 2} \biggl[ \frac{\pi}{2\cdot 3^{4}}\biggr] \biggl( \frac{G^3M_\mathrm{tot}^2}{K^3} \biggr)
\biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3} \xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
R_\mathrm{norm} \biggl[ \frac{\pi}{2^3\cdot 3^{7}}\biggr]^{1 / 2} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3} \xi
</math>
</td>
</tr>
</table>
</div>

<span id="m0">Also,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~m_0 \equiv M(r_0)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ 4\pi a_n^3 \rho_c \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2^2\pi \biggl\{ R_\mathrm{norm} \biggl[ \frac{\pi}{2^3\cdot 3^{7}}\biggr]^{1 / 2} \tilde\xi^{-6} (3 + {\tilde\xi}^2)^{3} \biggr\}^3
\biggl\{ \biggl[ \frac{2\cdot 3^{4}}{\pi}\biggr]^{ 5 / 2} \biggl[{\tilde\xi} (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{15} \biggl( \frac{K^3}{G^3M_\mathrm{tot}^2} \biggr)^{5/2} \biggr\}
\biggl\{ 3^{1 / 2} \xi^3 \biggl( 3 + \xi^2\biggr)^{-3/2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 3^{1 / 2} \biggl[ 2^4 \pi^2\biggr]^{1 / 2} \biggl[ \frac{\pi^3}{2^9\cdot 3^{21}}\biggr]^{1 / 2} \biggl[ \frac{2^5\cdot 3^{20}}{\pi^5}\biggr]^{ 1 / 2}
\biggl\{ \tilde\xi^{-6} (3 + {\tilde\xi}^2)^{3} \biggr\}^3
\biggl[{\tilde\xi} (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{15} \biggl( \frac{K^3}{G^3M_\mathrm{tot}^2} \biggr)^{5/2} R_\mathrm{norm}^3
\biggl\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{ \tilde\xi^{-3} (3 + {\tilde\xi}^2)^{3 / 2} \biggr\} M_\mathrm{tot}
\biggl\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~g_0 = \frac{Gm_0}{r_0^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{GM_\mathrm{tot}}{R_\mathrm{norm}^2}
\biggl\{ \tilde\xi^{-3} (3 + {\tilde\xi}^2)^{3 / 2} \biggr\}
\biggl\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \biggr\}
\biggl\{ \biggl[ \frac{\pi}{2^3\cdot 3^{7}}\biggr]^{1 / 2} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3} \xi \biggr\}^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{GM_\mathrm{tot}}{R_\mathrm{norm}^2}\biggl[ \frac{2^3\cdot 3^{7}}{\pi}\biggr]
\biggl[ \tilde\xi (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{9}
\xi ( 3 + \xi^2 )^{-3/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{g_0 }{r_0} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{GM_\mathrm{tot}}{R_\mathrm{norm}^3}\biggl[ \frac{2^3\cdot 3^{7}}{\pi}\biggr]
\biggl\{ \tilde\xi^{9} (3 + {\tilde\xi}^2)^{-9 / 2} \biggr\} \biggl\{ \biggl[ \frac{\pi}{2^3\cdot 3^{7}}\biggr]^{1 / 2} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3} \xi \biggr\}^{-1}
\xi ( 3 + \xi^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{GM_\mathrm{tot}}{R_\mathrm{norm}^3}\biggl[ \frac{2^3\cdot 3^{7}}{\pi}\biggr]^{3/2}
\biggl[ \tilde\xi (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{15}
( 3 + \xi^2 )^{-3/2}
\, ;
</math>
</td>
</tr>
</table>
</div>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\rho_0}{P_0} = \frac{\rho_0}{K\rho_0^{1+1/n}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[K^5 \rho_c \theta^5 \biggr]^{-1/5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \theta^{-1}
\biggl\{ K^5 \biggl[ \frac{2\cdot 3^{4}}{\pi}\biggr]^{ 5 / 2} \biggl[{\tilde\xi} (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{15} \biggl( \frac{K^3}{G^3M_\mathrm{tot}^2} \biggr)^{5/2}\biggr\}^{-1/5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ 3^{-1} ( 3 + \xi^2 ) \biggr]^{1/2}
\biggl\{ \biggl[ \frac{\pi}{2\cdot 3^{4}}\biggr]^{1 / 2} \cancelto{\mathrm{mistake}}{\biggl[{\tilde\xi}^{-3} (3 + {\tilde\xi}^2)^{3 / 2} \biggr]^{-3} } \biggl( \frac{G^3M_\mathrm{tot}^2}{K^5} \biggr)^{1/2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{G^3M_\mathrm{tot}^2}{K^5} \biggr)^{1 / 2}
\biggl[ \frac{\pi}{2\cdot 3^{5}}\biggr]^{1 / 2} \cancelto{\mathrm{mistake}}{\biggl[ {\tilde\xi} (3 + {\tilde\xi}^2)^{-1 / 2} \biggr]^{9} }
( 3 + \xi^2 )^{1 / 2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ 3^{-1} ( 3 + \xi^2 ) \biggr]^{1/2}
\biggl\{ \biggl[ \frac{\pi}{2\cdot 3^{4}}\biggr]^{1 / 2} \biggl[{\tilde\xi}^{-3} (3 + {\tilde\xi}^2)^{3 / 2} \biggr] \biggl( \frac{G^3M_\mathrm{tot}^2}{K^5} \biggr)^{1/2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{G^3M_\mathrm{tot}^2}{K^5} \biggr)^{1/2}\biggl[ \frac{\pi}{2\cdot 3^{5}}\biggr]^{1 / 2}
\biggl[ \frac{(3 + {\tilde\xi}^2)}{{\tilde\xi}^2 } \biggr]^{3 / 2} ( 3 + \xi^2 )^{1/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{g_0\rho_0}{P_0} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{G^3M_\mathrm{tot}^2}{K^5} \biggr)^{1/2}\biggl[ \frac{\pi}{2\cdot 3^{5}}\biggr]^{1 / 2}
\biggl[ \frac{(3 + {\tilde\xi}^2)}{{\tilde\xi}^2 } \biggr]^{3 / 2} ( 3 + \xi^2 )^{1/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~ \times ~
\biggl( \frac{G^2M_\mathrm{tot}^2}{R_\mathrm{norm}^4} \biggr)^{1 / 2}\biggl[ \frac{2^6\cdot 3^{14}}{\pi^2}\biggr]^{1 / 2}
\biggl[ \frac{(3 + {\tilde\xi}^2)}{{\tilde\xi}^2 } \biggr]^{-9 / 2}
\xi ( 3 + \xi^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1 / 2} R_\mathrm{norm}^{-2}
\biggl[ \frac{{\tilde\xi}^2 }{(3 + {\tilde\xi}^2)} \biggr]^{3} \biggl[ \frac{2^5\cdot 3^{9}}{\pi}\biggr]^{1 / 2} \xi ( 3 + \xi^2 )^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{K^5}{G^5 M_\mathrm{tot}^4} \biggr)^{1 / 2}
\biggl[ \frac{{\tilde\xi}^2 }{(3 + {\tilde\xi}^2)} \biggr]^{3} \biggl[ \frac{2^5\cdot 3^{9}}{\pi}\biggr]^{1 / 2}
\xi ( 3 + \xi^2 )^{-1}
\, .
</math>
</td>
</tr>
</table>
</div>

===The Wave Equation===

====Starting from our Key Adiabatic Wave Equation====

The [[#Adiabatic_.28Polytropic.29_Wave_Equation|adiabatic wave equation]] therefore becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0}
+ \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr_0^2} + \frac{1}{R_\mathrm{norm}} \biggl\{
\biggl[ \frac{2^7\cdot 3^{7}}{\pi} \biggr]^{1 / 2} \biggl[ \frac{ {\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{3} \frac{1}{\xi}
- \biggl[ \frac{{\tilde\xi}^2 }{(3 + {\tilde\xi}^2)} \biggr]^{3} \biggl[ \frac{2^5\cdot 3^{9}}{\pi}\biggr]^{1 / 2} \xi ( 3 + \xi^2 )^{-1}
\biggr\} \frac{dx}{dr_0}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{(4 - 3\gamma_\mathrm{g})}{\gamma_g R_\mathrm{norm}^2}
\biggl[ \frac{\pi}{2\cdot 3^{5}}\biggr]^{1 / 2} \biggl[ \frac{{\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{-3/2}
( 3 + \xi^2 )^{1 / 2}
\biggl\{ \frac{R_\mathrm{norm}^3}{GM_\mathrm{tot}} \biggl[\frac{\omega^2}{(4 - 3\gamma_\mathrm{g})} \biggr]
+ \biggl[ \frac{2^3\cdot 3^{7}}{\pi}\biggr]^{3/2}
\biggl[ \frac{{\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{15/2}
( 3 + \xi^2 )^{-3/2}
\biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr_0^2} + \frac{1}{R_\mathrm{norm}} \biggl[ \frac{2^3\cdot 3^{7}}{\pi} \biggr]^{1 / 2} \biggl[ \frac{ {\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{3} \biggl[
\frac{4}{\xi} - \frac{6 \xi}{ ( 3 + \xi^2 )}
\biggr] \frac{dx}{dr_0}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{6(4 - 3\gamma_\mathrm{g})}{\gamma_g R_\mathrm{norm}^2}
( 3 + \xi^2 )^{1 / 2} \biggl[ \frac{2^3\cdot 3^{7}}{\pi}\biggr] \biggl[ \frac{{\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{6}
\biggl\{ \frac{R_\mathrm{norm}^3}{GM_\mathrm{tot}} \biggl[\frac{\omega^2}{(4 - 3\gamma_\mathrm{g})} \biggr] \biggl[ \frac{2^3\cdot 3^{7}}{\pi}\biggr]^{-3/2} \biggl[ \frac{{\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{-15/2}
+
( 3 + \xi^2 )^{-3/2}
\biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr_0^2} + \frac{1}{R_*} \biggl[ \frac{ {\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{3} \biggl[
\frac{4}{\xi} - \frac{6 \xi}{ ( 3 + \xi^2 )}
\biggr] \frac{dx}{dr_0}
+ \frac{6}{\gamma_g R_*^2} \biggl[ \frac{{\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{6}
\biggl\{ \frac{\omega^2 R_*^3}{GM_\mathrm{tot}} \biggl[ \frac{{\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{-15/2}( 3 + \xi^2 )^{1 / 2}
+ \frac{(4 - 3\gamma_\mathrm{g})}{( 3 + \xi^2 ) }
\biggr\} x
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>R_* \equiv R_\mathrm{norm} \biggl[ \frac{\pi}{2^3 \cdot 3^7} \biggr]^{1/2} \, .</math>
</div>

Recognizing that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
R_* \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3} \xi \, ,
</math>
</td>
</tr>
</table>
</div>
we can write,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{R_*^2} \biggl[ \frac{ {\tilde\xi}^2}{(3 + {\tilde\xi}^2)} \biggr]^{6} \biggl\{
\frac{d^2x}{d\xi^2} + \biggl[
\frac{4}{\xi} - \frac{6 \xi}{ ( 3 + \xi^2 )}
\biggr] \frac{dx}{d\xi}
+ \frac{6}{\gamma_g }
\biggl[\sigma^2 ( 3 + \xi^2 )^{1 / 2}
+ \frac{(4 - 3\gamma_\mathrm{g})}{( 3 + \xi^2 ) }
\biggr] x \biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>

where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sigma^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \frac{\omega^2 R_*^3}{GM_\mathrm{tot}} \biggl( \frac{3 + {\tilde\xi}^2}{{\tilde\xi}^2} \biggr)^{15/2} = \frac{\sigma_c^2}{2\cdot 3^{3/2}}\, .</math>
</td>
</tr>
</table>
</div>

Finally, if — because we are specifically considering the case of <math>~n=5</math> — we set <math>~\gamma_\mathrm{g} = 1 + 1/n = 6/5</math>, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{d\xi^2} + \biggl[ \frac{4}{\xi} - \frac{6 \xi}{ ( 3 + \xi^2 )} \biggr] \frac{dx}{d\xi}
+ \biggl[\frac{5\sigma_c^2}{2\cdot 3^{3/2}} ( 3 + \xi^2 )^{1 / 2} + \frac{2}{( 3 + \xi^2 ) }\biggr] x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{( 3 + \xi^2 ) } \biggl\{ ( 3 + \xi^2 )\frac{d^2x}{d\xi^2}
+ \biggl[ \frac{2(6 - \xi^2) }{ \xi} \biggr] \frac{dx}{d\xi}
+ \biggl[\frac{5\sigma_c^2}{2\cdot 3^{3/2}} ( 3 + \xi^2 )^{3 / 2} + 2 \biggr] x \biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
which matches exactly the [[#n5LAWE|form of the LAWE derived above]], if in ''that'' expression, <math>~\gamma_g</math> is also forced to align with our specification of the polytropic index, that is, if <math>~\gamma_g = (n+1)/n = 6/5</math> and, in turn, <math>~\alpha = (3-4/\gamma) = -1/3</math>.

====Starting from the HRW66 Radial Pulsation Equation====
More directly, if we begin with the [[User:Tohline/SSC/Stability/Polytropes#HRW66excerpt|HRW66 radial pulsation equation]] that is already tuned to polytropic configurations, the wave equation appropriate to <math>~n=5</math> polytropes is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2 X}{d\xi^2} + \biggl[ \frac{4}{\xi} - \frac{6 (-\theta^'_5)}{\theta_5} \biggr]\frac{d X}{d\xi}
+ \frac{5(-\theta_5^') }{6\theta_5 \xi} \bigg[ \frac{\xi (s^')^2}{\theta^'_5} + \frac{12}{5} \biggr] X
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2 X}{d\xi^2} + \biggl[ \frac{4}{\xi} - \frac{6 \xi}{(3 + \xi^2)} \biggr]\frac{d X}{d\xi}
+ \frac{1}{(3 + \xi^2)} \bigg[ -\frac{5(s^')^2(3 + \xi^2)^{3 / 2}}{2 \cdot 3^{3 / 2}} + 2 \biggr] X
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(3+\xi^2)} \biggl\{
(3+\xi^2)\frac{d^2 X}{d\xi^2} + \biggl[ \frac{2(6-\xi^2)}{\xi}\biggr]\frac{d X}{d\xi}
+ \bigg[ -\frac{5(s^')^2}{2 \cdot 3^{3 / 2}} \cdot (3 + \xi^2)^{3 / 2} + 2 \biggr] X
\biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
which is identical to the brute-force derivation just presented, allowing for the mapping,
<div align="center">
<math>\sigma^2 ~~ \Leftrightarrow ~~ -\frac{(s^')^2}{2 \cdot 3^{3 / 2}} \, .</math>
</div>

Finally, remembering that the [[User:Tohline/SSC/Stability/Polytropes#HRW66frequency|HRW66 dimensionless frequency definition]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(s^')^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-\omega^2 \biggl[\frac{n+1 }{4\pi G \rho_c} \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
we recognize that, specifically for the case of <math>~n=5</math>, we can make the substitution, <math>~(s^')^2 \rightarrow -\sigma_c^2</math>, in which case the LAWE becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(3+\xi^2)} \biggl\{
(3+\xi^2)\frac{d^2 X}{d\xi^2} + \biggl[ \frac{2(6-\xi^2)}{\xi}\biggr]\frac{d X}{d\xi}
+ \bigg[ \frac{5\sigma_c^2}{2 \cdot 3^{3 / 2}} \cdot (3 + \xi^2)^{3 / 2} + 2 \biggr] X
\biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
which matches exactly the [[#n5LAWE|form of the LAWE derived above]], if in ''that'' expression, <math>~\gamma_g</math> is also forced to align with our specification of the polytropic index, that is, if <math>~\gamma_g = (n+1)/n = 6/5</math> and, in turn, <math>~\alpha = (3-4/\gamma) = -1/3</math>.

====New Independent Variable====
Guided by our [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation#Conjectures|conjecture]] regarding the proper shape of the radial eigenfunction, let's switch the dependent variable to,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~u \equiv 1 + \frac{3}{\xi^2}</math>
</td>
<td align="center">
        <math>~\Rightarrow</math>        
</td>
<td align="left">
<math>~3 + \xi^2 = \frac{3u}{(u-1)} \, ,</math>
</td>
<td align="center">
        and        
</td>
<td align="left">
<math>~\xi = 3^{1 / 2} (u-1)^{-1 / 2} \, .</math>
</td>
</tr>
</table>
</div>
This implies that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d}{d\xi}</math>
</td>
<td align="center">
<math>~~~\rightarrow ~~~</math>
</td>
<td align="left">
<math>~-\frac{2}{\sqrt{3}}(u-1)^{3 / 2} \frac{d}{du} \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2}{d\xi^2}</math>
</td>
<td align="center">
<math>~~~\rightarrow ~~~</math>
</td>
<td align="left">
<math>~\frac{4}{3}(u-1)^3 \frac{d^2}{du^2} + 2(u-1)^{2} \frac{d}{du} \, .</math>
</td>
</tr>
</table>
</div>
Hence, the governing wave equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~( 3 + \xi^2 )\frac{d^2x}{d\xi^2}
+ \biggl[ \frac{2(6 - \xi^2) }{ \xi} \biggr] \frac{dx}{d\xi}
+ \biggl[5\sigma^2 ( 3 + \xi^2 )^{3 / 2} + 2 \biggr] x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3u}{(u-1)} \biggl[\frac{4}{3}(u-1)^3 \frac{d^2x}{du^2} + 2(u-1)^{2} \frac{dx}{du}\biggr]
+ 4(2u-3)(u-1)\frac{dx}{du}
+ \biggl\{ 5\sigma^2 \biggl[ \frac{3u}{(u-1)} \biggr]^{3 / 2} + 2 \biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4u(u-1)^2 \frac{d^2x}{du^2}
+ (14u-12)(u-1)\frac{dx}{du}
+ \biggl\{ 5\sigma^2 \biggl[ \frac{3u}{(u-1)} \biggr]^{3 / 2} + 2 \biggr\} x \, .
</math>
</td>
</tr>
</table>
</div>

{{ SGFworkInProgress }}

If we ''assume'' that <math>~\sigma^2 = 0</math>, then the governing relation is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4u(u-1)^2 \frac{d^2x}{du^2}
+ (14u-12)(u-1)\frac{dx}{du}
+ 2 x \, .
</math>
</td>
</tr>
</table>
</div>
Now, again, guided by our [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation#Conjectures|conjecture]], let's guess an eigenfunction of the form:

=====First Guess (n5)=====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
A^3 (u - 1)^{1 / 2} (A u - 1 )^{-1 / 2} \, ,
</math>
</td>
</tr>
</table>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dx}{du}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{A^3}{2} \biggl[ (u - 1)^{-1 / 2} (A u - 1 )^{-1 / 2} - A(u - 1)^{1 / 2} (A u - 1 )^{-3 / 2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{A^3(A-1)}{2} \biggr] (u-1)^{-1 / 2} (Au-1)^{-3 / 2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{d^2x}{du^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{A^3(A-1)}{2} \biggr] \biggl\{
-\frac{1}{2}(u-1)^{-3 / 2} (Au-1)^{-3 / 2} -\frac{3A}{2} (u-1)^{-1 / 2} (Au-1)^{-5 / 2}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{1}{2} \biggl[ \frac{A^3(A-1)}{2} \biggr] (u-1)^{-3 / 2} (Au-1)^{-5 / 2}\biggl[
(Au-1) +3A (u-1)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{A^3(A-1)}{4} \biggr] (u-1)^{-3 / 2} (Au-1)^{-5 / 2}\biggl[(3A+1) - 4Au \biggr] \, .
</math>
</td>
</tr>
</table>
</div>



So the governing relation becomes:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4u(u-1)^2 \biggl\{ \biggl[ \frac{A^3(A-1)}{4} \biggr] (u-1)^{-3 / 2} (Au-1)^{-5 / 2}\biggl[(3A+1) - 4Au \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (14u-12)(u-1) \biggl\{ \biggl[ \frac{A^3(A-1)}{2} \biggr] (u-1)^{-1 / 2} (Au-1)^{-3 / 2} \biggr\}
+ 2 A^3 (u - 1)^{1 / 2} (A u - 1 )^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~u(u-1)^{1 / 2} A^3(A-1) (Au-1)^{-5 / 2}\biggl[(3A+1) - 4Au \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (7u-6)(u-1)^{1 / 2} A^3(A-1) (Au-1)^{-3 / 2}
+ 2 A^3 (u - 1)^{1 / 2} (A u - 1 )^{-1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(u-1)^{1 / 2} \biggl\{ uA^3(A-1) (Au-1)^{-5 / 2}\biggl[(3A+1) - 4Au \biggr]
+ (7u-6) A^3(A-1) (Au-1)^{-3 / 2}
+ 2 A^3 (A u - 1 )^{-1 / 2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A^3(u-1)^{1 / 2} (Au-1)^{-5 / 2} \biggl\{ u(A-1) \biggl[(3A+1) - 4Au \biggr]
+ (7u-6) (A-1) (Au-1)
+ 2 (A u - 1 )^{2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A^3(u-1)^{1 / 2} (Au-1)^{-5 / 2} \biggl\{ - 4u^2 A(A-1) + u(A-1) (3A+1)
+ (7u-6) [A(A-1)u +1 - A]
+ 2 (A^2u^2 - 2Au +1) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A^3(u-1)^{1 / 2} (Au-1)^{-5 / 2} \biggl\{ u^2 \biggl[ - 4A(A-1) +7A(A-1) +2A^2 \biggr]
+ u\biggl[ (A-1) (3A+1) - 7(A-1) -6A(A-1) - 4A \biggr] + 2(3A-2) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A^3(u-1)^{1 / 2} (Au-1)^{-5 / 2} \biggl\{ Au^2 \biggl[ 5A-3 \biggr]
+ u\biggl[ 3A^2-2A-1-7A+7 -6A^2+6A -4A \biggr] + 2(3A-2) \biggr\} \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A^3(u-1)^{1 / 2} (Au-1)^{-5 / 2} \biggl\{ Au^2 \biggl[ 5A-3 \biggr]
+ u\biggl[ -3A^2 -7A +6\biggr] + 2(3A-2) \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=====Second Guess (n5)=====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(u - 1)^{b / 2} (A u - 1 )^{-a / 2} \, ,
</math>
</td>
</tr>
</table>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dx}{du}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{b}{2}(u-1)^{b/2-1} (A u - 1 )^{-a / 2}
- \frac{aA}{2}(u - 1)^{b / 2} (A u - 1 )^{-a / 2-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~x \biggl[
\frac{b}{2}(u-1)^{-1}
- \frac{aA}{2} (A u - 1 )^{-1} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{(u-1)}{x} \frac{dx}{du}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ (A u - 1 )^{-1} \biggl[
\frac{b}{2} (A u - 1 )
- \frac{aA}{2} (u-1) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1 }{2(A u - 1 )} \biggl[
b (A u - 1 ) - aA (u-1) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1 }{2(A u - 1 )} \biggl[
(aA - b) + A(b - a)u \biggr] \, ;
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2x}{du^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b}{2}(u-1)^{-1} - \frac{aA}{2} (A u - 1 )^{-1} \biggr]\frac{dx}{du}
+ x \frac{d}{du}\biggl[ \frac{b}{2}(u-1)^{-1} - \frac{aA}{2} (A u - 1 )^{-1} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x\biggl[ \frac{b}{2}(u-1)^{-1} - \frac{aA}{2} (A u - 1 )^{-1} \biggr]^2
+ x \biggl[ -\frac{b}{2}(u-1)^{-2} + \frac{aA^2}{2} (A u - 1 )^{-2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{x}{4(u-1)^2 (Au-1)^2} \biggl\{
\biggl[ b(Au-1) - aA (u - 1 ) \biggr]^2
+ \biggl[ 2aA^2 (u-1)^{2} -2b (A u - 1 )^{2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{(1-u)^2}{x}\frac{d^2x}{du^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4 (Au-1)^2} \biggl\{
\biggl[ b(Au-1) - aA (u - 1 ) \biggr]^2
+ \biggl[ 2aA^2 (u-1)^{2} -2b (A u - 1 )^{2} \biggr]
\biggr\}
</math>
</td>
</tr>
</table>
</div>

Hence, the governing wave equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2u \biggl\{ \frac{(u-1)^2}{x} \frac{d^2x}{du^2} \biggr\}
+ (7u-6)\biggl\{ \frac{(u-1)}{x} \frac{dx}{du} \biggl\} + 1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2u}{4 (Au-1)^2} \biggl\{
\biggl[ (aA - b) + A(b - a)u \biggr]^2
+ \biggl[ 2aA^2 (u-1)^{2} -2b (A u - 1 )^{2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{(7u-6) }{2(A u - 1 )} \biggl[
(aA - b) + A(b - a)u \biggr]
+ 1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4 (Au-1)^2} \biggl\{
2u\biggl[ (aA - b)^2 + 2(aA - b)A(b - a)u + A^2(b - a)^2u^2 \biggr]
+ 2u\biggl[ 2aA^2 (u^2 - 2u + 1) -2b (A^2 u^2 - 2Au + 1 ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 2(A u - 1 )(7u-6) \biggl[
(aA - b) + A(b - a)u \biggr]
+ 4 (Au-1)^2 \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4 (Au-1)^2} \biggl\{
2u\biggl[ (aA - b)^2 + 2(aA - b)A(b - a)u + A^2(b - a)^2u^2 \biggr]
+ 2u\biggl[ 2A^2(a-b)u^2 + 4A(b - aA) u + 2(aA^2 -b) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 2\biggl[7Au^2 - (6A+7)u +6 \biggr]\biggl[
(aA - b) + A(b - a)u \biggr]
+ (4A^2u^2-8Au + 4) \biggr\}
</math>
</td>
</tr>
</table>
</div>

If <math>~b=a</math>,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2u\biggl[ (aA - b)^2 \biggr]
+ 2u\biggl[ 4A(b - aA) u + 2(aA^2 -b) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 2\biggl[7Au^2 - (6A+7)u +6 \biggr]\biggl[
(aA - b) \biggr] + (4A^2u^2-8Au + 4)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2a^2u (A - 1)^2
+ 2au [ 4A(1 - A) u + 2(A^2 -1) ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 2a(A - 1) \biggl[7Au^2 - (6A+7)u +6 \biggr] + (4A^2u^2-8Au + 4)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2Au^2 [4a (1 - A) + 7a(A - 1) + 2A] + 2u [ a^2 (A - 1)^2 + 2a(A^2 -1) - a(A - 1) (6A+7) - 4A] + 4[ 3a(A-1) + 1]
</math>
</td>
</tr>
</table>
</div>
This should then match the [[#First_Guess_.28n5.29|"first guess"]] algebraic condition if we set <math>~a=1</math>. Let's see.

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2Au^2 [4 (1 - A) + 7(A - 1) + 2A] + 2u [ (A - 1)^2 + 2(A^2 -1) - (A - 1) (6A+7) - 4A] + 4[ 3(A-1) + 1]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2Au^2 [4 - 4A + 7A - 7 + 2A] + 2u [ (A^2 - 2A + 1) + 2A^2 -2 + (1-A ) (6A+7) -4A] + 4[ 3A-2]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2Au^2 [5A - 3] + 2u [ - 3A^2 - 7A + 6 ] + 4[ 3A-2] \, .
</math>
</td>
</tr>
</table>
</div>
And we see that this expression ''does'' match the one derived earlier.

Going back a bit, before setting <math>~a=1</math>, we have the expression:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2Au^2 [4a (1 - A) + 7a(A - 1) + 2A] + 2u [ a^2 (A - 1)^2 + 2a(A^2 -1) - a(A - 1) (6A+7) - 4A] + 4[ 3a(A-1) + 1]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2Au^2 [ 3aA -3a + 2A] + 2u [ a^2 (A - 1)^2 + 2a(A^2 -1) - a(6A^2+A-7) - 4A] + 4[ 3a(A-1) + 1]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2Au^2 [ 3a(A - 1) + 2A] + 2u [ a^2 (A - 1)^2 + a( -4A^2-A+5) - 4A] + 4[ 3a(A-1) + 1] \, .
</math>
</td>
</tr>
</table>
</div>
Now, in order for all three expressions inside the square-bracket pairs to be zero, we need, first,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3a(A - 1) + 2A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ a</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2A}{3(1-A)} \, ;</math>
</td>
</tr>
</table>
</div>
and, third, by simple visual comparison with the first expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3a(A-1) + 1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3a(A-1) + 2A</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ a</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{3} \, ;</math>
</td>
</tr>
</table>
</div>
which forces the second expression to the value,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a^2 (A - 1)^2 + a( -4A^2-A+5) - 4A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{2}{3}\biggr)^2 \biggl(-\frac{1}{2} \biggr)^2 + \frac{2}{3}\biggl[ -1-\frac{1}{2} +5 \biggr] - 2</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{9} + \frac{7}{3} - 2</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4}{9} \, ,</math>
</td>
</tr>
</table>
</div>
which is ''not'' zero. Hence our pair of unknown parameters — <math>~a </math> and <math>~A</math> — do not simultaneously satisfy all three conditions. (Not really a surprise.)

==Setup Using Lagrangian Mass Coordinate==

===Alternative Terms===
Let's change the independent coordinate from <math>~r_0</math> to <math>~m_0</math>. In particular, the derivative operation will change as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d}{dr_0}</math>
</td>
<td align="center">
<math>~~\rightarrow~~</math>
</td>
<td align="left">
<math>~\biggl( \frac{dm_0}{dr_0} \biggr)\frac{d}{dm_0}
= \biggl( \frac{dm_0}{d\xi} \cdot \frac{d\xi}{dr_0} \biggr)\frac{d}{dm_0}
\, ,</math>
</td>
</tr>
</table>
</div>
so what is the expression for the leading coefficient? From [[#r0|above]], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
R_* \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3} \xi
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{R_*} \biggl[ \frac{ {\tilde\xi}^2}{(3 + {\tilde\xi}^2)}\biggr]^{3} r_0 \, .
</math>
</td>
</tr>
</table>
</div>

Also, from [[#m0|above]], we know that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~m_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
M_\mathrm{tot} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3 / 2}
\biggl\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{dm_0}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
M_\mathrm{tot} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3 / 2}
\biggl\{ 3\xi^2 ( 3 + \xi^2 )^{-3/2}
- 3 \xi^4 ( 3 + \xi^2 )^{-5/2}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
M_\mathrm{tot} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3 / 2} 3\xi^2 (3 + \xi^2)^{-5/2}
\biggl\{ ( 3 + \xi^2 )
- \xi^2 \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
M_\mathrm{tot} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3 / 2} 3^2\xi^2 (3 + \xi^2)^{-5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{dm_0}{dr_0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
M_\mathrm{tot} \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3 / 2} 3^2\xi^2 (3 + \xi^2)^{-5/2}
\frac{1}{R_*} \biggl[ \frac{ {\tilde\xi}^2}{(3 + {\tilde\xi}^2)}\biggr]^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{M_\mathrm{tot} }{R_*}
\biggl[ \frac{ {\tilde\xi}^2}{(3 + {\tilde\xi}^2)}\biggr]^{3 / 2} 3^2\xi^2 (3 + \xi^2)^{-5/2} \, .
</math>
</td>
</tr>
</table>
</div>

To simplify expressions, let's [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation#DefineTildeC|borrow from an accompanying derivation]] and define,
<div align="center">
<math>\tilde{C} \equiv \frac{3^2}{{\tilde\xi}^2} \biggl( 1 + \frac{ {\tilde\xi}^2}{3} \biggr) = 3 \biggl[ \frac{( 3 + {\tilde\xi}^2 )}{ {\tilde\xi}^2} \biggr] \, .</math>
</div>
Then we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{m_0}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{\tilde{C}}{ 3}\biggr]^{3 / 2}
\biggl[ \frac{\xi^2}{ ( 3 + \xi^2 )} \biggr]^{3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{ 3}{\tilde{C}}\biggr] \biggl[\frac{m_0}{M_\mathrm{tot}}\biggr]^{2 / 3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\xi^2}{ ( 3 + \xi^2 )}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~( 3 + \xi^2 )\biggl[ \frac{ 3}{\tilde{C}}\biggr] \biggl[\frac{m_0}{M_\mathrm{tot}}\biggr]^{2 / 3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~3 m_*</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \xi^2 (1-m_*)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\xi^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{3m_*}{(1-m_*)} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~m_* \equiv \biggl[ \frac{ 3}{\tilde{C}}\biggr] \biggl[\frac{m_0}{M_\mathrm{tot}}\biggr]^{2 / 3} \, .</math>
</div>

In summary:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{\xi^2}{ ( 3 + \xi^2 )} = m_* \, ;
</math>
</td>
<td align="center">
      while,      
</td>
<td align="left">
<math>~
\frac{ {\tilde\xi}^2}{ ( 3 + {\tilde\xi}^2 )} = \frac{3}{\tilde{C}} \, ;
</math>
</td>
</tr>
</table>
</div>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
R_* \biggl[ \frac{(3 + {\tilde\xi}^2)}{ {\tilde\xi}^2}\biggr]^{3} \xi
= R_* \biggl( \frac{ \tilde{C} }{ 3}\biggr)^{3} \biggr[ \frac{3m_*}{ (1-m_*) }\biggr]^{1 / 2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{g_0\rho_0}{P_0} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{6}{R_*} \biggl[ \frac{ {\tilde\xi}^2 }{ (3 + {\tilde\xi}^2) }\biggr]^{9} \frac{\xi}{ ( 3 + \xi^2 )}
=
\frac{6}{R_*} \biggl[ \frac{ 3 }{ \tilde{C} }\biggr]^{9} \frac{m_*}{ \xi }
=
\frac{6}{R_*} \biggl[ \frac{ 3 }{ \tilde{C} }\biggr]^{9} m_* \biggl[ \frac{(1-m_*)}{3m_*} \biggr]^{1 / 2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{g_0 }{r_0} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{GM_\mathrm{tot}}{R_*^3}
\biggl[ \frac{ {\tilde\xi}^2 }{ (3 + {\tilde\xi}^2)}\biggr]^{15/2} \frac{1}{\xi^3}
\biggl[ \frac{ \xi^2 }{ ( 3 + \xi^2 ) }\biggr]^{3/2}
=
\frac{GM_\mathrm{tot}}{R_*^3} \biggl[ \frac{3 }{ \tilde{C} }\biggr]^{15/2} (1-m_*)^{3 / 2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\rho_0}{\gamma_g P_0} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{6R_* }{\gamma_g GM_\mathrm{tot} }\biggl( \frac{ 3}{ \tilde{C} } \biggr)^{9 / 2} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2} \, .
</math>
</td>
</tr>
</table>
</div>
So, the wave equation may be written as,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0}
+ \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr_0^2}
+ \biggl\{ \frac{4}{R_*} \biggl( \frac{ 3}{ \tilde{C} }\biggr)^{3} \biggr[ \frac{ (1-m_*) }{3m_*}\biggr]^{1 / 2}
- \frac{6}{R_*} \biggl[ \frac{ 3 }{ \tilde{C} }\biggr]^{9} m_* \biggl[ \frac{(1-m_*)}{3m_*} \biggr]^{1 / 2} \biggr\} \frac{dx}{dr_0}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{6R_* }{\gamma_g GM_\mathrm{tot} }\biggl( \frac{ 3}{ \tilde{C} } \biggr)^{9 / 2} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}
\biggl\{ \omega^2 + (4 - 3\gamma_\mathrm{g})\frac{GM_\mathrm{tot}}{R_*^3} \biggl[ \frac{3 }{ \tilde{C} }\biggr]^{15/2} (1-m_*)^{3 / 2} \biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr_0^2}
+ \frac{1}{R_*} \biggl( \frac{ 3}{ \tilde{C} }\biggr)^{3} \biggl\{ 4
- 6\biggl[ \frac{ 3 }{ \tilde{C} }\biggr]^{6} m_* \biggr\} \biggr[ \frac{ (1-m_*) }{3m_*}\biggr]^{1 / 2}\frac{dx}{dr_0}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{6(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \cdot \frac{1 }{R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{3} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}
\biggl\{\sigma^2 + (1-m_*)^{3 / 2} \biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1 }{R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{3} \biggl\{
R_*^2 \biggl( \frac{ \tilde{C} }{3 }\biggr)^{3} \frac{d^2x}{dr_0^2}
+ R_* \biggl[ 4 - 6\biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* \biggr] \biggr[ \frac{ (1-m_*) }{3m_*}\biggr]^{1 / 2}\frac{dx}{dr_0}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{6(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \cdot \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}
\biggl[ \sigma^2 + (1-m_*)^{3 / 2} \biggr] x \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1 }{R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{3} \biggl[ \frac{1}{3m_*(1-m_*)}\biggr]^{1 / 2} \biggl\{ [ 3m_*(1-m_*) ]^{1 / 2}
R_*^2 \biggl( \frac{ \tilde{C} }{3 }\biggr)^{3} \frac{d^2x}{dr_0^2}
+ R_* \biggl[ 4 - 6\biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* \biggr] (1-m_*) \frac{dx}{dr_0}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{18(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \cdot m_*^{1 / 2}
\biggl[ \sigma^2 + (1-m_*)^{3 / 2} \biggr] x \biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>

where,
<div align="center">
<math>~\sigma^2 \equiv (4 - 3\gamma_\mathrm{g})^{-1} \frac{R_*^3}{GM_\mathrm{tot}} \biggl[ \frac{ \tilde{C} }{3 } \biggr]^{15/2} \omega^2 \, .</math>
</div>

Now, let's look at the differential operators, after defining.
<div align="center">
<math>~c_0 \equiv 3^{1 / 2} R_* \biggl( \frac{ \tilde{C} }{ 3}\biggr)^{3} ~~~~\Rightarrow ~~~~R_* = c_0 3^{-1 / 2} \biggl( \frac{ \tilde{C} }{ 3}\biggr)^{-3} \, .</math>
</div>

We find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~dr_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
c_0 ~d[ m_*^{1 / 2} (1-m_*)^{-1 / 2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
c_0 ~\biggl[\frac{1}{2} ~m_*^{-1 / 2}( 1 - m_*)^{-1 / 2} + \frac{1}{2} ~m_*^{1 / 2} (1 - m_*)^{-3 / 2}
\biggr] dm_*
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{c_0}{2} ~m_*^{-1 / 2}( 1 - m_*)^{-3 / 2}~ dm_*
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{d}{dr_0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{c_0} ~m_*^{1 / 2}( 1 - m_*)^{3 / 2}~ \frac{d}{dm_*}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ R_*\frac{dx}{dr_0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{3^{1 / 2}}\biggl( \frac{ \tilde{C} }{ 3}\biggr)^{-3} ~m_*^{1 / 2}( 1 - m_*)^{3 / 2}~ \frac{dx}{dm_*} \, .
</math>
</td>
</tr>
</table>
</div>

Also,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2}{dr_0^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2}{c_0} \biggr)^{2}~m_*^{1 / 2}( 1 - m_*)^{3 / 2}~ \frac{d}{dm_*} \biggl[ m_*^{1 / 2}( 1 - m_*)^{3 / 2}~ \frac{d}{dm_*} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2}{c_0} \biggr)^{2}~m_* ( 1 - m_*)^{3 }~ \frac{d^2}{dm_*^2}
+\biggl( \frac{2}{c_0} \biggr)^{2}~m_*^{1 / 2}( 1 - m_*)^{3 / 2} \biggl[ \frac{1}{2} m_*^{-1 / 2}( 1 - m_*)^{3 / 2} - \frac{3}{2} m_*^{1 / 2}( 1 - m_*)^{1 / 2}~
\biggr] ~ \frac{d}{dm_*}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2}{c_0} \biggr)^{2}~m_* ( 1 - m_*)^{3 }~ \frac{d^2}{dm_*^2} +\frac{1}{2} \biggl( \frac{2}{c_0} \biggr)^{2}~ ( 1 - m_*)^{2} ( 1 - 4m_*) \frac{d}{dm_*}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ R_*^2 \biggl( \frac{ \tilde{C} }{3 }\biggr)^{3} \frac{d^2x}{dr_0^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2^2}{3} \biggl(\frac{ \tilde{C} }{3} \biggr)^{-3} \biggr]
\biggl[ ~m_* ( 1 - m_*)^{3 }~ \frac{d^2x}{dm_*^2} +\frac{1}{2} ~ ( 1 - m_*)^{2} ( 1 - 4m_*) \frac{dx}{dm_*} \biggr]
</math>
</td>
</tr>
</table>
</div>

So, the wave equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1 }{R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{3} \biggl[ \frac{1}{3m_*(1-m_*)}\biggr]^{1 / 2} \biggl\{ [ 3m_*(1-m_*) ]^{1 / 2}
\biggl[\frac{2^2}{3} \biggl(\frac{ \tilde{C} }{3} \biggr)^{-3} \biggr]
\biggl[ ~m_* ( 1 - m_*)^{3 }~ \frac{d^2x}{dm_*^2} +\frac{1}{2} ~ ( 1 - m_*)^{2} ( 1 - 4m_*) \frac{dx}{dm_*} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ 4 - 6\biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* \biggr] (1-m_*) \biggl[ \frac{2}{3^{1 / 2}}\biggl( \frac{ \tilde{C} }{ 3}\biggr)^{-3} ~m_*^{1 / 2}( 1 - m_*)^{3 / 2}~ \frac{dx}{dm_*} \biggr]
+ \frac{18(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \cdot m_*^{1 / 2}
\biggl[ \sigma^2 + (1-m_*)^{3 / 2} \biggr] x \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1 }{R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{6} \biggl[ \frac{1}{3m_*(1-m_*)}\biggr]^{1 / 2} \biggl\{ [ 3m_*(1-m_*) ]^{1 / 2}
\biggl[\frac{2^2}{3} \biggr]
\biggl[ ~m_* ( 1 - m_*)^{3 }~ \frac{d^2x}{dm_*^2} +\frac{1}{2} ~ ( 1 - m_*)^{2} ( 1 - 4m_*) \frac{dx}{dm_*} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ 4 - 6\biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* \biggr] (1-m_*) \biggl[ \frac{2}{3^{1 / 2}} ~m_*^{1 / 2}( 1 - m_*)^{3 / 2}~ \frac{dx}{dm_*} \biggr]
+ \frac{18(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \biggl( \frac{ \tilde{C} }{ 3}\biggr)^{3} m_*^{1 / 2} \biggl[ \sigma^2 + (1-m_*)^{3 / 2} \biggr] x \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2 }{3R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{6}
\biggl\{ 2m_* ( 1 - m_*)^{3 }~ \frac{d^2x}{dm_*^2} + ( 1 - m_*)^{2} ( 1 - 4m_*) \frac{dx}{dm_*}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ 4 - 6\biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* \biggr] (1-m_*)^2 \frac{dx}{dm_*}
+ \frac{9(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \biggl( \frac{ \tilde{C} }{ 3}\biggr)^{3} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}\biggl[ \sigma^2 + (1-m_*)^{3 / 2} \biggr] x \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2 }{3R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{6}
\biggl\{ 2m_* ( 1 - m_*)^{3 }~ \frac{d^2x}{dm_*^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ 5 - 4m_* - 6\biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* \biggr] (1-m_*)^2 \frac{dx}{dm_*}
+ \frac{9(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \biggl( \frac{ \tilde{C} }{ 3}\biggr)^{3} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}\biggl[ \sigma^2 + (1-m_*)^{3 / 2} \biggr] x \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2 }{3R_*^2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{6} \biggl\{ 2m_* ( 1 - m_*)^{3 }~ \frac{d^2x}{dm_*^2}
+ (5 - \mathcal{A} m_*) (1-m_*)^2 \frac{dx}{dm_*}
+ \mathcal{B} \biggl[ \frac{\sigma^2}{(1-m_*)^{1 / 2}} + (1-m_*) \biggr] x \biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~4 + 6\biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathcal{B}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{3^{5/2}(4 - 3\gamma_\mathrm{g}) }{\gamma_g } \biggl( \frac{ \tilde{C} }{ 3}\biggr)^{3} \, .</math>
</td>
</tr>
</table>
</div>

==Try Again==

This time, let's adopt the notation used in a [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation#n_.3D_5_Mass-Radius_Relation|related chapter in our ''Ramblings'' appendix]]. Specifically, the parametric relationship between <math>~m_\xi</math> and <math>~r_\xi</math> in pressure-truncated, <math>~n=5</math> polytropes is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~m_\xi \equiv \frac{m_0}{ M_\mathrm{tot} } = \frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{\xi}{\tilde\xi}\biggr)^3 \biggl(3 + \xi^2 \biggr)^{-3/2}
\biggl(3 + {\tilde\xi}^2 \biggr)^{3/2} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{( 3+\tilde\xi^2)}{ {\tilde\xi}^2} \biggr]^{3 / 2}\biggl[ \frac{( 3+\xi^2)}{ {\xi}^2} \biggr]^{- 3 / 2}
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r_\xi \equiv \frac{r_0}{R_\mathrm{norm}} = \biggl(\frac{\xi}{\tilde\xi} \biggr) \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi \biggl\{
\biggl[ \frac{4\pi}{2^5\cdot 3}\biggr]^{1/2} \tilde\xi^{-6}
\biggl( 1+\frac{\tilde\xi^2}{3} \biggr)^{3}\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{\pi}{2^3\cdot 3^7}\biggr]^{1/2}
\biggl[ \frac{( 3+\tilde\xi^2)}{ {\tilde\xi}^2} \biggr]^{3} \xi \, .
</math>
</td>
</tr>
</table>
</div>

And we are in the fortunate situation of being able to eliminate <math>~\xi</math> to obtain the direct relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
r_\xi (m_\xi)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tilde{r}_\mathrm{edge}
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\tilde{C}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{3^2}{\tilde\xi^2}\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr)
= 3 \biggl[ \frac{( 3+\tilde\xi^2)}{ {\tilde\xi}^2} \biggr]
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\tilde{r}_\mathrm{edge}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} {\tilde\xi}^{-6} \biggl(1+\frac{\tilde\xi^2}{3}\biggr)^3
= \biggl[ \frac{\pi}{2^3\cdot 3^7}\biggr]^{1 / 2} \biggl[ \frac{\tilde{C}}{ 3} \biggr]^{3}
\, .
</math>
</td>
</tr>
</table>
</div>

If we furthermore define,
<div align="center">
<math>m_* \equiv \frac{3}{\tilde{C}} \cdot m_\xi^{2 / 3} \, ,</math>
</div>
then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
r_\xi (m_*)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3^{1 / 2} \tilde{r}_\mathrm{edge} \biggl[\frac{m_*}{1-m_*}\biggr]^{1/2}
\, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{dr_0}{R_\mathrm{norm}} = dr_\xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{1 / 2} \tilde{r}_\mathrm{edge} \biggl\{
\frac{1}{2} (1-m_*)^{- 1 / 2} m_*^{-1 / 2} + \frac{1}{2}m_*^{1 / 2}(1-m_*)^{-3 / 2}
\biggr\} dm_*
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( \frac{3^{1 / 2}}{2} \biggr) \tilde{r}_\mathrm{edge} m_*^{-1 / 2} (1-m_*)^{-3 / 2} dm_*
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ R_\mathrm{norm} \cdot \frac{d}{dr_0} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{ \tilde{r}_\mathrm{edge}} \biggl( \frac{2}{3^{1 / 2}} \biggr) m_*^{1 / 2} (1-m_*)^{3 / 2} \frac{d}{dm_*} \, .
</math>
</td>
</tr>
</table>
</div>
We therefore also have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ R^2_\mathrm{norm} \cdot \frac{d^2}{dr_0^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2^2}{3} \biggr) m_*^{1 / 2} (1-m_*)^{3 / 2} \frac{d}{dm_*}\biggl[ m_*^{1 / 2} (1-m_*)^{3 / 2} \frac{d}{dm_*}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2^2}{3} \biggr) m_*^{1 / 2} (1-m_*)^{3 / 2}
\biggl\{
\biggl[ m_*^{1 / 2} (1-m_*)^{3 / 2} \frac{d^2}{dm_*^2}\biggr]
+ \biggl[ \frac{1}{2} m_*^{-1 / 2} (1-m_*)^{3 / 2} + \frac{3}{2}m_*^{1 / 2} (1-m_*)^{1 / 2}\biggr] \frac{d}{dm_*}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2}{3} \biggr) \biggl\{
\biggl[ 2m_* (1-m_*)^{3} \frac{d^2}{dm_*^2}\biggr]
+ \biggl[ (1-m_*)^{3 } + 3m_* (1-m_*)^{2}\biggr] \frac{d}{dm_*}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2}{3} \biggr) \biggl\{
2m_* (1-m_*)^{3} \frac{d^2}{dm_*^2}
+ (1-m_*)^{2} ( 1 + 2m_* ) \frac{d}{dm_*} \biggr\}
\, .
</math>
</td>
</tr>
</table>
</div>

So the wave equation may be written,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
R_\mathrm{norm}^2 \cdot \frac{d^2x}{dr_0^2} + \biggl[\frac{4R_\mathrm{norm}}{r_0} - \biggl(\frac{g_0 \rho_0 R_\mathrm{norm}}{P_0}\biggr) \biggr] R_\mathrm{norm} \cdot \frac{dx}{dr_0}
+ \biggl(\frac{\rho_0 R_\mathrm{norm}}{\gamma_\mathrm{g} P_0} \biggr)\biggl[R_\mathrm{norm} \omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0 R_\mathrm{norm}}{r_0} \biggr] x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2}{3} \biggr) \biggl\{
2m_* (1-m_*)^{3} \frac{d^2x}{dm_*^2}
+ (1-m_*)^{2} ( 1 + 2m_* ) \frac{dx}{dm_*} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{1}{ \tilde{r}_\mathrm{edge}} \biggl( \frac{2}{3^{1 / 2}} \biggr) \biggl\{ \frac{4}{r_\xi}
- \biggl[\frac{6R_\mathrm{norm}}{R_*} \biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{9} m_* \biggl[ \frac{(1-m_*)}{3m_*} \biggr]^{1 / 2} \biggr] \biggr\}
m_*^{1 / 2} (1-m_*)^{3 / 2} \frac{dx}{dm_*}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{6R_* R_\mathrm{norm}}{\gamma_g GM_\mathrm{tot} }\biggl( \frac{ 3}{ \tilde{C} } \biggr)^{9 / 2} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}
\biggl\{
R_\mathrm{norm} \omega^2 + (4 - 3\gamma_\mathrm{g}) \frac{GM_\mathrm{tot} R_\mathrm{norm}}{R_*^3} \biggl[ \frac{3 }{ \tilde{C} }\biggr]^{15/2} (1-m_*)^{3 / 2}
\biggr\} x \, .
</math>
</td>
</tr>
</table>
</div>
Keeping in mind that,
<div align="center">
<math>~\frac{R_*}{R_\mathrm{norm}} = \biggl[ \frac{\pi}{2^3 \cdot 3^7} \biggr]^{1 / 2} = {\tilde{r}}_\mathrm{edge} \biggl( \frac{3}{\tilde{C}} \biggr)^3 \, ,</math>
</div>

we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2}{3} \biggr) \biggl\{
2m_* (1-m_*)^{3} \frac{d^2x}{dm_*^2}
+ (1-m_*)^{2} ( 1 + 2m_* ) \frac{dx}{dm_*} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{1}{ \tilde{r}_\mathrm{edge}} \biggl( \frac{2}{3^{1 / 2}} \biggr) \biggl\{ 4 \biggl[3^{1 / 2} \tilde{r}_\mathrm{edge} \biggl[\frac{m_*}{1-m_*}\biggr]^{1/2} \biggr]^{-1}
- 6 \biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{9} \biggl[{\tilde{r}}_\mathrm{edge} \biggl( \frac{3}{\tilde{C}} \biggr)^3 \biggr]^{-1} m_* \biggl[ \frac{(1-m_*)}{3m_*} \biggr]^{1 / 2} \biggr\}
m_*^{1 / 2} (1-m_*)^{3 / 2} \frac{dx}{dm_*}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 6 \biggl( \frac{ 3}{ \tilde{C} } \biggr)^{9 / 2} \biggl[{\tilde{r}}_\mathrm{edge} \biggl( \frac{3}{\tilde{C}} \biggr)^3 \biggr]^{-2} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}
\biggl\{
\biggl[ \frac{R_*^3}{\gamma_g GM_\mathrm{tot} } \biggr] \omega^2
+ \frac{(4 - 3\gamma_\mathrm{g})}{\gamma_g} \biggl[ \frac{3 }{ \tilde{C} }\biggr]^{15/2} (1-m_*)^{3 / 2}
\biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2}{3} \biggr) \biggl\{
2m_* (1-m_*)^{3} \frac{d^2x}{dm_*^2}
+ (1-m_*)^{2} ( 1 + 2m_* ) \frac{dx}{dm_*} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{1}{ \tilde{r}_\mathrm{edge}^2} \biggl( \frac{2^3}{3} \biggr)
\biggl[ 1 - \frac{3}{2} \biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* \biggr]
(1-m_*)^{2} \frac{dx}{dm_*}
+ \frac{6}{ {\tilde{r}}_\mathrm{edge}^2 }
\biggl( \frac{3 }{ \tilde{C} }\biggr)^{6} \biggl[ \frac{3}{(1-m_*)}\biggr]^{1 / 2}
\frac{(4 - 3\gamma_\mathrm{g})}{\gamma_g}
\biggl[ \sigma^2 + (1-m_*)^{3 / 2} \biggr] x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde{r}}^2_\mathrm{edge}} \biggl( \frac{2}{3} \biggr) \biggl\{
2m_* (1-m_*)^{3} \frac{d^2x}{dm_*^2}
+ \biggl[ 5 - 6 \biggl( \frac{ 3 }{ \tilde{C} }\biggr)^{6} m_* + 2m_* \biggr]
(1-m_*)^{2} \frac{dx}{dm_*}
+ 3^{5 / 2} \biggl( \frac{3 }{ \tilde{C} }\biggr)^{6}
\frac{(4 - 3\gamma_\mathrm{g})}{\gamma_g}
\biggl[ \frac{\sigma^2 }{(1-m_*)^{1 / 2}} + (1-m_*) \biggr] x \biggr\}
\, ,
</math>
</td>
</tr>
</table>
</div>
where, as before,
<div align="center">
<math>\sigma^2 \equiv \biggl( \frac{ \tilde{C} }{3 } \biggr)^{15/2} \biggl[ \frac{R_*^3}{(4 - 3\gamma_g) GM_\mathrm{tot} } \biggr] \omega^2 \, .</math>
</div>

==Take Another Approach Using Logarithmic Derivatives==

===Change Independent Variable===

Returning to the [[#n5LAWE|LAWE for n = 3 polytropes, as given, above]], and repeated here,

<table border="1" cellpadding="8" align="center">
<tr><th align="center">LAWE for <math>~n=5</math> Polytropes</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3+\xi^2) \frac{d^2x}{d\xi^2} + \biggl[12 - 2\xi^2 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} +
\biggl[\frac{\sigma_c^2}{3^{1/2} \gamma_g } \cdot (3+\xi^2)^{3/2} -
6\alpha \biggr] x </math>
</td>
</tr>
</table>
</td></tr></table>

let's make the substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~u \equiv (3 + \xi^2)^{1/2}</math>
</td>
<td align="center">
      <math>~\Rightarrow</math>     
</td>
<td align="left">
<math>~\xi^2 = u^2-3 \, .</math>
</td>
</tr>
</table>
</div>
We must therefore also make the operator substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{du}{d\xi} \cdot \frac{d}{du}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \xi (3+\xi^2)^{-1/2} \biggr] \frac{d}{du} = \biggl[ 1 - \frac{3}{u^2} \biggr]^{1/2} \frac{d}{du}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{1}{\xi} \cdot \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{u}\cdot \frac{dx}{du} \, ;</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2}{d\xi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ 1 - \frac{3}{u^2} \biggr]^{1/2} \frac{d}{du} \biggl\{ \biggl[ 1 - \frac{3}{u^2} \biggr]^{1/2} \frac{d}{du} \biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ 1 - \frac{3}{u^2} \biggr]^{1/2} \biggl\{ \frac{3}{u^3} \biggl[ 1 - \frac{3}{u^2} \biggr]^{-1/2} \frac{d}{du} + \biggl[ 1 - \frac{3}{u^2} \biggr]^{1/2} \frac{d^2}{du^2}\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{3}{u^3} \frac{d}{du} + \biggl[ 1 - \frac{3}{u^2} \biggr] \frac{d^2}{du^2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{d^2x}{d\xi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{3}{u^3} \frac{dx}{du} + \biggl[ 1 - \frac{3}{u^2} \biggr] \frac{d^2x}{du^2} \, .</math>
</td>
</tr>
</table>
</div>

The rewritten LAWE is therefore,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~u^2 \biggl\{ \frac{3}{u^3} \frac{dx}{du} + \biggl[ 1 - \frac{3}{u^2} \biggr] \frac{d^2x}{du^2} \biggr\}
+ 2\biggl[9 - u^2 \biggr] \frac{1}{u} \cdot \frac{dx}{du} +
\biggl[\Omega^2 u^3 -
6\alpha \biggr] x </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(u^2-3) \frac{d^2x}{du^2}
+ (21 - 2u^2 ) \frac{1}{u} \cdot \frac{dx}{du} +
(\Omega^2 u^3 - 6\alpha ) x \, ,</math>
</td>
</tr>
</table>
</div>
where we have adopted the shorthand notation,
<div align="center">
<math>~\Omega^2 \equiv \frac{\sigma_c^2}{3^{1/2} \gamma_g } \, .</math>
</div>

===Look at Logarithmic Derivative===

Multiplying through by <math>~(u^2/x)</math> gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(u^2-3) \frac{u^2}{x} \cdot \frac{d^2x}{du^2}
+ (21 - 2u^2 ) \frac{d\ln x}{d\ln u} +
(\Omega^2 u^5 - 6\alpha u^2 ) \, .</math>
</td>
</tr>
</table>
</div>
Now, in the context of a [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Idea_Involving_Logarithmic_Derivatives|separate derivation]], we showed that, quite generally we can make the substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{u^2}{x} \cdot \frac{d^2x}{du^2}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{d\ln u} \biggl[ \frac{d\ln x}{d\ln u} \biggr]
+ \biggl[ \frac{d\ln x}{d\ln u}-1 \biggr]\cdot \frac{d\ln x}{d\ln u} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, if we assume that the displacement function can be expressed as a power-law in <math>~u</math>, such that,
<div align="center">
<math>\frac{d\ln x}{d\ln u} = c_0 \, ,</math>
</div>
then the LAWE for <math>~n=5</math> polytropes simplifies as follows,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(u^2-3) c_0(c_0-1) + (21 - 2u^2 ) c_0 + (\Omega^2 u^5 - 6\alpha u^2 ) \, .</math>
</td>
</tr>
</table>
</div>
This polynomial equation will be satisfied if, simultaneously, we set:
<ul>
<li><math>\Omega^2 = 0 \, ;</math></li>
<li><math>c_0^2 -3c_0 -6\alpha = 0 </math>      <math>~\Rightarrow</math>      <math>c_0 = \frac{3}{2}\biggl[1 \pm \biggl(1+\frac{8\alpha}{3} \biggr)^{1/2} \biggl]\, ;</math></li>
<li><math>~\alpha = 20/3 \, .</math>
</ul>

This gives us some hope that a more general solution of the following form will work:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~u^{c_0} \biggl[ a + bu + cu^2 + du^3 + \cdots\biggr] \, .</math>
</td>
</tr>
</table>
</div>

This means that, for example,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dx}{du}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
c_0 u^{c_0-1} \biggl[ a + bu + cu^2 + du^3 \biggr]
+ u^{c_0} \biggl[ b + 2cu + 3du^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{d\ln x}{d\ln u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{c_0(a + bu + cu^2 + du^3) + bu + 2cu^2 + 3du^3}{a + bu + cu^2 + du^3}
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2x}{du^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
c_0(c_0-1) u^{c_0-2} \biggl[ a + bu + cu^2 + du^3 \biggr]
+ 2c_0 u^{c_0-1} \biggl[ b + 2cu + 3du^2 \biggr]
+ u^{c_0} \biggl[ 2c + 6du \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{u^2}{x} \cdot \frac{d^2x}{du^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{c_0(c_0-1) ( a + bu + cu^2 + du^3 ) + 2c_0 ( bu + 2cu^2 + 3du^3 ) + ( 2cu^2 + 6du^3 ) }{ a + bu + cu^2 + du^3}
</math>
</td>
</tr>
</table>
</div>
So the LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- (\Omega^2 u^5 - 6\alpha u^2 ) (a + bu + cu^2 + du^3)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(u^2-3) [c_0(c_0-1) ( a + bu + cu^2 + du^3 ) + 2c_0 ( bu + 2cu^2 + 3du^3 ) + ( 2cu^2 + 6du^3 )]
+ (21 - 2u^2 ) [c_0(a + bu + cu^2 + du^3) + bu + 2cu^2 + 3du^3] \,.
</math>
</td>
</tr>
</table>
</div>
This is cute, but I don't see any way that this approach will provide an avenue to cancel the <math>~\Omega^2 u^5</math> term.

==Yet Another Guess==
Let's try,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e^{a + b\ln\xi + c(\ln\xi)^2} \, ,</math>
</td>
</tr>
</table>
</div>
and examine the specific case of <math>~\sigma_c^2 = 0</math>, and, <math>~\gamma = (n+1)/n = 6/5 ~~\Rightarrow~~ \alpha = (3-20/6) = -1/3</math>. Under these conditions, the LAWE for <math>~n=5</math> polytropes becomes,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3+\xi^2) \frac{d^2x}{d\xi^2} + \biggl[12 - 2\xi^2 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 2x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3+\xi^2) \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} + \biggl[12 - 2\xi^2 \biggr] \frac{\xi}{x} \cdot \frac{dx}{d\xi} + 2\xi^2 \, .
</math>
</td>
</tr>
</table>

And the derivatives give,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~x \frac{d}{d\xi}\biggl[ a + b\ln\xi + c(\ln\xi)^2 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~x \biggl[ \frac{b}{\xi}+ \frac{2c\ln\xi}{\xi} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~b + 2c\ln\xi \, ;</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2x}{d\xi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ \frac{b}{\xi}+ \frac{2c\ln\xi}{\xi} \biggr] \frac{dx}{d\xi} + x \frac{d}{d\xi}\biggl[ \frac{b}{\xi}+ \frac{2c\ln\xi}{\xi} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \xi \biggl[ \frac{b}{\xi}+ \frac{2c\ln\xi}{\xi} \biggr] \frac{\xi}{x} \cdot \frac{dx}{d\xi} + \xi^2 \frac{d}{d\xi}\biggl[ \frac{b+ 2c\ln\xi}{\xi} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ b + 2c\ln\xi \biggr]^2 + \xi \frac{d}{d\xi}\biggl[ b+ 2c\ln\xi \biggr]
+ (b+ 2c\ln\xi) \xi^2 \biggl[ - \frac{1}{\xi^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ (b + 2c\ln\xi )^2 + 2c- (b+ 2c\ln\xi)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[ b^2 + 2c - b] + [4bc - 2c] \ln\xi+4c^2 (\ln\xi)^2 \, .
</math>
</td>
</tr>
</table>
</div>

Hence the "fundamental mode" LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3+\xi^2) \biggl[ ( b^2 + 2c - b ) + (4bc - 2c) \ln\xi+4c^2 (\ln\xi)^2 \biggr]
+ (12 - 2\xi^2 ) \biggl[ b + 2c\ln\xi \biggr] \, .
+ 2\xi^2
</math>
</td>
</tr>
</table>
</div>

Now, this expression cannot be satisfied for arbitrary <math>~\xi</math>. But, here we seek a solution only at the surface for the ''specific'' model, <math>~\xi = 3</math>. Plugging this value into the expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~12 \biggl[ ( b^2 + 2c - b ) + (4bc - 2c) \ln 3+4c^2 (\ln 3)^2 \biggr]
+ (12 - 18 ) \biggl[ b + 2c\ln 3 \biggr] + 18
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2 \biggl[ ( b^2 + 2c - b ) + (4bc - 2c) \ln 3+4c^2 (\ln 3)^2 \biggr]
-\biggl[ b + 2c\ln 3 \biggr] + 3 \, .
</math>
</td>
</tr>
</table>
</div>
It appears as though one perfectly satisfactory solution is, <math>~c = 0</math>, in which case, we need,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2 b^2 - 3b + 3
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~b</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{4}\biggl[1 \pm \sqrt{1-\frac{8}{3} } \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Thus, <math>~b</math> is an complex number.

=Related Discussions=
* Radial Oscillations of [[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density sphere]]
* Radial Oscillations of Isolated Polytropes
** [[User:Tohline/SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|Setup]]
** n = 1:  [[User:Tohline/SSC/Stability/n1PolytropeLAWE|Attempt at Formulating an Analytic Solution]]
** n = 3:  [[User:Tohline/SSC/Stability/n3PolytropeLAWE|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]
** n = 5:  [[User:Tohline/SSC/Stability/n5PolytropeLAWE|Attempt at Formulating an Analytic Solution]]

* In an accompanying [[User:Tohline/Appendix/Ramblings/SphericalWaveEquation#Playing_With_Spherical_Wave_Equation|Chapter within our "Ramblings" Appendix]], we have played with the adiabatic wave equation for polytropes, examining its form when the primary perturbation variable is an enthalpy-like quantity, rather than the radial displacement of a spherical mass shell. This was done in an effort to mimic the approach that has been taken in studies of the [[User:Tohline/Apps/ImamuraHadleyCollaboration#Papaloizou-Pringle_Tori|stability of Papaloizou-Pringle tori]].

* <math>~n=3</math> … [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941, ApJ, 94, 245)], ''Overtone Pulsations of the Standard Model'': This work is referenced in §38.3 of [<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>]. It contains an analysis of the radial modes of oscillation of <math>~n=3</math> polytropes, assuming various values of the adiabatic exponent.

* <math>~n=2</math> … [http://adsabs.harvard.edu/abs/1961MNRAS.122..409P C. Prasad & H. S. Gurm (1961, MNRAS, 122, 409)], ''Radial Pulsations of the Polytrope, n = 2''

* <math>~n=\tfrac{3}{2}</math> … D. Lucas (1953, Bul. Soc. Roy. Sci. Liege, 25, 585) … Citation obtained from the Prasad & Gurm (1961) article.

* <math>~n=1</math> … L. D. Chatterji (1951, Proc. Nat. Inst. Sci. [India], 17, 467) … Citation obtained from the Prasad & Gurm (1961) article.

* Composite Polytropes … [http://adsabs.harvard.edu/abs/1968MNRAS.140..235S M. Singh (1968, MNRAS, 140, 235-240)], ''Effect of Central Condensation on the Pulsation Characteristics''

* Summary of Known Analytic Solutions … [http://adsabs.harvard.edu/abs/1981MNRAS.197..351S R. Stothers (1981, MNRAS, 197, 351-361)], ''Analytic Solutions of the Radial Pulsation Equation for Rotating and Magnetic Star Models''

* Interesting Composite! … [http://adsabs.harvard.edu/abs/1948MNRAS.108..414P C. Prasad (1948, MNRAS, 108, 414-416)], ''Radial Oscillations of a Particular Stellar Model''

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<ul>
<li>Menu Tile:  [https://www.vistrails.org/index.php/User:Tohline/Apps/PapaloizouPringleTori Papaloizou-Pringle Tori (1984)] = '''Apps/PapaloizouPringleTori'''; this chapter focuses on equilibrium structure.<li>
<li>Menu Tile:  [https://www.vistrails.org/index.php/User:Tohline/Apps/PapaloizouPringle84 (Massless) Papaloizou-Pringle Tori] = '''Apps/PapaloizouPringle84'''; in mid-July of 2020, <font color="red">Blaes granted permission</font> to include in this chapter some reprinted material from his 1985 paper.</li>
<li>Menu Tile:  [https://www.vistrails.org/index.php/User:Tohline/Apps/ImamuraHadleyCollaboration Analytic Analysis by Blaes (1985)] = '''Apps/ImamuraHadleyCollaboration'''; compare Blaes85 against Imamura-Hadley Collaboration]; in mid-July of 2020, <font color="red">Blaes granted permission</font> to include in this chapter some reprinted material from his 1985 paper.</li>
<li>[https://www.vistrails.org/index.php/User:Tohline/Apps/Blaes85SlimLimit#Oscillations_of_PP_Tori_in_the_Slim_Torus_Limit Oscillations in the Slim Torus limit] = '''Apps/Blaes85SlimLimit'''</li>
</ul>
</li>
<li>[[PGE/ConservingMomentum|Earlier version of the PGE/Euler chapter]]</li>
<li>[[SSC/Structure/PolytropesASIDE1|Explains Whitworth's chosen scaling]]</li>
<li>[[SSC/Structure/StahlerMassRadius|Details regarding Stahler's (1983) derived mass-radius relation]]</li>
<li>[[SSC/Structure/BiPolytropes|Practice use of ''image map'']] (Image:SchonbergChandra1942.jpg)</li>
<li>[[SSC/Structure/LimitingMasses|Mass Upper Limits]]</li>
<li>Bipolytrope Generalization:
<ul>
<li>[[SSC/BipolytropeGeneralization|Bipolytrope Generalization]] (original)</li>
<li>[[SSC/BipolytropeGeneralizationVersion2|Bipolytrope Generalization (Version 2)]]</li>
</ul>
</li>
<li>Note that …Template: "SSCstructure" points to original "table of contents"</li>
<li>[[SSC/StabilityEulerianPerspective|Stability of Spherically Symmetric Configurations (Eulerian Perspective)]] <-- Includes ''scratch work'' with rarely used html characters, such as <font color="green" size="+1">⑧</font></li>
<li>[[SSC/FreeEnergy/Powerpoint|Supports Free-Energy PowerPoint Presentation]]</li>
<li>[[StabilityVariationalPrinciple|Free-Energy Stability Analysis]] (early version)</li>
</ul>

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SSCpt1/Virial/FormFactors

2021-08-01T20:35:13Z

174.64.8.247: /* See Also */

__FORCETOC__

=Structural Form Factors=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|<b>Structural<br />Form<br />Factors</b>]]</font>
|}
As has been defined in [[SSCpt1/Virial#Structural_Form_Factors|a companion, introductory discussion]], three key dimensionless structural form factors are:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>x \equiv r/R_\mathrm{limit}</math>, and the subscript "0" denotes central values. The principal purpose of this chapter is to carry out the integrations that are required to obtain expressions for these structural form factors, at least in the few cases where they can be determined analytically. These form-factor expressions will then be used to provide expressions for the two constants, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, that appear in the free-energy function and in the virial theorem, and to provide corresponding expressions for the normalized energies, <math>W_\mathrm{grav}/E_\mathrm{norm}</math> and <math>S_\mathrm{therm}/E_\mathrm{norm}</math>.
 <br />

==Synopsis==

<div align="center"><b>Summary of Derived Structural Form-Factors</b></div>
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> Polytropes <math>(n \ne 5)</math>
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> Polytropes <math>(n \ne 5)</math>
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 1 Polytrope<br /><math>\tilde\xi \rightarrow \xi_1 = \pi</math>
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 1 Polytropes<br /><math>0 < \tilde\xi < \pi</math>
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\pi^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3^2\cdot 5}{2^2 \pi^4}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2\pi^2}
</math>
</td>
</tr>
</table>
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 5 Polytrope
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 5 Polytropes
</th>
</tr>
<tr>
<td align="center">
 <br />
 <br />
 <br />
 <br />
 <br />
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 1 + \ell^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>
where,     <math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}}</math>
</td>
</tr>
</table>

==Expectation in Context of Pressure-Truncated Polytropes==
For pressure-truncated polytropic configurations, the normalized virial theorem states that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>2 \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr) + \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} \, .</math>
</td>
</tr>
</table>
</div>
This provides one mechanism by which the correctness of our form-factor expressions can be checked. Specifically, having determined <math>S_\mathrm{therm}</math> and <math>W_\mathrm{grav}</math> from the derived form factors, we can see whether the sum of these energies as specified on the lefthand-side of this virial theorem expression indeed match the normalized energy term involving the external pressure, as specified on the righthand side. In order to facilitate this "reality check" at the end of each example, below, we will use [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's detailed force-balanced solution of the equilibrium structure of embedded polytropes]] to provide an expression for the term on the righthand side of the virial theorem expression.

We begin by plugging our [[SSCpt1/Virial#Normalizations|general expression for <math>E_\mathrm{norm}</math>]] into this righthand-side term and grouping factors to facilitate insertion of Stahler's expressions.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{tot}^{(5-n)} \biggr]^{1/(n-3)} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)/(n-3)} P_e R_\mathrm{eq}^3
\biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)} \, . </math>
</td>
</tr>
</table>
</div>
From [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's equilibrium solution]], we have,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
R_\mathrm{eq}
</math>
</td>
<td align="center">
<math>=~</math>
</td>
<td align="left">
<math>
R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
~P_e R_\mathrm{eq}^3
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{1 + 3(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
M_\mathrm{limit}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
K^{-n} G^3 M_\mathrm{limit}^{(5-n)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3-3(5-n)/2} K_n^{-n +2n(5-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3(n-3)/2} K_n^{3n(3-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} \biggr\}^{(5-n)/(n-3)} G^{3/2} K_n^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>\times ~\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(5-n)/2}
\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^{3(n-3)}_{\tilde\xi} \biggl( \frac{n+1}{4\pi} \biggr)^{3(n-3)/2} \biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{
(n+1)^{3[(5-n)+(n-3)]/2} (4\pi)^{[(n-5)+(9-3n)]/2}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{[(n-3)(5-n) + 3(n-1)(n-3)]/2} \tilde\xi^{[2(5-n) + 3(n-3)]}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{ (n+1)^{3} (4\pi)^{(2-n)}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{(n+1)(n-3)} \tilde\xi^{(n+1)}
\biggr\}^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, the expectation based on Stahler's equilibrium models is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

As a cross-check, multiplying this expression through by <math>[(R_\mathrm{eq}/R_\mathrm{norm})(M_\mathrm{norm}/M_\mathrm{limit})^2]</math> — where the expression for <math>R_\mathrm{eq}/R_\mathrm{norm}</math> can be obtained from our [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|discussions of detailed force-balanced models]] — gives a related result that can be obtained directly from [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|Horedt's expressions]], namely,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{4\pi P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(n+1)( -\tilde\theta' )^{2}} \, .
</math>
</td>
</tr>
</table>
</div>

==Viala and Horedt (1974) Expressions==
===Presentation===

[http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have provided analytic expressions for the gravitational potential energy and the internal energy — which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively — that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres. [The same expression for <math>~\Omega</math> is also effectively provided in §1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
<div align="center">
<table border="1" align="center" cellpadding="8" width="90%">
<tr><td align="center">


Astronomy & Astrophysics, 33: 195-202, (1974)<br />
POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
Y. P. Viala & Gp. Horedt<br />
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>\Omega</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- G\int_0^M \frac{MdM}{r} = \frac{16\pi^2 G \rho_0^2 \alpha^5}{(5-n)} \biggl[
\mp \xi^3 \theta^{n+1} - 3\xi^3 (\theta')^2 - 3\xi^2 \theta (\theta')
\biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>U</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{\gamma - 1}\int_V pdV = \frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \int_0^\xi \theta^{n+1} 4\pi \alpha^2 \xi^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \cdot
\frac{4\pi \alpha^2(n+1)}{(5-n)}
\biggl[
\frac{2\xi^3 \theta^{n+1}}{n+1} \pm \xi^3 (\theta')^2 \pm \xi^2 \theta (\theta')
\biggr]_0^\xi \, .
</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
(the superior sign holds if <math>-1 < n < \infty</math>, the inferior if <math>-\infty < n < -1</math>)
</td>
</tr>
</table>

</td></tr>
<tr>
<td align="left">
A couple of key equations drawn directly from [http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have been shown here. As its title indicates, the paper includes discussion of — and accompanying equation derivations for — equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries: planar sheets, axisymmetric cylinders, and spheres. We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only. These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
</td>
</tr>
</table>
</div>
Rewriting these two expressions to accommodate our parameter notations — recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> — we have from Viala & Horedt's (VH74) work,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \mathfrak{S}_\mathrm{A} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n(4\pi)^2}{3(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

===First Reality Check===

As a quick reality check, let's see whether, when appropriately added together, these two energies satisfy the scalar virial theorem for isolated polytropes.

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
W_\mathrm{grav} + \frac{3}{n} \mathfrak{S}_A</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>+
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} - 1 \biggr] \tilde\xi^3 \tilde\theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(n+1)} \cdot G \rho_c^2 a_n^5 \tilde\xi^3 \tilde\theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>
For ''isolated polytropes'', <math>\tilde\theta \rightarrow 0</math>, so this sum of terms goes to zero, as it should if the system is in virial equilibrium.

===Renormalization===
Both of the energy-term expressions derived by Viala & Horedt are written in terms of <math>\rho_c</math> and
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>a_\mathrm{n}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n}\biggr]^{1/2}
</math>
</td>
</tr>
</table>
</div>
— that is, effectively in terms of <math>\rho_c</math> and {{ Template:Math/MP_PolytropicConstant}} — whereas, in the context of our discussions, we would prefer to express them in terms of [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Adopted_Normalizations|our generally adopted energy normalization]],
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_n^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>
In order to accomplish this, we need to replace the central density with the total mass of an ''isolated polytrope'', <math>M_\mathrm{tot}</math>, whose generic expression is (see, for example, equation 69 of Chandrasekhar),
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}^{n-3}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_n^n G^{-3}\biggl\{ (4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr\}^{n-5} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
K_n^{[2n + 3(n-5)]/2} G^{[-6-3(n-5)]/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
\rho_c^{(n-3)(5-n)/2n} K_n^{5(n-3)/2} G^{-3(n-3)/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)}
\rho_c^{(5-n)/2n} K_n^{5/2} G^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\rho_c^{[ - 4n +(5-n)]/2n} \biggl( \frac{K_n}{G}\biggr)^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{K_n}{G} \cdot \rho_c^{(1-n)/n}\biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{4\pi}{(n+1)} \cdot a_n^2 \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~(4\pi)^2 G\rho_c^2 a_n^5</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (4\pi)^2
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(5-n)/(n-3)}
\biggl[ \frac{(n+1)}{4\pi} \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)/(n-3)}
(4\pi)^{[-(n-3)-(5-n)]/2(n-3)} (n+1)^{[3(5-n)+5(n-3)]/2(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
So, employing our preferred normalization, the VH74 expressions become,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{\mathfrak{S}_\mathrm{A}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n}{3(5-n)}
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

===Second Reality Check===
If we now renormalize the sum of energy terms discussed in our [[SSCpt1/Virial/FormFactors#First_Reality_Check|first reality check, above]], we have,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>
\frac{1}{E_\mathrm{norm}} \biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}
= \frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(n+1)^{-1} \tilde\xi^3 \tilde\theta^{n+1} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

(This may or may not be useful!)

===Implication for Structural Form Factors===
On the other hand, our expressions for these two [[SSCpt1/Virial#Structural_Form_Factors|normalized energy components written in terms of the structural form factors]] are,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>
where, in equilibrium (see [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|here]] and [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|here]] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl\{ \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl\{ \biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-1)/(n-3)}
\biggr\} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
Hence, we deduce that,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> - \frac{5}{3} \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]
\chi_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \cdot \tilde\mathfrak{f}^2_M
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{5}{3} \biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggr\}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(n-1)-2(n-3)]/(n-3)}
\cdot \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr)^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{[(1-n)+2(n-3)]/(n-3)} \tilde\xi^{[-(n-3)+2(1-n)]/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{(n-5)/(n-3)} \tilde\xi^{(5-3n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
If we now adopt the VH74 expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\biggl\{~~~\biggr\}_\mathrm{VH74}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)}
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
(-\tilde\xi^2 \tilde\theta^')^{(5-n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Therefore,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_W \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)} \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\tilde\xi^{-5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>

Now, from [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|our earlier work]] we deduced that <math>\tilde\mathfrak{f}_A</math> is related to <math>\tilde\mathfrak{f}_W</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \tilde\mathfrak{f}_W\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \, .</math>
</td>
</tr>
</table>
</div>
Hence, we now have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_A \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \frac{(n+1)}{(5-n)}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
\, .
</math>
</td>
</tr>
</table>
</div>

Building on the work of VH74, we have, quite generally,

<div align="center" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>
</table>

</td>
</tr>
<tr>
<td align="left" colspan="2">
We should point out that [http://adsabs.harvard.edu/abs/1993ApJS...88..205L Lai, Rasio, & Shapiro (1993b, ApJS, 88, 205)] define a different set of dimensionless structure factors for ''isolated'' polytropic spheres — <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10) is used in the determination of the gravitational potential energy.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>k_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[ \frac{n(n+1)}{5-n} \biggr] \xi_1|\theta^'_1|</math>
</td>
</tr>

<tr>
<td align="right">
<math>k_2</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{5-n} \biggl[ \frac{4\pi |\theta^'_1|}{\xi_1} \biggr]^{1 / 3} </math>
</td>
</tr>
</table>
</div>
Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter. We note, as well, that for rotating configurations they define two additional dimensionless structure factors — <math>k_3</math> (their equation 3.17) is used in the determination of the rotational kinetic energy; and <math>\kappa_n</math> (their equation 3.14; also equation 7.4.9 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]) is used in the determination of the moment of inertia.
</td>
</tr>
</table>
</div>

The singularity that arises when <math>n = 5</math> leads us to suspect that these general expressions fail in that one specific case. Fortunately, as [[#Summary_.28n.3D5.29|we have shown in an accompanying discussion]], <math>\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, as well as <math>\mathfrak{f}_M</math>, can be determined by direct integration in this single case.

===Related Discussions===
* See [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Model_Sequences|our plot of, what Kimura (1981b) would refer to as, several <math>M_1</math> sequences]]

==First Detailed Example (n = 5)==

Here we complete these integrals to derive detailed expressions for the above subset of structural form factors in the case of spherically symmetric configurations that obey an <math>~n=5</math> polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should simplify the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases that cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|cores of bipolytropes]].

===Foundation (n = 5)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=5; ~\gamma=6/5)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{K^{10}}{G^{9} M_\mathrm{tot}^{6}} \biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl( \frac{K}{G} \biggr)^{15/2} M_\mathrm{tot}^{-5} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{K^5}{G^3} \biggr)^{1/2} M_\mathrm{tot}^{-1} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/BiPolytropes/Analytic5_1#Profile|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,
<div align="center">
<math>
a_{5} =\biggr[ \frac{3K}{2\pi G} \biggr]^{1/2} \rho_0^{-2/5} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{[K^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^{6/5}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{[K^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5}
~~~~\Rightarrow ~~~~
\rho_0^{1/5} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} M_\mathrm{tot}^{-1} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr) \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi
= \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3^4}{\pi} \biggr)^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr)^{1/2}
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
= \biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2}
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 5)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math>, and recognizing that,
<div align="center">
<math>dr^\dagger = \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} d\xi \, ,</math>
</div>
gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], [[Image:OnlineIntegral01.png|250px|right|Mathematica Integral]]
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{3/2} \biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2}
\int_0^{\xi} \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{1}{3} \biggr)^{3/2}
\biggl[ \frac{\xi^3}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-3/2} \biggr]_0^{\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\xi^2}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=5</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 5)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5} \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=5</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_5 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_5 \xi}{a_5 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=5</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{3/2} \biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[SSCpt1/Virial/FormFactors#NormalizedProfiles|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2} \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 5)===

From the above line of reasoning we appreciate that, for any spherically symmetric configuration, the ratio of the configuration's mean density to its central density can be obtained by setting the upper limit of our just-completed "Mass2" integration to <math>x=1</math>. That is to say, quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>
</table>
</div>
But the integral expression on the righthand side of this relation is also the definition of the structural form factor, <math>~\mathfrak{f}_M</math>, given at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]]. Hence, we can say, quite generally, that,
<div align="center">
<math>\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} \, .</math>
</div>
And, given that we have just completed this integral for the case of truncated <math>n=5</math> polytropic structures, we can state, specifically, that,
<div align="center">
<math>\mathfrak{f}_M\biggr|_{n=5} = \biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</div>

===Gravitational Potential Energy (n = 5)===
As presented at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral02.png|225px|right|Mathematica Integral]]Given that an expression for the normalized density profile, <math>\rho(x)/\rho_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been determined, above]], we can carry out the nested pair of integrals immediately. Indeed, the integral contained inside of the curly braces has already been completed [[SSCpt1/Virial/FormFactors#Mass2|in the "Mass2" subsection, above]], in order to determine the radial mass profile. Specifically, we have already determined that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ \int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\}
\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} x dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-4} x^4 dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \biggl( \frac{\tilde\xi^2}{3}\biggr)^{-5/2} \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-3}
\biggl\{
\biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggl[ 3\biggl( \frac{\tilde\xi^2}{3}\biggr)^2 - 8\biggl( \frac{\tilde\xi^2}{3}\biggr) - 3 \biggr]
+ 3\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr)^3\tan^{-1}\biggl[ \biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>\mathfrak{f}_M</math> and <math>\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>n=5</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{\pi}{2^3\cdot 3^7}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
In order to simplify typing, we will switch to the variable,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
in which case a summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>( 1 + \ell^2 )^{-3/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \cdot \ell^{-5} (1 + \ell^2)^{-3}
\biggl\{ \ell [ 3\ell^4 - 8\ell^2 - 3 ] + 3( 1 + \ell^2 )^3\tan^{-1}(\ell ) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{3}{5} \biggl(\frac{2^3\cdot 3^6}{\pi}\biggr)^{1/2} \frac{1}{\ell} \cdot (1 + \ell^2)^3
\mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5 \pi}\biggr)^{1/2} \cdot \ell^{-6} (1 + \ell^2)^3
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized gravitational potential energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>W_\mathrm{core}^*</math>.

Hence, also, as defined in the [[SSCpt1/Virial#Gathering_it_All_Together|accompanying introductory discussion]], the constant, <math>\mathcal{A}</math>, that appears in our general free-energy equation is (for <math>n=5</math> polytropic configurations),
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 5)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral03.png|225px|right|Mathematica Integral]]Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} x^2 dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{2^3}
\biggl\{\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-3/2} \tan^{-1}\biggl[ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
+ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-1}
- 2\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-2}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=5 (\gamma=6/5)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)^{1/5}
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{6/5}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>

If, as above, we adopt the simplifying variable notation,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
the various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} [ \ell^{-3} \tan^{-1}(\ell ) + \ell^{-2}(1+\ell^2)^{-1} - 2\ell^{-2}(1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, ;
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3/5} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2} \cdot \biggl[ \biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \biggr]^{-3/5}
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5} \ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{3^{10}}{2^{10}} \biggl(\frac{2^9\cdot 3^{18}}{\pi^3}\biggr)
\biggl(\frac{3^{12}}{2^{34} \pi^2} \biggr) \biggr]^{1/10} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^{8}}{2^{7}\pi}\biggr)^{1/2} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized thermal energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>S_\mathrm{core}^*</math>. Its similarity to the expression for the gravitational potential energy — which is relevant to the virial theorem — is more apparent if it is rewritten in the following form:
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2}
\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 5)===
In summary, for <math>n=5</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 1 + \ell^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 5)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} \biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Check (n = 5)===

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}\biggl\{ [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
-
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}
\biggl[\frac{8}{3}\ell^3 (1+\ell^2)^{-3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2 \cdot 3^{6}}{\pi}\biggr)^{1/2}
\biggl[\frac{\ell}{ (1+\ell^2)} \biggr]^3 \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=5</math> polytropes — see the [[#Mass1|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \biggl[ \ell^2(1+\ell^2)^{-1} \biggr]^{3/2}
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_5 = ( 1 + \ell^2 )^{-1/2}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_5}{d\xi} \biggr|_{\xi_e} = 3^{1/2} \ell ( 1 + \ell^2 )^{-3/2} \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2\cdot 3^3}{\pi}\biggr]^{1/2} ( 1 + \ell^2 )^{-3} (3^{1/2}\ell)^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{2\cdot 3^6}{\pi}\biggr)^{1/2} \biggl[ \frac{\ell}{( 1 + \ell^2 )} \biggr]^{3} \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors. This gives us confidence that our form-factor expressions are correct, at least in the case of embedded <math>n=5</math> polytropic structures.

==Second Detailed Example (n = 1)==

===Foundation (n = 1)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=1; ~\gamma=2)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl(\frac{K}{G}\biggr)^{1/2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G^3 M_\mathrm{tot}^2}{K^2}\biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot}^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi}\biggl( \frac{G}{K} \biggr)^{3/2} M_\mathrm{tot} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of the properties of ''isolated'' polytropes]], in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{1}</math>, where,
<div align="center">
<math>
a_{1} = \biggl( \frac{K}{2\pi G} \biggr)^{1/2} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{(K/G)^{1/2}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^{2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{(K/G)^{3/2}\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2}{\pi} \biggr)^{1/2} (\sin\xi - \xi \cos\xi) \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=1</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0
~~~~\Rightarrow ~~~~
\rho_0 = \biggr[ \frac{G^3}{2\pi K^3} \biggr]^{1/2} M_\mathrm{tot} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles1">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3 \pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\pi} \biggl( \frac{\sin\xi}{\xi}\biggr)^2 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi \cos\xi)
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 1)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math> gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \int_0^{\xi} \frac{\xi^2}{2\pi} \biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \cdot \frac{d\xi }{(2\pi)^{1/2}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3( 2\pi)^{-3/2}\biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \int_0^{\xi} \xi \sin\xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi\cos\xi) \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=1</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 1)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0 \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=1</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1 = \pi</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_1 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_1 \xi}{a_1 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx1">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin(\tilde\xi x)}{\tilde\xi x} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi} \int_0^{x} x \sin(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{3}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, . </math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=1</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
\biggl\{ \frac{[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ]}{( \sin \tilde\xi - \tilde\xi \cos \tilde\xi )} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] </math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[#NormalizedProfiles1|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\xi - \xi \cos\xi ) \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 1)===

[[SSCpt1/Virial/FormFactors#Mean-to-Central_Density|Following the line of reasoning provided above]], we can use the just-derived central-to-mean density ratio to specify one of the structural form factors. Specifically,
<div align="center">
<math>~\mathfrak{f}_M\biggr|_{n=1} = \frac{\bar\rho}{\rho_c} = \frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</div>

===Gravitational Potential Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
From the derivations already presented, above, for <math>~n=1</math> polytropic configurations, we know all of the functions under this integral. We know, for example, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \int_0^1 \biggl\{ [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \biggr\}
\sin(\tilde\xi x) dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \biggl\{ \frac{x}{2} - \frac{\sin(2\tilde\xi x)}{4\tilde\xi} - \tilde\xi\biggl[
\frac{\sin(2\tilde\xi x) - 2\tilde\xi x\cos(2\tilde\xi x)}{8\tilde\xi^2}
\biggr] \biggr\}_0^1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>~n=1</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- 3\mathcal{A} \chi^{-1} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\chi</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
A summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{5} \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^{2} \mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[#Foundation_2|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi}\biggr)^{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\sin(\tilde\xi x)}{(\tilde\xi x)}\biggr]^{2} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2} \int_0^1 \sin^2(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2}
\biggl[\frac{x}{2}- \frac{\sin(2\tilde\xi x)}{4\tilde\xi} \biggr]_0^1
= \frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=1 (\gamma=2)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>
The various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\xi^3}{3\pi} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr) \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^2
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3 \tilde\xi^3}{2^5\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
\biggl[ (2\pi)^{3/2} \tilde\xi^{-3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 1)===
In summary, for <math>~n=1</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 1)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Checks (n = 1)===
====Expectation from Stahler's Equilibrium Models====
If we add twice the thermal energy to the gravitational potential energy, we obtain,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} \biggl\{ \biggl[6\tilde\xi - 3\sin(2\tilde\xi ) \biggr]
- \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} 2\tilde\xi\biggl\{ 1-\cos(2\tilde\xi ) \biggr\}
= \biggl( \frac{1}{2\pi^3} \biggr)^{1/2} \tilde\xi \sin^2(\tilde\xi ) \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=1</math> polytropes — see the [[#Mass1_.28n_.3D_1.29|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D1.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_1 = \frac{\sin\tilde\xi}{\tilde\xi}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_1}{d\xi} \biggr|_{\tilde\xi} = \frac{1}{\tilde\xi^2}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2}{\pi} \biggr]^{-1/2} \biggl[ \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \tilde\xi^2(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)^{-1}\biggr]^{2}
\biggl( \frac{\sin\tilde\xi}{\tilde\xi}\biggr)^2 \tilde\xi^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi^3}\biggr)^{1/2} \tilde\xi \sin^2\tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors, giving us additional confidence that our form-factor expressions are correct.

====Compare With General Expressions Based on VH74 Work====
Based on the general expressions [[#PTtable|derived above]] in the context of VH74's work, for the specific case of <math>n=1</math> polytropic configurations, the three structural form factor should be,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\tilde\xi^2}
\biggl[\tilde\theta^{2} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ 3\tilde\theta^{2} +
3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
Also, remember that,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\theta</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~\theta^' \equiv \frac{d\theta}{d\xi}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\cos\xi}{\xi} - \frac{\sin\xi}{\xi^2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~(\theta^' )^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\xi^4} \biggl[ \xi\cos\xi - \sin\xi \biggr]^2
=\frac{1}{\xi^4} \biggl[ \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr] \, .
</math>
</td>
</tr>
</table>
Now, let's look at the structural form factors, one at a time. First, we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\xi^3} \biggl[\sin\xi - \xi\cos\xi \biggr]</math>
</td>
</tr>
</table>

which matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Next,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~ \frac{3\cdot 5}{4\xi^2} \biggl[ \frac{\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 + 2\xi^2\cos^2\xi - 3 \xi\sin\xi \cos\xi \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl\{ \xi^2 + \xi^2[1+\cos(2\xi)] - \frac{3}{2} \xi\sin(2\xi) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{8\xi^6} \biggl[ 4\xi^2 + 2\xi^2 \cos(2\xi) - 3 \xi\sin(2\xi) \biggr] \, ,
</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Finally,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ \frac{3\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi \biggr)
+ 3\xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2\xi^4} \biggl[ \xi^2 - \xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2^2\xi^3} \biggl[ 2\xi - \sin(2\xi) \biggr] \, ,</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. So this adds support to the deduction, above, that VH74 have provided us with the information necessary to develop general expressions for the three structural form factors.

==Fiddling Around==
NOTE (from Tohline on 17 March 2015): Chronologically, this "Fiddling Around" subsection was developed before our discovery of the VH74 derivations. It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres. But this subsection's conclusions are superseded by the VH74 work.

In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>. In the case of <math>n=1</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\sin\xi}{\xi}\biggr)^2
= \frac{1}{2\xi^2} \biggl[ 1 - \cos(2\xi) \biggr]
= \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\mathfrak{f}_A - \theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[6\xi - 3\sin(2\xi ) \biggr]
- \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[4\xi - 3\sin(2\xi ) + 2\xi\cos(2\xi) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^3 \xi^5}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ 4\xi - 3\sin(2\xi) + 2\xi \cos(2\xi ) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2 \xi^2}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

Similarly, in the case of <math>n = 5</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\biggl( \frac{2^3}{3} \ell^{3} \biggr) \biggl[ \mathfrak{f}_A - \theta^{n+1} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
[ \tan^{-1}(\ell ) + \ell (\ell^4-1) (1+\ell^2)^{-3} ] - \biggl( \frac{2^3}{3} \ell^{3} \biggr) (1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\tan^{-1}(\ell ) + \ell \biggl(\ell^4-\frac{8}{3}\ell^2 - 1 \biggr) (1+\ell^2)^{-3} \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^4}{5} \cdot \ell^{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2\cdot 3}{5} \cdot \ell^{2} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{2\xi^2}{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>
This is pretty amazing! Both examples produce almost exactly the same relationship between the two structural form factors, <math>\mathfrak{f}_A</math> and <math>\mathfrak{f}_W</math>. I think that we are well on our way toward nailing down the generic, analytic relationship and, in turn, a generally applicable mass-radius relationship for pressure-truncated polytropic configurations.

Okay … here is the final piece of information. In the case of isolated polytropes, we know that the correct expressions for the structural form factors are as summarized in the following table:
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi}
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>
</div>
We notice, from this, that the ratio,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} \cdot \frac{5-n}{3^2\cdot 5} \biggl[ \frac{\xi}{\Theta^'} \biggr]^{2}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(n+1)\tilde\xi^2 }{3\cdot 5} \, .
</math>
</td>
</tr>
</table>
Even in the case of the two pressure-truncated polytropes, analyzed above, this ratio proves to give the correct prefactor on <math>\mathfrak{f}_W</math>. So we ''suspect'' that the universal relationship between the two form factors is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{(n+1) \xi^2 }{3\cdot 5} \biggr] \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

=See Also=
<ul>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>

{{ SGFfooter }}

SSCpt1/PGE

2021-08-01T20:34:12Z

174.64.8.247: /* See Also */

__FORCETOC__


=PGE for Spherically Symmetric Configurations=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:lightgreen;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|One-Dimensional<br />PGEs]]</font>
|}
If the self-gravitating configuration that we wish to construct is spherically symmetric, then the coupled set of multidimensional, partial differential equations that serve as our [[PGE#Principal_Governing_Equations|principal governing equations]] can be simplified to a coupled set of one-dimensional, ordinary differential equations. This is accomplished by expressing each of the multidimensional spatial operators — gradient, divergence, and Laplacian — in spherical coordinates<sup>&dagger;</sup> <math>(r, \theta, \varphi)</math> then setting to zero all derivatives that are taken with respect to the angular coordinates <math>\theta</math> and <math>\varphi</math>. After making this simplification, our governing equations become,
 <br />
 <br />
 <br />

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr] = 0 </math><br />

<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </math><br />

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{ Template:Math/EQ_FirstLaw02 }}

<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math><br />
</div>

=Footnotes=

<sup>&dagger;</sup>See, for example, the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates].

=See Also=
<ul>
<li>
Part 2 of ''Spherically Symmetric Configurations'': Structure — [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Solution Strategies]]
</li>
<li>
Part 2 of ''Spherically Symmetric Configurations'': Stability — [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Linearization of Governing Equations]]
</li>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>

{{ SGFfooter }}

SSC/VariationalPrinciple

2021-08-01T20:16:49Z

174.64.8.247: /* Directly to n = 5 Polytropic Configurations */

__FORCETOC__


=Ledoux's Variational Principle=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>Variational<br />Principle</b>]]</font>
|}
All of the discussion in this chapter will build upon our [[SSC/Perturbations#2ndOrderODE|derivation elsewhere]] of the,
<div align="center" id="2ndOrderODE">
<font color="#770000">'''LAWE:   Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

We will draw heavily from the papers published by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964)], as well as from pp. 458-474 of the review by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] in explaining how the ''variational principle'' can be used to identify the eigenvector of the fundamental mode of radial oscillation in spherically symmetric configurations. In an associated "Ramblings" appendix, we provide [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29|various derivations that support]] this chapter's relatively abbreviated presentation.

==Ledoux and Pekeris (1941)==
Historically, by the 1940s, the LAWE was a relatively familiar one to astrophysicists. For example, the opening paragraph of a 1941 paper by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124), reads:
<div align="center">
<table border="1" cellpadding="5">
<tr><td align="center">
Paragraph extracted from [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941)]<p></p>
"''Radial Pulsations of Stars''"<p></p>
ApJ, vol. 94, pp. 124-135 © American Astronomical Society
</td></tr>
<tr>
<td>
[[File:LedouxPekeris1941.jpg|600px|center|Ledoux & Pekeris (1941, ApJ, 94, 124)]]
</td>
</tr>
</table>
</div>

If we divide their equation (1) through by <math>~Xr = \Gamma_1 P r</math> and recognize that,
<div align="center">
<math>
\frac{dX}{dr} = \frac{dX}{dm}\frac{dm}{dr} = - \Gamma_1 g_0 \rho \, ,
</math>
</div>

we obtain,

<div align="center">
<math>
\frac{d^2\xi}{dr^2} + \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} +\frac{\rho}{\Gamma_1 P} \biggl[ \sigma^2 + (4 - 3\Gamma_1) \frac{g_0}{r} \biggr] \xi = 0 \, .
</math>
</div>
Clearly, this 2<sup>nd</sup>-order, ordinary differential equation is the same as our derived LAWE, but with a more general definition of the adiabatic exponent that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.

Multiplying this last equation through by <math>~\Gamma_1 P r^4</math>, and recognizing that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(r^4 \Gamma_1 P)\frac{d^2\xi}{dr^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
</math>
</td>
</tr>
</table>
</div>

<span id="RewrittenLAWE">we can write,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
+ ( \Gamma_1 P r^4 ) \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr}
+\rho \biggl[ \sigma^2 r^4 + (4 - 3\Gamma_1) g_0 r^3\biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \biggl[4r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr} \biggr] \frac{d\xi}{dr}
+ \biggl[ 4 r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr}\biggr] \frac{d\xi}{dr}
+\biggl[ \sigma^2 \rho r^4 - (4 - 3\Gamma_1) r^3 \frac{dP}{dr} \biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
(checked for n = 5) ==>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
+\biggl[ \sigma^2 \rho r^4 + (3\Gamma_1 - 4) r^3 \frac{dP}{dr} \biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ \Gamma_1 P r^4 ~\frac{d\xi}{dr} \biggr]
+\biggl[ \sigma^2 r^4 \rho + 4 Gm (r ) r \rho + 3\Gamma_1 r^3 \frac{dP}{dr} \biggr] \xi \, .
</math>
</td>
</tr>
</table>
</div>
Assuming that <math>~\Gamma_1</math> is uniform throughout the configuration, this last expression is the same as equation (3) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)], while the next-to-last expression is identical to equation (58.1) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)].

==Stability Based on Variational Principle==

Here we derive the Lagrangian directly from the governing LAWE. We begin with the next-to-last derived form of the LAWE that [[#RewrittenLAWE|appears above]] in our review of the paper by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and, following the guidance provided at the top of p. 666 of [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964, ApJ, 139, 664)], we multiply the LAWE through by the fractional displacement, <math>~\xi</math>. This gives, what we will henceforth refer to as, the,

<div align="center" id="FoundationalVariationalRelation">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3"><font color="maroon"><b>Foundational Variational Relation</b></font></td>
</tr>
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\xi \cdot \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) \, .
</math>
</td>
</tr>
</table>
</div>

===Chandrasekhar's Approach===

Next, in an effort to adopt the notation used by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C Chandrasekhar (1964)], we make the substitution, <math>~\xi \rightarrow \psi/r^3</math>, and regroup terms to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\sigma^2 \rho \psi^2}{r^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d}{dr} \biggl( \frac{\psi}{r^3} \biggr) \biggr]
- (3\Gamma_1 - 4) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r \Gamma_1 P ~\frac{d\psi}{dr} -3 \Gamma_1 P \psi ~\biggr]
- (3\Gamma_1 - 4) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4-3\Gamma_1 ) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r \Gamma_1 P ~\frac{d\psi}{dr} \biggr]
+ 3 \Gamma_1 \biggl( \frac{\psi^2}{r^3}\biggr) \frac{dP}{dr}
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
- \biggl\{
\frac{d}{dr}\biggl[ r \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}\biggr] -r\Gamma_1 P ~\frac{d\psi}{dr} \cdot \frac{d}{dr}\biggl( \frac{\psi}{r^3}\biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
+ \frac{\Gamma_1 P}{r^2} \biggl[\frac{d\psi}{dr} \biggr]^2
- \biggl[\frac{3\Gamma_1 P\psi}{r^3}\biggr]\frac{d\psi}{dr}
- \frac{d}{dr}\biggl[\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+ \frac{\Gamma_1 P}{r^2} \biggl[\frac{d\psi}{dr} \biggr]^2
- \frac{d}{dr}\biggl[ \frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">
Let's check to see whether the terms in the RHS of this last expression sum to zero when we plug in the appropriate functions for the marginally unstable, n = 5 configuration. In particular (replacing <math>~\xi</math> with <math>~x</math>, and setting <math>~r = a_5\xi</math>), we start with knowing,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="left">
<math>~\theta_5 = \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2}</math>;        
</td>
<td align="left">
<math>~\frac{d\theta_5}{d\xi} = - \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}</math>;        
</td>
</tr>

<tr>
<td align="left">
<math>~x = \biggl(\frac{3\cdot 5 - \xi^2}{3\cdot 5} \biggr)</math>;        
</td>
<td align="left">
<math>~\frac{dx}{d\xi} = -\frac{2\xi}{3\cdot 5}</math>;        
</td>
</tr>

<tr>
<td align="left">
<math>~\psi = a_5^3 \xi^3 x</math>;        
</td>
<td align="left">
<math>~
\frac{d\psi}{d\xi} = a_5^3 \biggl[ 3\xi^2 x + \xi^3 \biggl(\frac{dx}{d\xi}\biggr)\biggr] = \frac{a_5^3 \xi^2}{3}\biggl( 3^2 - \xi^2 \biggr) \, .
</math>
</td>
</tr>
</table>
</div>

----

Then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathrm{RHS}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl[ \frac{a_5^6 \xi^6 x^2}{a_5^3 \xi^3} \biggr] \frac{P_c}{a_5} \cdot \frac{d\theta^6}{d\xi}
+ P_c \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6}{a_5^4 \xi^2} \biggl\{ \frac{d\psi}{d\xi} \biggr\}^2
- \frac{P_c}{a_5} \cdot \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6 a_5^3 \xi^3 x}{a_5^3 \xi^2} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{\mathrm{RHS}}{P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl[ \xi^3 x^2 \biggr] \frac{d\theta^6}{d\xi}
+ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6}{ \xi^2} \biggl\{ \frac{\xi^2}{3}\biggl( 3^2 - \xi^2 \biggr) \biggr\}^2
- \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6 \xi^3 x}{ \xi^2} \biggl[ \frac{\xi^2}{3}\biggl( 3^2 - \xi^2 \biggr)\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2^3\cdot 3 \xi^3 x^2 \biggl(\frac{3+\xi^2}{3}\biggr)^{-5 / 2} \biggl[- \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]
+ \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{\xi}{3}\biggr)^2 \biggl(\frac{3+\xi^2}{3}\biggr)^{-3} ( 3^2 - \xi^2 )^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr)\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \frac{\xi^3 x}{ 3} ( 3^2 - \xi^2) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2^3 \cdot 3^4 ~\xi^4 x^2 \biggl(\frac{1}{3+\xi^2}\biggr)^{4}
+ \biggl(\frac{2\cdot 3^2}{5}\biggr) \xi^2 \biggl[ \frac{( 3^2 - \xi^2 )^2}{(3+\xi^2)^3} \biggr]
- \biggl(\frac{2\cdot 3^3}{5}\biggr)\frac{d}{d\xi}\biggl\{ \xi^3 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5 (3+\xi^2)^4 [\mathrm{RHS} ]}{ 2\cdot 3^2 P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2^2 \cdot 3^2 \cdot 5~\xi^4 x^2
+ \xi^2 (3+\xi^2) ( 3^2 - \xi^2 )^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 3(3+\xi^2)^4 \biggl\{
\xi^3 \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr] \frac{dx}{d\xi}
+ 3\xi^2 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr]
+ \xi^3 x \biggl[ \frac{ -2\xi }{(3+\xi^2)^3} \biggr]
-2\cdot 3 \xi^4 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^4} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^2 (3+\xi^2) ( 3^2 - \xi^2 )^2
- 2^2 \cdot 3^2 \cdot 5~\xi^4 x^2
- 3\xi^3 ( 3^2 - \xi^2) (3+\xi^2) \biggl[ - \frac{2\xi}{ 3\cdot 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 3 \xi^2\biggl\{
3 ( 3^2 - \xi^2) (3+\xi^2)
-2 \xi^2 (3+\xi^2)
-2\cdot 3 \xi^2 ( 3^2 - \xi^2)
\biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5^2 (3+\xi^2)^4 [\mathrm{RHS} ]}{ 2\cdot 3^2 ~\xi^2 P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
5 (3+\xi^2) ( 3^2 - \xi^2 )^2
- 2^2~\xi^2 (15-\xi^2)^2
+ 2 \xi^2 ( 3^2 - \xi^2) (3+\xi^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[
2 \xi^2 (3+\xi^2)
+ 2\cdot 3 \xi^2 ( 3^2 - \xi^2)
- 3 ( 3^2 - \xi^2) (3+\xi^2)
\biggr] (15-\xi^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(3\cdot 5 + 5\xi^2) ( 3^4 - 2\cdot 3^2\xi^2 + \xi^4)
- 2^2~\xi^2 (3^2\cdot 5^2 - 2\cdot 3\cdot 5 \xi^2 + \xi^4)
+ 2 \xi^2 ( 3^3 + 2\cdot 3\xi^2 -\xi^4)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ 2\cdot 3 \xi^2 + 2\xi^4 + 2\cdot 3^3 \xi^2 - 2\cdot 3 \xi^4 - 3(3^3 + 2\cdot 3\xi^2 -\xi^4)
\biggr] (15-\xi^2)
</math>
</td>
</tr>
</table>
</div>

Coefficients of various powers of <math>~\xi</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi^0:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~3^5\cdot 5 -3^5\cdot 5 = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^2:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~-2\cdot 3^3\cdot 5 + 3^4\cdot 5 -2^2 \cdot 3^2 \cdot 5^2 +2\cdot 3^3 + 2\cdot 3^2\cdot 5 + 2\cdot 3^4\cdot 5 - 2\cdot 3^3\cdot 5 + 3^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
   
</td>
<td align="left">
<math>~= 3^2\cdot 5[-2\cdot 3 + 3^2 + 2 + 2\cdot 3^2 - 2\cdot 3] + 3^2[ 2\cdot 3 + 3^2 - 2^2 \cdot 5^2 ] = 3^2\cdot 5[17 ] - 3^2[5\cdot 17 ] = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^4:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~3\cdot 5 -2\cdot 3^2\cdot 5 +2^3\cdot 3\cdot 5 + 2^2\cdot 3 + 2\cdot 3 \cdot 5 - 2\cdot 3^2\cdot 5 - 2\cdot 3 -2\cdot 3^3 + 2\cdot 3^2 + 3^2\cdot 5</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
   
</td>
<td align="left">
<math>~= 3\cdot 5[1 -2\cdot 3 +2^3 + 2 - 2\cdot 3 + 3] + 2\cdot 3[2 - 1 - 3^2 + 3] = 2\cdot 3\cdot 5 - 2\cdot 3\cdot 5 = 0 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^6:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~5 - 2^2 -2 -2 +2\cdot 3 -3 = 0</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

<span id="ChandraEq49">Multiplying through by <math>~dr</math>, and integrating over the volume gives,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R (\sigma^2 \rho \psi^2)\frac{dr}{r^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl[ \Gamma_1 P \biggl(\frac{d\psi}{dr} \biggr)^2
+ \frac{4\psi^2}{r} \biggl( \frac{dP}{dr} \biggr) \biggr]\frac{dr}{r^2}
- \biggl[\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr]_0^R \, ,
</math>
</td>
</tr>
</table>
</div>
which is identical to equation (49) of [http://adsabs.harvard.edu/abs/1964ApJ...139..664C Chandrasekhar (1964)], if the last term — the difference of the central and surface boundary conditions — is set to zero.

Note that if we shift from the variable, <math>~\psi</math>, back to the fractional displacement function, <math>~\xi</math>, the last term in this expression may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r \xi \frac{d}{dr} \biggl[r^3 \xi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r \xi \biggl[3r^2 \xi + r^3 \frac{d\xi}{dr}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r^3 \xi^2 \biggl[3 + \frac{d\ln\xi}{d\ln r}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>
So, as is pointed out by [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)] in connection with their equation (57.31), setting this expression to zero at the surface of the configuration is equivalent to setting the variation of the pressure to zero at the surface. Quite generally, this can be accomplished by demanding that,
<div align="center" id="SufaceBC">
<math>~\frac{d\ln\xi}{d\ln r}\biggr|_\mathrm{surface} = -3 \, .</math>
</div>

(An [[SSC/Perturbations#Boundary_Conditions|accompanying chapter]] provides a broader discussion of this and other astrophysically reasonable boundary conditions that are associated with solutions to the LAWE.)

===Ledoux & Walraven Approach===
Returning to the above [[#FoundationalVariationalRelation|''Foundational Variational Relation'']], we can also write,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\xi \cdot \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \frac{d}{dr}\biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \int_0^R\sigma^2 \rho r^4 \xi^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]_0^R
</math>
</td>
</tr>
</table>
</div>

If the last term (boundary conditions) is set to zero, then we may also write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, if the radial profile of the pressure and the density is known throughout a spherically symmetric, equilibrium configuration, and if, furthermore, the eigenfunction, <math>~\xi(r)</math>, of a radial oscillation mode is specified precisely, then this expression will give the (square of the) ''eigenfrequency'' of that oscillation mode.

By using formal ''variational principle'' techniques to derive this same expression, [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)] are able to offer a broader interpretation, which is encapsulated by their equation (59.10), viz.,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma_0^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathrm{min}~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, if the exact radial eigenfunction, <math>~\xi(r)</math>, is not known, various approximate eigenfunctions can be tried. The trial eigenfunction that ''minimizes'' the righthand-side of this expression will give the (square of the) eigenfrequency of the ''fundamental'' mode of oscillation (subscript zero). Furthermore, via an evaluation of this righthand-side expression, any reasonable trial eigenfunction — for example, <math>~\xi</math> = constant — can provide an ''upper limit'' to <math>~\sigma_0^2</math>.

===Ledoux & Pekeris Approach===
Here we follow the lead of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)]. Returning to the integral expression just derived in our discussion of the ''Ledoux & Walraven approach'', and multiplying through by <math>~4\pi</math>, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R 4\pi \sigma^2 \rho r^4 \xi^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R 4\pi r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) 4\pi r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[4\pi r^3 \Gamma_1 P\xi^2 \biggl(\frac{d\ln \xi}{d\ln r}\biggr) \biggr]_0^R \, .
</math>
</td>
</tr>
</table>
</div>
If we acknowledge that:
* at the center of the configuration, <math>~r^3 = 0</math>;
* [[#SurfaceBC|as above]], the boundary condition at the surface is <math>~P = P_e</math> while <math>~(d\ln \xi/d\ln r) = -3</math>;
* the differential mass element is, <math>~dm = 4\pi r^2 \rho dr</math> and the corresponding differential volume element is, <math>~dV = 4\pi r^2 dr</math>; and
* a statement of detailed force balance is, <math>~dP/dr = - Gm\rho/r^2</math>,
this integral relation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \sigma^2 \int_0^R r^2 \xi^2 dm</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 \int_0^R \biggl[ r \biggl(\frac{d\xi}{dr}\biggr)\biggr]^2 P dV
+ (3\Gamma_1 - 4) \int_0^R \xi^2 \biggl( \frac{Gm}{r} \biggr) dm
- \biggr[\Gamma_1 \xi_\mathrm{surface}^2 (3P_e V) \biggl(-3\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Now, as we have [[SSCpt1/Virial#Wgrav|discussed separately]] — see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] — the gravitational potential energy of the unperturbed configuration is given by the integral,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - \int_0^{M} \biggl( \frac{Gm}{r_0} \biggr) dm \, ;</math>
</td>
</tr>
</table>
</div>

for adiabatic systems, the [[SSCpt1/Virial#Reservoir|internal energy]] is,
<div align="center">
<math>
U_\mathrm{int}
= \frac{1}{(\Gamma_1-1)} \int_0^R P_0 dV
\, ;</math>
</div>
and — see the text at the top of p. 126 of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] — the moment of inertia of the configuration about its center is,
<div align="center">
<math>
I = \int_0^M r_0^2 dm
\, .</math>
</div>
(Note that, defined in this way, <math>~I</math> is the same as [[VE#Standard_Presentation_.5Bthe_Virial_of_Clausius_.281870.29.5D|what we have referred to elsewhere]] as the ''scalar moment of inertia'', which is obtained by taking the trace of the [[VE#MOItensor|moment of inertia tensor]], <math>~I_{ij}</math>.)
<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3"><font color="maroon"><b>Variational Principle's Governing Integral Relation</b></font></td>
</tr>
<tr>
<td align="right">
<math>~ \sigma^2 \int_0^R \xi^2 dI</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 (\Gamma_1 - 1) \int_0^R \xi^2 \biggl[ \frac{d\ln\xi}{d\ln r}\biggr]^2 dU_\mathrm{int}
- (3\Gamma_1 - 4) \int_0^R \xi^2 dW_\mathrm{grav}
+ 3^2 \Gamma_1 P_e V \xi_\mathrm{surface}^2 \, .
</math>
</td>
</tr>
</table>
</div>

==Free-Energy Analysis==

<span id="Homologous">If we assume</span> the simplest approximation for the fundamental-mode eigenfunction, namely, <math>~\xi = \xi_0</math> = constant — that is, homologous expansion/contraction — then this last integral expression gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \sigma^2 I</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4 - 3\Gamma_1) W_\mathrm{grav}
+ 3^2 \Gamma_1 P_e V \, .
</math>
</td>
</tr>
</table>
</div>

Contrast this result with the following free-energy analysis:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV \, ,</math>
</td>
</tr>
</table>
</div>
where, in terms of the configuration's (generally non-equilibrium) dimensionless radius, <math>~\chi \equiv R/R_0</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-a\chi^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~U_\mathrm{int}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~b\chi^{3-3\Gamma_1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3} \chi^3 \, .</math>
</td>
</tr>
</table>
</div>
Then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial \mathfrak{G}}{\partial \chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~+a \chi^{-2} + 3(1-\Gamma_1) b \chi^{2-3\Gamma_1} + 4\pi P_e \chi^{2} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^{-1} \biggl[- W_\mathrm{grav} + 3(1-\Gamma_1) U_\mathrm{int} + 3 P_e V \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial^2 \mathfrak{G}}{\partial \chi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2a \chi^{-3} + 3(1-\Gamma_1)(2-3\Gamma_1) b \chi^{1-3\Gamma_1} + 8\pi P_e \chi </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^{-2} \biggl[ 2W_\mathrm{grav} + 3(1-\Gamma_1)(2-3\Gamma_1) U_\mathrm{int}+ 6 P_e V \biggr] \, .</math>
</td>
</tr>
</table>
</div>
The equilibrium condition occurs when <math>~\partial \mathfrak{G}/\partial \chi = 0</math>, that is, when,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3(1-\Gamma_1) U_\mathrm{int}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~W_\mathrm{grav} - 3 P_e V \, ,</math>
</td>
</tr>
</table>
</div>
in which case,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\chi^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial \chi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2W_\mathrm{grav} + (2-3\Gamma_1) (W_\mathrm{grav} - 3P_eV) + 6 P_e V </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(4-3\Gamma_1)W_\mathrm{grav} + 3^2 \Gamma_1 P_e V \, .</math>
</td>
</tr>
</table>
</div>

Fantastic! The righthand-side of this "free-energy-based" expression exactly matches the righthand-side of the [[#Homologous|above expression]] that has been derived from the variational principle, assuming homologous expansion/contraction (''i.e.,'' <math>~\xi</math> = constant). In this case, we can make the direct association,
<div align="center">
<math>~\sigma^2 I = \chi^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial \chi^2} \, .</math>
</div>
This also make sense in that the equilibrium configuration should be stable if <math>~\tfrac{\partial^2 \mathfrak{G}}{\partial \chi^2} > 0</math> — in which case, <math>~\sigma^2</math> is positive; whereas the equilibrium configuration should be ''unstable'' if <math>~\tfrac{\partial^2 \mathfrak{G}}{\partial \chi^2} < 0</math> — in which case, <math>~\sigma^2</math> is negative.

=Related, Exploratory Ideas=

==Logarithmic Derivatives==

Returning to our above discussion of the [[#Ledoux_.26_Walraven_Approach|Ledoux & Walraven approach]], we appreciate that the ''differential'' relation governing the Variational Principle is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \frac{d}{dr}\biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{d}{dr}\biggr[r^3 \Gamma_1 P\xi^2 \biggl(\frac{d\ln\xi}{d\ln r}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \sigma^2 \rho r^4 \xi^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi^2 \biggl\{
r^2 \Gamma_1 P \biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- (3\Gamma_1 - 4) r^3 \biggl( \frac{dP}{dr} \biggr)
- \sigma^2 \rho r^4
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(r \xi)^2 P \biggl\{
\Gamma_1 \biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- (3\Gamma_1 - 4) \biggl( \frac{d\ln P}{d\ln r} \biggr)
- \frac{\sigma^2 \rho r^2}{P}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Gamma_1 (r \xi)^2 P \biggl\{
\biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- \alpha \biggl( \frac{d\ln P}{d\ln r} \biggr)
- \frac{\sigma^2 \rho r^2}{\Gamma_1 P}
\biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\alpha \equiv \biggl(3 - \frac{4}{\Gamma_1}\biggr) \, .</math>
</div>

==Pressure-Truncated Polytropes==

Let's start with the integral expression derived in our discussion of the [[#Ledoux_.26_Walraven_Approach|Ledoux & Walraven approach]]; insert the variable, <math>~x</math>, in place of <math>~\xi</math>; and adopt the boundary conditions,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r = 0</math>   at the center,
</td>
<td align="center">
        along with        
</td>
<td align="left">
<math>~P = P_e~</math>,   and  <math>\frac{d\ln x}{d\ln r} = -3</math>   at the surface (r = R).
</td>
</tr>
</table>
</div>
That is, let's start with,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2 \, .
</math>
</td>
</tr>
</table>
</div>

===Via Generalized Normalization===
Next, we'll divide through by the [[StabilityVariationalPrinciple#Energies_and_Structural_Form_Factors|normalization energy, as defined in an accompanying discussion]],
<div align="center">
<math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3 = \frac{GM_\mathrm{tot}^2}{R_\mathrm{norm}} \, ,</math>
</div>
thereby making the integral relation dimensionless:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2} \biggr] \int_0^R \sigma^2 \rho r^4 x^2 dr
+\biggl[\frac{1}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \biggl[\frac{1}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+ \biggl[\frac{P_e R^3 }{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{R_\mathrm{norm} R^5 \rho_c^2}{M_\mathrm{tot}^2} \biggr] \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
+ \biggl[\frac{P_c R^3}{P_\mathrm{norm}R_\mathrm{norm}^3 } \biggr] \int_0^R \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[\frac{P_c R^3}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+ \biggl[\frac{P_e R^3 }{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^2 \chi^{-1} \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
+ \biggl[\frac{P_e }{P_\mathrm{norm}} \biggr] 3\Gamma_1 \chi^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[\frac{P_c}{P_\mathrm{norm} } \biggr] \chi^3 \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\chi \equiv \frac{R}{R_\mathrm{norm}} \, .</math>
</div>

Note that we will ultimately insert the relation,
<div align="center">
<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\Gamma_1} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\Gamma_1} \, .</math>
</div>
But, for the time being, dividing through by <math>~[P_c/P_\mathrm{norm}]\chi^3</math> gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{P_c}{P_\mathrm{norm} } \biggr]^{-1} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^2 \chi^{-4} \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[\frac{P_e }{P_c} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
+ \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R} \, ,
</math>
</td>
</tr>
</table>
</div>

Now let's focus on the second line of this integral energy relation, evaluating it for pressure-truncated polytropic configurations, in which case, <math>~\Gamma_1 \rightarrow (n+1)/n</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r}{R} \rightarrow \frac{\xi}{\tilde\xi}</math>
</td>
<td align="center">
        and        
</td>
<td align="left">
<math>~
\frac{P}{P_c} \rightarrow \theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>
We have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
Second line of relation
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
+ \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \biggl\{ \biggl( \frac{\xi}{\tilde\xi}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2 {\tilde\xi}^2
- \biggl(\frac{3-n}{n}\biggr) \biggl( \frac{\xi}{\tilde\xi} \biggr)^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d\xi} \biggr] \tilde\xi \biggr\}\frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{1}{n {\tilde\xi}^3}\int_0^{\tilde\xi} \biggl\{ (n+1) \xi^4 \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2
- (3-n) \xi^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d\xi} \biggr] \biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{1}{n {\tilde\xi}^3}\int_0^{\tilde\xi} \biggl\{ (n+1) \xi^4 \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2
- (n+1) (3-n) \xi^3 x^2 \theta^n \theta^' \biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{(n+1)}{n {\tilde\xi}^3}\int_0^{\tilde\xi}
\biggl(\frac{3}{2n}\biggr)^2\frac{\xi}{\theta^n} \biggl\{ \xi \theta \biggl[ \biggl( \frac{2n}{3}\biggr)\xi \theta^n \cdot \frac{dx}{d\xi}\biggr]^2
- (3-n) \biggl[ \biggl( \frac{2n}{3}\biggr) \xi \theta^n x\biggr]^2 \theta^' \biggr\}d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Now, let's examine how these terms combine if we ''guess'' the [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|analytically defined eigenfunction that applies to marginally unstable, pressure-truncated polytropic configurations]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^'}{\xi \theta^{n} } \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{2n}{3}\biggr) \xi \theta^n x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\frac{3(n-3)}{2n}\biggr] \biggl\{
\frac{\theta^{''}}{\xi \theta^{n}}
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n(\theta^')^2}{\xi \theta^{(n+1)}}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[\frac{3(n-3)}{2n}\biggr] \frac{1 }{\xi \theta^{n}} \biggl[
\theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{2n}{3}\biggr) \xi \theta^n\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3-n)
\biggl[ \theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta} \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>
Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
Second line of relation
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
{\tilde\theta}^{n+1} \biggl[ \frac{3( n+1)}{n} \biggr] \biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{ {\tilde\theta}^'}{\tilde\xi {\tilde\theta}^{n} } \biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)}{2^2n^3 {\tilde\xi}^3}\int_0^{\tilde\xi}
\frac{\xi}{\theta^n} \biggl\{
\xi \theta (3-n)\biggl[ \theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta} \biggr]^2
- \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \theta^'
\biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{{\tilde\xi}^2 {\tilde\theta}^{n+1}} \biggl[ \frac{3^3( n+1)}{2^2n^3} \biggr] \biggl[(n-1) \tilde\xi {\tilde\theta}^{n+1} + (n-3) \tilde\theta {\tilde\theta}^' \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)}{2^2n^3 {\tilde\xi}^3} \int_0^{\tilde\xi}
\frac{1}{\theta^{n+1}} \biggl\{
(3-n)\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]^2
- \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi \theta \theta^'
\biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{{\tilde\xi}^2 {\tilde\theta}^{n+1}} \biggl[ \frac{3^3( n+1)}{2^2n^3} \biggr] \biggl[(n-1) \tilde\xi {\tilde\theta}^{n+1} + (n-3) \tilde\theta {\tilde\theta}^' \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)^2}{2^2n^3 {\tilde\xi}^3} \int_0^{\tilde\xi}
\frac{1}{\theta^{n+1}} \biggl\{
\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]^2
+ \frac{1}{(n-3)} \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi \theta \theta^'
\biggr\}d\xi
</math>
</td>
</tr>
</table>
</div>

Note that, in this derivation, we have inserted the expressions:
<div align="center">
<math>~
\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr] =
\xi^2 \theta^{2(n+1)} + 6\xi \theta^{n+2}\theta^' + 2n\xi^2 \theta^{n+1} (\theta^')^2 + 6n\xi\theta (\theta^')^3 + n^2 \xi^2 (\theta^')^4
</math>
</div>
<div align="center">
<math>~
\frac{1}{(n-3)} \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi\theta (\theta^')=
\biggl[ \frac{(n-1)^2}{(n-3)}\biggr] \xi^3 \theta^{2n+1}(\theta^') + 2(n-1)\xi^2 \theta^{n+1} (\theta^' )^2 + (n-3) \xi\theta (\theta^')^3
</math>
</div>

===Directly to n = 5 Polytropic Configurations===

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#MoreStabilityAnalyses|<b>''Exact''<br />Demonstration<br />of<br />Variational<br />Principle</b>]]</font>
|}

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{1}{R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl(\frac{r}{R}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{P}{P_c}\biggr) \biggl[\frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
- \int_0^R \biggl[3\biggl(\frac{n+1}{n}\biggr) - 4\biggr] \biggl(\frac{r}{R}\biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+3\biggl(\frac{n+1}{n}\biggr) \biggl( \frac{P_e}{P_c}\biggr) x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \frac{6}{5} \biggl(\frac{\xi}{\tilde\xi}\biggr)^4 \theta^6 \biggl[\frac{dx}{d(\xi/\tilde\xi)}\biggr]^2 \frac{d\xi}{\tilde\xi}
- \int_0^{\tilde\xi} \biggl( - \frac{2}{5}\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^3 x^2 \biggl[ \frac{d\theta^{6}}{d(\xi/\tilde\xi)} \biggr] \frac{d\xi}{\tilde\xi}
+\biggl(\frac{18}{5}\biggr) {\tilde\theta}^6 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde\xi}^3} \int_0^{\tilde\xi} \biggl( \frac{6}{5}\biggr) \xi^4 \theta^6 \biggl[\frac{dx}{d\xi}\biggr]^2 d\xi
+ \frac{1}{ {\tilde\xi}^3} \int_0^{\tilde\xi} \biggl(\frac{2}{5}\biggr) \xi^3 x^2 \biggl[ \frac{d\theta^{6}}{d\xi} \biggr] d\xi
+\biggl(\frac{18}{5}\biggr) {\tilde\theta}^6 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5 {\tilde\xi}^3 }{2R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} 3\xi^4 \theta^6 \biggl[ - \frac{2\xi}{15} \biggr]^2 d\xi
+ \int_0^{\tilde\xi} 6\xi^3 \biggl[\frac{15-\xi^2}{15}\biggr]^2 \theta^5\biggl[ \frac{d\theta}{d\xi} \biggr] d\xi
+9 {\tilde\xi}^3 {\tilde\theta}^6 \biggl[\frac{15- {\tilde\xi}^2}{15}\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{ 2^2}{3\cdot 5^2 } \biggr) \int_0^{\tilde\xi} \xi^6 \biggl( \frac{3}{3+\xi^2}\biggr)^3 d\xi
+ \biggl(\frac{ 2}{3\cdot 5^2 } \biggr) \int_0^{\tilde\xi} \xi^3 \biggl[15-\xi^2\biggr]^2 \biggl( \frac{3}{3+\xi^2}\biggr)^{4} \biggl[- \frac{\xi}{3}\biggr] d\xi
+ \biggl( \frac{1}{5^2} \biggr) {\tilde\xi}^3 \biggl( \frac{3}{3+ {\tilde\xi}^2}\biggr)^3 \biggl[15- {\tilde\xi}^2\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{ 2^2\cdot 3^2}{5^2 } \biggr) \int_0^{\tilde\xi} \biggl[ \frac{\xi^6 }{(3+\xi^2)^3}\biggr] d\xi
~~- ~~ \biggl(\frac{ 2\cdot 3^2}{5^2 } \biggr) \int_0^{\tilde\xi} \biggl[ \frac{\xi^4 (15-\xi^2)^2}{(3+\xi^2)^4}\biggr] d\xi
~~ + ~~ \biggl( \frac{3^3}{5^2} \biggr) \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5^3 {\tilde\xi}^3 }{2\cdot 3^2R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl[ \frac{4\xi^6(3+\xi^2)-2\xi^4 (15-\xi^2)^2}{(3+\xi^2)^4}\biggr] d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl\{ \frac{2\xi^4 [6\xi^2 + 2\xi^4 -15^2 + 30\xi^2 - \xi^4] }{(3+\xi^2)^4}\biggr\} d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl\{ \frac{2\xi^4 [\xi^4 + 36\xi^2 -15^2 ] }{(3+\xi^2)^4}\biggr\} d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2\xi^5(\xi^2-15)}{(\xi^2+3)^3} \biggr]_0^{\tilde\xi}
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2{\tilde\xi}^5({\tilde\xi}^2-15)}{({\tilde\xi}^2+3)^3} \biggr]
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2{\tilde\xi}^5({\tilde\xi}^2-15) + 3{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{({\tilde\xi}^2+3)^3}
=
\frac{5{\tilde\xi}^7 - 120{\tilde\xi}^5 + 3^3\cdot 5^2{\tilde\xi}^3 }{({\tilde\xi}^2+3)^3} \, ,
</math>
</td>
</tr>
</table>
</div>
which equals zero if <math>~\tilde\xi = 3</math>. <font size="+1" color="red"><b>Hooray!!</b></font>

===For All Polytropic Indexes===

====Generalized Governing Integral Relation====
Given that the derivation just completed works for the special case of n = 5, let's generalize it to all polytropic indexes
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{R^5 \rho_c}{R^3 P_c}\int_0^R \sigma^2 \biggl( \frac{\rho}{\rho_c}\biggr) \biggl(\frac{r}{R}\biggr)^4 x^2 \frac{dr}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl(\frac{r}{R}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{P}{P_c}\biggr) \biggl[\frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
- \int_0^R \biggl[3\biggl(\frac{n+1}{n}\biggr) - 4\biggr] \biggl(\frac{r}{R}\biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+3\biggl(\frac{n+1}{n}\biggr) \biggl( \frac{P_e}{P_c}\biggr) x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{R^2 \rho_c}{P_c} \int_0^{\tilde\xi} \sigma^2 \theta^n \biggl(\frac{\xi}{\tilde\xi}\biggr)^4 x^2 \frac{d\xi}{\tilde\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl(\frac{\xi}{{\tilde\xi}}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \theta^{n+1} \biggl[\frac{dx}{d(\xi/\tilde\xi)}\biggr]^2 \frac{d\xi}{\tilde\xi}
~+ \int_0^{\tilde\xi} \biggl(\frac{n-3}{n}\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d(\xi/\tilde\xi)} \biggr] \frac{d\xi}{\tilde\xi}
~+~3\biggl(\frac{n+1}{n}\biggr) {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{n R^2\rho_c}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \sigma^2 \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^4 \theta^{n+1} \biggl[\frac{dx}{d\xi}\biggr]^2 d\xi
~+ \int_0^{\tilde\xi} (n-3) \xi^3 \theta^n x^2 \biggl[ \frac{d\theta}{d\xi} \biggr] d\xi
~+~3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{n R^2 G \rho_c^2}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 d\xi
~+ \int_0^{\tilde\xi} (n-3) \xi^2 \theta^{n+1} x^2 \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] d\xi
~+~3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl\{ \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 + (n-3) \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] \biggr\} d\xi
</math>
</td>
</tr>
</table>
</div>

For additional clarification, let's rewrite the leading coefficient on the lefthand-side of this expression.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
LHS
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{n R^2 G \rho_c^2}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr] \biggl[ \frac{G R_\mathrm{norm}^2}{P_\mathrm{norm}} \biggr]
\biggl(\frac{R}{R_\mathrm{norm}^2}\biggr) \biggl( \frac{\rho_c}{ {\bar\rho}}\biggr)^2
\biggl[ \frac{3M}{4\pi R^3}\biggr]^2
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{P_e}{P_c} \biggr) \biggl[ \frac{1}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr] \biggl[ \frac{G M_\mathrm{tot}^2}{P_\mathrm{norm}R_\mathrm{norm}^4} \biggr]
\biggl(\frac{R_\mathrm{norm}}{R}\biggr)^4 \biggl( \frac{\rho_c}{ {\bar\rho}}\biggr)^2
\biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{tot}}\biggr]^2
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{P_e}{P_c} \biggr) \biggl[ \frac{1}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr]
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{R_\mathrm{norm}}{R}\biggr)^4 \biggl( - \frac{\tilde\xi}{3 {\tilde\theta}^'}\biggr)^2
\biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{tot}}\biggr]^2
\biggl[ \frac{{\tilde\theta}^{n+1}}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>
</table>
</div>

Now, from an [[StabilityVariationalPrinciple#Test_Virial_Equilibrium_Condition|accompanying discussion]], we know that, in equilibrium,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\, ,
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl\{ \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)}
\tilde\theta_n^{(n+1)(n-3)}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)} \biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~\times
\biggl\{\biggl[(n+1)^{-n} ( 4\pi )\biggr] \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)}
\tilde\xi^{(n-3)} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)} \biggr\}^{4/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ \tilde\xi^{4} \tilde\theta_n^{(n+1)}
\biggl\{ (n+1)^{3(n+1)} ( 4\pi )^{(-n-1)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2n-2}
( -\tilde\xi^2 \tilde\theta' )^{2n+2} (n+1)^{-4n} ( 4\pi )^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(4n-4)}
( -\tilde\xi^2 \tilde\theta' )^{(4-4n)}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ \tilde\xi^{4} \tilde\theta_n^{(n+1)}
\biggl\{ (n+1)^{(3-n)} ( 4\pi )^{(3-n)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n-3)}
( -\tilde\xi^2 \tilde\theta' )^{2(3-n)}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
(n+1)^{-1} ( 4\pi )^{(-1)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2} \tilde\xi^{4} \tilde\theta_n^{(n+1)}( -\tilde\xi^2 \tilde\theta' )^{-2} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, in equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
LHS
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr]
\biggl\{ (n+1) ( 4\pi ) \tilde\xi^{-4} \tilde\theta_n^{-(n+1)}( -\tilde\xi^2 \tilde\theta' )^{2} \biggr\}
\biggl( - \frac{\tilde\xi}{3 {\tilde\theta}^'}\biggr)^2
\biggl(\frac{3}{4\pi}\biggr)^2
\biggl[ \frac{{\tilde\theta}^{n+1}}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi \, .</math>
</td>
</tr>
</table>
</div>

In summary, then, we have,

<div align="center" id="PolytropeRelation">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\int_0^{\tilde\xi} \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl\{ \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 + (n-3) \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] \biggr\} d\xi \, .
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

Perhaps this looks better if the terms are rearranged to give,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2\theta^{n+1} x^2 \biggl\{ \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \frac{\xi^2}{\theta}
- \biggl[ \biggl( \frac{d\ln x}{d\ln \xi}\biggr)^2 + (n-3) \biggl( \frac{d\ln\theta}{d\ln\xi} \biggr) \biggr] \biggr\} d\xi \, .
</math>
</td>
</tr>
</table>
</div>

====Plug in Known Marginally Unstable Solution====

As has been summarized in an [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|accompanying discussion]], we have found that, for marginally unstable pressure-truncated polytropic configurations, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma_c^2 = 0</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{2n}{3(n-1)} \biggr] \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \frac{d}{d\xi}\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl[
\frac{\theta^{''}}{\xi \theta^{n}}
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl[
- \frac{1}{\xi \theta^{n}} \biggl( \theta^n + \frac{2\theta^'}{\xi} \biggr)
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl[
\frac{1}{\xi }
+ \frac{3\theta^'}{\xi^2 \theta^{n}}
+ \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Hence, also,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl(\frac{n-3}{n-1}\biggr)^{-1} \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[\biggl(\frac{n-1}{n-3}\biggr) + \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[\biggl(\frac{n-1}{n-3}\biggr) + \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1} \, .
</math>
</td>
</tr>
</table>
</div>

Rather, let's try:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
\xi^2 x^2 \biggl[ \biggl( \frac{d\ln x}{d\ln \xi}\biggr)^2 + (n-3) \biggl( \frac{d\ln\theta}{d\ln\xi} \biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x^2 \xi^2 \biggl( \frac{\xi}{x}\cdot \frac{dx}{d\xi}\biggr)^2 + (n-3) x^2 \xi^2 \biggl( \frac{\xi}{\theta} \cdot \frac{d\theta}{d\xi} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^4 \biggl\{ \frac{dx}{d\xi}\biggr\}^2 + (n-3) \biggl[ \frac{\xi^3 \theta^'}{\theta} \biggr] x^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^4 \biggl\{ \frac{3(n-1)}{2n}\biggl(\frac{3-n}{n-1}\biggr) \biggl[
\frac{1}{\xi }
+ \frac{3\theta^'}{\xi^2 \theta^{n}}
+ \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]\biggr\}^2 + (n-3) \biggl[ \frac{\xi^3 \theta^'}{\theta} \biggr]
\biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi^2 (n-3) \biggl[ \frac{3}{2n} \biggr]^2\biggl\{
(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
\biggr\}
</math>
</td>
</tr>
</table>
</div>
Hence, after setting <math>~\sigma^2 = 0</math>, the [[#PolytropeRelation|above rearranged integral relation]] becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2 \theta^{n+1} \biggl\{
(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
\biggr\} d\xi
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">
Let's check to see whether the terms in this last expression balance out when we plug in the functions that are appropriate for the marginally unstable, n = 5 configuration, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="left">
<math>~\theta_5 = \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2}</math>,
</td>
<td align="center">
     and    
</td>
<td align="left">
<math>~\frac{d\theta_5}{d\xi} = - \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}</math>.
</td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
RHS Term 1
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(n-3) \int_0^{\tilde\xi} \xi^2 \theta^{n+1}
\biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \int_0^{\tilde\xi} \xi^2 \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2} \biggr]^{6} \biggl\{
1 - \biggl[ \xi \biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]\frac{1}{\xi }\biggl(\frac{3+\xi^2}{3}\biggr)^{5 / 2}
+5 \biggl[ \frac{\xi^2}{3^2}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \int_0^{\tilde\xi} \frac{3^3 \xi^2}{(3+\xi^2)^3} \biggl\{
1 - \biggl(\frac{3+\xi^2}{3}\biggr)
+ \frac{5\xi^2}{3^2}
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \int_0^{\tilde\xi} \frac{\xi^6 ~d\xi}{(3+\xi^2)^3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \biggl[
\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
\biggr]_0^{3} = \frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \biggr] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
RHS Term 2
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^3 \theta^{n} \theta^'
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \int_0^{\tilde\xi} \xi^3 \biggl(\frac{3+\xi^2}{3}\biggr)^{-5 / 2} \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}
\biggl\{ 4 - 2\frac{1}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}\biggl(\frac{3+\xi^2}{3}\biggr)^{5/2}
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{3}\int_0^{\tilde\xi} \biggl(\frac{3\xi}{3+\xi^2}\biggr)^{4}
\biggl\{ 4 - \frac{2}{3}\biggl(\frac{3+\xi^2}{3}\biggr)
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^3}{3}\int_0^{\tilde\xi} \frac{1}{2} \biggl(\frac{\xi}{3+\xi^2}\biggr)^{4}
\biggl\{ 15-\xi^2
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2^3}{3} \biggl[
\frac{123\xi}{8(3+\xi^2)} - \frac{243\xi}{4(3+\xi^2)^2} + \frac{162\xi}{2(3+\xi^2)^3} + \frac{\xi}{2} - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
\biggr]_0^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \cdot \frac{5}{2^5} \biggl[ 2^2\cdot 3^{1 / 2} \pi - 3^3\biggr] = -\frac{2^3}{3} \biggl[ \frac{3^3\cdot 5}{2^5} - \frac{3^{1 / 2} \cdot 5\pi}{2^3} \bigg] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~</math> RHS Total
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^3\cdot 5}{2^5} \biggr]
= \frac{3^2}{2^3} \biggl[ 3^2 - 2\cdot 5 \biggr] = - \frac{3^2}{2^3} \, .
</math>
</td>
</tr>

<tr>
<td align="right">
LHS
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^2 5^2}{2\cdot 3} \biggl[ 3^3 \biggl(\frac{3}{3+3^2}\biggr)^{3} \frac{2^2}{5^2} \biggr]
=
- \frac{2^3 }{3} \biggl[\biggl(\frac{3}{2^2}\biggr)^{3} \biggr]
=
- \frac{3^2 }{2^3} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, the LHS = RHS.   <font size="+1" color="red"><b>Hooray!</b></font>
</td></tr>
</table>

=See Also=

* [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29|Derivations that support this chapter's discussion of the Ledoux Variational Principle]]
* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">A variational principle of great power is derived. It is naturally adapted for computers, and may be used to determine the stability of any fluid flow including those in differentially-rotating, self-gravitating stars and galaxies. The method also provides a powerful theoretical tool for studying general properties of eigenfunctions, and the relationships between secular and ordinary stability. In particular we prove the anti-sprial theorem indicating that no stable (or steady( mode can have a spiral structure</font>.
</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1972ApJS...24..319S/abstract B. F. Schutz, Jr. (1972)], ApJSuppl., 24, 319: ''Linear Pulsations and Stability of Differentially Rotating Stellar Models. I. Newtonian Analysis''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">A systematic method is presented for deriving the Lagrangian governing the evolution of small perturbations of arbitrary flows of a self-gravitating perfect fluid. The method is applied to a differentially rotating stellar model; the result is a Lagrangian equivalent to that of [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)]. A sufficient condition for stability of rotating stars, derived from this Lagrangian, is simplified greatly by using as trial functions not the three components of the Lagrangian displacement vector, but three scalar functions … This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.

… we examine the special cases of (i) axially symmetric perturbations of a rotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..267C/abstract S. Chandrasekhar & N. R. Lebovitz 1968]); and (ii) perturbations of a nonrotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1964ApJ...140.1517C/abstract Chandrasekhar and Lebovitz 1964)]. We find that the stability criteria for those cases can also be simplified …</font>
</td></tr></table>

{{ SGFfooter }}

StabilityVariationalPrinciple

2021-08-01T20:10:16Z

174.64.8.247: /* Old Approach */

__FORCETOC__


=Free-Energy Stability Analysis=

==Most General Case==
Consider a free-energy function of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{-3/j} + \mathcal{G}_0 \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>~a, b, c,</math> and <math>~\mathcal{G}_0</math> are constants, and the dimensionless configuration radius,
<div align="center">
<math>~\chi \equiv \frac{R}{R_0} \, ,</math>
</div>
is defined in terms of a characteristic length, <math>~R_0</math>, which is likely to be different for each type of problem.

===Virial Equilibrium===
The first variation (first derivative) of this function with respect to the configuration's radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{G}}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~a\chi^{-2} - \biggl(\frac{3b}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 c}{j}\biggr) \chi^{-3/j -1} \, .</math>
</td>
</tr>
</table>
</div>
According to the virial theorem, the radius of an equilibrium configuration is obtained by setting <math>~d\mathcal{G}/d\chi = 0</math> and identifying the roots of the resulting equation. For example, identifying roots of the polynomial expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} - \biggl(\frac{1}{j}\biggr) \chi_\mathrm{eq}^{(j-3)/j } \, .</math>
</td>
</tr>
</table>
</div>

===Stability===
Let's rewrite the first variation of the free-energy function in terms of three coefficients <math>~(e,f,g)</math> which, in general, we will permit to have different values from the original three <math>~(a,b,c)</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{G}^'</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e\chi^{-2} - \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -1} \, .</math>
</td>
</tr>
</table>
</div>
The first variation (first derivative) of ''this'' function with respect to the configuration's radius — which, in effect, represents the second variation of the free-energy function — gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + \biggl(\frac{3}{j} + 1\biggr) \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -2} \, .</math>
</td>
</tr>
</table>
</div>
If we evaluate this function by setting <math>~\chi = \chi_\mathrm{eq}</math>, the sign of the resulting expression should indicate stability (positive) or dynamical instability (negative); and the marginally unstable configuration is identified by the value of <math>~\chi_\mathrm{eq}</math> for which <math>~d\mathcal{G}^'/d\chi = 0</math>.

==Pressure-Truncated Configurations==

===Expectations===
For pressure-truncated polytropes, we set <math>~j = -1</math> and let <math>~n</math> represent the chosen polytropic index. In this situation, then, we have,
<div align="center">
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">Free-energy expression:</td>
<td align="center">     </td>
<td align="right">
<math>~\mathcal{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{3} + \mathcal{G}_0 \, ;</math>
</td>
</tr>

<tr>
<td align="right">Virial equilibrium:</td>
<td align="center">     </td>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} + \chi_\mathrm{eq}^{4 } \, ;</math>
</td>
</tr>

<tr>
<td align="right">Stability indicator:</td>
<td align="center">     </td>
<td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + 6g \chi \, .</math>
</td>
</tr>
</table>
</div>
Hence, the (critical) equilibrium radius of the marginally unstable configuration is given by the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~6g \chi_\mathrm{eq}^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2e - \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} + \chi_\mathrm{eq}^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
6g \chi_\mathrm{eq}^4
+\biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\chi_\mathrm{eq}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\biggl[6g + \frac{3cf(n+3)}{nb} \biggr]\chi_\mathrm{eq}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2e - \biggl[\frac{af(n+3)}{nb} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2nbe -af(n+3)}{6nbg +3cf(n+3)} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Notice that, if <math>~(e,f,g) \rightarrow (a,b,c)</math>, this gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2nba -ab(n+3)}{6nbc +3cb(n+3)} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{3^2c}\biggl[\frac{n-3}{n+1} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>


===Energies and Structural Form Factors===
====Old Approach====

As has been developed in, for example, our [[SSCpt1/Virial/PolytropesEmbedded/FirstEffortAgain#Review|accompanying review]], we adopt the following normalizations:
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~R_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R^3_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{n/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~E_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>

Then, from separate summaries — both [[SSCpt1/Virial#Summary_of_Normalized_Expressions|here]] and [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors|here]] — we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{P_e V}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi^3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr] \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx\biggr\} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}} = \frac{U_\mathrm{int}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1} 3x^2 \biggl[ \frac{P(x)}{P_c} \biggr] dx \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>
where the [[SSCpt1/Virial#Structural_Form_Factors|structural form factors are defined]] as follows:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\mathfrak{f}_M </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \int_0^1 3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_A</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>
</td>
</tr>
</table>
</div>

This gives, specifically for [[SSCpt1/Virial/FormFactors#PTtable|specifically for pressure-truncated polytropic configurations]],
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

====New Approach====
In order to accommodate the structural integrals required by the Ledoux variational principle, let's re-derive some of these key energy and form-factor expressions. Basically, we will be repeating some [[SSCpt1/Virial#Expressions_for_Various_Energy_Terms|earlier derivations]].

=====Mass=====

Defining <math>~M_\mathrm{tot}</math> as the total mass of the ''isolated'' configuration, while <math>~M \le M_\mathrm{tot}</math> is the truncated configuration's mass; defining <math>~R</math> as the truncated configuration's (not necessarily ''equilibrium'') radius; and being careful to define the mean density of the truncated configuration such that,
<div align="center">
<math>~\bar\rho \equiv \frac{3M}{4\pi R^3} \, ,</math>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~M_r(r) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \int_0^r 4\pi r^2 \rho dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{M_r(r)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{3}{4\pi} \int_0^r 4\pi \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{\rho}{\rho_\mathrm{norm}}\biggr) \frac{dr}{R_\mathrm{norm}} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\rho_\mathrm{norm}}\biggr) \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^r 3\biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{M/R^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}
\int_0^\xi 3\xi^2 \theta^n d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Acknowledging that <math>~M_r \rightarrow M</math> when the upper integration limit goes to <math>~\tilde\xi</math>, we see that the "mass" form-factor is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
</td>
<td align="center">
<math>~\equiv </math>
</td>
<td align="left">
<math>~ {\tilde\xi}^{-3}\int_0^{\tilde\xi} 3\xi^2 \theta^n d\xi = \biggl( \frac{\bar\rho}{\rho_c}\biggr) \, .</math>
</td>
</tr>
</table>
</div>
Now, from the,
<div align="center">
Polytropic Lane-Emden Equation<p></p>
{{ Math/EQ_SSLaneEmden01 }}
</div>
we realize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d}{d\xi}\biggl(\xi^2 \theta^'\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \xi^2 \theta^n \, .</math>
</td>
</tr>
</table>
</div>
So these last two expressions may also be written as,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{M_r(r)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}\biggl[ - 3 \xi^2 \theta^' \biggr]
\, ;
</math>
</td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
</td>
<td align="center">
<math>~\equiv </math>
</td>
<td align="left">
<math>~\biggl[ -\frac{3\theta^'}{\xi} \biggr]_\tilde\xi \, .</math>
</td>
</tr>
</table>
</div>

=====Modified Internal Energy=====
Now we want to develop the appropriately scaled integral definition of a "variational" internal energy having the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{U_\Upsilon}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) }
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{P}{P_\mathrm{norm}}\biggr) \biggl( \frac{dr}{R_\mathrm{norm}}\biggr) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) } \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho}\biggr]^{\gamma_\mathrm{g}}
\biggl( \frac{M}{M_\mathrm{tot}}\biggr)^{\gamma_\mathrm{g}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma_\mathrm{g}}
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{P}{P_c}\biggr) \biggl( \frac{dr}{R}\biggr) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}\chi^{-3/n}
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \, . </math>
</td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~f</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f = \chi^{3/n}\biggl[ \frac{U_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} \biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \biggr\} \, ;</math>
</td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_U(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f \rightarrow b = \chi^{3/n}\biggl[ \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} {\tilde\mathfrak{f}}_A
\, ;</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

When <math>~\Upsilon_U(\xi) = 1</math>, then according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala & Horedt (1974)]], this integral over polytropic functions becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~{\tilde\mathfrak{f}}_A \equiv
{\tilde\xi}^{-3}\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr] \, ,
</math>
</td>
</tr>
</table>
</div>
which matches the expression for <math>~{\tilde\mathfrak{f}}_A</math> [[SSCpt1/Virial/FormFactors#PTtable|derived earlier]].

=====Modified Gravitational Potential Energy=====

Similarly, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{W_\Upsilon}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\int_0^R \Upsilon_W(r) \biggl(\frac{GM_r}{r}\biggr) 4\pi r^2 \rho dr
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{R_\mathrm{norm}\rho_c R^2}{M_\mathrm{tot}}\int_0^R 4\pi \Upsilon_W(r) \biggl(\frac{M_r}{M_\mathrm{tot}}\biggr) \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\chi^{-1}
\int_0^R 3\Upsilon_W(r) \biggl[\frac{M_r}{M_\mathrm{tot}}\biggr] \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 3\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~e</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e = -\chi \biggl[ \frac{W_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, ;
</math>
</td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_W(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e \rightarrow a = -\chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\frac{1}{{\tilde\mathfrak{f}}_M} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 ~{\tilde\mathfrak{f}}_W \, ;
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Now, according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala & Horedt (1974)]], when <math>~\Upsilon_W(\xi) = 1</math>, this integral over polytropic functions becomes,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
As we have [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors|detailed elsewhere]], from this, we have deduced that, for polytropic configurations,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
{\tilde\xi}^{-5}
\int_0^\tilde\xi 5 \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>

===Test Virial Equilibrium Condition===

If the correct normalized equilibrium radius, <math>~\chi_\mathrm{eq}</math>, is specified, our [[#Expectations|expectation regarding virial equilibrium]] is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3nc\chi_\mathrm{eq}^{4 } - 3b\chi_\mathrm{eq}^{(n-3)/n} + an</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 0\, .</math>
</td>
</tr>
</table>
</div>
Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair — <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — that has been given by [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Horedt (1970)]], namely,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\chi_\mathrm{eq} = \frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}} = \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{e}}{P_\mathrm{Horedt}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\, ,
</math>
</td>
</tr>
</table>
</div>

where we have taken into account the [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|shift in normalization factors]],
<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">Switch from [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Hoerdt's (1970)]] Normalization</th><tr>

<tr>
<td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-(n-1)/(n-3)}\frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}\biggr]^{1/(4-3\gamma)}
= \biggl[(n+1)^{-1} \biggl( 4\pi \biggr)^{1/n}\biggr]^{n/(n-3)}
= \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n+1)/(n-3)}
\frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{ \biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3} \biggl( \frac{1}{4\pi} \biggr) \biggr\}^{\gamma/(4-3\gamma)}
= \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>

We therefore have:
====First Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3n\biggl[\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggr]\chi_\mathrm{eq}^{4 }</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi n \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggr\}^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{4(n-1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi n \biggl[(n+1)^{[3(n+1)-4n]} ( 4\pi )^{[4-(n+1)]} \biggr]^{1/(n-3)}
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{[2(n+1)+ 4(1-n)]/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{[4(n-1)-2(n+1)]/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{n}{(n+1) }\biggr]
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^2 \, .
</math>
</td>
</tr>
</table>
</div>

====Second Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3b\chi_\mathrm{eq}^{(n-3)/n}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi ~n \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggr\}^{(n-3)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\pi ~n}{(5-n)} \biggl[ \frac{1}{4\pi} \biggl( - \frac{\tilde\xi}{\tilde\theta^'}\biggr) \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times (n+1)^{-1} ( 4\pi )^{1/n}
{\tilde\xi}^{(n-3)/n} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times {\tilde\xi}^{[(n-3)/n + 3(n+1)/n]} ( -\tilde\xi^2 \tilde\theta' )^{[(1-n)/n - (n+1)/n]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2} \, .
</math>
</td>
</tr>
</table>
</div>

====Third Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~an</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\biggl( - \frac{\tilde\xi}{3\tilde\theta^'} \biggr) \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2
\frac{3\cdot 5~n}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^2
\frac{n \tilde\xi^4}{(5-n)}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2} \, .
</math>
</td>
</tr>
</table>
</div>

====Combined====
Combining the three terms, the virial expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ (5-n)(n+1)\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^{-2} \biggl[ 3nc\chi_\mathrm{eq}^{4 } + an - 3b\chi_\mathrm{eq}^{(n-3)/n} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
n(5-n)
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2}
+ n(n+1)\tilde\xi^4
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-n \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~n( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{
(5-n)\tilde\theta_n^{n+1}
+ (n+1) \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~n(n+1) ( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{ 0 \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Q. E. D.

==The Ledoux Variational Principle==
Drawing from a [[SSC/VariationalPrinciple#Ledoux.27s_Expression|separate presentation of Ledoux's variational principle]], let's normalize his Lagrangian using the same normalizations that have been used, above. His expression is …

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~L </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\pi e^{2i\omega t} \biggl\{
- \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0
- \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
+ \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0
-\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Our normalization produces,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0
- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0
-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2\int_0^R x^2 \biggl[\frac{R_\mathrm{norm} R^5}{G\rho_c}\biggr]
\biggl[ \frac{3\rho_c}{4\pi \bar\rho R^3}\biggr]^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \omega^2 \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3 \rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R}
- 3 \gamma_\mathrm{g} x_\mathrm{surface}^2 \biggl[ \frac{R^3 }{ R_\mathrm{norm}^3} \frac{P_e}{P_\mathrm{norm}}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g}\biggl[\frac{P_c }{P_\mathrm{norm} }\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}
~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Given that,
<div align="center">
<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\gamma} \, ,</math>
</div>
and,
<div align="center">
<math>~\frac{M_r}{M} = \biggl(\frac{\rho_c}{\bar\rho} \biggr) \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \, ,</math>
</div>

this expression for the Lagrangian becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g}\biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \biggl( \frac{3}{4\pi}\biggr) \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl\{ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg\} \frac{dr_0}{R}
~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

In an effort to help identify the various terms in this expression as well as the relationship between the entire expression and our unperturbed free energy expression, let's ignore all terms involving the radial eigenfunction, <math>~x</math>, and its derivative. In this case, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}\biggr|_\mathrm{unperturbed}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\frac{\gamma_\mathrm{g} (\gamma_\mathrm{g} -1)}{4\pi} \biggl\{\frac{4\pi}{3(\gamma_\mathrm{g} -1)} \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
\int_0^R 3\biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (3\gamma_\mathrm{g} - 4) \biggl( \frac{1}{4\pi}\biggr)\biggl\{- \frac{3}{5} \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R 5\biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl[ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg] \frac{dr_0}{R} \biggr\}
~- ~\frac{3^2 \gamma_\mathrm{g}}{4\pi}\biggl\{\frac{4\pi}{3} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{4\pi L_{\{\}} }{E_\mathrm{norm}}\biggr|_\mathrm{unperturbed}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{4\pi \omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g} (\gamma_\mathrm{g} -1) \biggl\{ \frac{U_\mathrm{int}}{E_\mathrm{norm}} \biggr\}
+ (3\gamma_\mathrm{g} - 4) \biggl\{ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr\}
~- ~3^2 \gamma_\mathrm{g} \biggl\{ \frac{P_e V}{E_\mathrm{norm}} \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=See Also=

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StabilityVariationalPrinciple

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174.64.8.247:

__FORCETOC__


=Free-Energy Stability Analysis=

==Most General Case==
Consider a free-energy function of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{-3/j} + \mathcal{G}_0 \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>~a, b, c,</math> and <math>~\mathcal{G}_0</math> are constants, and the dimensionless configuration radius,
<div align="center">
<math>~\chi \equiv \frac{R}{R_0} \, ,</math>
</div>
is defined in terms of a characteristic length, <math>~R_0</math>, which is likely to be different for each type of problem.

===Virial Equilibrium===
The first variation (first derivative) of this function with respect to the configuration's radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{G}}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~a\chi^{-2} - \biggl(\frac{3b}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 c}{j}\biggr) \chi^{-3/j -1} \, .</math>
</td>
</tr>
</table>
</div>
According to the virial theorem, the radius of an equilibrium configuration is obtained by setting <math>~d\mathcal{G}/d\chi = 0</math> and identifying the roots of the resulting equation. For example, identifying roots of the polynomial expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} - \biggl(\frac{1}{j}\biggr) \chi_\mathrm{eq}^{(j-3)/j } \, .</math>
</td>
</tr>
</table>
</div>

===Stability===
Let's rewrite the first variation of the free-energy function in terms of three coefficients <math>~(e,f,g)</math> which, in general, we will permit to have different values from the original three <math>~(a,b,c)</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{G}^'</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e\chi^{-2} - \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -1} \, .</math>
</td>
</tr>
</table>
</div>
The first variation (first derivative) of ''this'' function with respect to the configuration's radius — which, in effect, represents the second variation of the free-energy function — gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + \biggl(\frac{3}{j} + 1\biggr) \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -2} \, .</math>
</td>
</tr>
</table>
</div>
If we evaluate this function by setting <math>~\chi = \chi_\mathrm{eq}</math>, the sign of the resulting expression should indicate stability (positive) or dynamical instability (negative); and the marginally unstable configuration is identified by the value of <math>~\chi_\mathrm{eq}</math> for which <math>~d\mathcal{G}^'/d\chi = 0</math>.

==Pressure-Truncated Configurations==

===Expectations===
For pressure-truncated polytropes, we set <math>~j = -1</math> and let <math>~n</math> represent the chosen polytropic index. In this situation, then, we have,
<div align="center">
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">Free-energy expression:</td>
<td align="center">     </td>
<td align="right">
<math>~\mathcal{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{3} + \mathcal{G}_0 \, ;</math>
</td>
</tr>

<tr>
<td align="right">Virial equilibrium:</td>
<td align="center">     </td>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} + \chi_\mathrm{eq}^{4 } \, ;</math>
</td>
</tr>

<tr>
<td align="right">Stability indicator:</td>
<td align="center">     </td>
<td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + 6g \chi \, .</math>
</td>
</tr>
</table>
</div>
Hence, the (critical) equilibrium radius of the marginally unstable configuration is given by the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~6g \chi_\mathrm{eq}^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2e - \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} + \chi_\mathrm{eq}^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
6g \chi_\mathrm{eq}^4
+\biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\chi_\mathrm{eq}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\biggl[6g + \frac{3cf(n+3)}{nb} \biggr]\chi_\mathrm{eq}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2e - \biggl[\frac{af(n+3)}{nb} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2nbe -af(n+3)}{6nbg +3cf(n+3)} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Notice that, if <math>~(e,f,g) \rightarrow (a,b,c)</math>, this gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2nba -ab(n+3)}{6nbc +3cb(n+3)} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{3^2c}\biggl[\frac{n-3}{n+1} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>


===Energies and Structural Form Factors===
====Old Approach====

As has been developed in, for example, our [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review|accompanying review]], we adopt the following normalizations:
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~R_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R^3_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{n/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~E_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>

Then, from separate summaries — both [[SSCpt1Virial#Summary_of_Normalized_Expressions|here]] and [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors|here]] — we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{P_e V}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi^3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr] \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx\biggr\} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}} = \frac{U_\mathrm{int}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1} 3x^2 \biggl[ \frac{P(x)}{P_c} \biggr] dx \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>
where the [[SSCpt1/Virial#Structural_Form_Factors|structural form factors are defined]] as follows:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\mathfrak{f}_M </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \int_0^1 3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_A</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>
</td>
</tr>
</table>
</div>

This gives, specifically for [[SSCpt1/Virial/FormFactors#PTtable|specifically for pressure-truncated polytropic configurations]],
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

====New Approach====
In order to accommodate the structural integrals required by the Ledoux variational principle, let's re-derive some of these key energy and form-factor expressions. Basically, we will be repeating some [[SSCpt1/Virial#Expressions_for_Various_Energy_Terms|earlier derivations]].

=====Mass=====

Defining <math>~M_\mathrm{tot}</math> as the total mass of the ''isolated'' configuration, while <math>~M \le M_\mathrm{tot}</math> is the truncated configuration's mass; defining <math>~R</math> as the truncated configuration's (not necessarily ''equilibrium'') radius; and being careful to define the mean density of the truncated configuration such that,
<div align="center">
<math>~\bar\rho \equiv \frac{3M}{4\pi R^3} \, ,</math>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~M_r(r) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \int_0^r 4\pi r^2 \rho dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{M_r(r)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{3}{4\pi} \int_0^r 4\pi \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{\rho}{\rho_\mathrm{norm}}\biggr) \frac{dr}{R_\mathrm{norm}} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\rho_\mathrm{norm}}\biggr) \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^r 3\biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{M/R^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}
\int_0^\xi 3\xi^2 \theta^n d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Acknowledging that <math>~M_r \rightarrow M</math> when the upper integration limit goes to <math>~\tilde\xi</math>, we see that the "mass" form-factor is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
</td>
<td align="center">
<math>~\equiv </math>
</td>
<td align="left">
<math>~ {\tilde\xi}^{-3}\int_0^{\tilde\xi} 3\xi^2 \theta^n d\xi = \biggl( \frac{\bar\rho}{\rho_c}\biggr) \, .</math>
</td>
</tr>
</table>
</div>
Now, from the,
<div align="center">
Polytropic Lane-Emden Equation<p></p>
{{ Math/EQ_SSLaneEmden01 }}
</div>
we realize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d}{d\xi}\biggl(\xi^2 \theta^'\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \xi^2 \theta^n \, .</math>
</td>
</tr>
</table>
</div>
So these last two expressions may also be written as,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{M_r(r)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}\biggl[ - 3 \xi^2 \theta^' \biggr]
\, ;
</math>
</td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
</td>
<td align="center">
<math>~\equiv </math>
</td>
<td align="left">
<math>~\biggl[ -\frac{3\theta^'}{\xi} \biggr]_\tilde\xi \, .</math>
</td>
</tr>
</table>
</div>

=====Modified Internal Energy=====
Now we want to develop the appropriately scaled integral definition of a "variational" internal energy having the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{U_\Upsilon}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) }
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{P}{P_\mathrm{norm}}\biggr) \biggl( \frac{dr}{R_\mathrm{norm}}\biggr) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) } \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho}\biggr]^{\gamma_\mathrm{g}}
\biggl( \frac{M}{M_\mathrm{tot}}\biggr)^{\gamma_\mathrm{g}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma_\mathrm{g}}
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{P}{P_c}\biggr) \biggl( \frac{dr}{R}\biggr) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}\chi^{-3/n}
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \, . </math>
</td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~f</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f = \chi^{3/n}\biggl[ \frac{U_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} \biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \biggr\} \, ;</math>
</td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_U(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f \rightarrow b = \chi^{3/n}\biggl[ \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} {\tilde\mathfrak{f}}_A
\, ;</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

When <math>~\Upsilon_U(\xi) = 1</math>, then according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala & Horedt (1974)]], this integral over polytropic functions becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~{\tilde\mathfrak{f}}_A \equiv
{\tilde\xi}^{-3}\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr] \, ,
</math>
</td>
</tr>
</table>
</div>
which matches the expression for <math>~{\tilde\mathfrak{f}}_A</math> [[SSCpt1/Virial/FormFactors#PTtable|derived earlier]].

=====Modified Gravitational Potential Energy=====

Similarly, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{W_\Upsilon}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\int_0^R \Upsilon_W(r) \biggl(\frac{GM_r}{r}\biggr) 4\pi r^2 \rho dr
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{R_\mathrm{norm}\rho_c R^2}{M_\mathrm{tot}}\int_0^R 4\pi \Upsilon_W(r) \biggl(\frac{M_r}{M_\mathrm{tot}}\biggr) \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\chi^{-1}
\int_0^R 3\Upsilon_W(r) \biggl[\frac{M_r}{M_\mathrm{tot}}\biggr] \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 3\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~e</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e = -\chi \biggl[ \frac{W_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, ;
</math>
</td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_W(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e \rightarrow a = -\chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\frac{1}{{\tilde\mathfrak{f}}_M} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 ~{\tilde\mathfrak{f}}_W \, ;
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Now, according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala & Horedt (1974)]], when <math>~\Upsilon_W(\xi) = 1</math>, this integral over polytropic functions becomes,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
As we have [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors|detailed elsewhere]], from this, we have deduced that, for polytropic configurations,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
{\tilde\xi}^{-5}
\int_0^\tilde\xi 5 \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>

===Test Virial Equilibrium Condition===

If the correct normalized equilibrium radius, <math>~\chi_\mathrm{eq}</math>, is specified, our [[#Expectations|expectation regarding virial equilibrium]] is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3nc\chi_\mathrm{eq}^{4 } - 3b\chi_\mathrm{eq}^{(n-3)/n} + an</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 0\, .</math>
</td>
</tr>
</table>
</div>
Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair — <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — that has been given by [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Horedt (1970)]], namely,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\chi_\mathrm{eq} = \frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}} = \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{e}}{P_\mathrm{Horedt}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\, ,
</math>
</td>
</tr>
</table>
</div>

where we have taken into account the [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|shift in normalization factors]],
<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">Switch from [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Hoerdt's (1970)]] Normalization</th><tr>

<tr>
<td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-(n-1)/(n-3)}\frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}\biggr]^{1/(4-3\gamma)}
= \biggl[(n+1)^{-1} \biggl( 4\pi \biggr)^{1/n}\biggr]^{n/(n-3)}
= \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n+1)/(n-3)}
\frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{ \biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3} \biggl( \frac{1}{4\pi} \biggr) \biggr\}^{\gamma/(4-3\gamma)}
= \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>

We therefore have:
====First Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3n\biggl[\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggr]\chi_\mathrm{eq}^{4 }</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi n \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggr\}^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{4(n-1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi n \biggl[(n+1)^{[3(n+1)-4n]} ( 4\pi )^{[4-(n+1)]} \biggr]^{1/(n-3)}
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{[2(n+1)+ 4(1-n)]/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{[4(n-1)-2(n+1)]/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{n}{(n+1) }\biggr]
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^2 \, .
</math>
</td>
</tr>
</table>
</div>

====Second Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3b\chi_\mathrm{eq}^{(n-3)/n}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi ~n \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggr\}^{(n-3)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\pi ~n}{(5-n)} \biggl[ \frac{1}{4\pi} \biggl( - \frac{\tilde\xi}{\tilde\theta^'}\biggr) \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times (n+1)^{-1} ( 4\pi )^{1/n}
{\tilde\xi}^{(n-3)/n} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times {\tilde\xi}^{[(n-3)/n + 3(n+1)/n]} ( -\tilde\xi^2 \tilde\theta' )^{[(1-n)/n - (n+1)/n]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2} \, .
</math>
</td>
</tr>
</table>
</div>

====Third Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~an</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\biggl( - \frac{\tilde\xi}{3\tilde\theta^'} \biggr) \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2
\frac{3\cdot 5~n}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^2
\frac{n \tilde\xi^4}{(5-n)}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2} \, .
</math>
</td>
</tr>
</table>
</div>

====Combined====
Combining the three terms, the virial expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ (5-n)(n+1)\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^{-2} \biggl[ 3nc\chi_\mathrm{eq}^{4 } + an - 3b\chi_\mathrm{eq}^{(n-3)/n} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
n(5-n)
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2}
+ n(n+1)\tilde\xi^4
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-n \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~n( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{
(5-n)\tilde\theta_n^{n+1}
+ (n+1) \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~n(n+1) ( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{ 0 \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Q. E. D.

==The Ledoux Variational Principle==
Drawing from a [[SSC/VariationalPrinciple#Ledoux.27s_Expression|separate presentation of Ledoux's variational principle]], let's normalize his Lagrangian using the same normalizations that have been used, above. His expression is …

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~L </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\pi e^{2i\omega t} \biggl\{
- \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0
- \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
+ \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0
-\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Our normalization produces,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0
- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0
-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2\int_0^R x^2 \biggl[\frac{R_\mathrm{norm} R^5}{G\rho_c}\biggr]
\biggl[ \frac{3\rho_c}{4\pi \bar\rho R^3}\biggr]^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \omega^2 \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3 \rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R}
- 3 \gamma_\mathrm{g} x_\mathrm{surface}^2 \biggl[ \frac{R^3 }{ R_\mathrm{norm}^3} \frac{P_e}{P_\mathrm{norm}}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g}\biggl[\frac{P_c }{P_\mathrm{norm} }\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}
~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Given that,
<div align="center">
<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\gamma} \, ,</math>
</div>
and,
<div align="center">
<math>~\frac{M_r}{M} = \biggl(\frac{\rho_c}{\bar\rho} \biggr) \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \, ,</math>
</div>

this expression for the Lagrangian becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g}\biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \biggl( \frac{3}{4\pi}\biggr) \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl\{ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg\} \frac{dr_0}{R}
~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

In an effort to help identify the various terms in this expression as well as the relationship between the entire expression and our unperturbed free energy expression, let's ignore all terms involving the radial eigenfunction, <math>~x</math>, and its derivative. In this case, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}\biggr|_\mathrm{unperturbed}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\frac{\gamma_\mathrm{g} (\gamma_\mathrm{g} -1)}{4\pi} \biggl\{\frac{4\pi}{3(\gamma_\mathrm{g} -1)} \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
\int_0^R 3\biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (3\gamma_\mathrm{g} - 4) \biggl( \frac{1}{4\pi}\biggr)\biggl\{- \frac{3}{5} \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R 5\biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl[ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg] \frac{dr_0}{R} \biggr\}
~- ~\frac{3^2 \gamma_\mathrm{g}}{4\pi}\biggl\{\frac{4\pi}{3} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{4\pi L_{\{\}} }{E_\mathrm{norm}}\biggr|_\mathrm{unperturbed}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{4\pi \omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g} (\gamma_\mathrm{g} -1) \biggl\{ \frac{U_\mathrm{int}}{E_\mathrm{norm}} \biggr\}
+ (3\gamma_\mathrm{g} - 4) \biggl\{ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr\}
~- ~3^2 \gamma_\mathrm{g} \biggl\{ \frac{P_e V}{E_\mathrm{norm}} \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=See Also=

{{ SGFfooter }}

StabilityVariationalPrinciple

2021-08-01T20:00:48Z

174.64.8.247: Created page with "__FORCETOC__  =Free-Energy Stability Analysis= ==Most General Case== Consider a free-energy function of the form, <div align="center">..."

__FORCETOC__


=Free-Energy Stability Analysis=

==Most General Case==
Consider a free-energy function of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{-3/j} + \mathcal{G}_0 \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>~a, b, c,</math> and <math>~\mathcal{G}_0</math> are constants, and the dimensionless configuration radius,
<div align="center">
<math>~\chi \equiv \frac{R}{R_0} \, ,</math>
</div>
is defined in terms of a characteristic length, <math>~R_0</math>, which is likely to be different for each type of problem.

===Virial Equilibrium===
The first variation (first derivative) of this function with respect to the configuration's radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{G}}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~a\chi^{-2} - \biggl(\frac{3b}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 c}{j}\biggr) \chi^{-3/j -1} \, .</math>
</td>
</tr>
</table>
</div>
According to the virial theorem, the radius of an equilibrium configuration is obtained by setting <math>~d\mathcal{G}/d\chi = 0</math> and identifying the roots of the resulting equation. For example, identifying roots of the polynomial expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} - \biggl(\frac{1}{j}\biggr) \chi_\mathrm{eq}^{(j-3)/j } \, .</math>
</td>
</tr>
</table>
</div>

===Stability===
Let's rewrite the first variation of the free-energy function in terms of three coefficients <math>~(e,f,g)</math> which, in general, we will permit to have different values from the original three <math>~(a,b,c)</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{G}^'</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e\chi^{-2} - \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -1} \, .</math>
</td>
</tr>
</table>
</div>
The first variation (first derivative) of ''this'' function with respect to the configuration's radius — which, in effect, represents the second variation of the free-energy function — gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + \biggl(\frac{3}{j} + 1\biggr) \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -2} \, .</math>
</td>
</tr>
</table>
</div>
If we evaluate this function by setting <math>~\chi = \chi_\mathrm{eq}</math>, the sign of the resulting expression should indicate stability (positive) or dynamical instability (negative); and the marginally unstable configuration is identified by the value of <math>~\chi_\mathrm{eq}</math> for which <math>~d\mathcal{G}^'/d\chi = 0</math>.

==Pressure-Truncated Configurations==

===Expectations===
For pressure-truncated polytropes, we set <math>~j = -1</math> and let <math>~n</math> represent the chosen polytropic index. In this situation, then, we have,
<div align="center">
<table border="0" cellpadding="3" align="center">

<tr>
<td align="right">Free-energy expression:</td>
<td align="center">     </td>
<td align="right">
<math>~\mathcal{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{3} + \mathcal{G}_0 \, ;</math>
</td>
</tr>

<tr>
<td align="right">Virial equilibrium:</td>
<td align="center">     </td>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} + \chi_\mathrm{eq}^{4 } \, ;</math>
</td>
</tr>

<tr>
<td align="right">Stability indicator:</td>
<td align="center">     </td>
<td align="right">
<math>~\frac{d\mathcal{G}^'}{d\chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + 6g \chi \, .</math>
</td>
</tr>
</table>
</div>
Hence, the (critical) equilibrium radius of the marginally unstable configuration is given by the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~6g \chi_\mathrm{eq}^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2e - \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} + \chi_\mathrm{eq}^4 \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
6g \chi_\mathrm{eq}^4
+\biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\chi_\mathrm{eq}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\biggl[6g + \frac{3cf(n+3)}{nb} \biggr]\chi_\mathrm{eq}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2e - \biggl[\frac{af(n+3)}{nb} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2nbe -af(n+3)}{6nbg +3cf(n+3)} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Notice that, if <math>~(e,f,g) \rightarrow (a,b,c)</math>, this gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\chi_\mathrm{eq}^4\biggr|_\mathrm{crit}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2nba -ab(n+3)}{6nbc +3cb(n+3)} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{3^2c}\biggl[\frac{n-3}{n+1} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>


===Energies and Structural Form Factors===
====Old Approach====

As has been developed in, for example, our [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review|accompanying review]], we adopt the following normalizations:
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~R_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R^3_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{n/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~E_\mathrm{norm}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>

Then, from separate summaries — both [[User:Tohline/SphericallySymmetricConfigurations/Virial#Summary_of_Normalized_Expressions|here]] and [[User:Tohline/SSC/Virial/FormFactors#Implication_for_Structural_Form_Factors|here]] — we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{P_e V}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi^3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr] \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx\biggr\} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}} = \frac{U_\mathrm{int}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1} 3x^2 \biggl[ \frac{P(x)}{P_c} \biggr] dx \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>
where the [[User:Tohline/SphericallySymmetricConfigurations/Virial#Structural_Form_Factors|structural form factors are defined]] as follows:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\mathfrak{f}_M </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \int_0^1 3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_A</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>
</td>
</tr>
</table>
</div>

This gives, specifically for [[User:Tohline/SSC/Virial/FormFactors#PTtable|specifically for pressure-truncated polytropic configurations]],
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

====New Approach====
In order to accommodate the structural integrals required by the Ledoux variational principle, let's re-derive some of these key energy and form-factor expressions. Basically, we will be repeating some [[User:Tohline/SphericallySymmetricConfigurations/Virial#Expressions_for_Various_Energy_Terms|earlier derivations]].

=====Mass=====

Defining <math>~M_\mathrm{tot}</math> as the total mass of the ''isolated'' configuration, while <math>~M \le M_\mathrm{tot}</math> is the truncated configuration's mass; defining <math>~R</math> as the truncated configuration's (not necessarily ''equilibrium'') radius; and being careful to define the mean density of the truncated configuration such that,
<div align="center">
<math>~\bar\rho \equiv \frac{3M}{4\pi R^3} \, ,</math>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~M_r(r) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \int_0^r 4\pi r^2 \rho dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{M_r(r)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{3}{4\pi} \int_0^r 4\pi \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{\rho}{\rho_\mathrm{norm}}\biggr) \frac{dr}{R_\mathrm{norm}} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\rho_\mathrm{norm}}\biggr) \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^r 3\biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl[ \frac{M/R^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr] \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^\xi 3\biggl( \frac{\xi}{\tilde\xi}\biggr)^2 \biggl( \frac{\rho}{\rho_c}\biggr) \frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}
\int_0^\xi 3\xi^2 \theta^n d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Acknowledging that <math>~M_r \rightarrow M</math> when the upper integration limit goes to <math>~\tilde\xi</math>, we see that the "mass" form-factor is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
</td>
<td align="center">
<math>~\equiv </math>
</td>
<td align="left">
<math>~ {\tilde\xi}^{-3}\int_0^{\tilde\xi} 3\xi^2 \theta^n d\xi = \biggl( \frac{\bar\rho}{\rho_c}\biggr) \, .</math>
</td>
</tr>
</table>
</div>
Now, from the,
<div align="center">
Polytropic Lane-Emden Equation<p></p>
{{ User:Tohline/Math/EQ_SSLaneEmden01 }}
</div>
we realize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d}{d\xi}\biggl(\xi^2 \theta^'\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \xi^2 \theta^n \, .</math>
</td>
</tr>
</table>
</div>
So these last two expressions may also be written as,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{M_r(r)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho_c}{\bar\rho}\biggr) \biggl( \frac{M}{M_\mathrm{tot}} \biggr) {\tilde\xi}^{-3}\biggl[ - 3 \xi^2 \theta^' \biggr]
\, ;
</math>
</td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_M</math>
</td>
<td align="center">
<math>~\equiv </math>
</td>
<td align="left">
<math>~\biggl[ -\frac{3\theta^'}{\xi} \biggr]_\tilde\xi \, .</math>
</td>
</tr>
</table>
</div>

=====Modified Internal Energy=====
Now we want to develop the appropriately scaled integral definition of a "variational" internal energy having the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{U_\Upsilon}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) }
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R_\mathrm{norm}}\biggr)^2 \biggl( \frac{P}{P_\mathrm{norm}}\biggr) \biggl( \frac{dr}{R_\mathrm{norm}}\biggr) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{(\gamma_\mathrm{g}-1) } \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho}\biggr]^{\gamma_\mathrm{g}}
\biggl( \frac{M}{M_\mathrm{tot}}\biggr)^{\gamma_\mathrm{g}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma_\mathrm{g}}
\int_0^R 4\pi \Upsilon_U(r) \biggl( \frac{r}{R}\biggr)^2 \biggl( \frac{P}{P_c}\biggr) \biggl( \frac{dr}{R}\biggr) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}\chi^{-3/n}
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \, . </math>
</td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~f</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f = \chi^{3/n}\biggl[ \frac{U_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} \biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \Upsilon_U(\xi) \xi^2 \theta^{n+1} d\xi \biggr\} \, ;</math>
</td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_U(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f \rightarrow b = \chi^{3/n}\biggl[ \frac{U_\mathrm{int}}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi ~n}{3} \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n} {\tilde\mathfrak{f}}_A
\, ;</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
{\tilde\xi}^{-3} \int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

When <math>~\Upsilon_U(\xi) = 1</math>, then according to [[User:Tohline/SSC/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala & Horedt (1974)]], this integral over polytropic functions becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~{\tilde\mathfrak{f}}_A \equiv
{\tilde\xi}^{-3}\int_0^\tilde\xi 3 \xi^2 \theta^{n+1} d\xi
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr] \, ,
</math>
</td>
</tr>
</table>
</div>
which matches the expression for <math>~{\tilde\mathfrak{f}}_A</math> [[User:Tohline/SSC/Virial/FormFactors#PTtable|derived earlier]].

=====Modified Gravitational Potential Energy=====

Similarly, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{W_\Upsilon}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\int_0^R \Upsilon_W(r) \biggl(\frac{GM_r}{r}\biggr) 4\pi r^2 \rho dr
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{R_\mathrm{norm}\rho_c R^2}{M_\mathrm{tot}}\int_0^R 4\pi \Upsilon_W(r) \biggl(\frac{M_r}{M_\mathrm{tot}}\biggr) \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\chi^{-1}
\int_0^R 3\Upsilon_W(r) \biggl[\frac{M_r}{M_\mathrm{tot}}\biggr] \biggl(\frac{\rho}{\rho_c}\biggr) \frac{ r dr}{R^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 3\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \chi^{-1} {\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Hence, the coefficient, <math>~e</math>, in the free-energy expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e = -\chi \biggl[ \frac{W_\Upsilon}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\frac{\rho_c}{\bar\rho} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 \biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\Upsilon_W(\xi) \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, ;
</math>
</td>
</tr>
</table>
</div>
or, if <math>~\Upsilon_W(\xi) = 1</math>, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e \rightarrow a = -\chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\frac{1}{{\tilde\mathfrak{f}}_M} \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2 ~{\tilde\mathfrak{f}}_W \, ;
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~{\tilde\mathfrak{f}}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl\{{\tilde\xi}^{-5}
\int_0^\tilde\xi 5\biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Now, according to [[User:Tohline/SSC/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|Viala & Horedt (1974)]], when <math>~\Upsilon_W(\xi) = 1</math>, this integral over polytropic functions becomes,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
As we have [[User:Tohline/SSC/Virial/FormFactors#Implication_for_Structural_Form_Factors|detailed elsewhere]], from this, we have deduced that, for polytropic configurations,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
{\tilde\xi}^{-5}
\int_0^\tilde\xi 5 \biggl[ - 3 \xi^2 \theta^' \biggr] \theta^n \xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>

===Test Virial Equilibrium Condition===

If the correct normalized equilibrium radius, <math>~\chi_\mathrm{eq}</math>, is specified, our [[#Expectations|expectation regarding virial equilibrium]] is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3nc\chi_\mathrm{eq}^{4 } - 3b\chi_\mathrm{eq}^{(n-3)/n} + an</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 0\, .</math>
</td>
</tr>
</table>
</div>
Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair — <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — that has been given by [[User:Tohline/SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Horedt (1970)]], namely,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\chi_\mathrm{eq} = \frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}} = \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{e}}{P_\mathrm{Horedt}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\, ,
</math>
</td>
</tr>
</table>
</div>

where we have taken into account the [[User:Tohline/SphericallySymmetricConfigurations/Virial#Choices_Made_by_Other_Researchers|shift in normalization factors]],
<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">Switch from [[User:Tohline/SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Hoerdt's (1970)]] Normalization</th><tr>

<tr>
<td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-(n-1)/(n-3)}\frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}\biggr]^{1/(4-3\gamma)}
= \biggl[(n+1)^{-1} \biggl( 4\pi \biggr)^{1/n}\biggr]^{n/(n-3)}
= \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n+1)/(n-3)}
\frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{ \biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3} \biggl( \frac{1}{4\pi} \biggr) \biggr\}^{\gamma/(4-3\gamma)}
= \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>

We therefore have:
====First Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3n\biggl[\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggr]\chi_\mathrm{eq}^{4 }</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi n \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggr\}^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{4(n-1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi n \biggl[(n+1)^{[3(n+1)-4n]} ( 4\pi )^{[4-(n+1)]} \biggr]^{1/(n-3)}
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{[2(n+1)+ 4(1-n)]/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{[4(n-1)-2(n+1)]/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{n}{(n+1) }\biggr]
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^2 \, .
</math>
</td>
</tr>
</table>
</div>

====Second Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3b\chi_\mathrm{eq}^{(n-3)/n}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi ~n \biggl[ \biggl(\frac{3}{4\pi}\biggr) \frac{1}{{\tilde\mathfrak{f}}_M} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}
\frac{(n+1)}{(5-n)} \biggl[\frac{6\tilde\theta^{n+1}}{(n+1)} + 3 (\tilde\theta^')^2 - {\tilde\mathfrak{f}}_M\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl\{ \biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggr\}^{(n-3)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\pi ~n}{(5-n)} \biggl[ \frac{1}{4\pi} \biggl( - \frac{\tilde\xi}{\tilde\theta^'}\biggr) \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{(n+1)/n}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times (n+1)^{-1} ( 4\pi )^{1/n}
{\tilde\xi}^{(n-3)/n} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/n} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/n}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times {\tilde\xi}^{[(n-3)/n + 3(n+1)/n]} ( -\tilde\xi^2 \tilde\theta' )^{[(1-n)/n - (n+1)/n]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n}{(5-n)(n+1)} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^{2}
\biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2} \, .
</math>
</td>
</tr>
</table>
</div>

====Third Term====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~an</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3}{5}\biggl[\biggl( - \frac{\tilde\xi}{3\tilde\theta^'} \biggr) \biggl(\frac{M}{M_\mathrm{tot}}\biggr)\biggr]^2
\frac{3\cdot 5~n}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^2
\frac{n \tilde\xi^4}{(5-n)}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2} \, .
</math>
</td>
</tr>
</table>
</div>

====Combined====
Combining the three terms, the virial expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ (5-n)(n+1)\biggl[\frac{M}{M_\mathrm{tot}}\biggr]^{-2} \biggl[ 3nc\chi_\mathrm{eq}^{4 } + an - 3b\chi_\mathrm{eq}^{(n-3)/n} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
n(5-n)
{\tilde\xi}^4 \tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{-2}
+ n(n+1)\tilde\xi^4
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] ( - \tilde\xi^2 \tilde\theta^')^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-n \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] {\tilde\xi}^{4} ( -\tilde\xi^2 \tilde\theta' )^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~n( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{
(5-n)\tilde\theta_n^{n+1}
+ (n+1) \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[6\tilde\theta^{n+1} + 3(n+1) (\tilde\theta^')^2 - (n+1) \biggl( - \frac{3\tilde\theta^'}{\tilde\xi}\biggr) \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~n(n+1) ( -\tilde\xi^2 \tilde\theta' )^{-2} {\tilde\xi}^4 \biggl\{ 0 \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Q. E. D.

==The Ledoux Variational Principle==
Drawing from a [[User:Tohline/SSC/VariationalPrinciple#Ledoux.27s_Expression|separate presentation of Ledoux's variational principle]], let's normalize his Lagrangian using the same normalizations that have been used, above. His expression is …

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~L </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\pi e^{2i\omega t} \biggl\{
- \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0
- \int_0^R \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
+ \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0
-\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Our normalization produces,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] \rho_0 \omega^2 r_0^4 x^2 dr_0
- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \int_0^R \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2}\biggr] r_0^3 \rho_0 x^2 \biggl(- \frac{1}{\rho_0} \frac{dP_0}{dr_0} \biggr) dr_0
-\biggl[ \frac{3 \gamma_\mathrm{g} }{P_\mathrm{norm} R_\mathrm{norm}^3} ~r_0^3 x^2 P_0\biggr]_0^{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2\int_0^R x^2 \biggl[\frac{R_\mathrm{norm} R^5}{G\rho_c}\biggr]
\biggl[ \frac{3\rho_c}{4\pi \bar\rho R^3}\biggr]^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \omega^2 \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
- \gamma_\mathrm{g} \int_0^R \biggl[\frac{1}{P_\mathrm{norm} R_\mathrm{norm}^3}\biggr] P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2 dr_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4)\biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \int_0^R x^2 \biggl[\frac{R_\mathrm{norm}R^2}{GM^2}\biggr] \biggl( \frac{r_0}{R}\biggr)^3 \rho_0 \biggl(\frac{GM_r R^2}{r^2_0} \biggr) \frac{dr_0}{R}
- 3 \gamma_\mathrm{g} x_\mathrm{surface}^2 \biggl[ \frac{R^3 }{ R_\mathrm{norm}^3} \frac{P_e}{P_\mathrm{norm}}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g}\biggl[\frac{P_c }{P_\mathrm{norm} }\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3
\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho }\biggr] \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl(\frac{M_r}{M} \biggr) \frac{dr_0}{R}
~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Given that,
<div align="center">
<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\gamma} \, ,</math>
</div>
and,
<div align="center">
<math>~\frac{M_r}{M} = \biggl(\frac{\rho_c}{\bar\rho} \biggr) \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \, ,</math>
</div>

this expression for the Lagrangian becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g}\biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
\int_0^R \biggl[ \biggl(\frac{r_0}{R}\biggr) \frac{\partial x}{\partial (r_0/R)}\biggr]^2 \biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- (3\gamma_\mathrm{g} - 4) \biggl( \frac{3}{4\pi}\biggr) \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R x^2 \biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl\{ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg\} \frac{dr_0}{R}
~- ~3 \gamma_\mathrm{g} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 x_\mathrm{surface}^2 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

In an effort to help identify the various terms in this expression as well as the relationship between the entire expression and our unperturbed free energy expression, let's ignore all terms involving the radial eigenfunction, <math>~x</math>, and its derivative. In this case, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{L_{\{\}} }{E_\mathrm{norm}}\biggr|_\mathrm{unperturbed}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{\omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\frac{\gamma_\mathrm{g} (\gamma_\mathrm{g} -1)}{4\pi} \biggl\{\frac{4\pi}{3(\gamma_\mathrm{g} -1)} \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\gamma} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{3-3\gamma}
\int_0^R 3\biggl(\frac{P_0}{P_c}\biggr) \biggl(\frac{r_0}{R}\biggr)^2 \frac{dr_0}{R} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (3\gamma_\mathrm{g} - 4) \biggl( \frac{1}{4\pi}\biggr)\biggl\{- \frac{3}{5} \biggl[ \frac{\rho_c}{\bar\rho }\biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R 5\biggl(\frac{\rho_0}{\rho_c} \biggr) \biggl( \frac{r_0}{R}\biggr) \biggl[ \int_0^R 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \frac{dr_0}{R} \bigg] \frac{dr_0}{R} \biggr\}
~- ~\frac{3^2 \gamma_\mathrm{g}}{4\pi}\biggl\{\frac{4\pi}{3} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \biggl[ \frac{P_e}{P_\mathrm{norm}}\biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{4\pi L_{\{\}} }{E_\mathrm{norm}}\biggr|_\mathrm{unperturbed}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\biggl( \frac{3}{4\pi }\biggr)\frac{\rho_c}{\bar\rho} \biggr]^2 \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\frac{4\pi \omega^2}{G\rho_c}\biggr] \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1}
\int_0^R \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl( \frac{r_0}{R}\biggr)^4 \frac{dr_0}{R}
~- ~\gamma_\mathrm{g} (\gamma_\mathrm{g} -1) \biggl\{ \frac{U_\mathrm{int}}{E_\mathrm{norm}} \biggr\}
+ (3\gamma_\mathrm{g} - 4) \biggl\{ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr\}
~- ~3^2 \gamma_\mathrm{g} \biggl\{ \frac{P_e V}{E_\mathrm{norm}} \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=See Also=

{{ SGFfooter }}

SSC/VariationalPrinciple

2021-08-01T19:59:03Z

174.64.8.247: /* Generalized Governing Integral Relation */

__FORCETOC__


=Ledoux's Variational Principle=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>Variational<br />Principle</b>]]</font>
|}
All of the discussion in this chapter will build upon our [[SSC/Perturbations#2ndOrderODE|derivation elsewhere]] of the,
<div align="center" id="2ndOrderODE">
<font color="#770000">'''LAWE:   Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

We will draw heavily from the papers published by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964)], as well as from pp. 458-474 of the review by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] in explaining how the ''variational principle'' can be used to identify the eigenvector of the fundamental mode of radial oscillation in spherically symmetric configurations. In an associated "Ramblings" appendix, we provide [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29|various derivations that support]] this chapter's relatively abbreviated presentation.

==Ledoux and Pekeris (1941)==
Historically, by the 1940s, the LAWE was a relatively familiar one to astrophysicists. For example, the opening paragraph of a 1941 paper by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124), reads:
<div align="center">
<table border="1" cellpadding="5">
<tr><td align="center">
Paragraph extracted from [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941)]<p></p>
"''Radial Pulsations of Stars''"<p></p>
ApJ, vol. 94, pp. 124-135 © American Astronomical Society
</td></tr>
<tr>
<td>
[[File:LedouxPekeris1941.jpg|600px|center|Ledoux & Pekeris (1941, ApJ, 94, 124)]]
</td>
</tr>
</table>
</div>

If we divide their equation (1) through by <math>~Xr = \Gamma_1 P r</math> and recognize that,
<div align="center">
<math>
\frac{dX}{dr} = \frac{dX}{dm}\frac{dm}{dr} = - \Gamma_1 g_0 \rho \, ,
</math>
</div>

we obtain,

<div align="center">
<math>
\frac{d^2\xi}{dr^2} + \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} +\frac{\rho}{\Gamma_1 P} \biggl[ \sigma^2 + (4 - 3\Gamma_1) \frac{g_0}{r} \biggr] \xi = 0 \, .
</math>
</div>
Clearly, this 2<sup>nd</sup>-order, ordinary differential equation is the same as our derived LAWE, but with a more general definition of the adiabatic exponent that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.

Multiplying this last equation through by <math>~\Gamma_1 P r^4</math>, and recognizing that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(r^4 \Gamma_1 P)\frac{d^2\xi}{dr^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
</math>
</td>
</tr>
</table>
</div>

<span id="RewrittenLAWE">we can write,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
+ ( \Gamma_1 P r^4 ) \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr}
+\rho \biggl[ \sigma^2 r^4 + (4 - 3\Gamma_1) g_0 r^3\biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \biggl[4r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr} \biggr] \frac{d\xi}{dr}
+ \biggl[ 4 r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr}\biggr] \frac{d\xi}{dr}
+\biggl[ \sigma^2 \rho r^4 - (4 - 3\Gamma_1) r^3 \frac{dP}{dr} \biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
(checked for n = 5) ==>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
+\biggl[ \sigma^2 \rho r^4 + (3\Gamma_1 - 4) r^3 \frac{dP}{dr} \biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ \Gamma_1 P r^4 ~\frac{d\xi}{dr} \biggr]
+\biggl[ \sigma^2 r^4 \rho + 4 Gm (r ) r \rho + 3\Gamma_1 r^3 \frac{dP}{dr} \biggr] \xi \, .
</math>
</td>
</tr>
</table>
</div>
Assuming that <math>~\Gamma_1</math> is uniform throughout the configuration, this last expression is the same as equation (3) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)], while the next-to-last expression is identical to equation (58.1) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)].

==Stability Based on Variational Principle==

Here we derive the Lagrangian directly from the governing LAWE. We begin with the next-to-last derived form of the LAWE that [[#RewrittenLAWE|appears above]] in our review of the paper by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and, following the guidance provided at the top of p. 666 of [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964, ApJ, 139, 664)], we multiply the LAWE through by the fractional displacement, <math>~\xi</math>. This gives, what we will henceforth refer to as, the,

<div align="center" id="FoundationalVariationalRelation">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3"><font color="maroon"><b>Foundational Variational Relation</b></font></td>
</tr>
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\xi \cdot \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) \, .
</math>
</td>
</tr>
</table>
</div>

===Chandrasekhar's Approach===

Next, in an effort to adopt the notation used by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C Chandrasekhar (1964)], we make the substitution, <math>~\xi \rightarrow \psi/r^3</math>, and regroup terms to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\sigma^2 \rho \psi^2}{r^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d}{dr} \biggl( \frac{\psi}{r^3} \biggr) \biggr]
- (3\Gamma_1 - 4) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r \Gamma_1 P ~\frac{d\psi}{dr} -3 \Gamma_1 P \psi ~\biggr]
- (3\Gamma_1 - 4) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4-3\Gamma_1 ) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r \Gamma_1 P ~\frac{d\psi}{dr} \biggr]
+ 3 \Gamma_1 \biggl( \frac{\psi^2}{r^3}\biggr) \frac{dP}{dr}
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
- \biggl\{
\frac{d}{dr}\biggl[ r \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}\biggr] -r\Gamma_1 P ~\frac{d\psi}{dr} \cdot \frac{d}{dr}\biggl( \frac{\psi}{r^3}\biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
+ \frac{\Gamma_1 P}{r^2} \biggl[\frac{d\psi}{dr} \biggr]^2
- \biggl[\frac{3\Gamma_1 P\psi}{r^3}\biggr]\frac{d\psi}{dr}
- \frac{d}{dr}\biggl[\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+ \frac{\Gamma_1 P}{r^2} \biggl[\frac{d\psi}{dr} \biggr]^2
- \frac{d}{dr}\biggl[ \frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">
Let's check to see whether the terms in the RHS of this last expression sum to zero when we plug in the appropriate functions for the marginally unstable, n = 5 configuration. In particular (replacing <math>~\xi</math> with <math>~x</math>, and setting <math>~r = a_5\xi</math>), we start with knowing,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="left">
<math>~\theta_5 = \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2}</math>;        
</td>
<td align="left">
<math>~\frac{d\theta_5}{d\xi} = - \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}</math>;        
</td>
</tr>

<tr>
<td align="left">
<math>~x = \biggl(\frac{3\cdot 5 - \xi^2}{3\cdot 5} \biggr)</math>;        
</td>
<td align="left">
<math>~\frac{dx}{d\xi} = -\frac{2\xi}{3\cdot 5}</math>;        
</td>
</tr>

<tr>
<td align="left">
<math>~\psi = a_5^3 \xi^3 x</math>;        
</td>
<td align="left">
<math>~
\frac{d\psi}{d\xi} = a_5^3 \biggl[ 3\xi^2 x + \xi^3 \biggl(\frac{dx}{d\xi}\biggr)\biggr] = \frac{a_5^3 \xi^2}{3}\biggl( 3^2 - \xi^2 \biggr) \, .
</math>
</td>
</tr>
</table>
</div>

----

Then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathrm{RHS}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl[ \frac{a_5^6 \xi^6 x^2}{a_5^3 \xi^3} \biggr] \frac{P_c}{a_5} \cdot \frac{d\theta^6}{d\xi}
+ P_c \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6}{a_5^4 \xi^2} \biggl\{ \frac{d\psi}{d\xi} \biggr\}^2
- \frac{P_c}{a_5} \cdot \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6 a_5^3 \xi^3 x}{a_5^3 \xi^2} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{\mathrm{RHS}}{P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl[ \xi^3 x^2 \biggr] \frac{d\theta^6}{d\xi}
+ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6}{ \xi^2} \biggl\{ \frac{\xi^2}{3}\biggl( 3^2 - \xi^2 \biggr) \biggr\}^2
- \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6 \xi^3 x}{ \xi^2} \biggl[ \frac{\xi^2}{3}\biggl( 3^2 - \xi^2 \biggr)\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2^3\cdot 3 \xi^3 x^2 \biggl(\frac{3+\xi^2}{3}\biggr)^{-5 / 2} \biggl[- \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]
+ \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{\xi}{3}\biggr)^2 \biggl(\frac{3+\xi^2}{3}\biggr)^{-3} ( 3^2 - \xi^2 )^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr)\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \frac{\xi^3 x}{ 3} ( 3^2 - \xi^2) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2^3 \cdot 3^4 ~\xi^4 x^2 \biggl(\frac{1}{3+\xi^2}\biggr)^{4}
+ \biggl(\frac{2\cdot 3^2}{5}\biggr) \xi^2 \biggl[ \frac{( 3^2 - \xi^2 )^2}{(3+\xi^2)^3} \biggr]
- \biggl(\frac{2\cdot 3^3}{5}\biggr)\frac{d}{d\xi}\biggl\{ \xi^3 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5 (3+\xi^2)^4 [\mathrm{RHS} ]}{ 2\cdot 3^2 P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2^2 \cdot 3^2 \cdot 5~\xi^4 x^2
+ \xi^2 (3+\xi^2) ( 3^2 - \xi^2 )^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 3(3+\xi^2)^4 \biggl\{
\xi^3 \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr] \frac{dx}{d\xi}
+ 3\xi^2 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr]
+ \xi^3 x \biggl[ \frac{ -2\xi }{(3+\xi^2)^3} \biggr]
-2\cdot 3 \xi^4 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^4} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^2 (3+\xi^2) ( 3^2 - \xi^2 )^2
- 2^2 \cdot 3^2 \cdot 5~\xi^4 x^2
- 3\xi^3 ( 3^2 - \xi^2) (3+\xi^2) \biggl[ - \frac{2\xi}{ 3\cdot 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 3 \xi^2\biggl\{
3 ( 3^2 - \xi^2) (3+\xi^2)
-2 \xi^2 (3+\xi^2)
-2\cdot 3 \xi^2 ( 3^2 - \xi^2)
\biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5^2 (3+\xi^2)^4 [\mathrm{RHS} ]}{ 2\cdot 3^2 ~\xi^2 P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
5 (3+\xi^2) ( 3^2 - \xi^2 )^2
- 2^2~\xi^2 (15-\xi^2)^2
+ 2 \xi^2 ( 3^2 - \xi^2) (3+\xi^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[
2 \xi^2 (3+\xi^2)
+ 2\cdot 3 \xi^2 ( 3^2 - \xi^2)
- 3 ( 3^2 - \xi^2) (3+\xi^2)
\biggr] (15-\xi^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(3\cdot 5 + 5\xi^2) ( 3^4 - 2\cdot 3^2\xi^2 + \xi^4)
- 2^2~\xi^2 (3^2\cdot 5^2 - 2\cdot 3\cdot 5 \xi^2 + \xi^4)
+ 2 \xi^2 ( 3^3 + 2\cdot 3\xi^2 -\xi^4)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ 2\cdot 3 \xi^2 + 2\xi^4 + 2\cdot 3^3 \xi^2 - 2\cdot 3 \xi^4 - 3(3^3 + 2\cdot 3\xi^2 -\xi^4)
\biggr] (15-\xi^2)
</math>
</td>
</tr>
</table>
</div>

Coefficients of various powers of <math>~\xi</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi^0:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~3^5\cdot 5 -3^5\cdot 5 = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^2:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~-2\cdot 3^3\cdot 5 + 3^4\cdot 5 -2^2 \cdot 3^2 \cdot 5^2 +2\cdot 3^3 + 2\cdot 3^2\cdot 5 + 2\cdot 3^4\cdot 5 - 2\cdot 3^3\cdot 5 + 3^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
   
</td>
<td align="left">
<math>~= 3^2\cdot 5[-2\cdot 3 + 3^2 + 2 + 2\cdot 3^2 - 2\cdot 3] + 3^2[ 2\cdot 3 + 3^2 - 2^2 \cdot 5^2 ] = 3^2\cdot 5[17 ] - 3^2[5\cdot 17 ] = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^4:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~3\cdot 5 -2\cdot 3^2\cdot 5 +2^3\cdot 3\cdot 5 + 2^2\cdot 3 + 2\cdot 3 \cdot 5 - 2\cdot 3^2\cdot 5 - 2\cdot 3 -2\cdot 3^3 + 2\cdot 3^2 + 3^2\cdot 5</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
   
</td>
<td align="left">
<math>~= 3\cdot 5[1 -2\cdot 3 +2^3 + 2 - 2\cdot 3 + 3] + 2\cdot 3[2 - 1 - 3^2 + 3] = 2\cdot 3\cdot 5 - 2\cdot 3\cdot 5 = 0 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^6:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~5 - 2^2 -2 -2 +2\cdot 3 -3 = 0</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

<span id="ChandraEq49">Multiplying through by <math>~dr</math>, and integrating over the volume gives,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R (\sigma^2 \rho \psi^2)\frac{dr}{r^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl[ \Gamma_1 P \biggl(\frac{d\psi}{dr} \biggr)^2
+ \frac{4\psi^2}{r} \biggl( \frac{dP}{dr} \biggr) \biggr]\frac{dr}{r^2}
- \biggl[\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr]_0^R \, ,
</math>
</td>
</tr>
</table>
</div>
which is identical to equation (49) of [http://adsabs.harvard.edu/abs/1964ApJ...139..664C Chandrasekhar (1964)], if the last term — the difference of the central and surface boundary conditions — is set to zero.

Note that if we shift from the variable, <math>~\psi</math>, back to the fractional displacement function, <math>~\xi</math>, the last term in this expression may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r \xi \frac{d}{dr} \biggl[r^3 \xi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r \xi \biggl[3r^2 \xi + r^3 \frac{d\xi}{dr}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r^3 \xi^2 \biggl[3 + \frac{d\ln\xi}{d\ln r}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>
So, as is pointed out by [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)] in connection with their equation (57.31), setting this expression to zero at the surface of the configuration is equivalent to setting the variation of the pressure to zero at the surface. Quite generally, this can be accomplished by demanding that,
<div align="center" id="SufaceBC">
<math>~\frac{d\ln\xi}{d\ln r}\biggr|_\mathrm{surface} = -3 \, .</math>
</div>

(An [[SSC/Perturbations#Boundary_Conditions|accompanying chapter]] provides a broader discussion of this and other astrophysically reasonable boundary conditions that are associated with solutions to the LAWE.)

===Ledoux & Walraven Approach===
Returning to the above [[#FoundationalVariationalRelation|''Foundational Variational Relation'']], we can also write,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\xi \cdot \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \frac{d}{dr}\biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \int_0^R\sigma^2 \rho r^4 \xi^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]_0^R
</math>
</td>
</tr>
</table>
</div>

If the last term (boundary conditions) is set to zero, then we may also write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, if the radial profile of the pressure and the density is known throughout a spherically symmetric, equilibrium configuration, and if, furthermore, the eigenfunction, <math>~\xi(r)</math>, of a radial oscillation mode is specified precisely, then this expression will give the (square of the) ''eigenfrequency'' of that oscillation mode.

By using formal ''variational principle'' techniques to derive this same expression, [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)] are able to offer a broader interpretation, which is encapsulated by their equation (59.10), viz.,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma_0^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathrm{min}~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, if the exact radial eigenfunction, <math>~\xi(r)</math>, is not known, various approximate eigenfunctions can be tried. The trial eigenfunction that ''minimizes'' the righthand-side of this expression will give the (square of the) eigenfrequency of the ''fundamental'' mode of oscillation (subscript zero). Furthermore, via an evaluation of this righthand-side expression, any reasonable trial eigenfunction — for example, <math>~\xi</math> = constant — can provide an ''upper limit'' to <math>~\sigma_0^2</math>.

===Ledoux & Pekeris Approach===
Here we follow the lead of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)]. Returning to the integral expression just derived in our discussion of the ''Ledoux & Walraven approach'', and multiplying through by <math>~4\pi</math>, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R 4\pi \sigma^2 \rho r^4 \xi^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R 4\pi r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) 4\pi r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[4\pi r^3 \Gamma_1 P\xi^2 \biggl(\frac{d\ln \xi}{d\ln r}\biggr) \biggr]_0^R \, .
</math>
</td>
</tr>
</table>
</div>
If we acknowledge that:
* at the center of the configuration, <math>~r^3 = 0</math>;
* [[#SurfaceBC|as above]], the boundary condition at the surface is <math>~P = P_e</math> while <math>~(d\ln \xi/d\ln r) = -3</math>;
* the differential mass element is, <math>~dm = 4\pi r^2 \rho dr</math> and the corresponding differential volume element is, <math>~dV = 4\pi r^2 dr</math>; and
* a statement of detailed force balance is, <math>~dP/dr = - Gm\rho/r^2</math>,
this integral relation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \sigma^2 \int_0^R r^2 \xi^2 dm</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 \int_0^R \biggl[ r \biggl(\frac{d\xi}{dr}\biggr)\biggr]^2 P dV
+ (3\Gamma_1 - 4) \int_0^R \xi^2 \biggl( \frac{Gm}{r} \biggr) dm
- \biggr[\Gamma_1 \xi_\mathrm{surface}^2 (3P_e V) \biggl(-3\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Now, as we have [[SSCpt1/Virial#Wgrav|discussed separately]] — see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] — the gravitational potential energy of the unperturbed configuration is given by the integral,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - \int_0^{M} \biggl( \frac{Gm}{r_0} \biggr) dm \, ;</math>
</td>
</tr>
</table>
</div>

for adiabatic systems, the [[SSCpt1/Virial#Reservoir|internal energy]] is,
<div align="center">
<math>
U_\mathrm{int}
= \frac{1}{(\Gamma_1-1)} \int_0^R P_0 dV
\, ;</math>
</div>
and — see the text at the top of p. 126 of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] — the moment of inertia of the configuration about its center is,
<div align="center">
<math>
I = \int_0^M r_0^2 dm
\, .</math>
</div>
(Note that, defined in this way, <math>~I</math> is the same as [[VE#Standard_Presentation_.5Bthe_Virial_of_Clausius_.281870.29.5D|what we have referred to elsewhere]] as the ''scalar moment of inertia'', which is obtained by taking the trace of the [[VE#MOItensor|moment of inertia tensor]], <math>~I_{ij}</math>.)
<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3"><font color="maroon"><b>Variational Principle's Governing Integral Relation</b></font></td>
</tr>
<tr>
<td align="right">
<math>~ \sigma^2 \int_0^R \xi^2 dI</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 (\Gamma_1 - 1) \int_0^R \xi^2 \biggl[ \frac{d\ln\xi}{d\ln r}\biggr]^2 dU_\mathrm{int}
- (3\Gamma_1 - 4) \int_0^R \xi^2 dW_\mathrm{grav}
+ 3^2 \Gamma_1 P_e V \xi_\mathrm{surface}^2 \, .
</math>
</td>
</tr>
</table>
</div>

==Free-Energy Analysis==

<span id="Homologous">If we assume</span> the simplest approximation for the fundamental-mode eigenfunction, namely, <math>~\xi = \xi_0</math> = constant — that is, homologous expansion/contraction — then this last integral expression gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \sigma^2 I</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4 - 3\Gamma_1) W_\mathrm{grav}
+ 3^2 \Gamma_1 P_e V \, .
</math>
</td>
</tr>
</table>
</div>

Contrast this result with the following free-energy analysis:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV \, ,</math>
</td>
</tr>
</table>
</div>
where, in terms of the configuration's (generally non-equilibrium) dimensionless radius, <math>~\chi \equiv R/R_0</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-a\chi^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~U_\mathrm{int}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~b\chi^{3-3\Gamma_1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3} \chi^3 \, .</math>
</td>
</tr>
</table>
</div>
Then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial \mathfrak{G}}{\partial \chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~+a \chi^{-2} + 3(1-\Gamma_1) b \chi^{2-3\Gamma_1} + 4\pi P_e \chi^{2} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^{-1} \biggl[- W_\mathrm{grav} + 3(1-\Gamma_1) U_\mathrm{int} + 3 P_e V \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial^2 \mathfrak{G}}{\partial \chi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2a \chi^{-3} + 3(1-\Gamma_1)(2-3\Gamma_1) b \chi^{1-3\Gamma_1} + 8\pi P_e \chi </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^{-2} \biggl[ 2W_\mathrm{grav} + 3(1-\Gamma_1)(2-3\Gamma_1) U_\mathrm{int}+ 6 P_e V \biggr] \, .</math>
</td>
</tr>
</table>
</div>
The equilibrium condition occurs when <math>~\partial \mathfrak{G}/\partial \chi = 0</math>, that is, when,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3(1-\Gamma_1) U_\mathrm{int}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~W_\mathrm{grav} - 3 P_e V \, ,</math>
</td>
</tr>
</table>
</div>
in which case,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\chi^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial \chi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2W_\mathrm{grav} + (2-3\Gamma_1) (W_\mathrm{grav} - 3P_eV) + 6 P_e V </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(4-3\Gamma_1)W_\mathrm{grav} + 3^2 \Gamma_1 P_e V \, .</math>
</td>
</tr>
</table>
</div>

Fantastic! The righthand-side of this "free-energy-based" expression exactly matches the righthand-side of the [[#Homologous|above expression]] that has been derived from the variational principle, assuming homologous expansion/contraction (''i.e.,'' <math>~\xi</math> = constant). In this case, we can make the direct association,
<div align="center">
<math>~\sigma^2 I = \chi^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial \chi^2} \, .</math>
</div>
This also make sense in that the equilibrium configuration should be stable if <math>~\tfrac{\partial^2 \mathfrak{G}}{\partial \chi^2} > 0</math> — in which case, <math>~\sigma^2</math> is positive; whereas the equilibrium configuration should be ''unstable'' if <math>~\tfrac{\partial^2 \mathfrak{G}}{\partial \chi^2} < 0</math> — in which case, <math>~\sigma^2</math> is negative.

=Related, Exploratory Ideas=

==Logarithmic Derivatives==

Returning to our above discussion of the [[#Ledoux_.26_Walraven_Approach|Ledoux & Walraven approach]], we appreciate that the ''differential'' relation governing the Variational Principle is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \frac{d}{dr}\biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{d}{dr}\biggr[r^3 \Gamma_1 P\xi^2 \biggl(\frac{d\ln\xi}{d\ln r}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \sigma^2 \rho r^4 \xi^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi^2 \biggl\{
r^2 \Gamma_1 P \biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- (3\Gamma_1 - 4) r^3 \biggl( \frac{dP}{dr} \biggr)
- \sigma^2 \rho r^4
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(r \xi)^2 P \biggl\{
\Gamma_1 \biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- (3\Gamma_1 - 4) \biggl( \frac{d\ln P}{d\ln r} \biggr)
- \frac{\sigma^2 \rho r^2}{P}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Gamma_1 (r \xi)^2 P \biggl\{
\biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- \alpha \biggl( \frac{d\ln P}{d\ln r} \biggr)
- \frac{\sigma^2 \rho r^2}{\Gamma_1 P}
\biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\alpha \equiv \biggl(3 - \frac{4}{\Gamma_1}\biggr) \, .</math>
</div>

==Pressure-Truncated Polytropes==

Let's start with the integral expression derived in our discussion of the [[#Ledoux_.26_Walraven_Approach|Ledoux & Walraven approach]]; insert the variable, <math>~x</math>, in place of <math>~\xi</math>; and adopt the boundary conditions,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r = 0</math>   at the center,
</td>
<td align="center">
        along with        
</td>
<td align="left">
<math>~P = P_e~</math>,   and  <math>\frac{d\ln x}{d\ln r} = -3</math>   at the surface (r = R).
</td>
</tr>
</table>
</div>
That is, let's start with,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2 \, .
</math>
</td>
</tr>
</table>
</div>

===Via Generalized Normalization===
Next, we'll divide through by the [[StabilityVariationalPrinciple#Energies_and_Structural_Form_Factors|normalization energy, as defined in an accompanying discussion]],
<div align="center">
<math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3 = \frac{GM_\mathrm{tot}^2}{R_\mathrm{norm}} \, ,</math>
</div>
thereby making the integral relation dimensionless:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2} \biggr] \int_0^R \sigma^2 \rho r^4 x^2 dr
+\biggl[\frac{1}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \biggl[\frac{1}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+ \biggl[\frac{P_e R^3 }{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{R_\mathrm{norm} R^5 \rho_c^2}{M_\mathrm{tot}^2} \biggr] \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
+ \biggl[\frac{P_c R^3}{P_\mathrm{norm}R_\mathrm{norm}^3 } \biggr] \int_0^R \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[\frac{P_c R^3}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+ \biggl[\frac{P_e R^3 }{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^2 \chi^{-1} \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
+ \biggl[\frac{P_e }{P_\mathrm{norm}} \biggr] 3\Gamma_1 \chi^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[\frac{P_c}{P_\mathrm{norm} } \biggr] \chi^3 \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\chi \equiv \frac{R}{R_\mathrm{norm}} \, .</math>
</div>

Note that we will ultimately insert the relation,
<div align="center">
<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\Gamma_1} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\Gamma_1} \, .</math>
</div>
But, for the time being, dividing through by <math>~[P_c/P_\mathrm{norm}]\chi^3</math> gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{P_c}{P_\mathrm{norm} } \biggr]^{-1} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^2 \chi^{-4} \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[\frac{P_e }{P_c} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
+ \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R} \, ,
</math>
</td>
</tr>
</table>
</div>

Now let's focus on the second line of this integral energy relation, evaluating it for pressure-truncated polytropic configurations, in which case, <math>~\Gamma_1 \rightarrow (n+1)/n</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r}{R} \rightarrow \frac{\xi}{\tilde\xi}</math>
</td>
<td align="center">
        and        
</td>
<td align="left">
<math>~
\frac{P}{P_c} \rightarrow \theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>
We have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
Second line of relation
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
+ \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \biggl\{ \biggl( \frac{\xi}{\tilde\xi}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2 {\tilde\xi}^2
- \biggl(\frac{3-n}{n}\biggr) \biggl( \frac{\xi}{\tilde\xi} \biggr)^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d\xi} \biggr] \tilde\xi \biggr\}\frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{1}{n {\tilde\xi}^3}\int_0^{\tilde\xi} \biggl\{ (n+1) \xi^4 \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2
- (3-n) \xi^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d\xi} \biggr] \biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{1}{n {\tilde\xi}^3}\int_0^{\tilde\xi} \biggl\{ (n+1) \xi^4 \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2
- (n+1) (3-n) \xi^3 x^2 \theta^n \theta^' \biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{(n+1)}{n {\tilde\xi}^3}\int_0^{\tilde\xi}
\biggl(\frac{3}{2n}\biggr)^2\frac{\xi}{\theta^n} \biggl\{ \xi \theta \biggl[ \biggl( \frac{2n}{3}\biggr)\xi \theta^n \cdot \frac{dx}{d\xi}\biggr]^2
- (3-n) \biggl[ \biggl( \frac{2n}{3}\biggr) \xi \theta^n x\biggr]^2 \theta^' \biggr\}d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Now, let's examine how these terms combine if we ''guess'' the [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|analytically defined eigenfunction that applies to marginally unstable, pressure-truncated polytropic configurations]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^'}{\xi \theta^{n} } \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{2n}{3}\biggr) \xi \theta^n x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\frac{3(n-3)}{2n}\biggr] \biggl\{
\frac{\theta^{''}}{\xi \theta^{n}}
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n(\theta^')^2}{\xi \theta^{(n+1)}}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[\frac{3(n-3)}{2n}\biggr] \frac{1 }{\xi \theta^{n}} \biggl[
\theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{2n}{3}\biggr) \xi \theta^n\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3-n)
\biggl[ \theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta} \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>
Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
Second line of relation
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
{\tilde\theta}^{n+1} \biggl[ \frac{3( n+1)}{n} \biggr] \biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{ {\tilde\theta}^'}{\tilde\xi {\tilde\theta}^{n} } \biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)}{2^2n^3 {\tilde\xi}^3}\int_0^{\tilde\xi}
\frac{\xi}{\theta^n} \biggl\{
\xi \theta (3-n)\biggl[ \theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta} \biggr]^2
- \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \theta^'
\biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{{\tilde\xi}^2 {\tilde\theta}^{n+1}} \biggl[ \frac{3^3( n+1)}{2^2n^3} \biggr] \biggl[(n-1) \tilde\xi {\tilde\theta}^{n+1} + (n-3) \tilde\theta {\tilde\theta}^' \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)}{2^2n^3 {\tilde\xi}^3} \int_0^{\tilde\xi}
\frac{1}{\theta^{n+1}} \biggl\{
(3-n)\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]^2
- \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi \theta \theta^'
\biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{{\tilde\xi}^2 {\tilde\theta}^{n+1}} \biggl[ \frac{3^3( n+1)}{2^2n^3} \biggr] \biggl[(n-1) \tilde\xi {\tilde\theta}^{n+1} + (n-3) \tilde\theta {\tilde\theta}^' \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)^2}{2^2n^3 {\tilde\xi}^3} \int_0^{\tilde\xi}
\frac{1}{\theta^{n+1}} \biggl\{
\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]^2
+ \frac{1}{(n-3)} \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi \theta \theta^'
\biggr\}d\xi
</math>
</td>
</tr>
</table>
</div>

Note that, in this derivation, we have inserted the expressions:
<div align="center">
<math>~
\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr] =
\xi^2 \theta^{2(n+1)} + 6\xi \theta^{n+2}\theta^' + 2n\xi^2 \theta^{n+1} (\theta^')^2 + 6n\xi\theta (\theta^')^3 + n^2 \xi^2 (\theta^')^4
</math>
</div>
<div align="center">
<math>~
\frac{1}{(n-3)} \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi\theta (\theta^')=
\biggl[ \frac{(n-1)^2}{(n-3)}\biggr] \xi^3 \theta^{2n+1}(\theta^') + 2(n-1)\xi^2 \theta^{n+1} (\theta^' )^2 + (n-3) \xi\theta (\theta^')^3
</math>
</div>

===Directly to n = 5 Polytropic Configurations===

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{1}{R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl(\frac{r}{R}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{P}{P_c}\biggr) \biggl[\frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
- \int_0^R \biggl[3\biggl(\frac{n+1}{n}\biggr) - 4\biggr] \biggl(\frac{r}{R}\biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+3\biggl(\frac{n+1}{n}\biggr) \biggl( \frac{P_e}{P_c}\biggr) x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \frac{6}{5} \biggl(\frac{\xi}{\tilde\xi}\biggr)^4 \theta^6 \biggl[\frac{dx}{d(\xi/\tilde\xi)}\biggr]^2 \frac{d\xi}{\tilde\xi}
- \int_0^{\tilde\xi} \biggl( - \frac{2}{5}\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^3 x^2 \biggl[ \frac{d\theta^{6}}{d(\xi/\tilde\xi)} \biggr] \frac{d\xi}{\tilde\xi}
+\biggl(\frac{18}{5}\biggr) {\tilde\theta}^6 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde\xi}^3} \int_0^{\tilde\xi} \biggl( \frac{6}{5}\biggr) \xi^4 \theta^6 \biggl[\frac{dx}{d\xi}\biggr]^2 d\xi
+ \frac{1}{ {\tilde\xi}^3} \int_0^{\tilde\xi} \biggl(\frac{2}{5}\biggr) \xi^3 x^2 \biggl[ \frac{d\theta^{6}}{d\xi} \biggr] d\xi
+\biggl(\frac{18}{5}\biggr) {\tilde\theta}^6 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5 {\tilde\xi}^3 }{2R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} 3\xi^4 \theta^6 \biggl[ - \frac{2\xi}{15} \biggr]^2 d\xi
+ \int_0^{\tilde\xi} 6\xi^3 \biggl[\frac{15-\xi^2}{15}\biggr]^2 \theta^5\biggl[ \frac{d\theta}{d\xi} \biggr] d\xi
+9 {\tilde\xi}^3 {\tilde\theta}^6 \biggl[\frac{15- {\tilde\xi}^2}{15}\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{ 2^2}{3\cdot 5^2 } \biggr) \int_0^{\tilde\xi} \xi^6 \biggl( \frac{3}{3+\xi^2}\biggr)^3 d\xi
+ \biggl(\frac{ 2}{3\cdot 5^2 } \biggr) \int_0^{\tilde\xi} \xi^3 \biggl[15-\xi^2\biggr]^2 \biggl( \frac{3}{3+\xi^2}\biggr)^{4} \biggl[- \frac{\xi}{3}\biggr] d\xi
+ \biggl( \frac{1}{5^2} \biggr) {\tilde\xi}^3 \biggl( \frac{3}{3+ {\tilde\xi}^2}\biggr)^3 \biggl[15- {\tilde\xi}^2\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{ 2^2\cdot 3^2}{5^2 } \biggr) \int_0^{\tilde\xi} \biggl[ \frac{\xi^6 }{(3+\xi^2)^3}\biggr] d\xi
~~- ~~ \biggl(\frac{ 2\cdot 3^2}{5^2 } \biggr) \int_0^{\tilde\xi} \biggl[ \frac{\xi^4 (15-\xi^2)^2}{(3+\xi^2)^4}\biggr] d\xi
~~ + ~~ \biggl( \frac{3^3}{5^2} \biggr) \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5^3 {\tilde\xi}^3 }{2\cdot 3^2R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl[ \frac{4\xi^6(3+\xi^2)-2\xi^4 (15-\xi^2)^2}{(3+\xi^2)^4}\biggr] d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl\{ \frac{2\xi^4 [6\xi^2 + 2\xi^4 -15^2 + 30\xi^2 - \xi^4] }{(3+\xi^2)^4}\biggr\} d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl\{ \frac{2\xi^4 [\xi^4 + 36\xi^2 -15^2 ] }{(3+\xi^2)^4}\biggr\} d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2\xi^5(\xi^2-15)}{(\xi^2+3)^3} \biggr]_0^{\tilde\xi}
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2{\tilde\xi}^5({\tilde\xi}^2-15)}{({\tilde\xi}^2+3)^3} \biggr]
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2{\tilde\xi}^5({\tilde\xi}^2-15) + 3{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{({\tilde\xi}^2+3)^3}
=
\frac{5{\tilde\xi}^7 - 120{\tilde\xi}^5 + 3^3\cdot 5^2{\tilde\xi}^3 }{({\tilde\xi}^2+3)^3} \, ,
</math>
</td>
</tr>
</table>
</div>
which equals zero if <math>~\tilde\xi = 3</math>. <font size="+1" color="red"><b>Hooray!!</b></font>

===For All Polytropic Indexes===

====Generalized Governing Integral Relation====
Given that the derivation just completed works for the special case of n = 5, let's generalize it to all polytropic indexes
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{R^5 \rho_c}{R^3 P_c}\int_0^R \sigma^2 \biggl( \frac{\rho}{\rho_c}\biggr) \biggl(\frac{r}{R}\biggr)^4 x^2 \frac{dr}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl(\frac{r}{R}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{P}{P_c}\biggr) \biggl[\frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
- \int_0^R \biggl[3\biggl(\frac{n+1}{n}\biggr) - 4\biggr] \biggl(\frac{r}{R}\biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+3\biggl(\frac{n+1}{n}\biggr) \biggl( \frac{P_e}{P_c}\biggr) x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{R^2 \rho_c}{P_c} \int_0^{\tilde\xi} \sigma^2 \theta^n \biggl(\frac{\xi}{\tilde\xi}\biggr)^4 x^2 \frac{d\xi}{\tilde\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl(\frac{\xi}{{\tilde\xi}}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \theta^{n+1} \biggl[\frac{dx}{d(\xi/\tilde\xi)}\biggr]^2 \frac{d\xi}{\tilde\xi}
~+ \int_0^{\tilde\xi} \biggl(\frac{n-3}{n}\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d(\xi/\tilde\xi)} \biggr] \frac{d\xi}{\tilde\xi}
~+~3\biggl(\frac{n+1}{n}\biggr) {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{n R^2\rho_c}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \sigma^2 \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^4 \theta^{n+1} \biggl[\frac{dx}{d\xi}\biggr]^2 d\xi
~+ \int_0^{\tilde\xi} (n-3) \xi^3 \theta^n x^2 \biggl[ \frac{d\theta}{d\xi} \biggr] d\xi
~+~3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{n R^2 G \rho_c^2}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 d\xi
~+ \int_0^{\tilde\xi} (n-3) \xi^2 \theta^{n+1} x^2 \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] d\xi
~+~3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl\{ \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 + (n-3) \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] \biggr\} d\xi
</math>
</td>
</tr>
</table>
</div>

For additional clarification, let's rewrite the leading coefficient on the lefthand-side of this expression.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
LHS
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{n R^2 G \rho_c^2}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr] \biggl[ \frac{G R_\mathrm{norm}^2}{P_\mathrm{norm}} \biggr]
\biggl(\frac{R}{R_\mathrm{norm}^2}\biggr) \biggl( \frac{\rho_c}{ {\bar\rho}}\biggr)^2
\biggl[ \frac{3M}{4\pi R^3}\biggr]^2
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{P_e}{P_c} \biggr) \biggl[ \frac{1}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr] \biggl[ \frac{G M_\mathrm{tot}^2}{P_\mathrm{norm}R_\mathrm{norm}^4} \biggr]
\biggl(\frac{R_\mathrm{norm}}{R}\biggr)^4 \biggl( \frac{\rho_c}{ {\bar\rho}}\biggr)^2
\biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{tot}}\biggr]^2
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{P_e}{P_c} \biggr) \biggl[ \frac{1}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr]
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{R_\mathrm{norm}}{R}\biggr)^4 \biggl( - \frac{\tilde\xi}{3 {\tilde\theta}^'}\biggr)^2
\biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{tot}}\biggr]^2
\biggl[ \frac{{\tilde\theta}^{n+1}}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>
</table>
</div>

Now, from an [[StabilityVariationalPrinciple#Test_Virial_Equilibrium_Condition|accompanying discussion]], we know that, in equilibrium,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\, ,
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl\{ \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)}
\tilde\theta_n^{(n+1)(n-3)}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)} \biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~\times
\biggl\{\biggl[(n+1)^{-n} ( 4\pi )\biggr] \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)}
\tilde\xi^{(n-3)} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)} \biggr\}^{4/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ \tilde\xi^{4} \tilde\theta_n^{(n+1)}
\biggl\{ (n+1)^{3(n+1)} ( 4\pi )^{(-n-1)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2n-2}
( -\tilde\xi^2 \tilde\theta' )^{2n+2} (n+1)^{-4n} ( 4\pi )^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(4n-4)}
( -\tilde\xi^2 \tilde\theta' )^{(4-4n)}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ \tilde\xi^{4} \tilde\theta_n^{(n+1)}
\biggl\{ (n+1)^{(3-n)} ( 4\pi )^{(3-n)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n-3)}
( -\tilde\xi^2 \tilde\theta' )^{2(3-n)}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
(n+1)^{-1} ( 4\pi )^{(-1)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2} \tilde\xi^{4} \tilde\theta_n^{(n+1)}( -\tilde\xi^2 \tilde\theta' )^{-2} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, in equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
LHS
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr]
\biggl\{ (n+1) ( 4\pi ) \tilde\xi^{-4} \tilde\theta_n^{-(n+1)}( -\tilde\xi^2 \tilde\theta' )^{2} \biggr\}
\biggl( - \frac{\tilde\xi}{3 {\tilde\theta}^'}\biggr)^2
\biggl(\frac{3}{4\pi}\biggr)^2
\biggl[ \frac{{\tilde\theta}^{n+1}}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi \, .</math>
</td>
</tr>
</table>
</div>

In summary, then, we have,

<div align="center" id="PolytropeRelation">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\int_0^{\tilde\xi} \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl\{ \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 + (n-3) \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] \biggr\} d\xi \, .
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

Perhaps this looks better if the terms are rearranged to give,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2\theta^{n+1} x^2 \biggl\{ \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \frac{\xi^2}{\theta}
- \biggl[ \biggl( \frac{d\ln x}{d\ln \xi}\biggr)^2 + (n-3) \biggl( \frac{d\ln\theta}{d\ln\xi} \biggr) \biggr] \biggr\} d\xi \, .
</math>
</td>
</tr>
</table>
</div>

====Plug in Known Marginally Unstable Solution====

As has been summarized in an [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|accompanying discussion]], we have found that, for marginally unstable pressure-truncated polytropic configurations, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma_c^2 = 0</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{2n}{3(n-1)} \biggr] \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \frac{d}{d\xi}\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl[
\frac{\theta^{''}}{\xi \theta^{n}}
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl[
- \frac{1}{\xi \theta^{n}} \biggl( \theta^n + \frac{2\theta^'}{\xi} \biggr)
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl[
\frac{1}{\xi }
+ \frac{3\theta^'}{\xi^2 \theta^{n}}
+ \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Hence, also,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl(\frac{n-3}{n-1}\biggr)^{-1} \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[\biggl(\frac{n-1}{n-3}\biggr) + \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[\biggl(\frac{n-1}{n-3}\biggr) + \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1} \, .
</math>
</td>
</tr>
</table>
</div>

Rather, let's try:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
\xi^2 x^2 \biggl[ \biggl( \frac{d\ln x}{d\ln \xi}\biggr)^2 + (n-3) \biggl( \frac{d\ln\theta}{d\ln\xi} \biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x^2 \xi^2 \biggl( \frac{\xi}{x}\cdot \frac{dx}{d\xi}\biggr)^2 + (n-3) x^2 \xi^2 \biggl( \frac{\xi}{\theta} \cdot \frac{d\theta}{d\xi} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^4 \biggl\{ \frac{dx}{d\xi}\biggr\}^2 + (n-3) \biggl[ \frac{\xi^3 \theta^'}{\theta} \biggr] x^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^4 \biggl\{ \frac{3(n-1)}{2n}\biggl(\frac{3-n}{n-1}\biggr) \biggl[
\frac{1}{\xi }
+ \frac{3\theta^'}{\xi^2 \theta^{n}}
+ \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]\biggr\}^2 + (n-3) \biggl[ \frac{\xi^3 \theta^'}{\theta} \biggr]
\biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi^2 (n-3) \biggl[ \frac{3}{2n} \biggr]^2\biggl\{
(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
\biggr\}
</math>
</td>
</tr>
</table>
</div>
Hence, after setting <math>~\sigma^2 = 0</math>, the [[#PolytropeRelation|above rearranged integral relation]] becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2 \theta^{n+1} \biggl\{
(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
\biggr\} d\xi
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">
Let's check to see whether the terms in this last expression balance out when we plug in the functions that are appropriate for the marginally unstable, n = 5 configuration, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="left">
<math>~\theta_5 = \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2}</math>,
</td>
<td align="center">
     and    
</td>
<td align="left">
<math>~\frac{d\theta_5}{d\xi} = - \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}</math>.
</td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
RHS Term 1
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(n-3) \int_0^{\tilde\xi} \xi^2 \theta^{n+1}
\biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \int_0^{\tilde\xi} \xi^2 \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2} \biggr]^{6} \biggl\{
1 - \biggl[ \xi \biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]\frac{1}{\xi }\biggl(\frac{3+\xi^2}{3}\biggr)^{5 / 2}
+5 \biggl[ \frac{\xi^2}{3^2}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \int_0^{\tilde\xi} \frac{3^3 \xi^2}{(3+\xi^2)^3} \biggl\{
1 - \biggl(\frac{3+\xi^2}{3}\biggr)
+ \frac{5\xi^2}{3^2}
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \int_0^{\tilde\xi} \frac{\xi^6 ~d\xi}{(3+\xi^2)^3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \biggl[
\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
\biggr]_0^{3} = \frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \biggr] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
RHS Term 2
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^3 \theta^{n} \theta^'
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \int_0^{\tilde\xi} \xi^3 \biggl(\frac{3+\xi^2}{3}\biggr)^{-5 / 2} \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}
\biggl\{ 4 - 2\frac{1}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}\biggl(\frac{3+\xi^2}{3}\biggr)^{5/2}
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{3}\int_0^{\tilde\xi} \biggl(\frac{3\xi}{3+\xi^2}\biggr)^{4}
\biggl\{ 4 - \frac{2}{3}\biggl(\frac{3+\xi^2}{3}\biggr)
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^3}{3}\int_0^{\tilde\xi} \frac{1}{2} \biggl(\frac{\xi}{3+\xi^2}\biggr)^{4}
\biggl\{ 15-\xi^2
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2^3}{3} \biggl[
\frac{123\xi}{8(3+\xi^2)} - \frac{243\xi}{4(3+\xi^2)^2} + \frac{162\xi}{2(3+\xi^2)^3} + \frac{\xi}{2} - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
\biggr]_0^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \cdot \frac{5}{2^5} \biggl[ 2^2\cdot 3^{1 / 2} \pi - 3^3\biggr] = -\frac{2^3}{3} \biggl[ \frac{3^3\cdot 5}{2^5} - \frac{3^{1 / 2} \cdot 5\pi}{2^3} \bigg] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~</math> RHS Total
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^3\cdot 5}{2^5} \biggr]
= \frac{3^2}{2^3} \biggl[ 3^2 - 2\cdot 5 \biggr] = - \frac{3^2}{2^3} \, .
</math>
</td>
</tr>

<tr>
<td align="right">
LHS
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^2 5^2}{2\cdot 3} \biggl[ 3^3 \biggl(\frac{3}{3+3^2}\biggr)^{3} \frac{2^2}{5^2} \biggr]
=
- \frac{2^3 }{3} \biggl[\biggl(\frac{3}{2^2}\biggr)^{3} \biggr]
=
- \frac{3^2 }{2^3} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, the LHS = RHS.   <font size="+1" color="red"><b>Hooray!</b></font>
</td></tr>
</table>

=See Also=

* [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29|Derivations that support this chapter's discussion of the Ledoux Variational Principle]]
* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">A variational principle of great power is derived. It is naturally adapted for computers, and may be used to determine the stability of any fluid flow including those in differentially-rotating, self-gravitating stars and galaxies. The method also provides a powerful theoretical tool for studying general properties of eigenfunctions, and the relationships between secular and ordinary stability. In particular we prove the anti-sprial theorem indicating that no stable (or steady( mode can have a spiral structure</font>.
</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1972ApJS...24..319S/abstract B. F. Schutz, Jr. (1972)], ApJSuppl., 24, 319: ''Linear Pulsations and Stability of Differentially Rotating Stellar Models. I. Newtonian Analysis''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">A systematic method is presented for deriving the Lagrangian governing the evolution of small perturbations of arbitrary flows of a self-gravitating perfect fluid. The method is applied to a differentially rotating stellar model; the result is a Lagrangian equivalent to that of [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)]. A sufficient condition for stability of rotating stars, derived from this Lagrangian, is simplified greatly by using as trial functions not the three components of the Lagrangian displacement vector, but three scalar functions … This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.

… we examine the special cases of (i) axially symmetric perturbations of a rotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..267C/abstract S. Chandrasekhar & N. R. Lebovitz 1968]); and (ii) perturbations of a nonrotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1964ApJ...140.1517C/abstract Chandrasekhar and Lebovitz 1964)]. We find that the stability criteria for those cases can also be simplified …</font>
</td></tr></table>

{{ SGFfooter }}

SSC/VariationalPrinciple

2021-08-01T19:58:14Z

174.64.8.247: /* Via Generalized Normalization */

__FORCETOC__


=Ledoux's Variational Principle=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>Variational<br />Principle</b>]]</font>
|}
All of the discussion in this chapter will build upon our [[SSC/Perturbations#2ndOrderODE|derivation elsewhere]] of the,
<div align="center" id="2ndOrderODE">
<font color="#770000">'''LAWE:   Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

We will draw heavily from the papers published by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964)], as well as from pp. 458-474 of the review by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] in explaining how the ''variational principle'' can be used to identify the eigenvector of the fundamental mode of radial oscillation in spherically symmetric configurations. In an associated "Ramblings" appendix, we provide [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29|various derivations that support]] this chapter's relatively abbreviated presentation.

==Ledoux and Pekeris (1941)==
Historically, by the 1940s, the LAWE was a relatively familiar one to astrophysicists. For example, the opening paragraph of a 1941 paper by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124), reads:
<div align="center">
<table border="1" cellpadding="5">
<tr><td align="center">
Paragraph extracted from [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941)]<p></p>
"''Radial Pulsations of Stars''"<p></p>
ApJ, vol. 94, pp. 124-135 © American Astronomical Society
</td></tr>
<tr>
<td>
[[File:LedouxPekeris1941.jpg|600px|center|Ledoux & Pekeris (1941, ApJ, 94, 124)]]
</td>
</tr>
</table>
</div>

If we divide their equation (1) through by <math>~Xr = \Gamma_1 P r</math> and recognize that,
<div align="center">
<math>
\frac{dX}{dr} = \frac{dX}{dm}\frac{dm}{dr} = - \Gamma_1 g_0 \rho \, ,
</math>
</div>

we obtain,

<div align="center">
<math>
\frac{d^2\xi}{dr^2} + \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} +\frac{\rho}{\Gamma_1 P} \biggl[ \sigma^2 + (4 - 3\Gamma_1) \frac{g_0}{r} \biggr] \xi = 0 \, .
</math>
</div>
Clearly, this 2<sup>nd</sup>-order, ordinary differential equation is the same as our derived LAWE, but with a more general definition of the adiabatic exponent that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.

Multiplying this last equation through by <math>~\Gamma_1 P r^4</math>, and recognizing that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(r^4 \Gamma_1 P)\frac{d^2\xi}{dr^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
</math>
</td>
</tr>
</table>
</div>

<span id="RewrittenLAWE">we can write,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
+ ( \Gamma_1 P r^4 ) \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr}
+\rho \biggl[ \sigma^2 r^4 + (4 - 3\Gamma_1) g_0 r^3\biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \biggl[4r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr} \biggr] \frac{d\xi}{dr}
+ \biggl[ 4 r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr}\biggr] \frac{d\xi}{dr}
+\biggl[ \sigma^2 \rho r^4 - (4 - 3\Gamma_1) r^3 \frac{dP}{dr} \biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
(checked for n = 5) ==>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
+\biggl[ \sigma^2 \rho r^4 + (3\Gamma_1 - 4) r^3 \frac{dP}{dr} \biggr] \xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d}{dr}\biggl[ \Gamma_1 P r^4 ~\frac{d\xi}{dr} \biggr]
+\biggl[ \sigma^2 r^4 \rho + 4 Gm (r ) r \rho + 3\Gamma_1 r^3 \frac{dP}{dr} \biggr] \xi \, .
</math>
</td>
</tr>
</table>
</div>
Assuming that <math>~\Gamma_1</math> is uniform throughout the configuration, this last expression is the same as equation (3) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)], while the next-to-last expression is identical to equation (58.1) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)].

==Stability Based on Variational Principle==

Here we derive the Lagrangian directly from the governing LAWE. We begin with the next-to-last derived form of the LAWE that [[#RewrittenLAWE|appears above]] in our review of the paper by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and, following the guidance provided at the top of p. 666 of [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964, ApJ, 139, 664)], we multiply the LAWE through by the fractional displacement, <math>~\xi</math>. This gives, what we will henceforth refer to as, the,

<div align="center" id="FoundationalVariationalRelation">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3"><font color="maroon"><b>Foundational Variational Relation</b></font></td>
</tr>
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\xi \cdot \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) \, .
</math>
</td>
</tr>
</table>
</div>

===Chandrasekhar's Approach===

Next, in an effort to adopt the notation used by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C Chandrasekhar (1964)], we make the substitution, <math>~\xi \rightarrow \psi/r^3</math>, and regroup terms to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\sigma^2 \rho \psi^2}{r^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d}{dr} \biggl( \frac{\psi}{r^3} \biggr) \biggr]
- (3\Gamma_1 - 4) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r \Gamma_1 P ~\frac{d\psi}{dr} -3 \Gamma_1 P \psi ~\biggr]
- (3\Gamma_1 - 4) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4-3\Gamma_1 ) \biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
- \biggl( \frac{\psi}{r^3}\biggr) \frac{d}{dr}\biggl[ r \Gamma_1 P ~\frac{d\psi}{dr} \biggr]
+ 3 \Gamma_1 \biggl( \frac{\psi^2}{r^3}\biggr) \frac{dP}{dr}
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
- \biggl\{
\frac{d}{dr}\biggl[ r \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}\biggr] -r\Gamma_1 P ~\frac{d\psi}{dr} \cdot \frac{d}{dr}\biggl( \frac{\psi}{r^3}\biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+3 \Gamma_1 P \biggl( \frac{\psi}{r^3}\biggr) \frac{d\psi}{dr}
+ \frac{\Gamma_1 P}{r^2} \biggl[\frac{d\psi}{dr} \biggr]^2
- \biggl[\frac{3\Gamma_1 P\psi}{r^3}\biggr]\frac{d\psi}{dr}
- \frac{d}{dr}\biggl[\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{\psi^2}{r^3} \biggr) \biggl( \frac{dP}{dr} \biggr)
+ \frac{\Gamma_1 P}{r^2} \biggl[\frac{d\psi}{dr} \biggr]^2
- \frac{d}{dr}\biggl[ \frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">
Let's check to see whether the terms in the RHS of this last expression sum to zero when we plug in the appropriate functions for the marginally unstable, n = 5 configuration. In particular (replacing <math>~\xi</math> with <math>~x</math>, and setting <math>~r = a_5\xi</math>), we start with knowing,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="left">
<math>~\theta_5 = \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2}</math>;        
</td>
<td align="left">
<math>~\frac{d\theta_5}{d\xi} = - \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}</math>;        
</td>
</tr>

<tr>
<td align="left">
<math>~x = \biggl(\frac{3\cdot 5 - \xi^2}{3\cdot 5} \biggr)</math>;        
</td>
<td align="left">
<math>~\frac{dx}{d\xi} = -\frac{2\xi}{3\cdot 5}</math>;        
</td>
</tr>

<tr>
<td align="left">
<math>~\psi = a_5^3 \xi^3 x</math>;        
</td>
<td align="left">
<math>~
\frac{d\psi}{d\xi} = a_5^3 \biggl[ 3\xi^2 x + \xi^3 \biggl(\frac{dx}{d\xi}\biggr)\biggr] = \frac{a_5^3 \xi^2}{3}\biggl( 3^2 - \xi^2 \biggr) \, .
</math>
</td>
</tr>
</table>
</div>

----

Then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathrm{RHS}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl[ \frac{a_5^6 \xi^6 x^2}{a_5^3 \xi^3} \biggr] \frac{P_c}{a_5} \cdot \frac{d\theta^6}{d\xi}
+ P_c \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6}{a_5^4 \xi^2} \biggl\{ \frac{d\psi}{d\xi} \biggr\}^2
- \frac{P_c}{a_5} \cdot \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6 a_5^3 \xi^3 x}{a_5^3 \xi^2} \biggl( \frac{d\psi}{d\xi} \biggr) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{\mathrm{RHS}}{P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl[ \xi^3 x^2 \biggr] \frac{d\theta^6}{d\xi}
+ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6}{ \xi^2} \biggl\{ \frac{\xi^2}{3}\biggl( 3^2 - \xi^2 \biggr) \biggr\}^2
- \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr) \frac{\theta^6 \xi^3 x}{ \xi^2} \biggl[ \frac{\xi^2}{3}\biggl( 3^2 - \xi^2 \biggr)\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2^3\cdot 3 \xi^3 x^2 \biggl(\frac{3+\xi^2}{3}\biggr)^{-5 / 2} \biggl[- \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]
+ \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{\xi}{3}\biggr)^2 \biggl(\frac{3+\xi^2}{3}\biggr)^{-3} ( 3^2 - \xi^2 )^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \frac{d}{d\xi}\biggl\{ \biggl(\frac{n+1}{n}\biggr)\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \frac{\xi^3 x}{ 3} ( 3^2 - \xi^2) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2^3 \cdot 3^4 ~\xi^4 x^2 \biggl(\frac{1}{3+\xi^2}\biggr)^{4}
+ \biggl(\frac{2\cdot 3^2}{5}\biggr) \xi^2 \biggl[ \frac{( 3^2 - \xi^2 )^2}{(3+\xi^2)^3} \biggr]
- \biggl(\frac{2\cdot 3^3}{5}\biggr)\frac{d}{d\xi}\biggl\{ \xi^3 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5 (3+\xi^2)^4 [\mathrm{RHS} ]}{ 2\cdot 3^2 P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2^2 \cdot 3^2 \cdot 5~\xi^4 x^2
+ \xi^2 (3+\xi^2) ( 3^2 - \xi^2 )^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 3(3+\xi^2)^4 \biggl\{
\xi^3 \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr] \frac{dx}{d\xi}
+ 3\xi^2 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^3} \biggr]
+ \xi^3 x \biggl[ \frac{ -2\xi }{(3+\xi^2)^3} \biggr]
-2\cdot 3 \xi^4 x \biggl[ \frac{( 3^2 - \xi^2)}{(3+\xi^2)^4} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^2 (3+\xi^2) ( 3^2 - \xi^2 )^2
- 2^2 \cdot 3^2 \cdot 5~\xi^4 x^2
- 3\xi^3 ( 3^2 - \xi^2) (3+\xi^2) \biggl[ - \frac{2\xi}{ 3\cdot 5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 3 \xi^2\biggl\{
3 ( 3^2 - \xi^2) (3+\xi^2)
-2 \xi^2 (3+\xi^2)
-2\cdot 3 \xi^2 ( 3^2 - \xi^2)
\biggr\} x
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5^2 (3+\xi^2)^4 [\mathrm{RHS} ]}{ 2\cdot 3^2 ~\xi^2 P_c a_5^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
5 (3+\xi^2) ( 3^2 - \xi^2 )^2
- 2^2~\xi^2 (15-\xi^2)^2
+ 2 \xi^2 ( 3^2 - \xi^2) (3+\xi^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[
2 \xi^2 (3+\xi^2)
+ 2\cdot 3 \xi^2 ( 3^2 - \xi^2)
- 3 ( 3^2 - \xi^2) (3+\xi^2)
\biggr] (15-\xi^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(3\cdot 5 + 5\xi^2) ( 3^4 - 2\cdot 3^2\xi^2 + \xi^4)
- 2^2~\xi^2 (3^2\cdot 5^2 - 2\cdot 3\cdot 5 \xi^2 + \xi^4)
+ 2 \xi^2 ( 3^3 + 2\cdot 3\xi^2 -\xi^4)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ 2\cdot 3 \xi^2 + 2\xi^4 + 2\cdot 3^3 \xi^2 - 2\cdot 3 \xi^4 - 3(3^3 + 2\cdot 3\xi^2 -\xi^4)
\biggr] (15-\xi^2)
</math>
</td>
</tr>
</table>
</div>

Coefficients of various powers of <math>~\xi</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi^0:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~3^5\cdot 5 -3^5\cdot 5 = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^2:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~-2\cdot 3^3\cdot 5 + 3^4\cdot 5 -2^2 \cdot 3^2 \cdot 5^2 +2\cdot 3^3 + 2\cdot 3^2\cdot 5 + 2\cdot 3^4\cdot 5 - 2\cdot 3^3\cdot 5 + 3^4</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
   
</td>
<td align="left">
<math>~= 3^2\cdot 5[-2\cdot 3 + 3^2 + 2 + 2\cdot 3^2 - 2\cdot 3] + 3^2[ 2\cdot 3 + 3^2 - 2^2 \cdot 5^2 ] = 3^2\cdot 5[17 ] - 3^2[5\cdot 17 ] = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^4:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~3\cdot 5 -2\cdot 3^2\cdot 5 +2^3\cdot 3\cdot 5 + 2^2\cdot 3 + 2\cdot 3 \cdot 5 - 2\cdot 3^2\cdot 5 - 2\cdot 3 -2\cdot 3^3 + 2\cdot 3^2 + 3^2\cdot 5</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
   
</td>
<td align="left">
<math>~= 3\cdot 5[1 -2\cdot 3 +2^3 + 2 - 2\cdot 3 + 3] + 2\cdot 3[2 - 1 - 3^2 + 3] = 2\cdot 3\cdot 5 - 2\cdot 3\cdot 5 = 0 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^6:</math>
</td>
<td align="center">
   
</td>
<td align="left">
<math>~5 - 2^2 -2 -2 +2\cdot 3 -3 = 0</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

<span id="ChandraEq49">Multiplying through by <math>~dr</math>, and integrating over the volume gives,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R (\sigma^2 \rho \psi^2)\frac{dr}{r^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl[ \Gamma_1 P \biggl(\frac{d\psi}{dr} \biggr)^2
+ \frac{4\psi^2}{r} \biggl( \frac{dP}{dr} \biggr) \biggr]\frac{dr}{r^2}
- \biggl[\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr) \biggr]_0^R \, ,
</math>
</td>
</tr>
</table>
</div>
which is identical to equation (49) of [http://adsabs.harvard.edu/abs/1964ApJ...139..664C Chandrasekhar (1964)], if the last term — the difference of the central and surface boundary conditions — is set to zero.

Note that if we shift from the variable, <math>~\psi</math>, back to the fractional displacement function, <math>~\xi</math>, the last term in this expression may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\Gamma_1 P \psi}{r^2} \biggl( \frac{d\psi}{dr} \biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r \xi \frac{d}{dr} \biggl[r^3 \xi\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r \xi \biggl[3r^2 \xi + r^3 \frac{d\xi}{dr}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 P r^3 \xi^2 \biggl[3 + \frac{d\ln\xi}{d\ln r}\biggr] \, .
</math>
</td>
</tr>
</table>
</div>
So, as is pointed out by [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)] in connection with their equation (57.31), setting this expression to zero at the surface of the configuration is equivalent to setting the variation of the pressure to zero at the surface. Quite generally, this can be accomplished by demanding that,
<div align="center" id="SufaceBC">
<math>~\frac{d\ln\xi}{d\ln r}\biggr|_\mathrm{surface} = -3 \, .</math>
</div>

(An [[SSC/Perturbations#Boundary_Conditions|accompanying chapter]] provides a broader discussion of this and other astrophysically reasonable boundary conditions that are associated with solutions to the LAWE.)

===Ledoux & Walraven Approach===
Returning to the above [[#FoundationalVariationalRelation|''Foundational Variational Relation'']], we can also write,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\xi \cdot \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr]
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \frac{d}{dr}\biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \int_0^R\sigma^2 \rho r^4 \xi^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]_0^R
</math>
</td>
</tr>
</table>
</div>

If the last term (boundary conditions) is set to zero, then we may also write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, if the radial profile of the pressure and the density is known throughout a spherically symmetric, equilibrium configuration, and if, furthermore, the eigenfunction, <math>~\xi(r)</math>, of a radial oscillation mode is specified precisely, then this expression will give the (square of the) ''eigenfrequency'' of that oscillation mode.

By using formal ''variational principle'' techniques to derive this same expression, [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)] are able to offer a broader interpretation, which is encapsulated by their equation (59.10), viz.,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma_0^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathrm{min}~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, if the exact radial eigenfunction, <math>~\xi(r)</math>, is not known, various approximate eigenfunctions can be tried. The trial eigenfunction that ''minimizes'' the righthand-side of this expression will give the (square of the) eigenfrequency of the ''fundamental'' mode of oscillation (subscript zero). Furthermore, via an evaluation of this righthand-side expression, any reasonable trial eigenfunction — for example, <math>~\xi</math> = constant — can provide an ''upper limit'' to <math>~\sigma_0^2</math>.

===Ledoux & Pekeris Approach===
Here we follow the lead of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)]. Returning to the integral expression just derived in our discussion of the ''Ledoux & Walraven approach'', and multiplying through by <math>~4\pi</math>, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R 4\pi \sigma^2 \rho r^4 \xi^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R 4\pi r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) 4\pi r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[4\pi r^3 \Gamma_1 P\xi^2 \biggl(\frac{d\ln \xi}{d\ln r}\biggr) \biggr]_0^R \, .
</math>
</td>
</tr>
</table>
</div>
If we acknowledge that:
* at the center of the configuration, <math>~r^3 = 0</math>;
* [[#SurfaceBC|as above]], the boundary condition at the surface is <math>~P = P_e</math> while <math>~(d\ln \xi/d\ln r) = -3</math>;
* the differential mass element is, <math>~dm = 4\pi r^2 \rho dr</math> and the corresponding differential volume element is, <math>~dV = 4\pi r^2 dr</math>; and
* a statement of detailed force balance is, <math>~dP/dr = - Gm\rho/r^2</math>,
this integral relation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \sigma^2 \int_0^R r^2 \xi^2 dm</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 \int_0^R \biggl[ r \biggl(\frac{d\xi}{dr}\biggr)\biggr]^2 P dV
+ (3\Gamma_1 - 4) \int_0^R \xi^2 \biggl( \frac{Gm}{r} \biggr) dm
- \biggr[\Gamma_1 \xi_\mathrm{surface}^2 (3P_e V) \biggl(-3\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Now, as we have [[SSCpt1/Virial#Wgrav|discussed separately]] — see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] — the gravitational potential energy of the unperturbed configuration is given by the integral,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - \int_0^{M} \biggl( \frac{Gm}{r_0} \biggr) dm \, ;</math>
</td>
</tr>
</table>
</div>

for adiabatic systems, the [[SSCpt1/Virial#Reservoir|internal energy]] is,
<div align="center">
<math>
U_\mathrm{int}
= \frac{1}{(\Gamma_1-1)} \int_0^R P_0 dV
\, ;</math>
</div>
and — see the text at the top of p. 126 of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] — the moment of inertia of the configuration about its center is,
<div align="center">
<math>
I = \int_0^M r_0^2 dm
\, .</math>
</div>
(Note that, defined in this way, <math>~I</math> is the same as [[VE#Standard_Presentation_.5Bthe_Virial_of_Clausius_.281870.29.5D|what we have referred to elsewhere]] as the ''scalar moment of inertia'', which is obtained by taking the trace of the [[VE#MOItensor|moment of inertia tensor]], <math>~I_{ij}</math>.)
<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3"><font color="maroon"><b>Variational Principle's Governing Integral Relation</b></font></td>
</tr>
<tr>
<td align="right">
<math>~ \sigma^2 \int_0^R \xi^2 dI</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Gamma_1 (\Gamma_1 - 1) \int_0^R \xi^2 \biggl[ \frac{d\ln\xi}{d\ln r}\biggr]^2 dU_\mathrm{int}
- (3\Gamma_1 - 4) \int_0^R \xi^2 dW_\mathrm{grav}
+ 3^2 \Gamma_1 P_e V \xi_\mathrm{surface}^2 \, .
</math>
</td>
</tr>
</table>
</div>

==Free-Energy Analysis==

<span id="Homologous">If we assume</span> the simplest approximation for the fundamental-mode eigenfunction, namely, <math>~\xi = \xi_0</math> = constant — that is, homologous expansion/contraction — then this last integral expression gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \sigma^2 I</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4 - 3\Gamma_1) W_\mathrm{grav}
+ 3^2 \Gamma_1 P_e V \, .
</math>
</td>
</tr>
</table>
</div>

Contrast this result with the following free-energy analysis:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV \, ,</math>
</td>
</tr>
</table>
</div>
where, in terms of the configuration's (generally non-equilibrium) dimensionless radius, <math>~\chi \equiv R/R_0</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-a\chi^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~U_\mathrm{int}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~b\chi^{3-3\Gamma_1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3} \chi^3 \, .</math>
</td>
</tr>
</table>
</div>
Then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial \mathfrak{G}}{\partial \chi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~+a \chi^{-2} + 3(1-\Gamma_1) b \chi^{2-3\Gamma_1} + 4\pi P_e \chi^{2} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^{-1} \biggl[- W_\mathrm{grav} + 3(1-\Gamma_1) U_\mathrm{int} + 3 P_e V \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial^2 \mathfrak{G}}{\partial \chi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2a \chi^{-3} + 3(1-\Gamma_1)(2-3\Gamma_1) b \chi^{1-3\Gamma_1} + 8\pi P_e \chi </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^{-2} \biggl[ 2W_\mathrm{grav} + 3(1-\Gamma_1)(2-3\Gamma_1) U_\mathrm{int}+ 6 P_e V \biggr] \, .</math>
</td>
</tr>
</table>
</div>
The equilibrium condition occurs when <math>~\partial \mathfrak{G}/\partial \chi = 0</math>, that is, when,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3(1-\Gamma_1) U_\mathrm{int}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~W_\mathrm{grav} - 3 P_e V \, ,</math>
</td>
</tr>
</table>
</div>
in which case,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\chi^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial \chi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2W_\mathrm{grav} + (2-3\Gamma_1) (W_\mathrm{grav} - 3P_eV) + 6 P_e V </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(4-3\Gamma_1)W_\mathrm{grav} + 3^2 \Gamma_1 P_e V \, .</math>
</td>
</tr>
</table>
</div>

Fantastic! The righthand-side of this "free-energy-based" expression exactly matches the righthand-side of the [[#Homologous|above expression]] that has been derived from the variational principle, assuming homologous expansion/contraction (''i.e.,'' <math>~\xi</math> = constant). In this case, we can make the direct association,
<div align="center">
<math>~\sigma^2 I = \chi^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial \chi^2} \, .</math>
</div>
This also make sense in that the equilibrium configuration should be stable if <math>~\tfrac{\partial^2 \mathfrak{G}}{\partial \chi^2} > 0</math> — in which case, <math>~\sigma^2</math> is positive; whereas the equilibrium configuration should be ''unstable'' if <math>~\tfrac{\partial^2 \mathfrak{G}}{\partial \chi^2} < 0</math> — in which case, <math>~\sigma^2</math> is negative.

=Related, Exploratory Ideas=

==Logarithmic Derivatives==

Returning to our above discussion of the [[#Ledoux_.26_Walraven_Approach|Ledoux & Walraven approach]], we appreciate that the ''differential'' relation governing the Variational Principle is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma^2 \rho r^4 \xi^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \frac{d}{dr}\biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{d}{dr}\biggr[r^3 \Gamma_1 P\xi^2 \biggl(\frac{d\ln\xi}{d\ln r}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr)
- \sigma^2 \rho r^4 \xi^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi^2 \biggl\{
r^2 \Gamma_1 P \biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- (3\Gamma_1 - 4) r^3 \biggl( \frac{dP}{dr} \biggr)
- \sigma^2 \rho r^4
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(r \xi)^2 P \biggl\{
\Gamma_1 \biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- (3\Gamma_1 - 4) \biggl( \frac{d\ln P}{d\ln r} \biggr)
- \frac{\sigma^2 \rho r^2}{P}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Gamma_1 (r \xi)^2 P \biggl\{
\biggl(\frac{d\ln\xi}{d\ln r}\biggr)^2
- \alpha \biggl( \frac{d\ln P}{d\ln r} \biggr)
- \frac{\sigma^2 \rho r^2}{\Gamma_1 P}
\biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\alpha \equiv \biggl(3 - \frac{4}{\Gamma_1}\biggr) \, .</math>
</div>

==Pressure-Truncated Polytropes==

Let's start with the integral expression derived in our discussion of the [[#Ledoux_.26_Walraven_Approach|Ledoux & Walraven approach]]; insert the variable, <math>~x</math>, in place of <math>~\xi</math>; and adopt the boundary conditions,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r = 0</math>   at the center,
</td>
<td align="center">
        along with        
</td>
<td align="left">
<math>~P = P_e~</math>,   and  <math>\frac{d\ln x}{d\ln r} = -3</math>   at the surface (r = R).
</td>
</tr>
</table>
</div>
That is, let's start with,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2 \, .
</math>
</td>
</tr>
</table>
</div>

===Via Generalized Normalization===
Next, we'll divide through by the [[StabilityVariationalPrinciple#Energies_and_Structural_Form_Factors|normalization energy, as defined in an accompanying discussion]],
<div align="center">
<math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3 = \frac{GM_\mathrm{tot}^2}{R_\mathrm{norm}} \, ,</math>
</div>
thereby making the integral relation dimensionless:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{R_\mathrm{norm}}{GM_\mathrm{tot}^2} \biggr] \int_0^R \sigma^2 \rho r^4 x^2 dr
+\biggl[\frac{1}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \biggl[\frac{1}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+ \biggl[\frac{P_e R^3 }{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{R_\mathrm{norm} R^5 \rho_c^2}{M_\mathrm{tot}^2} \biggr] \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
+ \biggl[\frac{P_c R^3}{P_\mathrm{norm}R_\mathrm{norm}^3 } \biggr] \int_0^R \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[\frac{P_c R^3}{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] \int_0^R (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+ \biggl[\frac{P_e R^3 }{P_\mathrm{norm}R_\mathrm{norm}^3} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^2 \chi^{-1} \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
+ \biggl[\frac{P_e }{P_\mathrm{norm}} \biggr] 3\Gamma_1 \chi^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[\frac{P_c}{P_\mathrm{norm} } \biggr] \chi^3 \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\chi \equiv \frac{R}{R_\mathrm{norm}} \, .</math>
</div>

Note that we will ultimately insert the relation,
<div align="center">
<math>~\frac{P_c}{P_\mathrm{norm}} = \biggl[\biggl( \frac{3}{4\pi}\biggr) \frac{\rho_c}{\bar\rho} \biggl( \frac{M}{M_\mathrm{tot}}\biggr)\biggr]^{\Gamma_1} \biggl( \frac{R}{R_\mathrm{norm}}\biggr)^{-3\Gamma_1} \, .</math>
</div>
But, for the time being, dividing through by <math>~[P_c/P_\mathrm{norm}]\chi^3</math> gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{P_c}{P_\mathrm{norm} } \biggr]^{-1} \biggl[ \frac{M}{M_\mathrm{tot}}\biggr]^2 \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^2 \chi^{-4} \int_0^R x^2 \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \biggl( \frac{\rho}{\rho_c} \biggr) \biggl(\frac{r}{R}\biggr)^4 \frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[\frac{P_e }{P_c} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
+ \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R} \, ,
</math>
</td>
</tr>
</table>
</div>

Now let's focus on the second line of this integral energy relation, evaluating it for pressure-truncated polytropic configurations, in which case, <math>~\Gamma_1 \rightarrow (n+1)/n</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r}{R} \rightarrow \frac{\xi}{\tilde\xi}</math>
</td>
<td align="center">
        and        
</td>
<td align="left">
<math>~
\frac{P}{P_c} \rightarrow \theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>
We have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
Second line of relation
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] 3\Gamma_1 x_\mathrm{surface}^2
+ \int_0^R \biggl\{ \biggl( \frac{r}{R}\biggr)^4 \Gamma_1\biggl(\frac{ P }{P_c} \biggr) \biggl[ \frac{dx}{d(r/R)}\biggr]^2
- (3\Gamma_1 - 4) \biggl( \frac{r}{R} \biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \biggr\}\frac{dr}{R}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \biggl\{ \biggl( \frac{\xi}{\tilde\xi}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2 {\tilde\xi}^2
- \biggl(\frac{3-n}{n}\biggr) \biggl( \frac{\xi}{\tilde\xi} \biggr)^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d\xi} \biggr] \tilde\xi \biggr\}\frac{d\xi}{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{1}{n {\tilde\xi}^3}\int_0^{\tilde\xi} \biggl\{ (n+1) \xi^4 \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2
- (3-n) \xi^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d\xi} \biggr] \biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{1}{n {\tilde\xi}^3}\int_0^{\tilde\xi} \biggl\{ (n+1) \xi^4 \theta^{n+1} \biggl[ \frac{dx}{d\xi}\biggr]^2
- (n+1) (3-n) \xi^3 x^2 \theta^n \theta^' \biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{P_e }{P_c} \biggr] \biggl[ \frac{3( n+1)}{n} \biggr] x_\mathrm{surface}^2
+ \frac{(n+1)}{n {\tilde\xi}^3}\int_0^{\tilde\xi}
\biggl(\frac{3}{2n}\biggr)^2\frac{\xi}{\theta^n} \biggl\{ \xi \theta \biggl[ \biggl( \frac{2n}{3}\biggr)\xi \theta^n \cdot \frac{dx}{d\xi}\biggr]^2
- (3-n) \biggl[ \biggl( \frac{2n}{3}\biggr) \xi \theta^n x\biggr]^2 \theta^' \biggr\}d\xi \, .
</math>
</td>
</tr>
</table>
</div>

Now, let's examine how these terms combine if we ''guess'' the [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|analytically defined eigenfunction that applies to marginally unstable, pressure-truncated polytropic configurations]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^'}{\xi \theta^{n} } \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{2n}{3}\biggr) \xi \theta^n x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\frac{3(n-3)}{2n}\biggr] \biggl\{
\frac{\theta^{''}}{\xi \theta^{n}}
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n(\theta^')^2}{\xi \theta^{(n+1)}}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[\frac{3(n-3)}{2n}\biggr] \frac{1 }{\xi \theta^{n}} \biggl[
\theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{2n}{3}\biggr) \xi \theta^n\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3-n)
\biggl[ \theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta} \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>
Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
Second line of relation
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
{\tilde\theta}^{n+1} \biggl[ \frac{3( n+1)}{n} \biggr] \biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{ {\tilde\theta}^'}{\tilde\xi {\tilde\theta}^{n} } \biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)}{2^2n^3 {\tilde\xi}^3}\int_0^{\tilde\xi}
\frac{\xi}{\theta^n} \biggl\{
\xi \theta (3-n)\biggl[ \theta^n + \frac{3\theta^'}{\xi} + \frac{n(\theta^')^2}{\theta} \biggr]^2
- \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \theta^'
\biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{{\tilde\xi}^2 {\tilde\theta}^{n+1}} \biggl[ \frac{3^3( n+1)}{2^2n^3} \biggr] \biggl[(n-1) \tilde\xi {\tilde\theta}^{n+1} + (n-3) \tilde\theta {\tilde\theta}^' \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)}{2^2n^3 {\tilde\xi}^3} \int_0^{\tilde\xi}
\frac{1}{\theta^{n+1}} \biggl\{
(3-n)\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]^2
- \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi \theta \theta^'
\biggr\}d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{{\tilde\xi}^2 {\tilde\theta}^{n+1}} \biggl[ \frac{3^3( n+1)}{2^2n^3} \biggr] \biggl[(n-1) \tilde\xi {\tilde\theta}^{n+1} + (n-3) \tilde\theta {\tilde\theta}^' \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3^2 (n+1)(3-n)^2}{2^2n^3 {\tilde\xi}^3} \int_0^{\tilde\xi}
\frac{1}{\theta^{n+1}} \biggl\{
\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]^2
+ \frac{1}{(n-3)} \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi \theta \theta^'
\biggr\}d\xi
</math>
</td>
</tr>
</table>
</div>

Note that, in this derivation, we have inserted the expressions:
<div align="center">
<math>~
\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr]\biggl[ \xi \theta^{n+1} + 3\theta \theta^' + n\xi (\theta^')^2 \biggr] =
\xi^2 \theta^{2(n+1)} + 6\xi \theta^{n+2}\theta^' + 2n\xi^2 \theta^{n+1} (\theta^')^2 + 6n\xi\theta (\theta^')^3 + n^2 \xi^2 (\theta^')^4
</math>
</div>
<div align="center">
<math>~
\frac{1}{(n-3)} \biggl[(n-1)\xi \theta^n + (n-3) \theta^' \biggr]^2 \xi\theta (\theta^')=
\biggl[ \frac{(n-1)^2}{(n-3)}\biggr] \xi^3 \theta^{2n+1}(\theta^') + 2(n-1)\xi^2 \theta^{n+1} (\theta^' )^2 + (n-3) \xi\theta (\theta^')^3
</math>
</div>

===Directly to n = 5 Polytropic Configurations===

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{1}{R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl(\frac{r}{R}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{P}{P_c}\biggr) \biggl[\frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
- \int_0^R \biggl[3\biggl(\frac{n+1}{n}\biggr) - 4\biggr] \biggl(\frac{r}{R}\biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+3\biggl(\frac{n+1}{n}\biggr) \biggl( \frac{P_e}{P_c}\biggr) x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \frac{6}{5} \biggl(\frac{\xi}{\tilde\xi}\biggr)^4 \theta^6 \biggl[\frac{dx}{d(\xi/\tilde\xi)}\biggr]^2 \frac{d\xi}{\tilde\xi}
- \int_0^{\tilde\xi} \biggl( - \frac{2}{5}\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^3 x^2 \biggl[ \frac{d\theta^{6}}{d(\xi/\tilde\xi)} \biggr] \frac{d\xi}{\tilde\xi}
+\biggl(\frac{18}{5}\biggr) {\tilde\theta}^6 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{ {\tilde\xi}^3} \int_0^{\tilde\xi} \biggl( \frac{6}{5}\biggr) \xi^4 \theta^6 \biggl[\frac{dx}{d\xi}\biggr]^2 d\xi
+ \frac{1}{ {\tilde\xi}^3} \int_0^{\tilde\xi} \biggl(\frac{2}{5}\biggr) \xi^3 x^2 \biggl[ \frac{d\theta^{6}}{d\xi} \biggr] d\xi
+\biggl(\frac{18}{5}\biggr) {\tilde\theta}^6 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5 {\tilde\xi}^3 }{2R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} 3\xi^4 \theta^6 \biggl[ - \frac{2\xi}{15} \biggr]^2 d\xi
+ \int_0^{\tilde\xi} 6\xi^3 \biggl[\frac{15-\xi^2}{15}\biggr]^2 \theta^5\biggl[ \frac{d\theta}{d\xi} \biggr] d\xi
+9 {\tilde\xi}^3 {\tilde\theta}^6 \biggl[\frac{15- {\tilde\xi}^2}{15}\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{ 2^2}{3\cdot 5^2 } \biggr) \int_0^{\tilde\xi} \xi^6 \biggl( \frac{3}{3+\xi^2}\biggr)^3 d\xi
+ \biggl(\frac{ 2}{3\cdot 5^2 } \biggr) \int_0^{\tilde\xi} \xi^3 \biggl[15-\xi^2\biggr]^2 \biggl( \frac{3}{3+\xi^2}\biggr)^{4} \biggl[- \frac{\xi}{3}\biggr] d\xi
+ \biggl( \frac{1}{5^2} \biggr) {\tilde\xi}^3 \biggl( \frac{3}{3+ {\tilde\xi}^2}\biggr)^3 \biggl[15- {\tilde\xi}^2\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{ 2^2\cdot 3^2}{5^2 } \biggr) \int_0^{\tilde\xi} \biggl[ \frac{\xi^6 }{(3+\xi^2)^3}\biggr] d\xi
~~- ~~ \biggl(\frac{ 2\cdot 3^2}{5^2 } \biggr) \int_0^{\tilde\xi} \biggl[ \frac{\xi^4 (15-\xi^2)^2}{(3+\xi^2)^4}\biggr] d\xi
~~ + ~~ \biggl( \frac{3^3}{5^2} \biggr) \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{5^3 {\tilde\xi}^3 }{2\cdot 3^2R^3 P_c}\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl[ \frac{4\xi^6(3+\xi^2)-2\xi^4 (15-\xi^2)^2}{(3+\xi^2)^4}\biggr] d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl\{ \frac{2\xi^4 [6\xi^2 + 2\xi^4 -15^2 + 30\xi^2 - \xi^4] }{(3+\xi^2)^4}\biggr\} d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl\{ \frac{2\xi^4 [\xi^4 + 36\xi^2 -15^2 ] }{(3+\xi^2)^4}\biggr\} d\xi
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2\xi^5(\xi^2-15)}{(\xi^2+3)^3} \biggr]_0^{\tilde\xi}
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2{\tilde\xi}^5({\tilde\xi}^2-15)}{({\tilde\xi}^2+3)^3} \biggr]
~ + ~ 3 \biggl[ \frac{{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{(3+ {\tilde\xi}^2)^3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2{\tilde\xi}^5({\tilde\xi}^2-15) + 3{\tilde\xi}^3(15- {\tilde\xi}^2)^2}{({\tilde\xi}^2+3)^3}
=
\frac{5{\tilde\xi}^7 - 120{\tilde\xi}^5 + 3^3\cdot 5^2{\tilde\xi}^3 }{({\tilde\xi}^2+3)^3} \, ,
</math>
</td>
</tr>
</table>
</div>
which equals zero if <math>~\tilde\xi = 3</math>. <font size="+1" color="red"><b>Hooray!!</b></font>

===For All Polytropic Indexes===

====Generalized Governing Integral Relation====
Given that the derivation just completed works for the special case of n = 5, let's generalize it to all polytropic indexes
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_0^R \sigma^2 \rho r^4 x^2 dr</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
+3\Gamma_1 P_e R^3 x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{R^5 \rho_c}{R^3 P_c}\int_0^R \sigma^2 \biggl( \frac{\rho}{\rho_c}\biggr) \biggl(\frac{r}{R}\biggr)^4 x^2 \frac{dr}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^R \biggl(\frac{r}{R}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \biggl(\frac{P}{P_c}\biggr) \biggl[\frac{dx}{d(r/R)}\biggr]^2 \frac{dr}{R}
- \int_0^R \biggl[3\biggl(\frac{n+1}{n}\biggr) - 4\biggr] \biggl(\frac{r}{R}\biggr)^3 x^2 \biggl[ \frac{d(P/P_c)}{d(r/R)} \biggr] \frac{dr}{R}
+3\biggl(\frac{n+1}{n}\biggr) \biggl( \frac{P_e}{P_c}\biggr) x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{R^2 \rho_c}{P_c} \int_0^{\tilde\xi} \sigma^2 \theta^n \biggl(\frac{\xi}{\tilde\xi}\biggr)^4 x^2 \frac{d\xi}{\tilde\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl(\frac{\xi}{{\tilde\xi}}\biggr)^4 \biggl(\frac{n+1}{n}\biggr) \theta^{n+1} \biggl[\frac{dx}{d(\xi/\tilde\xi)}\biggr]^2 \frac{d\xi}{\tilde\xi}
~+ \int_0^{\tilde\xi} \biggl(\frac{n-3}{n}\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^3 x^2 \biggl[ \frac{d\theta^{n+1}}{d(\xi/\tilde\xi)} \biggr] \frac{d\xi}{\tilde\xi}
~+~3\biggl(\frac{n+1}{n}\biggr) {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{n R^2\rho_c}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \sigma^2 \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^4 \theta^{n+1} \biggl[\frac{dx}{d\xi}\biggr]^2 d\xi
~+ \int_0^{\tilde\xi} (n-3) \xi^3 \theta^n x^2 \biggl[ \frac{d\theta}{d\xi} \biggr] d\xi
~+~3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{n R^2 G \rho_c^2}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 d\xi
~+ \int_0^{\tilde\xi} (n-3) \xi^2 \theta^{n+1} x^2 \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] d\xi
~+~3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl\{ \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 + (n-3) \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] \biggr\} d\xi
</math>
</td>
</tr>
</table>
</div>

For additional clarification, let's rewrite the leading coefficient on the lefthand-side of this expression.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
LHS
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{n R^2 G \rho_c^2}{(n+1){\tilde\xi}^2 P_c}\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr] \biggl[ \frac{G R_\mathrm{norm}^2}{P_\mathrm{norm}} \biggr]
\biggl(\frac{R}{R_\mathrm{norm}^2}\biggr) \biggl( \frac{\rho_c}{ {\bar\rho}}\biggr)^2
\biggl[ \frac{3M}{4\pi R^3}\biggr]^2
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{P_e}{P_c} \biggr) \biggl[ \frac{1}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr] \biggl[ \frac{G M_\mathrm{tot}^2}{P_\mathrm{norm}R_\mathrm{norm}^4} \biggr]
\biggl(\frac{R_\mathrm{norm}}{R}\biggr)^4 \biggl( \frac{\rho_c}{ {\bar\rho}}\biggr)^2
\biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{tot}}\biggr]^2
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{P_e}{P_c} \biggr) \biggl[ \frac{1}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr]
\biggl(\frac{P_\mathrm{norm}}{P_e} \biggr)
\biggl(\frac{R_\mathrm{norm}}{R}\biggr)^4 \biggl( - \frac{\tilde\xi}{3 {\tilde\theta}^'}\biggr)^2
\biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{tot}}\biggr]^2
\biggl[ \frac{{\tilde\theta}^{n+1}}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>
</table>
</div>

Now, from an [[StabilityVariationalPrincipal#Test_Virial_Equilibrium_Condition|accompanying discussion]], we know that, in equilibrium,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{-n} ( 4\pi )\biggr]^{1/(n-3)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)/(n-3)}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)/(n-3)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\, ,
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
\biggl\{ \biggl[(n+1)^{3} ( 4\pi )^{-1} \biggr]^{(n+1)}\biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2(n+1)}
\tilde\theta_n^{(n+1)(n-3)}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)} \biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~\times
\biggl\{\biggl[(n+1)^{-n} ( 4\pi )\biggr] \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(n-1)}
\tilde\xi^{(n-3)} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)} \biggr\}^{4/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ \tilde\xi^{4} \tilde\theta_n^{(n+1)}
\biggl\{ (n+1)^{3(n+1)} ( 4\pi )^{(-n-1)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{-2n-2}
( -\tilde\xi^2 \tilde\theta' )^{2n+2} (n+1)^{-4n} ( 4\pi )^4 \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{(4n-4)}
( -\tilde\xi^2 \tilde\theta' )^{(4-4n)}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~ \tilde\xi^{4} \tilde\theta_n^{(n+1)}
\biggl\{ (n+1)^{(3-n)} ( 4\pi )^{(3-n)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2(n-3)}
( -\tilde\xi^2 \tilde\theta' )^{2(3-n)}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>~
(n+1)^{-1} ( 4\pi )^{(-1)} \biggl[\frac{M}{M_\mathrm{tot}} \biggr]^{2} \tilde\xi^{4} \tilde\theta_n^{(n+1)}( -\tilde\xi^2 \tilde\theta' )^{-2} \, .
</math>
</td>
</tr>
</table>
</div>
This means that, in equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
LHS
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n}{(n+1)} \biggr]
\biggl\{ (n+1) ( 4\pi ) \tilde\xi^{-4} \tilde\theta_n^{-(n+1)}( -\tilde\xi^2 \tilde\theta' )^{2} \biggr\}
\biggl( - \frac{\tilde\xi}{3 {\tilde\theta}^'}\biggr)^2
\biggl(\frac{3}{4\pi}\biggr)^2
\biggl[ \frac{{\tilde\theta}^{n+1}}{{\tilde\xi}^2} \biggr]
\int_0^{\tilde\xi} \biggl( \frac{\sigma^2}{G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi \, .</math>
</td>
</tr>
</table>
</div>

In summary, then, we have,

<div align="center" id="PolytropeRelation">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\int_0^{\tilde\xi} \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \theta^n \xi^4 x^2 d\xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
+ \int_0^{\tilde\xi} \xi^2 \theta^{n+1} x^2 \biggl\{ \biggl[\frac{\xi}{x} \cdot \frac{dx}{d\xi}\biggr]^2 + (n-3) \biggl[\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr] \biggr\} d\xi \, .
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

Perhaps this looks better if the terms are rearranged to give,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
3 {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2\theta^{n+1} x^2 \biggl\{ \biggl( \frac{n \sigma^2}{4\pi G\rho_c}\biggr) \frac{\xi^2}{\theta}
- \biggl[ \biggl( \frac{d\ln x}{d\ln \xi}\biggr)^2 + (n-3) \biggl( \frac{d\ln\theta}{d\ln\xi} \biggr) \biggr] \biggr\} d\xi \, .
</math>
</td>
</tr>
</table>
</div>

====Plug in Known Marginally Unstable Solution====

As has been summarized in an [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|accompanying discussion]], we have found that, for marginally unstable pressure-truncated polytropic configurations, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sigma_c^2 = 0</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{2n}{3(n-1)} \biggr] \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \frac{d}{d\xi}\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl[
\frac{\theta^{''}}{\xi \theta^{n}}
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl[
- \frac{1}{\xi \theta^{n}} \biggl( \theta^n + \frac{2\theta^'}{\xi} \biggr)
- \frac{\theta^'}{\xi^2 \theta^{n}}
- \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl[
\frac{1}{\xi }
+ \frac{3\theta^'}{\xi^2 \theta^{n}}
+ \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Hence, also,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl(\frac{n-3}{n-1}\biggr)^{-1} \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[\biggl(\frac{n-1}{n-3}\biggr) + \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[1
+ \frac{3\theta^'}{\xi \theta^{n}}
+ \frac{n (\theta^')^2}{\theta^{n+1}}
\biggr] \biggl[\biggl(\frac{n-1}{n-3}\biggr) + \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^{-1} \, .
</math>
</td>
</tr>
</table>
</div>

Rather, let's try:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
\xi^2 x^2 \biggl[ \biggl( \frac{d\ln x}{d\ln \xi}\biggr)^2 + (n-3) \biggl( \frac{d\ln\theta}{d\ln\xi} \biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x^2 \xi^2 \biggl( \frac{\xi}{x}\cdot \frac{dx}{d\xi}\biggr)^2 + (n-3) x^2 \xi^2 \biggl( \frac{\xi}{\theta} \cdot \frac{d\theta}{d\xi} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^4 \biggl\{ \frac{dx}{d\xi}\biggr\}^2 + (n-3) \biggl[ \frac{\xi^3 \theta^'}{\theta} \biggr] x^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi^4 \biggl\{ \frac{3(n-1)}{2n}\biggl(\frac{3-n}{n-1}\biggr) \biggl[
\frac{1}{\xi }
+ \frac{3\theta^'}{\xi^2 \theta^{n}}
+ \frac{n (\theta^')^2}{\xi \theta^{n+1}}
\biggr]\biggr\}^2 + (n-3) \biggl[ \frac{\xi^3 \theta^'}{\theta} \biggr]
\biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr] \biggr\}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi^2 (n-3) \biggl[ \frac{3}{2n} \biggr]^2\biggl\{
(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
\biggr\}
</math>
</td>
</tr>
</table>
</div>
Hence, after setting <math>~\sigma^2 = 0</math>, the [[#PolytropeRelation|above rearranged integral relation]] becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^2 \theta^{n+1} \biggl\{
(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
\biggr\} d\xi
</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">
Let's check to see whether the terms in this last expression balance out when we plug in the functions that are appropriate for the marginally unstable, n = 5 configuration, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="left">
<math>~\theta_5 = \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2}</math>,
</td>
<td align="center">
     and    
</td>
<td align="left">
<math>~\frac{d\theta_5}{d\xi} = - \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}</math>.
</td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
RHS Term 1
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(n-3) \int_0^{\tilde\xi} \xi^2 \theta^{n+1}
\biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \int_0^{\tilde\xi} \xi^2 \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2} \biggr]^{6} \biggl\{
1 - \biggl[ \xi \biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]\frac{1}{\xi }\biggl(\frac{3+\xi^2}{3}\biggr)^{5 / 2}
+5 \biggl[ \frac{\xi^2}{3^2}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \int_0^{\tilde\xi} \frac{3^3 \xi^2}{(3+\xi^2)^3} \biggl\{
1 - \biggl(\frac{3+\xi^2}{3}\biggr)
+ \frac{5\xi^2}{3^2}
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \int_0^{\tilde\xi} \frac{\xi^6 ~d\xi}{(3+\xi^2)^3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \biggl[
\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
\biggr]_0^{3} = \frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \biggr] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
RHS Term 2
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int_0^{\tilde\xi} \xi^3 \theta^{n} \theta^'
\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \int_0^{\tilde\xi} \xi^3 \biggl(\frac{3+\xi^2}{3}\biggr)^{-5 / 2} \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}
\biggl\{ 4 - 2\frac{1}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}\biggl(\frac{3+\xi^2}{3}\biggr)^{5/2}
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{3}\int_0^{\tilde\xi} \biggl(\frac{3\xi}{3+\xi^2}\biggr)^{4}
\biggl\{ 4 - \frac{2}{3}\biggl(\frac{3+\xi^2}{3}\biggr)
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^3}{3}\int_0^{\tilde\xi} \frac{1}{2} \biggl(\frac{\xi}{3+\xi^2}\biggr)^{4}
\biggl\{ 15-\xi^2
\biggr\}^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2^3}{3} \biggl[
\frac{123\xi}{8(3+\xi^2)} - \frac{243\xi}{4(3+\xi^2)^2} + \frac{162\xi}{2(3+\xi^2)^3} + \frac{\xi}{2} - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
\biggr]_0^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \cdot \frac{5}{2^5} \biggl[ 2^2\cdot 3^{1 / 2} \pi - 3^3\biggr] = -\frac{2^3}{3} \biggl[ \frac{3^3\cdot 5}{2^5} - \frac{3^{1 / 2} \cdot 5\pi}{2^3} \bigg] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~</math> RHS Total
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^3\cdot 5}{2^5} \biggr]
= \frac{3^2}{2^3} \biggl[ 3^2 - 2\cdot 5 \biggr] = - \frac{3^2}{2^3} \, .
</math>
</td>
</tr>

<tr>
<td align="right">
LHS
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2^2 5^2}{2\cdot 3} \biggl[ 3^3 \biggl(\frac{3}{3+3^2}\biggr)^{3} \frac{2^2}{5^2} \biggr]
=
- \frac{2^3 }{3} \biggl[\biggl(\frac{3}{2^2}\biggr)^{3} \biggr]
=
- \frac{3^2 }{2^3} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, the LHS = RHS.   <font size="+1" color="red"><b>Hooray!</b></font>
</td></tr>
</table>

=See Also=

* [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29|Derivations that support this chapter's discussion of the Ledoux Variational Principle]]
* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">A variational principle of great power is derived. It is naturally adapted for computers, and may be used to determine the stability of any fluid flow including those in differentially-rotating, self-gravitating stars and galaxies. The method also provides a powerful theoretical tool for studying general properties of eigenfunctions, and the relationships between secular and ordinary stability. In particular we prove the anti-sprial theorem indicating that no stable (or steady( mode can have a spiral structure</font>.
</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1972ApJS...24..319S/abstract B. F. Schutz, Jr. (1972)], ApJSuppl., 24, 319: ''Linear Pulsations and Stability of Differentially Rotating Stellar Models. I. Newtonian Analysis''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">A systematic method is presented for deriving the Lagrangian governing the evolution of small perturbations of arbitrary flows of a self-gravitating perfect fluid. The method is applied to a differentially rotating stellar model; the result is a Lagrangian equivalent to that of [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)]. A sufficient condition for stability of rotating stars, derived from this Lagrangian, is simplified greatly by using as trial functions not the three components of the Lagrangian displacement vector, but three scalar functions … This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.

… we examine the special cases of (i) axially symmetric perturbations of a rotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..267C/abstract S. Chandrasekhar & N. R. Lebovitz 1968]); and (ii) perturbations of a nonrotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1964ApJ...140.1517C/abstract Chandrasekhar and Lebovitz 1964)]. We find that the stability criteria for those cases can also be simplified …</font>
</td></tr></table>

{{ SGFfooter }}

SSCpt1/Virial/FormFactors

2021-07-29T21:02:32Z

174.64.8.247: /* Synopsis */

__FORCETOC__

=Structural Form Factors=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|<b>Structural<br />Form<br />Factors</b>]]</font>
|}
As has been defined in [[SSCpt1/Virial#Structural_Form_Factors|a companion, introductory discussion]], three key dimensionless structural form factors are:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>x \equiv r/R_\mathrm{limit}</math>, and the subscript "0" denotes central values. The principal purpose of this chapter is to carry out the integrations that are required to obtain expressions for these structural form factors, at least in the few cases where they can be determined analytically. These form-factor expressions will then be used to provide expressions for the two constants, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, that appear in the free-energy function and in the virial theorem, and to provide corresponding expressions for the normalized energies, <math>W_\mathrm{grav}/E_\mathrm{norm}</math> and <math>S_\mathrm{therm}/E_\mathrm{norm}</math>.
 <br />

==Synopsis==

<div align="center"><b>Summary of Derived Structural Form-Factors</b></div>
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> Polytropes <math>(n \ne 5)</math>
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> Polytropes <math>(n \ne 5)</math>
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 1 Polytrope<br /><math>\tilde\xi \rightarrow \xi_1 = \pi</math>
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 1 Polytropes<br /><math>0 < \tilde\xi < \pi</math>
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\pi^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 5 Polytrope
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 5 Polytropes
</th>
</tr>
<tr>
<td align="center">
 <br />
 <br />
 <br />
 <br />
 <br />
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 1 + \ell^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>
where,     <math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}}</math>
</td>
</tr>
</table>

==Expectation in Context of Pressure-Truncated Polytropes==
For pressure-truncated polytropic configurations, the normalized virial theorem states that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>2 \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr) + \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} \, .</math>
</td>
</tr>
</table>
</div>
This provides one mechanism by which the correctness of our form-factor expressions can be checked. Specifically, having determined <math>S_\mathrm{therm}</math> and <math>W_\mathrm{grav}</math> from the derived form factors, we can see whether the sum of these energies as specified on the lefthand-side of this virial theorem expression indeed match the normalized energy term involving the external pressure, as specified on the righthand side. In order to facilitate this "reality check" at the end of each example, below, we will use [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's detailed force-balanced solution of the equilibrium structure of embedded polytropes]] to provide an expression for the term on the righthand side of the virial theorem expression.

We begin by plugging our [[SSCpt1/Virial#Normalizations|general expression for <math>E_\mathrm{norm}</math>]] into this righthand-side term and grouping factors to facilitate insertion of Stahler's expressions.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{tot}^{(5-n)} \biggr]^{1/(n-3)} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)/(n-3)} P_e R_\mathrm{eq}^3
\biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)} \, . </math>
</td>
</tr>
</table>
</div>
From [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's equilibrium solution]], we have,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
R_\mathrm{eq}
</math>
</td>
<td align="center">
<math>=~</math>
</td>
<td align="left">
<math>
R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
~P_e R_\mathrm{eq}^3
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{1 + 3(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
M_\mathrm{limit}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
K^{-n} G^3 M_\mathrm{limit}^{(5-n)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3-3(5-n)/2} K_n^{-n +2n(5-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3(n-3)/2} K_n^{3n(3-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} \biggr\}^{(5-n)/(n-3)} G^{3/2} K_n^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>\times ~\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(5-n)/2}
\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^{3(n-3)}_{\tilde\xi} \biggl( \frac{n+1}{4\pi} \biggr)^{3(n-3)/2} \biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{
(n+1)^{3[(5-n)+(n-3)]/2} (4\pi)^{[(n-5)+(9-3n)]/2}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{[(n-3)(5-n) + 3(n-1)(n-3)]/2} \tilde\xi^{[2(5-n) + 3(n-3)]}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{ (n+1)^{3} (4\pi)^{(2-n)}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{(n+1)(n-3)} \tilde\xi^{(n+1)}
\biggr\}^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, the expectation based on Stahler's equilibrium models is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

As a cross-check, multiplying this expression through by <math>[(R_\mathrm{eq}/R_\mathrm{norm})(M_\mathrm{norm}/M_\mathrm{limit})^2]</math> — where the expression for <math>R_\mathrm{eq}/R_\mathrm{norm}</math> can be obtained from our [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|discussions of detailed force-balanced models]] — gives a related result that can be obtained directly from [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|Horedt's expressions]], namely,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{4\pi P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(n+1)( -\tilde\theta' )^{2}} \, .
</math>
</td>
</tr>
</table>
</div>

==Viala and Horedt (1974) Expressions==
===Presentation===

[http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have provided analytic expressions for the gravitational potential energy and the internal energy — which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively — that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres. [The same expression for <math>~\Omega</math> is also effectively provided in §1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
<div align="center">
<table border="1" align="center" cellpadding="8" width="90%">
<tr><td align="center">


Astronomy & Astrophysics, 33: 195-202, (1974)<br />
POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
Y. P. Viala & Gp. Horedt<br />
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>\Omega</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- G\int_0^M \frac{MdM}{r} = \frac{16\pi^2 G \rho_0^2 \alpha^5}{(5-n)} \biggl[
\mp \xi^3 \theta^{n+1} - 3\xi^3 (\theta')^2 - 3\xi^2 \theta (\theta')
\biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>U</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{\gamma - 1}\int_V pdV = \frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \int_0^\xi \theta^{n+1} 4\pi \alpha^2 \xi^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \cdot
\frac{4\pi \alpha^2(n+1)}{(5-n)}
\biggl[
\frac{2\xi^3 \theta^{n+1}}{n+1} \pm \xi^3 (\theta')^2 \pm \xi^2 \theta (\theta')
\biggr]_0^\xi \, .
</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
(the superior sign holds if <math>-1 < n < \infty</math>, the inferior if <math>-\infty < n < -1</math>)
</td>
</tr>
</table>

</td></tr>
<tr>
<td align="left">
A couple of key equations drawn directly from [http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have been shown here. As its title indicates, the paper includes discussion of — and accompanying equation derivations for — equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries: planar sheets, axisymmetric cylinders, and spheres. We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only. These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
</td>
</tr>
</table>
</div>
Rewriting these two expressions to accommodate our parameter notations — recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> — we have from Viala & Horedt's (VH74) work,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \mathfrak{S}_\mathrm{A} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n(4\pi)^2}{3(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

===First Reality Check===

As a quick reality check, let's see whether, when appropriately added together, these two energies satisfy the scalar virial theorem for isolated polytropes.

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
W_\mathrm{grav} + \frac{3}{n} \mathfrak{S}_A</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>+
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} - 1 \biggr] \tilde\xi^3 \tilde\theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(n+1)} \cdot G \rho_c^2 a_n^5 \tilde\xi^3 \tilde\theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>
For ''isolated polytropes'', <math>\tilde\theta \rightarrow 0</math>, so this sum of terms goes to zero, as it should if the system is in virial equilibrium.

===Renormalization===
Both of the energy-term expressions derived by Viala & Horedt are written in terms of <math>\rho_c</math> and
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>a_\mathrm{n}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n}\biggr]^{1/2}
</math>
</td>
</tr>
</table>
</div>
— that is, effectively in terms of <math>\rho_c</math> and {{ Template:Math/MP_PolytropicConstant}} — whereas, in the context of our discussions, we would prefer to express them in terms of [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Adopted_Normalizations|our generally adopted energy normalization]],
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_n^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>
In order to accomplish this, we need to replace the central density with the total mass of an ''isolated polytrope'', <math>M_\mathrm{tot}</math>, whose generic expression is (see, for example, equation 69 of Chandrasekhar),
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}^{n-3}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_n^n G^{-3}\biggl\{ (4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr\}^{n-5} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
K_n^{[2n + 3(n-5)]/2} G^{[-6-3(n-5)]/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
\rho_c^{(n-3)(5-n)/2n} K_n^{5(n-3)/2} G^{-3(n-3)/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)}
\rho_c^{(5-n)/2n} K_n^{5/2} G^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\rho_c^{[ - 4n +(5-n)]/2n} \biggl( \frac{K_n}{G}\biggr)^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{K_n}{G} \cdot \rho_c^{(1-n)/n}\biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{4\pi}{(n+1)} \cdot a_n^2 \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~(4\pi)^2 G\rho_c^2 a_n^5</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (4\pi)^2
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(5-n)/(n-3)}
\biggl[ \frac{(n+1)}{4\pi} \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)/(n-3)}
(4\pi)^{[-(n-3)-(5-n)]/2(n-3)} (n+1)^{[3(5-n)+5(n-3)]/2(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
So, employing our preferred normalization, the VH74 expressions become,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{\mathfrak{S}_\mathrm{A}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n}{3(5-n)}
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

===Second Reality Check===
If we now renormalize the sum of energy terms discussed in our [[SSCpt1/Virial/FormFactors#First_Reality_Check|first reality check, above]], we have,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>
\frac{1}{E_\mathrm{norm}} \biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}
= \frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(n+1)^{-1} \tilde\xi^3 \tilde\theta^{n+1} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

(This may or may not be useful!)

===Implication for Structural Form Factors===
On the other hand, our expressions for these two [[SSCpt1/Virial#Structural_Form_Factors|normalized energy components written in terms of the structural form factors]] are,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>
where, in equilibrium (see [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|here]] and [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|here]] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl\{ \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl\{ \biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-1)/(n-3)}
\biggr\} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
Hence, we deduce that,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> - \frac{5}{3} \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]
\chi_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \cdot \tilde\mathfrak{f}^2_M
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{5}{3} \biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggr\}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(n-1)-2(n-3)]/(n-3)}
\cdot \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr)^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{[(1-n)+2(n-3)]/(n-3)} \tilde\xi^{[-(n-3)+2(1-n)]/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{(n-5)/(n-3)} \tilde\xi^{(5-3n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
If we now adopt the VH74 expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\biggl\{~~~\biggr\}_\mathrm{VH74}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)}
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
(-\tilde\xi^2 \tilde\theta^')^{(5-n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Therefore,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_W \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)} \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\tilde\xi^{-5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>

Now, from [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|our earlier work]] we deduced that <math>\tilde\mathfrak{f}_A</math> is related to <math>\tilde\mathfrak{f}_W</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \tilde\mathfrak{f}_W\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \, .</math>
</td>
</tr>
</table>
</div>
Hence, we now have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_A \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \frac{(n+1)}{(5-n)}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
\, .
</math>
</td>
</tr>
</table>
</div>

Building on the work of VH74, we have, quite generally,

<div align="center" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>
</table>

</td>
</tr>
<tr>
<td align="left" colspan="2">
We should point out that [http://adsabs.harvard.edu/abs/1993ApJS...88..205L Lai, Rasio, & Shapiro (1993b, ApJS, 88, 205)] define a different set of dimensionless structure factors for ''isolated'' polytropic spheres — <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10) is used in the determination of the gravitational potential energy.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>k_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[ \frac{n(n+1)}{5-n} \biggr] \xi_1|\theta^'_1|</math>
</td>
</tr>

<tr>
<td align="right">
<math>k_2</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{5-n} \biggl[ \frac{4\pi |\theta^'_1|}{\xi_1} \biggr]^{1 / 3} </math>
</td>
</tr>
</table>
</div>
Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter. We note, as well, that for rotating configurations they define two additional dimensionless structure factors — <math>k_3</math> (their equation 3.17) is used in the determination of the rotational kinetic energy; and <math>\kappa_n</math> (their equation 3.14; also equation 7.4.9 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]) is used in the determination of the moment of inertia.
</td>
</tr>
</table>
</div>

The singularity that arises when <math>n = 5</math> leads us to suspect that these general expressions fail in that one specific case. Fortunately, as [[#Summary_.28n.3D5.29|we have shown in an accompanying discussion]], <math>\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, as well as <math>\mathfrak{f}_M</math>, can be determined by direct integration in this single case.

===Related Discussions===
* See [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Model_Sequences|our plot of, what Kimura (1981b) would refer to as, several <math>M_1</math> sequences]]

==First Detailed Example (n = 5)==

Here we complete these integrals to derive detailed expressions for the above subset of structural form factors in the case of spherically symmetric configurations that obey an <math>~n=5</math> polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should simplify the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases that cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|cores of bipolytropes]].

===Foundation (n = 5)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=5; ~\gamma=6/5)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{K^{10}}{G^{9} M_\mathrm{tot}^{6}} \biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl( \frac{K}{G} \biggr)^{15/2} M_\mathrm{tot}^{-5} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{K^5}{G^3} \biggr)^{1/2} M_\mathrm{tot}^{-1} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/BiPolytropes/Analytic5_1#Profile|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,
<div align="center">
<math>
a_{5} =\biggr[ \frac{3K}{2\pi G} \biggr]^{1/2} \rho_0^{-2/5} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{[K^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^{6/5}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{[K^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5}
~~~~\Rightarrow ~~~~
\rho_0^{1/5} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} M_\mathrm{tot}^{-1} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr) \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi
= \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3^4}{\pi} \biggr)^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr)^{1/2}
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
= \biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2}
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 5)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math>, and recognizing that,
<div align="center">
<math>dr^\dagger = \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} d\xi \, ,</math>
</div>
gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], [[Image:OnlineIntegral01.png|250px|right|Mathematica Integral]]
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{3/2} \biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2}
\int_0^{\xi} \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{1}{3} \biggr)^{3/2}
\biggl[ \frac{\xi^3}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-3/2} \biggr]_0^{\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\xi^2}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=5</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 5)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5} \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=5</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_5 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_5 \xi}{a_5 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=5</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{3/2} \biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[SSCpt1/Virial/FormFactors#NormalizedProfiles|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2} \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 5)===

From the above line of reasoning we appreciate that, for any spherically symmetric configuration, the ratio of the configuration's mean density to its central density can be obtained by setting the upper limit of our just-completed "Mass2" integration to <math>x=1</math>. That is to say, quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>
</table>
</div>
But the integral expression on the righthand side of this relation is also the definition of the structural form factor, <math>~\mathfrak{f}_M</math>, given at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]]. Hence, we can say, quite generally, that,
<div align="center">
<math>\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} \, .</math>
</div>
And, given that we have just completed this integral for the case of truncated <math>n=5</math> polytropic structures, we can state, specifically, that,
<div align="center">
<math>\mathfrak{f}_M\biggr|_{n=5} = \biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</div>

===Gravitational Potential Energy (n = 5)===
As presented at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral02.png|225px|right|Mathematica Integral]]Given that an expression for the normalized density profile, <math>\rho(x)/\rho_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been determined, above]], we can carry out the nested pair of integrals immediately. Indeed, the integral contained inside of the curly braces has already been completed [[SSCpt1/Virial/FormFactors#Mass2|in the "Mass2" subsection, above]], in order to determine the radial mass profile. Specifically, we have already determined that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ \int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\}
\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} x dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-4} x^4 dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \biggl( \frac{\tilde\xi^2}{3}\biggr)^{-5/2} \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-3}
\biggl\{
\biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggl[ 3\biggl( \frac{\tilde\xi^2}{3}\biggr)^2 - 8\biggl( \frac{\tilde\xi^2}{3}\biggr) - 3 \biggr]
+ 3\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr)^3\tan^{-1}\biggl[ \biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>\mathfrak{f}_M</math> and <math>\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>n=5</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{\pi}{2^3\cdot 3^7}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
In order to simplify typing, we will switch to the variable,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
in which case a summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>( 1 + \ell^2 )^{-3/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \cdot \ell^{-5} (1 + \ell^2)^{-3}
\biggl\{ \ell [ 3\ell^4 - 8\ell^2 - 3 ] + 3( 1 + \ell^2 )^3\tan^{-1}(\ell ) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{3}{5} \biggl(\frac{2^3\cdot 3^6}{\pi}\biggr)^{1/2} \frac{1}{\ell} \cdot (1 + \ell^2)^3
\mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5 \pi}\biggr)^{1/2} \cdot \ell^{-6} (1 + \ell^2)^3
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized gravitational potential energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>W_\mathrm{core}^*</math>.

Hence, also, as defined in the [[SSCpt1/Virial#Gathering_it_All_Together|accompanying introductory discussion]], the constant, <math>\mathcal{A}</math>, that appears in our general free-energy equation is (for <math>n=5</math> polytropic configurations),
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 5)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral03.png|225px|right|Mathematica Integral]]Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} x^2 dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{2^3}
\biggl\{\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-3/2} \tan^{-1}\biggl[ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
+ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-1}
- 2\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-2}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=5 (\gamma=6/5)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)^{1/5}
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{6/5}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>

If, as above, we adopt the simplifying variable notation,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
the various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} [ \ell^{-3} \tan^{-1}(\ell ) + \ell^{-2}(1+\ell^2)^{-1} - 2\ell^{-2}(1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, ;
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3/5} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2} \cdot \biggl[ \biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \biggr]^{-3/5}
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5} \ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{3^{10}}{2^{10}} \biggl(\frac{2^9\cdot 3^{18}}{\pi^3}\biggr)
\biggl(\frac{3^{12}}{2^{34} \pi^2} \biggr) \biggr]^{1/10} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^{8}}{2^{7}\pi}\biggr)^{1/2} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized thermal energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>S_\mathrm{core}^*</math>. Its similarity to the expression for the gravitational potential energy — which is relevant to the virial theorem — is more apparent if it is rewritten in the following form:
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2}
\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 5)===
In summary, for <math>n=5</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 1 + \ell^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 5)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} \biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Check (n = 5)===

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}\biggl\{ [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
-
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}
\biggl[\frac{8}{3}\ell^3 (1+\ell^2)^{-3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2 \cdot 3^{6}}{\pi}\biggr)^{1/2}
\biggl[\frac{\ell}{ (1+\ell^2)} \biggr]^3 \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=5</math> polytropes — see the [[#Mass1|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \biggl[ \ell^2(1+\ell^2)^{-1} \biggr]^{3/2}
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_5 = ( 1 + \ell^2 )^{-1/2}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_5}{d\xi} \biggr|_{\xi_e} = 3^{1/2} \ell ( 1 + \ell^2 )^{-3/2} \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2\cdot 3^3}{\pi}\biggr]^{1/2} ( 1 + \ell^2 )^{-3} (3^{1/2}\ell)^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{2\cdot 3^6}{\pi}\biggr)^{1/2} \biggl[ \frac{\ell}{( 1 + \ell^2 )} \biggr]^{3} \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors. This gives us confidence that our form-factor expressions are correct, at least in the case of embedded <math>n=5</math> polytropic structures.

==Second Detailed Example (n = 1)==

===Foundation (n = 1)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=1; ~\gamma=2)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl(\frac{K}{G}\biggr)^{1/2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G^3 M_\mathrm{tot}^2}{K^2}\biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot}^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi}\biggl( \frac{G}{K} \biggr)^{3/2} M_\mathrm{tot} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of the properties of ''isolated'' polytropes]], in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{1}</math>, where,
<div align="center">
<math>
a_{1} = \biggl( \frac{K}{2\pi G} \biggr)^{1/2} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{(K/G)^{1/2}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^{2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{(K/G)^{3/2}\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2}{\pi} \biggr)^{1/2} (\sin\xi - \xi \cos\xi) \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=1</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0
~~~~\Rightarrow ~~~~
\rho_0 = \biggr[ \frac{G^3}{2\pi K^3} \biggr]^{1/2} M_\mathrm{tot} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles1">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3 \pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\pi} \biggl( \frac{\sin\xi}{\xi}\biggr)^2 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi \cos\xi)
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 1)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math> gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \int_0^{\xi} \frac{\xi^2}{2\pi} \biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \cdot \frac{d\xi }{(2\pi)^{1/2}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3( 2\pi)^{-3/2}\biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \int_0^{\xi} \xi \sin\xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi\cos\xi) \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=1</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 1)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0 \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=1</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1 = \pi</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_1 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_1 \xi}{a_1 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx1">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin(\tilde\xi x)}{\tilde\xi x} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi} \int_0^{x} x \sin(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{3}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, . </math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=1</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
\biggl\{ \frac{[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ]}{( \sin \tilde\xi - \tilde\xi \cos \tilde\xi )} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] </math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[#NormalizedProfiles1|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\xi - \xi \cos\xi ) \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 1)===

[[SSCpt1/Virial/FormFactors#Mean-to-Central_Density|Following the line of reasoning provided above]], we can use the just-derived central-to-mean density ratio to specify one of the structural form factors. Specifically,
<div align="center">
<math>~\mathfrak{f}_M\biggr|_{n=1} = \frac{\bar\rho}{\rho_c} = \frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</div>

===Gravitational Potential Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
From the derivations already presented, above, for <math>~n=1</math> polytropic configurations, we know all of the functions under this integral. We know, for example, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \int_0^1 \biggl\{ [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \biggr\}
\sin(\tilde\xi x) dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \biggl\{ \frac{x}{2} - \frac{\sin(2\tilde\xi x)}{4\tilde\xi} - \tilde\xi\biggl[
\frac{\sin(2\tilde\xi x) - 2\tilde\xi x\cos(2\tilde\xi x)}{8\tilde\xi^2}
\biggr] \biggr\}_0^1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>~n=1</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- 3\mathcal{A} \chi^{-1} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\chi</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
A summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{5} \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^{2} \mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[#Foundation_2|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi}\biggr)^{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\sin(\tilde\xi x)}{(\tilde\xi x)}\biggr]^{2} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2} \int_0^1 \sin^2(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2}
\biggl[\frac{x}{2}- \frac{\sin(2\tilde\xi x)}{4\tilde\xi} \biggr]_0^1
= \frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=1 (\gamma=2)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>
The various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\xi^3}{3\pi} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr) \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^2
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3 \tilde\xi^3}{2^5\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
\biggl[ (2\pi)^{3/2} \tilde\xi^{-3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 1)===
In summary, for <math>~n=1</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 1)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Checks (n = 1)===
====Expectation from Stahler's Equilibrium Models====
If we add twice the thermal energy to the gravitational potential energy, we obtain,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} \biggl\{ \biggl[6\tilde\xi - 3\sin(2\tilde\xi ) \biggr]
- \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} 2\tilde\xi\biggl\{ 1-\cos(2\tilde\xi ) \biggr\}
= \biggl( \frac{1}{2\pi^3} \biggr)^{1/2} \tilde\xi \sin^2(\tilde\xi ) \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=1</math> polytropes — see the [[#Mass1_.28n_.3D_1.29|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D1.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_1 = \frac{\sin\tilde\xi}{\tilde\xi}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_1}{d\xi} \biggr|_{\tilde\xi} = \frac{1}{\tilde\xi^2}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2}{\pi} \biggr]^{-1/2} \biggl[ \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \tilde\xi^2(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)^{-1}\biggr]^{2}
\biggl( \frac{\sin\tilde\xi}{\tilde\xi}\biggr)^2 \tilde\xi^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi^3}\biggr)^{1/2} \tilde\xi \sin^2\tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors, giving us additional confidence that our form-factor expressions are correct.

====Compare With General Expressions Based on VH74 Work====
Based on the general expressions [[#PTtable|derived above]] in the context of VH74's work, for the specific case of <math>n=1</math> polytropic configurations, the three structural form factor should be,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\tilde\xi^2}
\biggl[\tilde\theta^{2} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ 3\tilde\theta^{2} +
3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
Also, remember that,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\theta</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~\theta^' \equiv \frac{d\theta}{d\xi}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\cos\xi}{\xi} - \frac{\sin\xi}{\xi^2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~(\theta^' )^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\xi^4} \biggl[ \xi\cos\xi - \sin\xi \biggr]^2
=\frac{1}{\xi^4} \biggl[ \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr] \, .
</math>
</td>
</tr>
</table>
Now, let's look at the structural form factors, one at a time. First, we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\xi^3} \biggl[\sin\xi - \xi\cos\xi \biggr]</math>
</td>
</tr>
</table>

which matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Next,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~ \frac{3\cdot 5}{4\xi^2} \biggl[ \frac{\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 + 2\xi^2\cos^2\xi - 3 \xi\sin\xi \cos\xi \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl\{ \xi^2 + \xi^2[1+\cos(2\xi)] - \frac{3}{2} \xi\sin(2\xi) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{8\xi^6} \biggl[ 4\xi^2 + 2\xi^2 \cos(2\xi) - 3 \xi\sin(2\xi) \biggr] \, ,
</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Finally,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ \frac{3\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi \biggr)
+ 3\xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2\xi^4} \biggl[ \xi^2 - \xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2^2\xi^3} \biggl[ 2\xi - \sin(2\xi) \biggr] \, ,</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. So this adds support to the deduction, above, that VH74 have provided us with the information necessary to develop general expressions for the three structural form factors.

==Fiddling Around==
NOTE (from Tohline on 17 March 2015): Chronologically, this "Fiddling Around" subsection was developed before our discovery of the VH74 derivations. It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres. But this subsection's conclusions are superseded by the VH74 work.

In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>. In the case of <math>n=1</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\sin\xi}{\xi}\biggr)^2
= \frac{1}{2\xi^2} \biggl[ 1 - \cos(2\xi) \biggr]
= \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\mathfrak{f}_A - \theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[6\xi - 3\sin(2\xi ) \biggr]
- \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[4\xi - 3\sin(2\xi ) + 2\xi\cos(2\xi) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^3 \xi^5}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ 4\xi - 3\sin(2\xi) + 2\xi \cos(2\xi ) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2 \xi^2}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

Similarly, in the case of <math>n = 5</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\biggl( \frac{2^3}{3} \ell^{3} \biggr) \biggl[ \mathfrak{f}_A - \theta^{n+1} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
[ \tan^{-1}(\ell ) + \ell (\ell^4-1) (1+\ell^2)^{-3} ] - \biggl( \frac{2^3}{3} \ell^{3} \biggr) (1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\tan^{-1}(\ell ) + \ell \biggl(\ell^4-\frac{8}{3}\ell^2 - 1 \biggr) (1+\ell^2)^{-3} \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^4}{5} \cdot \ell^{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2\cdot 3}{5} \cdot \ell^{2} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{2\xi^2}{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>
This is pretty amazing! Both examples produce almost exactly the same relationship between the two structural form factors, <math>\mathfrak{f}_A</math> and <math>\mathfrak{f}_W</math>. I think that we are well on our way toward nailing down the generic, analytic relationship and, in turn, a generally applicable mass-radius relationship for pressure-truncated polytropic configurations.

Okay … here is the final piece of information. In the case of isolated polytropes, we know that the correct expressions for the structural form factors are as summarized in the following table:
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi}
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>
</div>
We notice, from this, that the ratio,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} \cdot \frac{5-n}{3^2\cdot 5} \biggl[ \frac{\xi}{\Theta^'} \biggr]^{2}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(n+1)\tilde\xi^2 }{3\cdot 5} \, .
</math>
</td>
</tr>
</table>
Even in the case of the two pressure-truncated polytropes, analyzed above, this ratio proves to give the correct prefactor on <math>\mathfrak{f}_W</math>. So we ''suspect'' that the universal relationship between the two form factors is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{(n+1) \xi^2 }{3\cdot 5} \biggr] \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

=See Also=
<ul>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>

{{ SGFfooter }}

SSCpt1/Virial/FormFactors

2021-07-29T20:44:28Z

174.64.8.247: /* Synopsis */

__FORCETOC__

=Structural Form Factors=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|<b>Structural<br />Form<br />Factors</b>]]</font>
|}
As has been defined in [[SSCpt1/Virial#Structural_Form_Factors|a companion, introductory discussion]], three key dimensionless structural form factors are:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>x \equiv r/R_\mathrm{limit}</math>, and the subscript "0" denotes central values. The principal purpose of this chapter is to carry out the integrations that are required to obtain expressions for these structural form factors, at least in the few cases where they can be determined analytically. These form-factor expressions will then be used to provide expressions for the two constants, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, that appear in the free-energy function and in the virial theorem, and to provide corresponding expressions for the normalized energies, <math>W_\mathrm{grav}/E_\mathrm{norm}</math> and <math>S_\mathrm{therm}/E_\mathrm{norm}</math>.
 <br />

==Synopsis==

<div align="center"><b>Summary of Derived Structural Form-Factors</b></div>
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> Polytropes <math>(n \ne 5)</math>
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> Polytropes <math>(n \ne 5)</math>
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 1 Polytrope
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 1 Polytropes
</th>
</tr>
<tr>
<td align="center">
 <br />
 <br />
 <br />
 <br />
 <br />
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 5 Polytrope
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 5 Polytropes
</th>
</tr>
<tr>
<td align="center">
 <br />
 <br />
 <br />
 <br />
 <br />
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 1 + \ell^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>
where,     <math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}}</math>
</td>
</tr>
</table>

==Expectation in Context of Pressure-Truncated Polytropes==
For pressure-truncated polytropic configurations, the normalized virial theorem states that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>2 \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr) + \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} \, .</math>
</td>
</tr>
</table>
</div>
This provides one mechanism by which the correctness of our form-factor expressions can be checked. Specifically, having determined <math>S_\mathrm{therm}</math> and <math>W_\mathrm{grav}</math> from the derived form factors, we can see whether the sum of these energies as specified on the lefthand-side of this virial theorem expression indeed match the normalized energy term involving the external pressure, as specified on the righthand side. In order to facilitate this "reality check" at the end of each example, below, we will use [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's detailed force-balanced solution of the equilibrium structure of embedded polytropes]] to provide an expression for the term on the righthand side of the virial theorem expression.

We begin by plugging our [[SSCpt1/Virial#Normalizations|general expression for <math>E_\mathrm{norm}</math>]] into this righthand-side term and grouping factors to facilitate insertion of Stahler's expressions.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{tot}^{(5-n)} \biggr]^{1/(n-3)} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)/(n-3)} P_e R_\mathrm{eq}^3
\biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)} \, . </math>
</td>
</tr>
</table>
</div>
From [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's equilibrium solution]], we have,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
R_\mathrm{eq}
</math>
</td>
<td align="center">
<math>=~</math>
</td>
<td align="left">
<math>
R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
~P_e R_\mathrm{eq}^3
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{1 + 3(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
M_\mathrm{limit}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
K^{-n} G^3 M_\mathrm{limit}^{(5-n)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3-3(5-n)/2} K_n^{-n +2n(5-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3(n-3)/2} K_n^{3n(3-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} \biggr\}^{(5-n)/(n-3)} G^{3/2} K_n^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>\times ~\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(5-n)/2}
\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^{3(n-3)}_{\tilde\xi} \biggl( \frac{n+1}{4\pi} \biggr)^{3(n-3)/2} \biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{
(n+1)^{3[(5-n)+(n-3)]/2} (4\pi)^{[(n-5)+(9-3n)]/2}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{[(n-3)(5-n) + 3(n-1)(n-3)]/2} \tilde\xi^{[2(5-n) + 3(n-3)]}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{ (n+1)^{3} (4\pi)^{(2-n)}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{(n+1)(n-3)} \tilde\xi^{(n+1)}
\biggr\}^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, the expectation based on Stahler's equilibrium models is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

As a cross-check, multiplying this expression through by <math>[(R_\mathrm{eq}/R_\mathrm{norm})(M_\mathrm{norm}/M_\mathrm{limit})^2]</math> — where the expression for <math>R_\mathrm{eq}/R_\mathrm{norm}</math> can be obtained from our [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|discussions of detailed force-balanced models]] — gives a related result that can be obtained directly from [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|Horedt's expressions]], namely,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{4\pi P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(n+1)( -\tilde\theta' )^{2}} \, .
</math>
</td>
</tr>
</table>
</div>

==Viala and Horedt (1974) Expressions==
===Presentation===

[http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have provided analytic expressions for the gravitational potential energy and the internal energy — which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively — that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres. [The same expression for <math>~\Omega</math> is also effectively provided in §1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
<div align="center">
<table border="1" align="center" cellpadding="8" width="90%">
<tr><td align="center">


Astronomy & Astrophysics, 33: 195-202, (1974)<br />
POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
Y. P. Viala & Gp. Horedt<br />
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>\Omega</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- G\int_0^M \frac{MdM}{r} = \frac{16\pi^2 G \rho_0^2 \alpha^5}{(5-n)} \biggl[
\mp \xi^3 \theta^{n+1} - 3\xi^3 (\theta')^2 - 3\xi^2 \theta (\theta')
\biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>U</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{\gamma - 1}\int_V pdV = \frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \int_0^\xi \theta^{n+1} 4\pi \alpha^2 \xi^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \cdot
\frac{4\pi \alpha^2(n+1)}{(5-n)}
\biggl[
\frac{2\xi^3 \theta^{n+1}}{n+1} \pm \xi^3 (\theta')^2 \pm \xi^2 \theta (\theta')
\biggr]_0^\xi \, .
</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
(the superior sign holds if <math>-1 < n < \infty</math>, the inferior if <math>-\infty < n < -1</math>)
</td>
</tr>
</table>

</td></tr>
<tr>
<td align="left">
A couple of key equations drawn directly from [http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have been shown here. As its title indicates, the paper includes discussion of — and accompanying equation derivations for — equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries: planar sheets, axisymmetric cylinders, and spheres. We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only. These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
</td>
</tr>
</table>
</div>
Rewriting these two expressions to accommodate our parameter notations — recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> — we have from Viala & Horedt's (VH74) work,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \mathfrak{S}_\mathrm{A} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n(4\pi)^2}{3(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

===First Reality Check===

As a quick reality check, let's see whether, when appropriately added together, these two energies satisfy the scalar virial theorem for isolated polytropes.

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
W_\mathrm{grav} + \frac{3}{n} \mathfrak{S}_A</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>+
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} - 1 \biggr] \tilde\xi^3 \tilde\theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(n+1)} \cdot G \rho_c^2 a_n^5 \tilde\xi^3 \tilde\theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>
For ''isolated polytropes'', <math>\tilde\theta \rightarrow 0</math>, so this sum of terms goes to zero, as it should if the system is in virial equilibrium.

===Renormalization===
Both of the energy-term expressions derived by Viala & Horedt are written in terms of <math>\rho_c</math> and
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>a_\mathrm{n}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n}\biggr]^{1/2}
</math>
</td>
</tr>
</table>
</div>
— that is, effectively in terms of <math>\rho_c</math> and {{ Template:Math/MP_PolytropicConstant}} — whereas, in the context of our discussions, we would prefer to express them in terms of [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Adopted_Normalizations|our generally adopted energy normalization]],
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_n^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>
In order to accomplish this, we need to replace the central density with the total mass of an ''isolated polytrope'', <math>M_\mathrm{tot}</math>, whose generic expression is (see, for example, equation 69 of Chandrasekhar),
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}^{n-3}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_n^n G^{-3}\biggl\{ (4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr\}^{n-5} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
K_n^{[2n + 3(n-5)]/2} G^{[-6-3(n-5)]/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
\rho_c^{(n-3)(5-n)/2n} K_n^{5(n-3)/2} G^{-3(n-3)/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)}
\rho_c^{(5-n)/2n} K_n^{5/2} G^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\rho_c^{[ - 4n +(5-n)]/2n} \biggl( \frac{K_n}{G}\biggr)^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{K_n}{G} \cdot \rho_c^{(1-n)/n}\biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{4\pi}{(n+1)} \cdot a_n^2 \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~(4\pi)^2 G\rho_c^2 a_n^5</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (4\pi)^2
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(5-n)/(n-3)}
\biggl[ \frac{(n+1)}{4\pi} \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)/(n-3)}
(4\pi)^{[-(n-3)-(5-n)]/2(n-3)} (n+1)^{[3(5-n)+5(n-3)]/2(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
So, employing our preferred normalization, the VH74 expressions become,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{\mathfrak{S}_\mathrm{A}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n}{3(5-n)}
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

===Second Reality Check===
If we now renormalize the sum of energy terms discussed in our [[SSCpt1/Virial/FormFactors#First_Reality_Check|first reality check, above]], we have,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>
\frac{1}{E_\mathrm{norm}} \biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}
= \frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(n+1)^{-1} \tilde\xi^3 \tilde\theta^{n+1} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

(This may or may not be useful!)

===Implication for Structural Form Factors===
On the other hand, our expressions for these two [[SSCpt1/Virial#Structural_Form_Factors|normalized energy components written in terms of the structural form factors]] are,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>
where, in equilibrium (see [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|here]] and [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|here]] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl\{ \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl\{ \biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-1)/(n-3)}
\biggr\} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
Hence, we deduce that,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> - \frac{5}{3} \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]
\chi_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \cdot \tilde\mathfrak{f}^2_M
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{5}{3} \biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggr\}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(n-1)-2(n-3)]/(n-3)}
\cdot \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr)^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{[(1-n)+2(n-3)]/(n-3)} \tilde\xi^{[-(n-3)+2(1-n)]/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{(n-5)/(n-3)} \tilde\xi^{(5-3n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
If we now adopt the VH74 expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\biggl\{~~~\biggr\}_\mathrm{VH74}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)}
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
(-\tilde\xi^2 \tilde\theta^')^{(5-n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Therefore,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_W \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)} \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\tilde\xi^{-5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>

Now, from [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|our earlier work]] we deduced that <math>\tilde\mathfrak{f}_A</math> is related to <math>\tilde\mathfrak{f}_W</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \tilde\mathfrak{f}_W\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \, .</math>
</td>
</tr>
</table>
</div>
Hence, we now have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_A \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \frac{(n+1)}{(5-n)}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
\, .
</math>
</td>
</tr>
</table>
</div>

Building on the work of VH74, we have, quite generally,

<div align="center" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>
</table>

</td>
</tr>
<tr>
<td align="left" colspan="2">
We should point out that [http://adsabs.harvard.edu/abs/1993ApJS...88..205L Lai, Rasio, & Shapiro (1993b, ApJS, 88, 205)] define a different set of dimensionless structure factors for ''isolated'' polytropic spheres — <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10) is used in the determination of the gravitational potential energy.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>k_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[ \frac{n(n+1)}{5-n} \biggr] \xi_1|\theta^'_1|</math>
</td>
</tr>

<tr>
<td align="right">
<math>k_2</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{5-n} \biggl[ \frac{4\pi |\theta^'_1|}{\xi_1} \biggr]^{1 / 3} </math>
</td>
</tr>
</table>
</div>
Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter. We note, as well, that for rotating configurations they define two additional dimensionless structure factors — <math>k_3</math> (their equation 3.17) is used in the determination of the rotational kinetic energy; and <math>\kappa_n</math> (their equation 3.14; also equation 7.4.9 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]) is used in the determination of the moment of inertia.
</td>
</tr>
</table>
</div>

The singularity that arises when <math>n = 5</math> leads us to suspect that these general expressions fail in that one specific case. Fortunately, as [[#Summary_.28n.3D5.29|we have shown in an accompanying discussion]], <math>\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, as well as <math>\mathfrak{f}_M</math>, can be determined by direct integration in this single case.

===Related Discussions===
* See [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Model_Sequences|our plot of, what Kimura (1981b) would refer to as, several <math>M_1</math> sequences]]

==First Detailed Example (n = 5)==

Here we complete these integrals to derive detailed expressions for the above subset of structural form factors in the case of spherically symmetric configurations that obey an <math>~n=5</math> polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should simplify the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases that cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|cores of bipolytropes]].

===Foundation (n = 5)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=5; ~\gamma=6/5)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{K^{10}}{G^{9} M_\mathrm{tot}^{6}} \biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl( \frac{K}{G} \biggr)^{15/2} M_\mathrm{tot}^{-5} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{K^5}{G^3} \biggr)^{1/2} M_\mathrm{tot}^{-1} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/BiPolytropes/Analytic5_1#Profile|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,
<div align="center">
<math>
a_{5} =\biggr[ \frac{3K}{2\pi G} \biggr]^{1/2} \rho_0^{-2/5} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{[K^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^{6/5}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{[K^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5}
~~~~\Rightarrow ~~~~
\rho_0^{1/5} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} M_\mathrm{tot}^{-1} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr) \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi
= \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3^4}{\pi} \biggr)^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr)^{1/2}
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
= \biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2}
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 5)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math>, and recognizing that,
<div align="center">
<math>dr^\dagger = \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} d\xi \, ,</math>
</div>
gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], [[Image:OnlineIntegral01.png|250px|right|Mathematica Integral]]
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{3/2} \biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2}
\int_0^{\xi} \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{1}{3} \biggr)^{3/2}
\biggl[ \frac{\xi^3}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-3/2} \biggr]_0^{\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\xi^2}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=5</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 5)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5} \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=5</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_5 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_5 \xi}{a_5 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=5</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{3/2} \biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[SSCpt1/Virial/FormFactors#NormalizedProfiles|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2} \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 5)===

From the above line of reasoning we appreciate that, for any spherically symmetric configuration, the ratio of the configuration's mean density to its central density can be obtained by setting the upper limit of our just-completed "Mass2" integration to <math>x=1</math>. That is to say, quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>
</table>
</div>
But the integral expression on the righthand side of this relation is also the definition of the structural form factor, <math>~\mathfrak{f}_M</math>, given at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]]. Hence, we can say, quite generally, that,
<div align="center">
<math>\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} \, .</math>
</div>
And, given that we have just completed this integral for the case of truncated <math>n=5</math> polytropic structures, we can state, specifically, that,
<div align="center">
<math>\mathfrak{f}_M\biggr|_{n=5} = \biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</div>

===Gravitational Potential Energy (n = 5)===
As presented at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral02.png|225px|right|Mathematica Integral]]Given that an expression for the normalized density profile, <math>\rho(x)/\rho_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been determined, above]], we can carry out the nested pair of integrals immediately. Indeed, the integral contained inside of the curly braces has already been completed [[SSCpt1/Virial/FormFactors#Mass2|in the "Mass2" subsection, above]], in order to determine the radial mass profile. Specifically, we have already determined that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ \int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\}
\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} x dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-4} x^4 dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \biggl( \frac{\tilde\xi^2}{3}\biggr)^{-5/2} \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-3}
\biggl\{
\biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggl[ 3\biggl( \frac{\tilde\xi^2}{3}\biggr)^2 - 8\biggl( \frac{\tilde\xi^2}{3}\biggr) - 3 \biggr]
+ 3\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr)^3\tan^{-1}\biggl[ \biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>\mathfrak{f}_M</math> and <math>\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>n=5</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{\pi}{2^3\cdot 3^7}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
In order to simplify typing, we will switch to the variable,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
in which case a summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>( 1 + \ell^2 )^{-3/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \cdot \ell^{-5} (1 + \ell^2)^{-3}
\biggl\{ \ell [ 3\ell^4 - 8\ell^2 - 3 ] + 3( 1 + \ell^2 )^3\tan^{-1}(\ell ) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{3}{5} \biggl(\frac{2^3\cdot 3^6}{\pi}\biggr)^{1/2} \frac{1}{\ell} \cdot (1 + \ell^2)^3
\mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5 \pi}\biggr)^{1/2} \cdot \ell^{-6} (1 + \ell^2)^3
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized gravitational potential energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>W_\mathrm{core}^*</math>.

Hence, also, as defined in the [[SSCpt1/Virial#Gathering_it_All_Together|accompanying introductory discussion]], the constant, <math>\mathcal{A}</math>, that appears in our general free-energy equation is (for <math>n=5</math> polytropic configurations),
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 5)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral03.png|225px|right|Mathematica Integral]]Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} x^2 dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{2^3}
\biggl\{\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-3/2} \tan^{-1}\biggl[ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
+ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-1}
- 2\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-2}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=5 (\gamma=6/5)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)^{1/5}
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{6/5}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>

If, as above, we adopt the simplifying variable notation,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
the various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} [ \ell^{-3} \tan^{-1}(\ell ) + \ell^{-2}(1+\ell^2)^{-1} - 2\ell^{-2}(1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, ;
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3/5} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2} \cdot \biggl[ \biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \biggr]^{-3/5}
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5} \ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{3^{10}}{2^{10}} \biggl(\frac{2^9\cdot 3^{18}}{\pi^3}\biggr)
\biggl(\frac{3^{12}}{2^{34} \pi^2} \biggr) \biggr]^{1/10} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^{8}}{2^{7}\pi}\biggr)^{1/2} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized thermal energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>S_\mathrm{core}^*</math>. Its similarity to the expression for the gravitational potential energy — which is relevant to the virial theorem — is more apparent if it is rewritten in the following form:
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2}
\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 5)===
In summary, for <math>n=5</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 1 + \ell^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 5)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} \biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Check (n = 5)===

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}\biggl\{ [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
-
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}
\biggl[\frac{8}{3}\ell^3 (1+\ell^2)^{-3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2 \cdot 3^{6}}{\pi}\biggr)^{1/2}
\biggl[\frac{\ell}{ (1+\ell^2)} \biggr]^3 \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=5</math> polytropes — see the [[#Mass1|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \biggl[ \ell^2(1+\ell^2)^{-1} \biggr]^{3/2}
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_5 = ( 1 + \ell^2 )^{-1/2}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_5}{d\xi} \biggr|_{\xi_e} = 3^{1/2} \ell ( 1 + \ell^2 )^{-3/2} \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2\cdot 3^3}{\pi}\biggr]^{1/2} ( 1 + \ell^2 )^{-3} (3^{1/2}\ell)^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{2\cdot 3^6}{\pi}\biggr)^{1/2} \biggl[ \frac{\ell}{( 1 + \ell^2 )} \biggr]^{3} \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors. This gives us confidence that our form-factor expressions are correct, at least in the case of embedded <math>n=5</math> polytropic structures.

==Second Detailed Example (n = 1)==

===Foundation (n = 1)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=1; ~\gamma=2)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl(\frac{K}{G}\biggr)^{1/2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G^3 M_\mathrm{tot}^2}{K^2}\biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot}^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi}\biggl( \frac{G}{K} \biggr)^{3/2} M_\mathrm{tot} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of the properties of ''isolated'' polytropes]], in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{1}</math>, where,
<div align="center">
<math>
a_{1} = \biggl( \frac{K}{2\pi G} \biggr)^{1/2} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{(K/G)^{1/2}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^{2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{(K/G)^{3/2}\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2}{\pi} \biggr)^{1/2} (\sin\xi - \xi \cos\xi) \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=1</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0
~~~~\Rightarrow ~~~~
\rho_0 = \biggr[ \frac{G^3}{2\pi K^3} \biggr]^{1/2} M_\mathrm{tot} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles1">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3 \pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\pi} \biggl( \frac{\sin\xi}{\xi}\biggr)^2 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi \cos\xi)
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 1)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math> gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \int_0^{\xi} \frac{\xi^2}{2\pi} \biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \cdot \frac{d\xi }{(2\pi)^{1/2}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3( 2\pi)^{-3/2}\biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \int_0^{\xi} \xi \sin\xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi\cos\xi) \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=1</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 1)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0 \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=1</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1 = \pi</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_1 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_1 \xi}{a_1 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx1">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin(\tilde\xi x)}{\tilde\xi x} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi} \int_0^{x} x \sin(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{3}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, . </math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=1</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
\biggl\{ \frac{[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ]}{( \sin \tilde\xi - \tilde\xi \cos \tilde\xi )} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] </math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[#NormalizedProfiles1|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\xi - \xi \cos\xi ) \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 1)===

[[SSCpt1/Virial/FormFactors#Mean-to-Central_Density|Following the line of reasoning provided above]], we can use the just-derived central-to-mean density ratio to specify one of the structural form factors. Specifically,
<div align="center">
<math>~\mathfrak{f}_M\biggr|_{n=1} = \frac{\bar\rho}{\rho_c} = \frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</div>

===Gravitational Potential Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
From the derivations already presented, above, for <math>~n=1</math> polytropic configurations, we know all of the functions under this integral. We know, for example, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \int_0^1 \biggl\{ [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \biggr\}
\sin(\tilde\xi x) dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \biggl\{ \frac{x}{2} - \frac{\sin(2\tilde\xi x)}{4\tilde\xi} - \tilde\xi\biggl[
\frac{\sin(2\tilde\xi x) - 2\tilde\xi x\cos(2\tilde\xi x)}{8\tilde\xi^2}
\biggr] \biggr\}_0^1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>~n=1</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- 3\mathcal{A} \chi^{-1} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\chi</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
A summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{5} \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^{2} \mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[#Foundation_2|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi}\biggr)^{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\sin(\tilde\xi x)}{(\tilde\xi x)}\biggr]^{2} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2} \int_0^1 \sin^2(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2}
\biggl[\frac{x}{2}- \frac{\sin(2\tilde\xi x)}{4\tilde\xi} \biggr]_0^1
= \frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=1 (\gamma=2)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>
The various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\xi^3}{3\pi} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr) \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^2
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3 \tilde\xi^3}{2^5\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
\biggl[ (2\pi)^{3/2} \tilde\xi^{-3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 1)===
In summary, for <math>~n=1</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 1)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Checks (n = 1)===
====Expectation from Stahler's Equilibrium Models====
If we add twice the thermal energy to the gravitational potential energy, we obtain,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} \biggl\{ \biggl[6\tilde\xi - 3\sin(2\tilde\xi ) \biggr]
- \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} 2\tilde\xi\biggl\{ 1-\cos(2\tilde\xi ) \biggr\}
= \biggl( \frac{1}{2\pi^3} \biggr)^{1/2} \tilde\xi \sin^2(\tilde\xi ) \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=1</math> polytropes — see the [[#Mass1_.28n_.3D_1.29|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D1.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_1 = \frac{\sin\tilde\xi}{\tilde\xi}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_1}{d\xi} \biggr|_{\tilde\xi} = \frac{1}{\tilde\xi^2}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2}{\pi} \biggr]^{-1/2} \biggl[ \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \tilde\xi^2(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)^{-1}\biggr]^{2}
\biggl( \frac{\sin\tilde\xi}{\tilde\xi}\biggr)^2 \tilde\xi^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi^3}\biggr)^{1/2} \tilde\xi \sin^2\tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors, giving us additional confidence that our form-factor expressions are correct.

====Compare With General Expressions Based on VH74 Work====
Based on the general expressions [[#PTtable|derived above]] in the context of VH74's work, for the specific case of <math>n=1</math> polytropic configurations, the three structural form factor should be,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\tilde\xi^2}
\biggl[\tilde\theta^{2} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ 3\tilde\theta^{2} +
3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
Also, remember that,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\theta</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~\theta^' \equiv \frac{d\theta}{d\xi}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\cos\xi}{\xi} - \frac{\sin\xi}{\xi^2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~(\theta^' )^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\xi^4} \biggl[ \xi\cos\xi - \sin\xi \biggr]^2
=\frac{1}{\xi^4} \biggl[ \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr] \, .
</math>
</td>
</tr>
</table>
Now, let's look at the structural form factors, one at a time. First, we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\xi^3} \biggl[\sin\xi - \xi\cos\xi \biggr]</math>
</td>
</tr>
</table>

which matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Next,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~ \frac{3\cdot 5}{4\xi^2} \biggl[ \frac{\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 + 2\xi^2\cos^2\xi - 3 \xi\sin\xi \cos\xi \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl\{ \xi^2 + \xi^2[1+\cos(2\xi)] - \frac{3}{2} \xi\sin(2\xi) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{8\xi^6} \biggl[ 4\xi^2 + 2\xi^2 \cos(2\xi) - 3 \xi\sin(2\xi) \biggr] \, ,
</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Finally,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ \frac{3\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi \biggr)
+ 3\xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2\xi^4} \biggl[ \xi^2 - \xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2^2\xi^3} \biggl[ 2\xi - \sin(2\xi) \biggr] \, ,</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. So this adds support to the deduction, above, that VH74 have provided us with the information necessary to develop general expressions for the three structural form factors.

==Fiddling Around==
NOTE (from Tohline on 17 March 2015): Chronologically, this "Fiddling Around" subsection was developed before our discovery of the VH74 derivations. It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres. But this subsection's conclusions are superseded by the VH74 work.

In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>. In the case of <math>n=1</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\sin\xi}{\xi}\biggr)^2
= \frac{1}{2\xi^2} \biggl[ 1 - \cos(2\xi) \biggr]
= \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\mathfrak{f}_A - \theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[6\xi - 3\sin(2\xi ) \biggr]
- \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[4\xi - 3\sin(2\xi ) + 2\xi\cos(2\xi) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^3 \xi^5}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ 4\xi - 3\sin(2\xi) + 2\xi \cos(2\xi ) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2 \xi^2}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

Similarly, in the case of <math>n = 5</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\biggl( \frac{2^3}{3} \ell^{3} \biggr) \biggl[ \mathfrak{f}_A - \theta^{n+1} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
[ \tan^{-1}(\ell ) + \ell (\ell^4-1) (1+\ell^2)^{-3} ] - \biggl( \frac{2^3}{3} \ell^{3} \biggr) (1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\tan^{-1}(\ell ) + \ell \biggl(\ell^4-\frac{8}{3}\ell^2 - 1 \biggr) (1+\ell^2)^{-3} \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^4}{5} \cdot \ell^{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2\cdot 3}{5} \cdot \ell^{2} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{2\xi^2}{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>
This is pretty amazing! Both examples produce almost exactly the same relationship between the two structural form factors, <math>\mathfrak{f}_A</math> and <math>\mathfrak{f}_W</math>. I think that we are well on our way toward nailing down the generic, analytic relationship and, in turn, a generally applicable mass-radius relationship for pressure-truncated polytropic configurations.

Okay … here is the final piece of information. In the case of isolated polytropes, we know that the correct expressions for the structural form factors are as summarized in the following table:
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi}
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>
</div>
We notice, from this, that the ratio,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} \cdot \frac{5-n}{3^2\cdot 5} \biggl[ \frac{\xi}{\Theta^'} \biggr]^{2}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(n+1)\tilde\xi^2 }{3\cdot 5} \, .
</math>
</td>
</tr>
</table>
Even in the case of the two pressure-truncated polytropes, analyzed above, this ratio proves to give the correct prefactor on <math>\mathfrak{f}_W</math>. So we ''suspect'' that the universal relationship between the two form factors is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{(n+1) \xi^2 }{3\cdot 5} \biggr] \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

=See Also=
<ul>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>

{{ SGFfooter }}

SSC/Structure/IsothermalSphere

2021-07-28T17:23:46Z

174.64.8.247: /* Isothermal Sphere */

__FORCETOC__


=Isothermal Sphere=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Isothermal<br />Sphere</b>]]</font>
|}
Here we supplement the [[SSCpt1/PGE|simplified set of principal governing equations]] with an isothermal equation of state, that is, {{Math/VAR_Pressure01}} is related to {{Math/VAR_Density01}} through the relation,
<div align="center">
<math>P = c_s^2 \rho \, ,</math>
</div>
where, <math>~c_s</math> is the isothermal sound speed. Comparing this {{Math/VAR_Pressure01}}-{{Math/VAR_Density01}} relationship to
 <br />
 <br />
 <br />
 <br />

<div align="center">
<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br />
of the Ideal Gas Equation of State,

{{Math/EQ_EOSideal0A}}
</div>
we see that,
<div align="center">
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
</div>
where, {{Math/C_GasConstant}}, {{Math/C_BoltzmannConstant}}, {{Math/C_AtomicMassUnit}}, and {{Math/MP_MeanMolecularWeight}} are all defined in the accompanying [[Appendix/VariablesTemplates|variables appendix]]. It will be useful to note that, for an isothermal gas, the enthalpy, {{Math/VAR_Enthalpy01}}, is related to {{Math/VAR_Density01}} via the expression,

<div align="center">
<math>
dH = \frac{dP}{\rho} = c_s^2 d\ln\rho \, .
</math>
</div>

==Governing Relations==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following second-order ODE relating the two unknown functions, {{Math/VAR_Density01}} and {{Math/VAR_Enthalpy01}}:

<div align="center">
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \rho</math> .
</div>

Using the {{Math/VAR_Enthalpy01}}-{{Math/VAR_Density01}} relationship for an isothermal gas presented above, this can be rewritten entirely in terms of the density as,

<div align="center">
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =- \frac{4\pi G}{c_s^2} \rho \, ,</math>
</div>

<span id="keyExpression">or, equivalently,</span>
<div align="center">
<math>
\frac{d^2\ln\rho}{dr^2} +\frac{2}{r} \frac{d\ln\rho}{dr} + \beta^2 \rho = 0 \, ,
</math>
</div>

where,

<div align="center">
<math>
\beta^2 \equiv \frac{4\pi G}{c_s^2} \, .
</math>
</div>

This matches the governing ODE whose derivation was published on p. 131 of [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Robert Emden's (1907) book titled, ''Gaskugeln''].

<div align="center">
<table border="1" cellpadding="4">
<tr>
<td colspan="2" align="center">
Derivation by [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (edited)
</td>
</tr>
<tr>
<td align="center" bgcolor="black">
[[File:EmdenBookCover1907.jpg|240px|center|Emden (1907)]]
</td>
<td align="center">
[[File:EmdenIsothermalDerivation.jpg|500px|center|Emden (1907)]]
</td>
</tr>
<tr>
<td align="center" colspan="2">
<font size="-1">
Note that, in Emden's derivation, <math>H</math> is not enthalpy but, rather, the effective gas constant, <math>H = c_s^2/T</math>.
</font>
</td>
</tr>
</table>
</div>

By adopting the following dimensionless variables,

<div align="center">
<math>
\mathfrak{r}_1 \equiv \rho_c^{1/2} \beta r \, , ~~~~\mathrm{and}~~~~v_1 \equiv \ln(\rho/\rho_c) \, ,
</math>
</div>

where <math>~\rho_c</math> is the configuration's central density, the governing ODE can be rewritten in dimensionless form as,

<div align="center">
<math>
\frac{d^2v_1}{d\mathfrak{r}_1^2} +\frac{2}{\mathfrak{r}_1} \frac{dv_1}{d\mathfrak{r}_1} + e^{v_1} = 0 \, ,
</math>
</div>

which is exactly the equation numbered (II"a) that can be found on p. 133 of [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)].
Emden numerically determined the behavior of the function <math>~v_1(\mathfrak{r}_1)</math>, its first derivative with respect to <math>~\mathfrak{r}_1</math>, <math>~v_1'</math>, along with <math>~e^{v_1}</math> and several other useful products, and published his results as Table 14, on p. 135 of his book. This table has been reproduced [[#Emden.27s_Numerical_Solution|immediately below]], primarily for historical purposes.

Note that a somewhat more extensive tabulation of the structural properties of isothermal spheres is provided by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar & Wares (1949, ApJ, 109, 551)]. In this published work as well as in §22 of Chapter IV in [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chandrasekhar has written the governing ODE in a form that we will refer to as the,
<div align="center" id="Chandrasekhar">
<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font><br />
{{ Math/EQ_SSLaneEmden02 }}
</div>
It is straightforward to show that this is identical to Emden's governing expression after making the variable substitutions:
<div align="center">
<math>~\mathfrak{r}_1 \rightarrow \xi</math>        and         <math>~v_1 \rightarrow -\psi </math>.
</div>
Across the astrophysics community, Chadrasekhar's notation has been widely — although not universally — adopted as the standard.

==Emden's Numerical Solution==
<div align="center">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<td align="center" valign="top" rowspan="1">
[[File:EmdenTable14.jpg|600px|center|Emden's (1907) Table 14]]
</td>
</tr>
<tr>
<td align="center">
<font size="-1">
Note: The entry highlighted in blue in the <math>3^\mathrm{rd}</math> column must be a typesetting error.
</font>
</td>
</tr>
<tr>
<td align="left">
<font size="-1">
A more recent and more extensive tabulation of the structural properties of isothermal spheres is provided by:
* [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar & Wares (1949, ApJ, 109, 551)]:   ''The Isothermal Function''
* [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H G. P. Horedt (1986, Astrophys. & Space Science, 126, 357-408)]:  ''Seven-Digit Tables of Lane-Emden Functions'' — See, in particular, pp. 405-406 (Sphere of polytropic index <math>~n = \pm \infty</math>).

An analytic — but ''approximate'' — solution to the isothermal Lane-Emden equation can be found:
* [http://adsabs.harvard.edu/abs/1996MNRAS.281.1197L F. K. Liu (1996, MNRAS, 281, 1197-1205)]:   ''Polytropic Gas Spheres: An Approximate Analytic Solution of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1997MNRAS.286..268N Priyamvada Natarajan & Donald Lynden-Bell (1997, MNRAS, 286, 268-270)]:   '' An Analytic Approximation to the Isothermal Sphere''
* [http://adsabs.harvard.edu/abs/2013RMxAA..49...63R A. C. Raga, J. C. Rodríguez-Ramírez, M. Villasante, A. Rodríguez-González, & V. Lora (2013, Revista Mexicana de Astronomía y Astrofísica, 49, 63-69)]:   ''A New Analytic Approximation to the Isothermal, Self-Gravitating Sphere''

</font>
</td>
</tr>
</table>
</div>

A plot of <math>v_1</math> versus <math>\ln\mathfrak{r}_1</math>, as shown below in Figure 1a, translates into a log-log plot of the equilibrium configuration's <math>~\rho(r)</math> density profile. Notice that this isolated isothermal configuration extends to infinity and that, at large radii, the density profile displays a simple power-law behavior — specifically, <math>~ \rho \propto r^{-2}</math>. This is consistent with our general discussion, [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|presented elsewhere]], of power-law density distributions as solutions of the Lane-Emden equation.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2">
'''Figure 1: Emden's Numerical Solution'''
</td>
</tr>
<tr>
<td align="center" valign="top">
[[File:IsothermalDensityPlot.jpg|350px|center|Plotted from Emden's (1907) tabulated data]]
</td>
<td align="center" valign="top">
[[File:EmdenMassProfile.jpg|350px|center|Plotted from Emden's (1907) tabulated data]]
</td>
</tr>
<tr>
<td valign="top">
(a) The <math>~(x,y)</math> locations of the data points plotted in blue are drawn directly from column 1 and column 3 of Emden's Table 14 — specifically, <math>~x = \ln(\mathfrak{r}_1)</math> and <math>~y = v_1</math>. The dashed red line has a slope of <math>~-2</math> and serves to illustrate that, at large radii, the [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|isothermal density profile tends toward a <math>~\rho \propto r^{-2}</math> distribution]].
</td>
<td valign="top">
(b) The <math>~(x,y)</math> locations of the data points plotted in purple are drawn directly from column 1 and column 7 of Emden's Table 14 — specifically, <math>~x = \ln(\mathfrak{r}_1)</math> and <math>~y = \mathfrak{r}_1^2 v_1'</math>. The dashed green line has a slope of <math>~+1</math> and serves to illustrate that, at large radii, the isothermal <math>~M(r)</math> distribution tends toward a <math>~M_r \propto r</math> distribution.
</td>
</tr>
</table>
</div>

==Mass Profile==
The mass enclosed within a given radius, <math>~M_r</math>, can be determined by performing an appropriate volume-weighted integral over the density distribution. Specifically, based on the key expression for,

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Mass Conservation'''</font></span><br />

{{Math/EQ_SSmassConservation01}}
</div>
in spherically symmetric configurations, the relevant integral is,
<div align="center">
<math>
~M_r = \int_0^r 4\pi r^2 \rho(r) dr \, .
</math>
</div>
But <math>~M_r</math> also can be determined from the information provided in column 7 of Emden's Table 14 — that is, from knowledge of the first derivative of <math>~v_1</math>. The appropriate expression can be obtained from the mathematical prescription for

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

{{Math/EQ_SShydrostaticBalance01}}
</div>
in a spherically symmetric configuration. Since, for an isothermal equation of state (see above),

<div align="center">
<math>
~\frac{dP}{\rho} = c_s^2 {d\ln\rho} \, ,
</math>
</div>

the statement of hydrostatic balance can be rewritten as,

<div align="center">
<math>
~M_r = \frac{c_s^2}{G} \biggl[ - r^2 \frac{d\ln\rho}{dr} \biggr] = \frac{c_s^2}{G \rho_c^{1/2} \beta} \biggl[ - \mathfrak{r}_1^2 \frac{dv_1}{d\mathfrak{r}_1} \biggr]
= \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ - \mathfrak{r}_1^2 v_1' \biggr] \, .
</math>
</div>

The quantity tabulated in column 7 of Emden's Table 14 is precisely the dimensionless term inside the square brackets of this last expression; having units of mass, the coefficient out front sets the mass scale of the equilibrium configuration and depends only on the choice of central density and isothermal sound speed. Hence, a plot of <math>~\ln(\mathfrak{r}_1^2 v_1')</math> versus <math>~\ln\mathfrak{r}_1</math>, as shown above in Figure 1b, translates into a log-log plot of the equilibrium configuration's <math>~M_r</math> mass profile. Notice that, along with the radius, the mass of this isolated isothermal configuration extends to infinity and that, at large radii, the mass profile displays a simple power-law behavior — specifically, <math> ~M_r \propto r^{+1}</math>.

As was realized independently by [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert (1955)] and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)], a spherically symmetric isothermal equilibrium configuration of finite radius and finite mass can be constructed if the system is embedded in a pressure-confining external medium. We discuss their findings [[SSC/Structure/BonnorEbert|elsewhere]].

<font color="darkblue">

==Summary==
</font>

Based on the above derivations, the internal structural properties of an equilibrium isothermal sphere can be described in terms of the tabulated quantities provided in Emden's Table 14 as follows:
* <font color="red">Radial Coordinate Position</font>:
: Given the isothermal sound speed, <math>c_s</math>, and the central density, <math>\rho_c</math>, the radial coordinate is,
<div align="center">
<math>r = ( \rho_c \beta^2 )^{-1/2} \mathfrak{r}_1 = \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \mathfrak{r}_1 </math> .
</div>

* <font color="red">Density & Pressure</font>:
: As a function of the radial coordinate, <math>r(\mathfrak{r}_1)</math>, the density profile is,
<div align="center">
<math>\rho(r(\mathfrak{r}_1))= \rho_c e^{v_1(\mathfrak{r}_1)}</math>;
</div>
: and the pressure profile is,
<div align="center">
<math>P(r(\mathfrak{r}_1))= (c_s^2 \rho_c) e^{v_1(\mathfrak{r}_1)}</math>.
</div>
: As has been explicitly pointed out in the above discussion associated with Figure 1a, the density profile — and, hence, also the pressure profile — extends to infinity and, at large radii, behaves as a power law; specifically, <math>\rho \propto r^{-2}</math>.

* <font color="red">Mass</font>:
: Given <math>c_s</math> and <math>\rho_c</math>, the natural mass scale is,
<div align="center">
<math>M_0 \equiv \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} </math> ;
</div>
: and, expressed in terms of <math>M_0</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>
M_r = M_0 [ - \mathfrak{r}_1^2 v_1' ] \, .
</math>
</div>
: As discussed above in the context of Figure 1b, at large radii, the mass increases linearly with <math>r</math>. Because the density and pressure profiles extend to infinity, this means that the mass of an isolated isothermal sphere is infinite.

* <font color="red">Enthalpy & Gravitational Potential</font>:
: To within an additive constant, the enthalpy distribution is,
<div align="center">
<math>H(r(\mathfrak{r}_1))= c_s^2 [- v_1(\mathfrak{r}_1)]</math>;
</div>
: and the gravitational potential is,
<div align="center">
<math>\Phi(r(\mathfrak{r}_1)) = - H(r(\mathfrak{r}_1))= c_s^2 v_1(\mathfrak{r}_1)</math>.
</div>

* <font color="red">Mean-to-Local Density Ratio</font>:
: The ratio of the configuration's mean density, inside a given radius, to its local density at that radius is,
<div align="center">
<math>\frac{\bar{\rho}}{\rho} = \frac{3M_r}{4\pi r^3 \rho} = 3\biggl[- \frac{v_1'}{\mathfrak{r}_1 e^{v_1}} \biggr] </math> .
</div>
: As Figure 2 shows, at large <math>r</math> this density ratio goes to the value of 3, which means that the term inside the square brackets goes to unity at large <math>r</math>. This behavior is consistent with the limiting power-law behavior of both <math>M_r</math> and <math>\rho</math>, discussed above.

<div align="center">
<table border="0" cellpadding="5" width="360">
<tr>
<td align="center">
'''Figure 2: From Emden's Tabulated Data'''
</td>
</tr>
<tr>
<td align="center">
[[File:PlotMeanToLocalDensity.jpg|350px|center|Plot based on data from Emden's (1907) Table 14]]
</td>
</tr>
<tr>
<td align="left">
The blue curve displays an evaluation of the density ratio, <math>[- 3v_1'/(\mathfrak{r}_1 e^{v_1}) ]</math>, as a function of <math>\ln (\mathfrak{r}_1)</math>, as determined from the data presented in Emden's Table 14, shown above.
</td>
</tr>
</table>
</div>

=Our Numerical Integration=
The [[#keyExpression|above governing relation]] — see especially [[#Chandrasekhar|Chandrasekhar's notation]] — may be rewritten as (see also, for example, §19.8, eq. 19.35 of [<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>]),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e^{-w} \, ,</math>
</td>
</tr>
</table>
</div>
where we appreciate that,
<div align="center">
<math>~w \equiv \ln\biggl(\frac{\rho}{\rho_c}\biggr) \, .</math>
</div>

We'll adopt the following finite-difference approximations for the first and second derivatives on a grid of radial spacing, <math>~\Delta_r</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_i'</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{w_+ - w_-}{2\Delta_r}</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_i''</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{w_+ - 2w_i +w_-}{\Delta_r^2} \, .</math>
</td>
</tr>
</table>
</div>
Our finite-difference approximation of the governing equation is, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_i \biggl[ \frac{w_+ - 2w_i +w_-}{\Delta_r^2} \biggr] + 2\biggl[ \frac{w_+ - w_-}{2\Delta_r} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_i e^{-w_i} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ r_i [ w_+ - 2w_i +w_- ] + \Delta_r [ w_+ - w_- ]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Delta_r^2 r_i e^{-w_i} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~w_+
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\Delta_r^2 r_i e^{-w_i} + 2r_i w_i + w_- (\Delta_r - r_i)}{( \Delta_r + r_i)} \, .</math>
</td>
</tr>
</table>
</div>

Now, for the first two steps away from the center — where, <math>~w_i = w_0 = 0</math> and <math>~r_i = r_0 = 0</math> — we will use the following [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalLaneEmden|power-series expansion]] (see, for example, eq. 377 from §22 in Chapter IV of [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>]) to determine the value of <math>~w_i</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_1
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\Delta_r^2}{6} - \frac{\Delta_r^4}{120} + \frac{\Delta_r^6}{1890} \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(2\Delta_r)^2}{6} - \frac{(2\Delta_r)^4}{120} + \frac{(2\Delta_r)^6}{1890} \, .</math>
</td>
</tr>
</table>
</div>

=Related Discussions=
==Journal Articles==
* [http://adsabs.harvard.edu/abs/1870AmJS...50...57L J. H. Lane (1870)], '''American Journal of Science''', ''On the Theoretical Temperature of the Sun''
* [https://archive.org/details/mobot31753002152772/page/56 J. H. Lane (1870)], '''The American Journal of Science and Arts''', Vol. 50, pp. 57 - 74: ''On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining Its Volume by Its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment''

==Wikipedia==
* [https://en.wikipedia.org/wiki/Robert_Emden Robert Emden]
* [https://en.wikipedia.org/wiki/Emden–Chandrasekhar_equation Emden-Chandrasekhar equation]
* [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane Jonathan Home Lane]
* [https://en.wikipedia.org/wiki/Lane–Emden_equation Lane-Emden equation]

{{ SGFfooter }}

SSC/Structure/IsothermalSphere

2021-07-28T17:17:42Z

174.64.8.247: /* Isothermal Sphere */

__FORCETOC__


=Isothermal Sphere=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Isothermal<br />Sphere</b>]]</font>
|}
Here we supplement the [[SSCpt1/PGE|simplified set of principal governing equations]] with an isothermal equation of state, that is, {{Math/VAR_Pressure01}} is related to {{Math/VAR_Density01}} through the relation,
<div align="center">
<math>P = c_s^2 \rho \, ,</math>
</div>
where, <math>~c_s</math> is the isothermal sound speed. Comparing this {{Math/VAR_Pressure01}}-{{Math/VAR_Density01}} relationship to
 <br />
 <br />
 <br />

<div align="center">
<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br />
of the Ideal Gas Equation of State,

{{Math/EQ_EOSideal0A}}
</div>
we see that,
<div align="center">
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
</div>
where, {{Math/C_GasConstant}}, {{Math/C_BoltzmannConstant}}, {{Math/C_AtomicMassUnit}}, and {{Math/MP_MeanMolecularWeight}} are all defined in the accompanying [[Appendix/VariablesTemplates|variables appendix]]. It will be useful to note that, for an isothermal gas, the enthalpy, {{Math/VAR_Enthalpy01}}, is related to {{Math/VAR_Density01}} via the expression,

<div align="center">
<math>
dH = \frac{dP}{\rho} = c_s^2 d\ln\rho \, .
</math>
</div>

==Governing Relations==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following second-order ODE relating the two unknown functions, {{Math/VAR_Density01}} and {{Math/VAR_Enthalpy01}}:

<div align="center">
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \rho</math> .
</div>

Using the {{Math/VAR_Enthalpy01}}-{{Math/VAR_Density01}} relationship for an isothermal gas presented above, this can be rewritten entirely in terms of the density as,

<div align="center">
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =- \frac{4\pi G}{c_s^2} \rho \, ,</math>
</div>

<span id="keyExpression">or, equivalently,</span>
<div align="center">
<math>
\frac{d^2\ln\rho}{dr^2} +\frac{2}{r} \frac{d\ln\rho}{dr} + \beta^2 \rho = 0 \, ,
</math>
</div>

where,

<div align="center">
<math>
\beta^2 \equiv \frac{4\pi G}{c_s^2} \, .
</math>
</div>

This matches the governing ODE whose derivation was published on p. 131 of [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Robert Emden's (1907) book titled, ''Gaskugeln''].

<div align="center">
<table border="1" cellpadding="4">
<tr>
<td colspan="2" align="center">
Derivation by [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (edited)
</td>
</tr>
<tr>
<td align="center" bgcolor="black">
[[File:EmdenBookCover1907.jpg|240px|center|Emden (1907)]]
</td>
<td align="center">
[[File:EmdenIsothermalDerivation.jpg|500px|center|Emden (1907)]]
</td>
</tr>
<tr>
<td align="center" colspan="2">
<font size="-1">
Note that, in Emden's derivation, <math>H</math> is not enthalpy but, rather, the effective gas constant, <math>H = c_s^2/T</math>.
</font>
</td>
</tr>
</table>
</div>

By adopting the following dimensionless variables,

<div align="center">
<math>
\mathfrak{r}_1 \equiv \rho_c^{1/2} \beta r \, , ~~~~\mathrm{and}~~~~v_1 \equiv \ln(\rho/\rho_c) \, ,
</math>
</div>

where <math>~\rho_c</math> is the configuration's central density, the governing ODE can be rewritten in dimensionless form as,

<div align="center">
<math>
\frac{d^2v_1}{d\mathfrak{r}_1^2} +\frac{2}{\mathfrak{r}_1} \frac{dv_1}{d\mathfrak{r}_1} + e^{v_1} = 0 \, ,
</math>
</div>

which is exactly the equation numbered (II"a) that can be found on p. 133 of [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)].
Emden numerically determined the behavior of the function <math>~v_1(\mathfrak{r}_1)</math>, its first derivative with respect to <math>~\mathfrak{r}_1</math>, <math>~v_1'</math>, along with <math>~e^{v_1}</math> and several other useful products, and published his results as Table 14, on p. 135 of his book. This table has been reproduced [[#Emden.27s_Numerical_Solution|immediately below]], primarily for historical purposes.

Note that a somewhat more extensive tabulation of the structural properties of isothermal spheres is provided by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar & Wares (1949, ApJ, 109, 551)]. In this published work as well as in §22 of Chapter IV in [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chandrasekhar has written the governing ODE in a form that we will refer to as the,
<div align="center" id="Chandrasekhar">
<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font><br />
{{ Math/EQ_SSLaneEmden02 }}
</div>
It is straightforward to show that this is identical to Emden's governing expression after making the variable substitutions:
<div align="center">
<math>~\mathfrak{r}_1 \rightarrow \xi</math>        and         <math>~v_1 \rightarrow -\psi </math>.
</div>
Across the astrophysics community, Chadrasekhar's notation has been widely — although not universally — adopted as the standard.

==Emden's Numerical Solution==
<div align="center">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<td align="center" valign="top" rowspan="1">
[[File:EmdenTable14.jpg|600px|center|Emden's (1907) Table 14]]
</td>
</tr>
<tr>
<td align="center">
<font size="-1">
Note: The entry highlighted in blue in the <math>3^\mathrm{rd}</math> column must be a typesetting error.
</font>
</td>
</tr>
<tr>
<td align="left">
<font size="-1">
A more recent and more extensive tabulation of the structural properties of isothermal spheres is provided by:
* [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar & Wares (1949, ApJ, 109, 551)]:   ''The Isothermal Function''
* [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H G. P. Horedt (1986, Astrophys. & Space Science, 126, 357-408)]:  ''Seven-Digit Tables of Lane-Emden Functions'' — See, in particular, pp. 405-406 (Sphere of polytropic index <math>~n = \pm \infty</math>).

An analytic — but ''approximate'' — solution to the isothermal Lane-Emden equation can be found:
* [http://adsabs.harvard.edu/abs/1996MNRAS.281.1197L F. K. Liu (1996, MNRAS, 281, 1197-1205)]:   ''Polytropic Gas Spheres: An Approximate Analytic Solution of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1997MNRAS.286..268N Priyamvada Natarajan & Donald Lynden-Bell (1997, MNRAS, 286, 268-270)]:   '' An Analytic Approximation to the Isothermal Sphere''
* [http://adsabs.harvard.edu/abs/2013RMxAA..49...63R A. C. Raga, J. C. Rodríguez-Ramírez, M. Villasante, A. Rodríguez-González, & V. Lora (2013, Revista Mexicana de Astronomía y Astrofísica, 49, 63-69)]:   ''A New Analytic Approximation to the Isothermal, Self-Gravitating Sphere''

</font>
</td>
</tr>
</table>
</div>

A plot of <math>v_1</math> versus <math>\ln\mathfrak{r}_1</math>, as shown below in Figure 1a, translates into a log-log plot of the equilibrium configuration's <math>~\rho(r)</math> density profile. Notice that this isolated isothermal configuration extends to infinity and that, at large radii, the density profile displays a simple power-law behavior — specifically, <math>~ \rho \propto r^{-2}</math>. This is consistent with our general discussion, [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|presented elsewhere]], of power-law density distributions as solutions of the Lane-Emden equation.

<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
<td align="center" colspan="2">
'''Figure 1: Emden's Numerical Solution'''
</td>
</tr>
<tr>
<td align="center" valign="top">
[[File:IsothermalDensityPlot.jpg|350px|center|Plotted from Emden's (1907) tabulated data]]
</td>
<td align="center" valign="top">
[[File:EmdenMassProfile.jpg|350px|center|Plotted from Emden's (1907) tabulated data]]
</td>
</tr>
<tr>
<td valign="top">
(a) The <math>~(x,y)</math> locations of the data points plotted in blue are drawn directly from column 1 and column 3 of Emden's Table 14 — specifically, <math>~x = \ln(\mathfrak{r}_1)</math> and <math>~y = v_1</math>. The dashed red line has a slope of <math>~-2</math> and serves to illustrate that, at large radii, the [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|isothermal density profile tends toward a <math>~\rho \propto r^{-2}</math> distribution]].
</td>
<td valign="top">
(b) The <math>~(x,y)</math> locations of the data points plotted in purple are drawn directly from column 1 and column 7 of Emden's Table 14 — specifically, <math>~x = \ln(\mathfrak{r}_1)</math> and <math>~y = \mathfrak{r}_1^2 v_1'</math>. The dashed green line has a slope of <math>~+1</math> and serves to illustrate that, at large radii, the isothermal <math>~M(r)</math> distribution tends toward a <math>~M_r \propto r</math> distribution.
</td>
</tr>
</table>
</div>

==Mass Profile==
The mass enclosed within a given radius, <math>~M_r</math>, can be determined by performing an appropriate volume-weighted integral over the density distribution. Specifically, based on the key expression for,

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Mass Conservation'''</font></span><br />

{{Math/EQ_SSmassConservation01}}
</div>
in spherically symmetric configurations, the relevant integral is,
<div align="center">
<math>
~M_r = \int_0^r 4\pi r^2 \rho(r) dr \, .
</math>
</div>
But <math>~M_r</math> also can be determined from the information provided in column 7 of Emden's Table 14 — that is, from knowledge of the first derivative of <math>~v_1</math>. The appropriate expression can be obtained from the mathematical prescription for

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

{{Math/EQ_SShydrostaticBalance01}}
</div>
in a spherically symmetric configuration. Since, for an isothermal equation of state (see above),

<div align="center">
<math>
~\frac{dP}{\rho} = c_s^2 {d\ln\rho} \, ,
</math>
</div>

the statement of hydrostatic balance can be rewritten as,

<div align="center">
<math>
~M_r = \frac{c_s^2}{G} \biggl[ - r^2 \frac{d\ln\rho}{dr} \biggr] = \frac{c_s^2}{G \rho_c^{1/2} \beta} \biggl[ - \mathfrak{r}_1^2 \frac{dv_1}{d\mathfrak{r}_1} \biggr]
= \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ - \mathfrak{r}_1^2 v_1' \biggr] \, .
</math>
</div>

The quantity tabulated in column 7 of Emden's Table 14 is precisely the dimensionless term inside the square brackets of this last expression; having units of mass, the coefficient out front sets the mass scale of the equilibrium configuration and depends only on the choice of central density and isothermal sound speed. Hence, a plot of <math>~\ln(\mathfrak{r}_1^2 v_1')</math> versus <math>~\ln\mathfrak{r}_1</math>, as shown above in Figure 1b, translates into a log-log plot of the equilibrium configuration's <math>~M_r</math> mass profile. Notice that, along with the radius, the mass of this isolated isothermal configuration extends to infinity and that, at large radii, the mass profile displays a simple power-law behavior — specifically, <math> ~M_r \propto r^{+1}</math>.

As was realized independently by [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert (1955)] and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)], a spherically symmetric isothermal equilibrium configuration of finite radius and finite mass can be constructed if the system is embedded in a pressure-confining external medium. We discuss their findings [[SSC/Structure/BonnorEbert|elsewhere]].

<font color="darkblue">

==Summary==
</font>

Based on the above derivations, the internal structural properties of an equilibrium isothermal sphere can be described in terms of the tabulated quantities provided in Emden's Table 14 as follows:
* <font color="red">Radial Coordinate Position</font>:
: Given the isothermal sound speed, <math>c_s</math>, and the central density, <math>\rho_c</math>, the radial coordinate is,
<div align="center">
<math>r = ( \rho_c \beta^2 )^{-1/2} \mathfrak{r}_1 = \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \mathfrak{r}_1 </math> .
</div>

* <font color="red">Density & Pressure</font>:
: As a function of the radial coordinate, <math>r(\mathfrak{r}_1)</math>, the density profile is,
<div align="center">
<math>\rho(r(\mathfrak{r}_1))= \rho_c e^{v_1(\mathfrak{r}_1)}</math>;
</div>
: and the pressure profile is,
<div align="center">
<math>P(r(\mathfrak{r}_1))= (c_s^2 \rho_c) e^{v_1(\mathfrak{r}_1)}</math>.
</div>
: As has been explicitly pointed out in the above discussion associated with Figure 1a, the density profile — and, hence, also the pressure profile — extends to infinity and, at large radii, behaves as a power law; specifically, <math>\rho \propto r^{-2}</math>.

* <font color="red">Mass</font>:
: Given <math>c_s</math> and <math>\rho_c</math>, the natural mass scale is,
<div align="center">
<math>M_0 \equiv \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} </math> ;
</div>
: and, expressed in terms of <math>M_0</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>
M_r = M_0 [ - \mathfrak{r}_1^2 v_1' ] \, .
</math>
</div>
: As discussed above in the context of Figure 1b, at large radii, the mass increases linearly with <math>r</math>. Because the density and pressure profiles extend to infinity, this means that the mass of an isolated isothermal sphere is infinite.

* <font color="red">Enthalpy & Gravitational Potential</font>:
: To within an additive constant, the enthalpy distribution is,
<div align="center">
<math>H(r(\mathfrak{r}_1))= c_s^2 [- v_1(\mathfrak{r}_1)]</math>;
</div>
: and the gravitational potential is,
<div align="center">
<math>\Phi(r(\mathfrak{r}_1)) = - H(r(\mathfrak{r}_1))= c_s^2 v_1(\mathfrak{r}_1)</math>.
</div>

* <font color="red">Mean-to-Local Density Ratio</font>:
: The ratio of the configuration's mean density, inside a given radius, to its local density at that radius is,
<div align="center">
<math>\frac{\bar{\rho}}{\rho} = \frac{3M_r}{4\pi r^3 \rho} = 3\biggl[- \frac{v_1'}{\mathfrak{r}_1 e^{v_1}} \biggr] </math> .
</div>
: As Figure 2 shows, at large <math>r</math> this density ratio goes to the value of 3, which means that the term inside the square brackets goes to unity at large <math>r</math>. This behavior is consistent with the limiting power-law behavior of both <math>M_r</math> and <math>\rho</math>, discussed above.

<div align="center">
<table border="0" cellpadding="5" width="360">
<tr>
<td align="center">
'''Figure 2: From Emden's Tabulated Data'''
</td>
</tr>
<tr>
<td align="center">
[[File:PlotMeanToLocalDensity.jpg|350px|center|Plot based on data from Emden's (1907) Table 14]]
</td>
</tr>
<tr>
<td align="left">
The blue curve displays an evaluation of the density ratio, <math>[- 3v_1'/(\mathfrak{r}_1 e^{v_1}) ]</math>, as a function of <math>\ln (\mathfrak{r}_1)</math>, as determined from the data presented in Emden's Table 14, shown above.
</td>
</tr>
</table>
</div>

=Our Numerical Integration=
The [[#keyExpression|above governing relation]] — see especially [[#Chandrasekhar|Chandrasekhar's notation]] — may be rewritten as (see also, for example, §19.8, eq. 19.35 of [<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>]),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e^{-w} \, ,</math>
</td>
</tr>
</table>
</div>
where we appreciate that,
<div align="center">
<math>~w \equiv \ln\biggl(\frac{\rho}{\rho_c}\biggr) \, .</math>
</div>

We'll adopt the following finite-difference approximations for the first and second derivatives on a grid of radial spacing, <math>~\Delta_r</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_i'</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{w_+ - w_-}{2\Delta_r}</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_i''</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{w_+ - 2w_i +w_-}{\Delta_r^2} \, .</math>
</td>
</tr>
</table>
</div>
Our finite-difference approximation of the governing equation is, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_i \biggl[ \frac{w_+ - 2w_i +w_-}{\Delta_r^2} \biggr] + 2\biggl[ \frac{w_+ - w_-}{2\Delta_r} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_i e^{-w_i} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ r_i [ w_+ - 2w_i +w_- ] + \Delta_r [ w_+ - w_- ]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Delta_r^2 r_i e^{-w_i} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~w_+
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\Delta_r^2 r_i e^{-w_i} + 2r_i w_i + w_- (\Delta_r - r_i)}{( \Delta_r + r_i)} \, .</math>
</td>
</tr>
</table>
</div>

Now, for the first two steps away from the center — where, <math>~w_i = w_0 = 0</math> and <math>~r_i = r_0 = 0</math> — we will use the following [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalLaneEmden|power-series expansion]] (see, for example, eq. 377 from §22 in Chapter IV of [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>]) to determine the value of <math>~w_i</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_1
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\Delta_r^2}{6} - \frac{\Delta_r^4}{120} + \frac{\Delta_r^6}{1890} \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w_2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(2\Delta_r)^2}{6} - \frac{(2\Delta_r)^4}{120} + \frac{(2\Delta_r)^6}{1890} \, .</math>
</td>
</tr>
</table>
</div>

=Related Discussions=
==Journal Articles==
* [http://adsabs.harvard.edu/abs/1870AmJS...50...57L J. H. Lane (1870)], '''American Journal of Science''', ''On the Theoretical Temperature of the Sun''
* [https://archive.org/details/mobot31753002152772/page/56 J. H. Lane (1870)], '''The American Journal of Science and Arts''', Vol. 50, pp. 57 - 74: ''On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining Its Volume by Its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment''

==Wikipedia==
* [https://en.wikipedia.org/wiki/Robert_Emden Robert Emden]
* [https://en.wikipedia.org/wiki/Emden–Chandrasekhar_equation Emden-Chandrasekhar equation]
* [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane Jonathan Home Lane]
* [https://en.wikipedia.org/wiki/Lane–Emden_equation Lane-Emden equation]

{{ SGFfooter }}

SSC/Structure/UniformDensity

2021-07-28T17:13:55Z

174.64.8.247: /* Isolated Uniform-Density Sphere */

=Isolated Uniform-Density Sphere=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Uniform-Density<br />Sphere</b>]]</font>
|}
Here we derive the interior structural properties of an isolated uniform-density sphere using all three [[SSCpt2/SolutionStrategies#Solution_Strategies|solution strategies]]. While deriving essentially the same solution three different ways might seem like a bit of overkill, this approach proves to be instructive because (a) it forces us to examine the structural behavior of a number of different physical parameters, and (b) it illustrates how to work through the different solution strategies for one model whose structure can in fact be derived analytically using any of the techniques. As we shall see when studying other self-gravitating configurations, the three strategies are not always equally fruitful.
 <br />
 <br />
 <br />
==Solution Technique 1==

Adopting [[SSCpt2/SolutionStrategies#Technique_1|solution technique #1]], we need to solve the integro-differential equation,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>
appreciating that,
<div align="center">
<math> M_r \equiv \int_0^r 4\pi r^2 \rho dr </math> .
</div>
For a uniform-density configuration, {{Math/VAR_Density01}} = <math>~\rho_c</math> = constant, so the density can be pulled outside the mass integral and the integral can be completed immediately to give,
<div align="center">
<math> M_r = \frac{4\pi}{3}\rho_c r^3 </math> .
</div>
Hence, the differential equation describing hydrostatic balance becomes,
<div align="center">
<math> \frac{dP}{dr} = - \frac{4\pi G}{3} \rho_c^2 r </math> .
</div>
Integrating this from the center of the configuration — where <math>~r=0</math> and <math>~P = P_c</math> — out to an arbitrary radius <math>~r</math> that is still inside the configuration, we obtain,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \int_{P_c}^P dP </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4\pi G}{3} \rho_c^2 \int_0^r r dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~ P </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</td>
</tr>
</table>

We expect the pressure to drop to zero at the surface of our spherical configuration — that is, at <math>~r=R</math> — so the central pressure must be,

<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr)</math> ,
</div>

where <math>~M</math> is the total mass of the configuration. Finally, then, we have,

<div align="center">
<math>P(r) = P_c\biggl[1 - \biggl(\frac{r}{R}\biggr)^2 \biggr] </math> .
</div>

==Solution Technique 3==

Adopting [[SSCpt2/SolutionStrategies#Technique_3|solution technique #3]], we need to solve the ''algebraic'' expression,
<div align="center">
<math>~H + \Phi = C_\mathrm{B}</math> .
</div>
in conjunction with the Poisson equation,
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .
</div>
Appreciating that, as shown above, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the Poisson equation once to give,
<div align="center">
<math> \frac{d\Phi}{dr} = \frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface, we find that,
<div align="center">
<math> \int_{\Phi(r)}^{\Phi_\mathrm{surf}} d\Phi = \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ \Phi_\mathrm{surf} - \Phi(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math>
</div>
Turning to the above algebraic condition, we will adopt the convention that {{Math/VAR_Enthalpy01}} is set to zero at the surface of a barotropic configuration, in which case the constant, <math>C_\mathrm{B}</math>, must be,
<div align="center">
<math>~C_\mathrm{B} = (H + \Phi)_\mathrm{surf} = \Phi_\mathrm{surf}</math> .
</div>
Therefore, everywhere inside the configuration {{Math/VAR_Enthalpy01}} must be given by the expression,
<div align="center">
<math>~H(r) = \Phi_\mathrm{surf} - \Phi(r)</math> .
</div>
Matching this with our solution of the Poisson equation, we conclude that, throughout the configuration,
<div align="center">
<math> H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>
Comparing this result with the result we obtained using solution technique #1, it is clear that throughout a uniform-density, self-gravitating sphere,
<div align="center">
<math>\frac{P}{H} = \rho</math> .
</div>

==Solution Technique 2==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following single, <math>2^\mathrm{nd}</math>-order ODE:
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d H}{dr} \biggr) = - 4\pi G \rho </math> .
</div>
Appreciating again that, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the <math>2^\mathrm{nd}</math>-order ODE once to give,
<div align="center">
<math> \frac{dH}{dr} = -\frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface — where, again, we adopt the convention that {{Math/VAR_Enthalpy01}} = 0 — we find that,
<div align="center">
<math> \int_{H(r)}^{0} dH = - \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

<font color="darkblue">
==Summary==
</font>

From the above derivations, we can describe the properties of a uniform-density, self-gravitating sphere as follows:
* <font color="red">Mass</font>:
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<div align="center">
<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
</div>

: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
</div>
* <font color="red">Pressure</font>:
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;
</div>

: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<div align="center">
<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSphereEnthalpy"><font color="red">Enthalpy</font>:</span>
: Throughout the configuration, the enthalpy is given by the relation,
<div align="center">
<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSpherePotential"><font color="red">Gravitational potential</font>: </span>
: Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,
<div align="center">
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>
: Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,
<div align="center">
<math>\Phi(r) = - \frac{GM}{r} </math> .
</div>
: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
<div align="center">
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<div align="center">
<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
</div>

* <font color="red">Central- to Mean-Density Ratio</font>:
: Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
<div align="center">
<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
</div>

=Uniform-Density Sphere Embedded in an External Medium=
For the ''isolated'' uniform-density sphere, discussed above, the surface of the configuration was identified as the radial location where the pressure drops to zero. Here we embed the sphere in a hot, tenuous medium that exerts a confining "external" pressure, <math>~P_e</math>, and ask how the configuration's equilibrium radius, <math>~R_e</math>, changes in response to this applied external pressure, for a given (fixed) total mass and central pressure, <math>~P_c</math>.

Following [[SSC/Structure/UniformDensity#Solution_Technique_1|solution technique #1]], the derivation remains the same up through the integration of the hydrostatic balance equation to obtain the relation,
<div align="center">
<math>P = P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</div>
Now we set <math>~P = P_e</math> at the surface of our spherical configuration — that is, at <math>~r=R_e</math> — so we can write,
<div align="center">
<math>P_c - P_e = \frac{2\pi G}{3} \rho_c^2 R_e^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr)</math>

<math>\Rightarrow ~~~~~ P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr) \, ,</math>
</div>
where <math>M</math> is the total mass of the configuration. Solving for the equilibrium radius, we have,
<div align="center">
<math> R_e = \biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .</math>
</div>
As it should, when the ratio <math>~P_e/P_c \rightarrow 0</math>, this relation reduces to the one obtained, above, for the isolated uniform-density sphere, namely,
<div align="center">
<math> R_e^4 = \biggl( \frac{3}{8\pi} \biggr) \frac{G M^2}{P_c} \, .</math>
</div>

{{ SGFfooter }}

SSC/Structure/UniformDensity

2021-07-28T17:03:51Z

174.64.8.247: /* Isolated Uniform-Density Sphere */

=Isolated Uniform-Density Sphere=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Uniform-Density<br />Sphere</b>]]</font>
|}
Here we derive the interior structural properties of an isolated uniform-density sphere using all three [[SSCpt2/SolutionStrategies#Solution_Strategies|solution strategies]]. While deriving essentially the same solution three different ways might seem like a bit of overkill, this approach proves to be instructive because (a) it forces us to examine the structural behavior of a number of different physical parameters, and (b) it illustrates how to work through the different solution strategies for one model whose structure can in fact be derived analytically using any of the techniques. As we shall see when studying other self-gravitating configurations, the three strategies are not always equally fruitful.
 <br />
 <br />
 <br />

==Solution Technique 1==

Adopting [[SSCpt2/SolutionStrategies#Technique_1|solution technique #1]], we need to solve the integro-differential equation,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>
appreciating that,
<div align="center">
<math> M_r \equiv \int_0^r 4\pi r^2 \rho dr </math> .
</div>
For a uniform-density configuration, {{Math/VAR_Density01}} = <math>~\rho_c</math> = constant, so the density can be pulled outside the mass integral and the integral can be completed immediately to give,
<div align="center">
<math> M_r = \frac{4\pi}{3}\rho_c r^3 </math> .
</div>
Hence, the differential equation describing hydrostatic balance becomes,
<div align="center">
<math> \frac{dP}{dr} = - \frac{4\pi G}{3} \rho_c^2 r </math> .
</div>
Integrating this from the center of the configuration — where <math>~r=0</math> and <math>~P = P_c</math> — out to an arbitrary radius <math>~r</math> that is still inside the configuration, we obtain,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \int_{P_c}^P dP </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4\pi G}{3} \rho_c^2 \int_0^r r dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~ P </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</td>
</tr>
</table>

We expect the pressure to drop to zero at the surface of our spherical configuration — that is, at <math>~r=R</math> — so the central pressure must be,

<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr)</math> ,
</div>

where <math>~M</math> is the total mass of the configuration. Finally, then, we have,

<div align="center">
<math>P(r) = P_c\biggl[1 - \biggl(\frac{r}{R}\biggr)^2 \biggr] </math> .
</div>

==Solution Technique 3==

Adopting [[SSCpt2/SolutionStrategies#Technique_3|solution technique #3]], we need to solve the ''algebraic'' expression,
<div align="center">
<math>~H + \Phi = C_\mathrm{B}</math> .
</div>
in conjunction with the Poisson equation,
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .
</div>
Appreciating that, as shown above, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the Poisson equation once to give,
<div align="center">
<math> \frac{d\Phi}{dr} = \frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface, we find that,
<div align="center">
<math> \int_{\Phi(r)}^{\Phi_\mathrm{surf}} d\Phi = \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ \Phi_\mathrm{surf} - \Phi(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math>
</div>
Turning to the above algebraic condition, we will adopt the convention that {{Math/VAR_Enthalpy01}} is set to zero at the surface of a barotropic configuration, in which case the constant, <math>C_\mathrm{B}</math>, must be,
<div align="center">
<math>~C_\mathrm{B} = (H + \Phi)_\mathrm{surf} = \Phi_\mathrm{surf}</math> .
</div>
Therefore, everywhere inside the configuration {{Math/VAR_Enthalpy01}} must be given by the expression,
<div align="center">
<math>~H(r) = \Phi_\mathrm{surf} - \Phi(r)</math> .
</div>
Matching this with our solution of the Poisson equation, we conclude that, throughout the configuration,
<div align="center">
<math> H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>
Comparing this result with the result we obtained using solution technique #1, it is clear that throughout a uniform-density, self-gravitating sphere,
<div align="center">
<math>\frac{P}{H} = \rho</math> .
</div>

==Solution Technique 2==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following single, <math>2^\mathrm{nd}</math>-order ODE:
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d H}{dr} \biggr) = - 4\pi G \rho </math> .
</div>
Appreciating again that, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the <math>2^\mathrm{nd}</math>-order ODE once to give,
<div align="center">
<math> \frac{dH}{dr} = -\frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface — where, again, we adopt the convention that {{Math/VAR_Enthalpy01}} = 0 — we find that,
<div align="center">
<math> \int_{H(r)}^{0} dH = - \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

<font color="darkblue">
==Summary==
</font>

From the above derivations, we can describe the properties of a uniform-density, self-gravitating sphere as follows:
* <font color="red">Mass</font>:
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<div align="center">
<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
</div>

: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
</div>
* <font color="red">Pressure</font>:
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;
</div>

: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<div align="center">
<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSphereEnthalpy"><font color="red">Enthalpy</font>:</span>
: Throughout the configuration, the enthalpy is given by the relation,
<div align="center">
<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSpherePotential"><font color="red">Gravitational potential</font>: </span>
: Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,
<div align="center">
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>
: Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,
<div align="center">
<math>\Phi(r) = - \frac{GM}{r} </math> .
</div>
: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
<div align="center">
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<div align="center">
<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
</div>

* <font color="red">Central- to Mean-Density Ratio</font>:
: Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
<div align="center">
<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
</div>

=Uniform-Density Sphere Embedded in an External Medium=
For the ''isolated'' uniform-density sphere, discussed above, the surface of the configuration was identified as the radial location where the pressure drops to zero. Here we embed the sphere in a hot, tenuous medium that exerts a confining "external" pressure, <math>~P_e</math>, and ask how the configuration's equilibrium radius, <math>~R_e</math>, changes in response to this applied external pressure, for a given (fixed) total mass and central pressure, <math>~P_c</math>.

Following [[SSC/Structure/UniformDensity#Solution_Technique_1|solution technique #1]], the derivation remains the same up through the integration of the hydrostatic balance equation to obtain the relation,
<div align="center">
<math>P = P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</div>
Now we set <math>~P = P_e</math> at the surface of our spherical configuration — that is, at <math>~r=R_e</math> — so we can write,
<div align="center">
<math>P_c - P_e = \frac{2\pi G}{3} \rho_c^2 R_e^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr)</math>

<math>\Rightarrow ~~~~~ P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr) \, ,</math>
</div>
where <math>M</math> is the total mass of the configuration. Solving for the equilibrium radius, we have,
<div align="center">
<math> R_e = \biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .</math>
</div>
As it should, when the ratio <math>~P_e/P_c \rightarrow 0</math>, this relation reduces to the one obtained, above, for the isolated uniform-density sphere, namely,
<div align="center">
<math> R_e^4 = \biggl( \frac{3}{8\pi} \biggr) \frac{G M^2}{P_c} \, .</math>
</div>

{{ SGFfooter }}

SSC/SynopsisStyleSheet

2021-07-28T16:57:44Z

174.64.8.247: /* Pointers to Relevant Chapters */

__FORCETOC__ 

=Spherically Symmetric Configurations Synopsis (Using Style Sheet)=

==Structure==
===Tabular Overview===

{| class="Synopsis1A" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|+ style="height:30px;" | <font size="+1">'''Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion'''</font> — adiabatic index, <math>\gamma</math>
|-
! colspan="2" |

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>dV = 4\pi r^2 dr</math>
</td>
<td align="center">
and
</td>
<td align="left">
   <math>dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>
</td>
</tr>
<tr>
<td align="right">
<math>W_\mathrm{grav}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>
</td>
</tr>
<tr>
<td align="right">
<math>U_\mathrm{int}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>
</td>
</tr>
</table>

|-
! style="background-color:lightgreen;" colspan="2"|<b><font size="+1">Equilibrium Structure</font></b>
|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">①</font></b>   <b>Detailed Force Balance</b>
! style="text-align:center; background-color:lightblue" |<b><font color="maroon" size="+1">③</font></b>   <b>Free-Energy Identification of Equilibria</b>
|-
! style="vertical-align:top; text-align:left;" |Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of
<div align="center">
<font color="maroon"><b>Hydrostatic Balance</b></font><br />
{{ Math/EQ_SShydrostaticBalance01 }}
</div>
for the radial density distribution, <math>\rho(r)</math>.
! style="vertical-align:top; text-align:left;" rowspan="3"|The Free-Energy is,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{G}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-a \biggl(\frac{R}{R_0}\biggr)^{-1} + b\biggl(\frac{R}{R_0}\biggr)^{3-3\gamma}+ c\biggl(\frac{R}{R_0}\biggr)^3 \, .</math>
</td>
</tr>
</table>
Therefore, also,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>a\biggl(\frac{R}{R_0}\biggr)^{-2} +(3-3\gamma)b\biggl(\frac{R}{R_0}\biggr)^{2-3\gamma} + 3c\biggl(\frac{R}{R_0}\biggr)^2</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>
</td>
</tr>
</table>
Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>d\mathfrak{G}/dR = 0</math>. Hence, equilibria are defined by the condition,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>
</td>
</tr>
</table>
|-
! style="text-align:center; background-color:#ffff99;" |<b><font color="maroon" size="+1">②</font></b>   <b>Virial Equilibrium</b>
|-
! style="vertical-align:top; text-align:left;" |
Multiply the hydrostatic-balance equation through by <math>rdV</math> and integrate over the volume:
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-\int_0^R r\biggl(\frac{dP}{dr}\biggr)dV - \int_0^R r\biggl(\frac{GM_r \rho}{r^2}\biggr)dV</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-\int_0^R 4\pi r^3 \biggl(\frac{dP}{dr}\biggr) dr - \int_0^R \biggl(\frac{GM_r}{r}\biggr)dM_r</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-\int_0^R\biggl[ \frac{d}{dr}\biggl( 4\pi r^3P \biggr) - 12\pi r^2 P\biggr] dr + W_\mathrm{grav}</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3(\gamma-1)U_\mathrm{int} + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .</math>
</td>
</tr>
</table>
|}

===Pointers to Relevant Chapters===



<font size="+1" color="maroon"><b>⓪ </b></font> Background Material:
{| class="Synopsis1B" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|·
|[[PGE#Principal_Governing_Equations|Principal Governing Equations]] (PGEs) in most general form being considered throughout this H_Book
|-
! width="30px" style="text-align:right; vertical-align:top; "|·
|PGEs in a form that is relevant to a study of the ''Structure, Stability, & Dynamics'' of [[SSCpt1/PGE|spherically symmetric systems]]
|-
! width="30px" style="text-align:right; vertical-align:top; "|·
|[[SR#Supplemental_Relations|Supplemental relations]] — see, especially, [[SR#Barotropic_Structure|barotropic equations of state]]
|}



<font size="+1" color="maroon"><b>① </b></font> Detailed Force Balance:
{| class="Synopsis1C" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|·
|[[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Derivation of the equation of Hydrostatic Balance]], and a description of several standard strategies that are used to determine its solution — see, especially, what we refer to as [[SSCpt2/SolutionStrategies#Technique_1|Technique 1]]
|}



<font size="+1" color="maroon"><b>② </b></font> Virial Equilibrium:
{| class="Synopsis1D" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|·
|Formal derivation of the multi-dimensional, [[VE#Second-Order_Tensor_Virial_Equations|2<sup>nd</sup>-order tensor virial equations]]
|-
! width="30px" style="text-align:right; vertical-align:top; "|·
|[[VE#Scalar_Virial_Theorem|Scalar Virial Theorem]], as appropriate for spherically symmetric configurations
|-
! width="30px" style="text-align:right; vertical-align:top; "|·
|[[VE#Generalization|Generalization]] of scalar virial theorem to include the bounding effects of a hot, tenuous external medium
|}

==Stability==
===Isolated & Pressure-Truncated Configurations===

{| class="Synopsis1E" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|-
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis:   Applicable to Isolated & Pressure-Truncated Configurations</b></font>
|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">④</font></b>   <b>Perturbation Theory</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">⑦</font></b>   <b>Free-Energy Analysis of Stability</b>

|-
! style="vertical-align:top; text-align:left;" |
Given the radial profile of the density and pressure in the equilibrium configuration, solve the [[SSC/VariationalPrinciple#Ledoux_and_Pekeris_.281941.29|eigenvalue problem defined]] by the,
<div align="center">
<font color="#770000">'''LAWE:   Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{d}{dr}\biggl[ r^4 \gamma P ~\frac{dx}{dr} \biggr]
+\biggl[ \omega^2 \rho r^4 + (3\gamma - 4) r^3 \frac{dP}{dr} \biggr] x
</math>
</td>
</tr>
<tr><td align="center" colspan="3">
[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, §3.7.1, p. 174, Eq. (3.145)
</td></tr>
</table>
</div>
to find one or more radially dependent, radial-displacement eigenvectors, <math>x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>\omega^2</math>.
! style="vertical-align:top; text-align:left;" rowspan="5"|
The second derivative of the free-energy function is,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-2a\biggl(\frac{R}{R_0}\biggr)^{-3} + (3-3\gamma)(2-3\gamma)b \biggl(\frac{R}{R_0}\biggr)^{1-3\gamma} + 6c\biggl(\frac{R}{R_0}\biggr)
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{R_0}{R} \biggr)^2\biggl[
2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V
\biggr] \, .
</math>
</td>
</tr>
</table>
Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>3(\gamma-1)U_\mathrm{int}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3P_e V - W_\mathrm{grav} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav} \biggr] + 6P_e V
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, .
</math>
</td>
</tr>
</table>
Note the similarity with <b><font color="maroon" size="+1">⑥</font></b>.
----

Alternatively, recalling that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3(\gamma - 1)U_\mathrm{int}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2S_\mathrm{therm} \, ,
</math>
</td>
</tr>
</table>
the conditions for virial equilibrium and stability, may be written respectively as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>3P_e V</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2S_\mathrm{therm}+ W_\mathrm{grav} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
4W_\mathrm{grav} + 6\gamma S_\mathrm{therm} \, .
</math>
</td>
</tr>
</table>

|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑤</font></b>   <b>Variational Principle</b>

|-
! style="vertical-align:top; text-align:left;" |
Multiply the LAWE through by <math>4\pi x dr</math>, and integrate over the volume of the configuration gives the,
<div align="center">
<font color="#770000">'''Governing Variational Relation</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int_0^R x^2 \biggl(\frac{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P dr
- \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r} \biggr) 4\pi \rho r^2 dr
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
</math>
</td>
</tr>
</table>
</div>
Now, by setting <math>(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>P = P_e</math> at the surface, in which case this relation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\omega^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int}
- \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav}
+ 3^2 \gamma x^2 P_eV}{ \int_0^R x^2 r^2 dM_r}
</math>
</td>
</tr>
</table>
</div>

|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑥</font></b>   <b>Approximation:   Homologous Expansion/Contraction</b>

|-
! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>x</math> = constant, and the governing variational relation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\omega^2 \int_0^R r^2 dM_r</math>
</td>
<td align="center">
<math>\leq</math>
</td>
<td align="left">
<math>
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, .
</math>
</td>
</tr>
</table>
</div>

|}

===Bipolytropes===

{| class="Synopsis1F" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|-
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis:   Applicable to Bipolytropic Configurations</b></font>
|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑧</font></b>   <b>Variational Principle</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">⑩</font></b>   <b>Free-Energy Analysis of Stability</b>

|-
! style="vertical-align:top; text-align:left;" |
<div align="center">
<font color="#770000">'''Governing Variational Relation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2 dM_r^* </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}} x^2 dW^*_\mathrm{grav}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_e - 4) \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^* \, .
</math>
</td>
</tr>
</table>
</div>
! style="vertical-align:top; text-align:left;" rowspan="3"|
As we have detailed in an [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives|accompanying discussion]], the first derivative of the relevant free-energy expression is,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R ~\frac{\partial \mathfrak{G}}{\partial R}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2S_\mathrm{tot} + W_\mathrm{tot}
\, ,
</math>
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math>
</td>
<td align="center">
    and    
</td>
<td align="left">
<math>W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math>
</td>
</tr>
</table>
and the second derivative of that free-energy function is,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2\biggl[
W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env}
\biggr] \, .
</math>
</td>
</tr>
</table>

----

This stability criterion may be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2[(3\gamma_c -4) S_\mathrm{core}
+ (3\gamma_e -4) S_\mathrm{env} ] \, .
</math>
</td>
</tr>
</table>
Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{core}}{S_\mathrm{env}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, .
</math>
</td>
</tr>
</table>

See the [[SSC/Stability/BiPolytropes#What_to_Expect_for_Equilibrium_Configurations|accompanying discussion]].

----

If — based for example on <b><font color="maroon" size="+1">⑦</font></b> — we make the reasonable assumption that, in equilibrium, the statements,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>
</td>
<td align="center">
    and    
</td>
<td align="left">
<math>2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>
</td>
</tr>
</table>
hold separately, then we satisfy the virial equilibrium condition, namely,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math>
</td>
</tr>
</table>
and the second derivative of the relevant free-energy function can be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2(W_\mathrm{core} + W_\mathrm{env})
+ (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core})
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env})
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e)
+ (4-3\gamma_c ) W_\mathrm{core}
+ (4-3\gamma_e)W_\mathrm{env} \, .
</math>
</td>
</tr>
</table>

Note the similarity with <b><font color="maroon" size="+1">⑨</font></b> — temporarily, see [[SSC/Stability/BiPolytropes#Revised_Free-Energy_Analysis|this discussion]].

|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑨</font></b>   <b>Approximation:   Homologous Expansion/Contraction</b>

|-
! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R r^2 dM_r</math>
</td>
<td align="center">
<math>\leq</math>
</td>
<td align="left">
<math>
(4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e) P_i V_\mathrm{core} \, .
</math>
</td>
</tr>
</table>
</div>

|}

=See Also=

{{ SGFfooter }}

VE

2021-07-28T16:51:27Z

174.64.8.247: /* Scalar Virial Theorem */

__FORCETOC__ 

=Global Energy Considerations=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Context|<b>Global Energy<br />Considerations</b>]]</font>
|}
==Preface==
The astrophysics community relies heavily on the virial equations — most often in the context of the [[#Scalar_Virial_Theorem|scalar virial theorem]] — to ascertain the basic properties of equilibrium systems. As is described below, fundamentally the virial equations are obtained by taking moments of the Euler equation. By examining the balance among various relevant energy reservoirs, the mathematical expression that defines virial equilibrium provides a means by which, for example, the radius of a configuration can be estimated, given a total system mass and mean system temperature. It can also be used to estimate a system's maximum allowed rotation frequency and whether or not the properties of the equilibrium configuration will be significantly modified if the system is embedded in a hot tenuous external medium.

As is also discussed, below, it can be even more informative to examine how a system's global, Gibbs-like free energy, <math>\mathfrak{G}</math>, varies under contraction or expansion. Extrema in the free energy identify equilibrium configurations, for example. For spherically symmetric systems, in particular, the [[#Scalar_Virial_Theorem|scalar virial theorem]] is "derived" by identifying under what conditions <math>d\mathfrak{G}/dR = 0</math>. Furthermore, the sign of the second derivative, <math>d^2\mathfrak{G}/dR^2</math>, tells whether or not the equilibrium state is stable or unstable. Here we define relevant energy reservoirs that contribute to a system's global free energy. In separate chapters we use the free energy function to help identify the properties of equilibrium systems and to examine their relative stability.

<table border="1" cellpadding="5" align="center" width="80%">
<tr>
<td align="center">
'''Why Bother?'''<br />

<font size="-1">Excerpts drawn from the introductory chapter (p. 3) of [http://ads.harvard.edu/books/1978vtsa.book/ ''The Virial Theorem in Stellar Astrophysics'' (2003)], <br />
by George W. Collins, II</font>
</td>
</tr>
<tr><td align="left">
<font color="#770000">'''Question'''</font>:   Why bother introducing the virial theorem and its allied free-energy expression, given that the astrophysical systems we are interested in analyzing can be fully described by solutions of the set of [[PGE#Principal_Governing_Equations|Principal Governing Equations]]?

<font color="#770000">'''Answer'''</font>:   The [[PGE#Principal_Governing_Equations|Principal Governing Equations]] are, in general, <font color="#008899">non-linear, second-order, vector differential equations which exhibit closed form solutions only in special cases. Although additional cases may be solved numerically, insight into the behavior of systems in general is very difficult to obtain in this manner. The virial theorem</font> and its associated free-energy expression <font color="#008899">generally deals in scalar quantities and usually is applied on a global scale. This reduction in complexity — from a vector description to a scalar one — frequently enables us to solve the resulting equations</font> in closed form and to ascertain more straightforwardly what physical processes are most responsible for defining properties of the solution.

<font color="#770000">'''Caution'''</font>:   We should always keep in mind that <font color="#008899">this reduction in complexity results in a concomitant loss of information and we cannot expect to obtain as complete a description of a physical system as would be possible from a full solution of the</font> [[PGE#Principal_Governing_Equations|Principal Governing Equations]].
</td></tr>
</table>

==Virial Equations (Inertial Frame)==
Most of the material presented here has been drawn from Chandrasekhar's ''Ellipsoidal Figures of Equilibrium'' — hereafter [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] — first published in 1969. Relying heavily on [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>'s] in-depth treatment of the topic, our aim is to highlight key aspects of the tensor-virial equations and to present them in a form that serves as a foundation for our separate discussions of the equilibrium and stability of self-gravitating fluid systems. Strong parallels are drawn between the [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] presentation and our own so that it will be relatively straightforward for the reader to consult the [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] publication to obtain details of the various derivations. Text that appears in a green font has been drawn ''verbatim'' from this reference.

===Setting the Stage===

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, §8, p. 15] <font color="#007700">A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations. The ''virial method'' … is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account. The ''virial equations'' of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations.</font> In this context, Chandrasekhar's focus is on two of the four [[PGE#Principal_Governing_Equations|principal governing equations]] that serve as the foundation of our entire H_Book, namely, the

<div align="center">
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />
('''Momentum Conservation''')

{{ Template:Math/EQ_Euler01 }}
</div>
and the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Template:Math/EQ_Poisson01}}
[[File:OriginButton.jpg|125px|link=PGE/PoissonOrigin]]
</div>

In [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the Euler equation first appears in §11 (p. 20) as equation (38) and is written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\rho \frac{du_i}{dt}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{\partial p}{\partial x_i} + \rho \frac{\partial \mathfrak{B}}{\partial x_i} \, ,</math>
</td>
</tr>
</table>
</div>
and the Poisson equation appears in §10 (p. 20) — specifically, the left-most component of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>'s] equation (37) — as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\nabla^2 \mathfrak{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- 4\pi G \rho \, .</math>
</td>
</tr>
</table>
</div>
It is clear, therefore, that Chandrasekhar uses the variable <math>\vec{u}</math> instead of <math>\vec{v}</math> to represent the inertial velocity field. More importantly, he adopts a different variable name ''and a different sign convention'' to represent the gravitational potential, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>- \Phi = \mathfrak{B} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>
</td>
</tr>
</table>
</div>
Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] into this H_Book.

===First-Order Virial Equations===
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, §11(a), p. 21] <font color="#007700">The [virial] equations of the first order are obtained by simply integrating [the Euler equation] over the instantaneous volume, <math>V</math>, occupied by the fluid</font>. Specifically, using our H_Book variable notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\int\limits_V \rho \frac{dv_i}{dt} d^3x</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \int\limits_V \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>
</td>
</tr>
</table>
</div>
leads to (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\frac{d^2 I_i}{dt^2} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>0 \, ,</math>
</td>
</tr>
</table>
</div>
where the <span id="MomentOfInertia">moments of inertia</span> about the three separate principal axes <math>(i = 1,2,3)</math> are defined by the expressions,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>I_i</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int\limits_V \rho x_i d^3x \, .</math>
</td>
</tr>
</table>
</div>
Thus, the first-order virial equation(s) <font color="#007700">expresses the uniform motion of the center of mass of the system</font>.

===Second-Order Tensor Virial Equations===
In discussing the origin of the second-order (tensor) virial equation, [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] will continue to serve as our primary reference. However, in §4.3 of their widely referenced textbook titled, "Galactic Dyamics," Binney & Tremaine (1987) — hereafter [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] — also present a detailed derivation of the second-order virial equation, which they refer to as the ''tensor virial theorem.'' Because their presentation is set in the context of discussions of the structure of ''stellar dynamic'' systems, the [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] derivation fundamentally originates from the collisionless Boltzmann equation. In what follows we will identify where various key equations appear in [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], as well as in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], because it can sometimes be useful to compare derivations made from the stellar-dynamic versus the fluid-dynamic perspective.

====Derivation====
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, §11(b), p. 22] The second-order (tensor) virial equations <font color="#007700">are obtained by multiplying [the Euler equation] by <math>x_j</math> and integrating over the volume, <math>V</math></font>. Specifically, again using our H_Book variable notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\int\limits_V \rho \frac{dv_i}{dt} x_j d^3x</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \int\limits_V x_j \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho x_j \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 211, Eq. (4-72)
</td>
</tr>
</table>
</div>
or, separating the term on the left-hand side into two physically distinguishable components — see equation 44 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] — this can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\frac{d}{dt} \int\limits_V \rho v_i x_j d^3x - 2 \mathfrak{T}_{ij}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\delta_{ij}\Pi + \mathfrak{W}_{ij} \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 22, Eq. (47)
</td>
</tr>
</table>
</div>
where, by definition,
<div align="center">
<table border="0" cellpadding="2" align="center">
<tr>
<td colspan="6">
 
</td>
<th align="center">
References
</th>
</tr>

<tr>
<td align="right">
<math>\mathfrak{T}_{ij}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{2} \int\limits_V \rho v_i v_j d^3x </math>
</td>
<td align="center">
  …  
</td>
<td align="left">
is the (ordered) kinetic energy tensor
</td>
<td align="center">
  …  
</td>
<td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (9)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
</td>
</tr>

<tr>
<td align="right">
<math>\Pi</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int\limits_V P d^3x </math>
</td>
<td align="center">
  …  
</td>
<td align="left">
is ⅔ of the total thermal (''i.e.,'' random kinetic) energy
</td>
<td align="center">
  …  
</td>
<td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (7)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{W}_{ij}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{2} \int\limits_V \rho \Phi_{ij} d^3x </math>
</td>
<td align="center">
    
</td>
<td align="left">
 
</td>
<td align="center">
  …  
</td>
<td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (15)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 68, Eq. (2-126)
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_j} d^3x </math>
</td>
<td align="center">
  …  
</td>
<td align="left">
is the gravitational potential energy tensor
</td>
<td align="center">
  …  
</td>
<td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 18, Eq. (18)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 67, Eq. (2-123)
</td>
</tr>
</table>
</div>
Note that, in the definition of the gravitational potential energy tensor, Chandrasekhar has introduced a tensor generalization of the gravitational potential [see his Eq. (14), p. 17], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>- \Phi_{ij} = \mathfrak{B}_{ij}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>G\int\limits_V \rho(\vec{x}^') \frac{ (x_i - x_i^')(x_j - x_j^') }{|\vec{x} - \vec{x}^{~'}|^3} d^3x^' \, ;</math>
</td>
</tr>
</table>
</div>
this same potential energy tensor appears explicitly as part of the expression for <math>\mathfrak{W}_{ij}</math> that is presented as Equation (2-126), on p. 67 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].

The antisymmetric part of this tensor expression gives (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\frac{d}{dt} \int\limits_V \rho (v_ix_j - v_j x_i) d^3x</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>0 \, ,</math>
</td>
</tr>
</table>
</div>
which <font color="#007700">expresses simply the conservation of the angular momentum of the system</font>. The symmetric part of the tensor expression gives what is generally referred to as (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details) the,
<div align="center">
<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 23, Eq. (51)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-78)
</td>
</tr>
</table>
</div>
<span id="MOItensor">where,</span>
<div align="center">
<table border="0" cellpadding="2" align="center">
<tr>
<td colspan="6">
 
</td>
<th align="center">
References
</th>
</tr>

<tr>
<td align="right">
<math>I_{ij}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int\limits_V \rho x_i x_j d^3x </math>
</td>
<td align="center">
  …  
</td>
<td align="left">
is the moment of inertia tensor
</td>
<td align="center">
  …  
</td>
<td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (4)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-76)
</td>
</tr>
</table>
</div>

====Steady State (Virial Equilibrium)====
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \delta_{ij}\Pi \, .</math>
</td>
</tr>
</table>
</div>
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Scalar<br />Virial<br />Theorem</b>]]</font>
|}
===Scalar Virial Theorem===
====Standard Presentation [the Virial of Clausius (1870)]====
The trace of the tensor virial equation (TVE), which is obtained by identifying the trace of each term in the TVE, produces the scalar virial equation, which is widely referenced and used by the astrophysics community. More specifically, setting,
<div align="center">
<table border="0" cellpadding="2" align="center">
<tr>
<td colspan="4">
 
</td>
<th align="center">
Description
</th>
<td colspan="1">
 
</td>
<th align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]
Reference
</th>
</tr>

<tr>
<td align="right">
<math>I = \sum\limits_{i=1,3} I_{ii}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int\limits_V \rho (\vec{x}) |\vec{x}|^2 d^3x </math>
</td>
<td align="center">
=
</td>
<td align="left">
scalar moment of inertia
</td>
<td align="center">
  …  
</td>
<td align="center">
[Eqs. (3) & (5), p. 16]
</td>
</tr>

<tr>
<td align="right">
<math>T_\mathrm{kin} = \sum\limits_{i=1,3} \mathfrak{T}_{ii}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \int\limits_V \rho |\vec{v}|^2 d^3x </math>
</td>
<td align="center">
=
</td>
<td align="left">
total (ordered) kinetic energy
</td>
<td align="center">
  …  
</td>
<td align="center">
[Eq. (8), p. 16]
</td>
</tr>

<tr>
<td align="right">
<math>W_\mathrm{grav} = \sum\limits_{i=1,3} \mathfrak{W}_{ii}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_i} d^3x </math>
</td>
<td align="center">
=
</td>
<td align="left">
gravitational potential energy
</td>
<td align="center">
  …  
</td>
<td align="center">
[Eq. (18), p. 18]
</td>
</tr>

<tr>
<td align="right">
<math>S_\mathrm{therm} = \frac{1}{2} \sum\limits_{i=1,3} \delta_{ii}\Pi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2} \int\limits_V P d^3x </math>
</td>
<td align="center">
=
</td>
<td align="left">
total thermal (random kinetic) energy
</td>
<td align="center">
  …  
</td>
<td align="center">
[Eq. (7), p. 16]
</td>
</tr>
</table>
</div>
the scalar virial equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\frac{1}{2} \frac{d^2 I}{dt^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) + W_\mathrm{grav} \, ;</math>
</td>
</tr>
</table>
</div>
and, for a stationary state, we have the equilibrium condition that is broadly referred to as the,
<div align="center">
<span id="TVE"><font color="#770000">'''Scalar Virial Theorm'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) + W_\mathrm{grav} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>0 \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-79)
</td>
</tr>
</table>
</div>
(In a footnote to their Equation 4-79, [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] point out that the ''scalar virial theorem'' was first proved by R. Clausius in 1870; see various links to this work under our [[VE#Related_Discussions|"Related Discussions" subsection, below]].)

====Generalization====
Chapter 24 in Volume II (''Gas Dynamics'') of [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] presents a generalization of the scalar virial theorem that includes the effects of (a) a magnetic field that threads through a self-gravitating fluid system, and (b) an imposed surface pressure, <math>P_e</math>, when the configuration is embedded in a hot, tenuous external medium. Text that appears in an orange font in the following paragraph has been drawn ''verbatim'' from this reference.

[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] begins by adding a term to the Euler equation that accounts for <font color="orange">the Maxwell stress tensor, <math>T_{ik}</math>, associated with the ambient magnetic field</font>, <math>~\vec{B}</math>, where,
<div align="center">
<math>
T_{ik} = \frac{B_i B_k}{4\pi} - \frac{|\vec{B}|^2}{8\pi} \delta_{ik} \, .
</math>
<br /> <br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 329, Eq. (24.3)
</div>
Drawing from Equation (24.1), the associated modified Euler equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\rho \frac{dv_i}{dt} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{\partial P}{\partial x_i} - \rho \frac{\partial \Phi}{\partial x_i} + \frac{\partial T_{ik}}{\partial x_k} \, .</math>
</td>
</tr>
</table>
</div>
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>, Vol. II, pp. 329-330] <font color="orange">If we were to multiply [this modified Euler equation] by <math>~x_m</math> and integrate over volume <math>V</math>, we would get the [appropriately modified] ''tensor virial theorem'', the off-diagonal elements of which carry information concerning angular-momentum conservation (see </font>[<b>[[Appendix/References#EFE|EFE]]</b>] <font color="orange">for an exposition). [Here] we shall be more interested in the trace of the tensor equation, which we may derive by simply multiplying [the modified Euler equation] by <math>~x_i</math> (with an implicit summation over repeated indices) and integrating over <math>V</math></font>. The resulting relation governing the equilibrium of stationary states (see [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] for derivation details), as we shall reference it, is the
<div align="center">
<span id="GenTVE"><font color="#770000">'''Generalized Scalar Virial Theorem'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) + W_\mathrm{grav} + \mathcal{M}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> P_e \oint \vec{x}\cdot \hat{n} dA - \oint \vec{x}\cdot \overrightarrow{T}\hat{n} dA \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 331, Eq. (24.12)
</td>
</tr>
</table>
</div>
<font color="orange">where <math>\mathcal{M}</math> equals the magnetic energy contained in volume <math>V</math></font>,
<div align="center">
<math>
\mathcal{M} \equiv \int\limits_V \frac{|\vec{B}|^2}{8\pi} d^3x \, .
</math>
<br /> <br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], p. 165, Eq. (7.1.18)<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 330, Eq. (24.9)
</div>

[It should be noted that [http://adsabs.harvard.edu/abs/1953ApJ...118..116C Chandrasekhar & Fermi] (1953, ApJ, 118, 116) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..503M Mestel & Spitzer] (1956, MNRAS, 116, 503) provide early discussions of virial equilibrium conditions that take into account the energy associated with a magnetic field.]

==Virial Equations (Rotating Frame)==

As we have [[PGE/RotatingFrame#Euler_Equation_.28rotating_frame.29|explained elsewhere]], when examining the equilibrium, stability, and dynamical behavior of configurations that are rotating with angular velocity, <math>\vec\Omega_f</math>, it is useful to reference the

<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) \, .</math>
</div>

Chandrasekhar also adopts this tactic. In [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the equivalent expression first appears in §12 as equation (62) and has the form,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\rho \frac{du_i}{dt}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{\partial p}{\partial x_i} + \rho \frac{\partial \mathfrak{B}}{\partial x_i}
+ 2\rho \epsilon_{i \ell m}u_\ell \Omega_m + \frac{1}{2} \rho \frac{\partial}{\partial x_i}|\vec\Omega \times \vec{x}|^2 \, ,</math>
</td>
</tr>
</table>
</div>
where, as noted in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> §12, p. 25], the terms <math>|\vec\Omega \times \vec{x}|^2/2</math> and <math>2\vec{u} \times \vec\Omega</math> <font color="#007700">represent the centrifugal potential and the Coriolis acceleration, respectively</font> — also see [[PGE/RotatingFrame#Centrifugal_and_Coriolis_Accelerations|our related discussion of the centrifugal and Coriolis accelerations]]. As Chandrasekhar details, the Coriolis and centrifugal contributions introduce additional terms to the second-order virial, as follows:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{d}{dt} \int\limits_V \rho v_i x_j d^3x </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2 \mathfrak{T}_{ij} + \delta_{ij}\Pi + \mathfrak{W}_{ij} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>+ 2\epsilon_{i \ell m} \Omega_m \int\limits_V \rho v_\ell x_j d^3x
+ \Omega^2I_{ij} - \Omega_i \Omega_k I_{kj}\, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], § 11a, p. 25, Eq. (63) and Epilogue, p. 244, Eq. (1)
</td>
</tr>
</table>
</div>


In his discussion of the ''Oscillation and Collapse of Interstellar Clouds,'' [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber (1976, ApJ, 208, 113)] begins with this form of the second-order virial, but adds to it a contribution due to pressure-confinement by an external medium, as [[VE#Generalization|introduced above in the context of Shu's generalization]].
Specifically, Weber opens up his discussion with the following form of the tensor virial equations:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{dL_{ij}}{dt} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2 \mathfrak{T}_{ij} + \delta_{ij}\Pi + \mathfrak{W}_{ij} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>+ 2\epsilon_{i \ell m} \Omega_m L_{\ell j} + I_{jm}(|\vec\Omega|^2 \delta_{im} - \Omega_i\Omega_m)
- \oint P_e x_j n_i dS \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber (1976)], Eq. (1)
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="2" align="center">

<tr>
<td align="right">
<math>L_{ij}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int\limits_V \rho v_i x_j d^3x \, .</math>
</td>
</tr>
</table>
</div>

==Free Energy Expression==
Associated with any isolated, self-gravitating, gaseous configuration we can identify a total Gibbs-like free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots
</math>
</div>
Here, we have explicitly included the gravitational potential energy, <math>W_\mathrm{grav}</math>, the ordered kinetic energy, <math>T_\mathrm{kin}</math>, a term that accounts for surface effects if the configuration of volume <math>V</math> is embedded in an external medium of pressure <math>P_e</math>, and <math>\mathfrak{S}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts. Our above discussion of the [[#Scalar_Virial_Theorem|scalar virial theorem]] provides mathematical definitions of each of these energy terms, except for <math>\mathfrak{S}_\mathrm{therm}</math>, which we discuss now.

===Reservoir of Thermodynamic Energy===
<math>\mathfrak{S}_\mathrm{therm}</math> derives from the differential, "PdV" work that is often discussed in the context of thermodynamic systems. It should be made clear that, here, "dV" refers to the differential volume ''per unit mass,'' so it should be written as "<math>~d(\rho^{-1})</math>", to be consistent with the notation used throughout this H_Book. Therefore, the differential thermodynamic work is,
<div align="center">
<math>d\mathfrak{w} = Pd(1/\rho) = - \biggl( \frac{P}{\rho^2} \biggr) d\rho \, .</math>
</div>
After an ''evolutionary'' equation of state has been adopted, this differential relationship can be integrated to give an expression for the energy per unit mass, <math>\mathfrak{w}</math>, that is potentially available for work. Then we define the thermodynamic energy reservoir as,
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} \equiv - \int \mathfrak{w} ~dm \, .</math>
</div>

====Isothermal Systems====
If each element of gas maintains its temperature when the system undergoes compression or expansion — that is, if the compression/expansion is isothermal — then, the relevant evolutionary equation of state is,
<div align="center">
<math>P = c_s^2 \rho \, ,</math>
</div>
where the constant, <math>c_s</math>, is the isothermal sound speed. In this case, the expression for the differential thermodynamic work becomes,
<div align="center">
<math>d\mathfrak{w} = - \biggl( \frac{c_s^2}{\rho} \biggr) d\rho = - c_s^2 d\ln\rho \, .</math>
</div>
Hence, to within an additive constant, we have,
<div align="center">
<math>\mathfrak{w} = - c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) \, ,</math>
</div>
where, <math>\rho_0</math> is a (as yet unspecified) reference density, and integration throughout the configuration gives (for the isothermal case),
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = + \int c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) dm \, .</math>
</div>

====Adiabatic Systems====
If, upon compression or expansion, the gaseous configuration evolves adiabatically, the pressure will vary with density as,
<div align="center">
<math>P = K \rho^{\gamma_g} \, ,</math>
</div>
where, <math>K</math> specifies the specific entropy of the gas and {{ Template:Math/MP_AdiabaticIndex }} is the ratio of specific heats that is relevant to the phase of compression/expansion. In this case, the expression for the differential thermodynamic work becomes,
<div align="center">
<math>d\mathfrak{w} = - K \rho^{{\gamma_g}-2} d\rho = - \frac{K}{({\gamma_g}-1)} d\rho^{{\gamma_g}-1} \, .</math>
</div>
Hence, to within an additive constant, we have,
<div align="center">
<math>\mathfrak{w} = - \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) \, ,</math>
</div>
and integration throughout the configuration gives (for the adiabatic case),
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = + \int \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) dm
= \frac{2}{3({\gamma_g}-1)} \int \frac{3}{2} \biggl( \frac{P}{\rho} \biggr) dm
= \frac{2}{3({\gamma_g}-1)} S_\mathrm{therm} \, ,</math>
</div>
where, as introduced in our above discussion of the [[#Scalar_Virial_Theorem|scalar virial theorem]], <math>S_\mathrm{therm}</math> is the system's total thermal (''i.e.,'' random kinetic) energy.

====Relationship to the System's Internal Energy====
It is instructive to tie this introductory material to the classic discussion of thermodynamic systems, which relates a change in the system's internal energy per unit mass, <math>\Delta u_\mathrm{int}</math>, to the differential work, <math>\Delta \mathfrak{w}</math>, via the expression,
<div align="center">
<math>\Delta u_\mathrm{int} = \Delta Q - \Delta \mathfrak{w} \, ,</math>
</div>
where, <math>\Delta Q</math> is the change in heat content of the system.

'''Isothermal Evolutions''': Because the internal energy is only a function of the temperature, we can set <math>\Delta u_\mathrm{int} = 0</math> for expansions or contractions that occur isothermally. Hence, for isothermal evolutions the change in heat content can immediately be deduced from the expression derived for the differential work; specifically, <math>\Delta Q = \Delta \mathfrak{w}</math>.

'''Adiabatic Evolutions''': By definition, <math>\Delta Q = 0</math> for adiabatic evolutions, in which case we find <math>\Delta u_\mathrm{int} = - \Delta \mathfrak{w}</math>. The definition of the thermodynamic energy reservoir can therefore be rewritten as,
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = - \int \mathfrak{w} ~dm = + \int u_\mathrm{int} ~dm = U_\mathrm{int} \, .</math>
</div>
Quite generally, then — in sync with the above derivation — we can replace <math>\mathfrak{S}_\mathrm{therm}</math> by,
<div align="center">
<math>~U_\mathrm{int} = \frac{2}{3(\gamma_g-1)} S_\mathrm{therm} \, ,</math>
</div>
in the expression for the free energy when analyzing adiabatic evolutions.

===Illustration===
As is derived in [[SSCpt1/Virial#Virial_Equilibrium|an accompanying discussion]], for a uniform-density, uniformly rotating, spherically symmetric configuration of mass <math>M</math> and radius <math>R</math>,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
W_\mathrm{grav}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \frac{GM^2}{R_0} \biggl( \frac{R}{R_0} \biggr)^{-1} \, ,
</math>
</td>
</tr>
<tr>
<td align="right">
<math>
T_\mathrm{kin}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{4} \frac{J^2}{MR_0^2} \biggl( \frac{R}{R_0} \biggr)^{-2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
V
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4}{3} \pi R_0^3 \biggl( \frac{R}{R_0} \biggr)^{3} \, ,
</math>
</td>
</tr>

</table>
</div>
where, <math>J</math> is the system's total angular momentum and <math>R_0</math> is a reference length scale.

'''Adiabatic Systems''': If, upon compression or expansion, the gaseous configuration behaves adiabatically, the reservoir of thermodynamic energy is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\mathfrak{S}_\mathrm{therm} = U_\mathrm{int} = \frac{M K \rho^{\gamma_g-1}}{(\gamma_g - 1)}
= \frac{M K }{(\gamma_g - 1)} \biggl( \frac{3M}{4\pi R_0^3} \biggr)^{\gamma_g-1} \biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} \, .
</math>
</td>
</tr>

</table>
</div>
Hence, the adiabatic free energy can be written as,
<div align="center">
<math>
\mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
where,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{5} \frac{GM^2}{R_0} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>B</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K}{(\gamma_g-1)} \biggl( \frac{3}{4\pi R_0^3} \biggr)^{\gamma_g - 1} \biggr] M^{\gamma_g} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>C</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{5J^2}{4MR_0^2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>D</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{4}{3} \pi R_0^3 P_e \, .
</math>
</td>
</tr>
</table>
</div>

'''Isothermal Systems''': If, upon compression or expansion, the configuration remains isothermal, [see, also, Appendix A of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler] (1983, ApJ, 268, 16)], the reservoir of thermal energy is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\mathfrak{S}_\mathrm{therm}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M c_s^2\ln \biggl( \frac{\rho}{\rho_0} \biggr) = - 3 M c_s^2 \biggl( \frac{R}{R_0} \biggr) \, .
</math>
</td>
</tr>

</table>
</div>
Hence, the isothermal free energy can be written as,
<div align="center">
<math>
\mathfrak{G} = -A \biggl( \frac{R}{R_0} \biggr)^{-1} - B_I \ln \biggl( \frac{R}{R_0} \biggr) + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
where, aside from the coefficient definitions provided above in association with the adiabatic case,
<div align="center">
<table border="0">
<tr>
<td align="right">
<math>B_I</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
3Mc_s^2 \, .
</math>
</td>
</tr>

</table>
</div>

'''Summary''': We can combine the two cases — adiabatic and isothermal — into a single expression for <math>\mathfrak{G}</math> through a strategic use of the Kroniker delta function, <math>\delta_{1\gamma_g}</math>, as follows:
<div align="center">
<math>
\mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +~ (1-\delta_{1\gamma_g})B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} -~ \delta_{1\gamma_g} B_I \ln \biggl( \frac{R}{R_0} \biggr) +~ C \biggl( \frac{R}{R_0} \biggr)^{-2} +~ D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M</math>), angular momentum (<math>J</math>), and specific entropy (via <math>K</math>) — or, in the isothermal case, sound speed (<math>c_s</math>) — have been specified, the values of all of the coefficients are known and this algebraic expression for <math>\mathfrak{G}</math> describes how the free energy of the configuration will vary with the configuration's relative size (<math>R/R_0</math>) for a given choice of <math>\gamma_g</math>.

==Whitworth (1981) and Stahler (1983)==
The above formulation of a [[#Free_Energy_Expression|Gibbs-like free energy]] has been motivated by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler's (1983, ApJ, 268, 16)] analysis of the stability of isothermal gas clouds, and it closely parallels [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981, MNRAS, 195, 967)] discussion of the "global gravitational stability for one-dimensional polytropes." Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"
<div align="center">
<table border="0">
<tr>
<td align="right">
<math>
\mathfrak{u}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\mathfrak{g} + \mathfrak{B}_\mathrm{in} + \mathfrak{B}_\mathrm{ex}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
 
<math>
- \frac{3}{5} \frac{GM_0^2}{R_0} \biggl(\frac{R}{R_0} \biggr)^{-1}
+ (1-\delta_{1\eta})\biggl[ \frac{KM_0^\eta}{(\eta - 1)} V_0^{(1-\eta)} \biggr] \biggl(\frac{R}{R_0}\biggr)^{3(1-\eta)}
- \delta_{1\eta} \biggl[ 3KM_0 \ln\biggl(\frac{R}{R_0} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
 
<math>
+ P_\mathrm{ex} V_0 \biggl( \frac{R}{R_0} \biggr)^{3}
</math>
</td>
</tr>

</table>
</div>
Clearly Whitworth's global potential function, <math>\mathfrak{u}</math>, is what we have referred to as the configuration's Gibbs-like free energy, with <math>\eta</math> being used rather than <math>\gamma_g</math> to represent the ratio of specific heats in the adiabatic case. Our expression for <math>\mathfrak{G}</math> would precisely match his expression for <math>\mathfrak{u}</math> if we chose to examine the free energy of a nonrotating configuration, that is, if we set <math>C=J=0</math>.

=See Also=
<ul>
<li>Clausius, R. J. E. (1870). "On a Mechanical Theorem Applicable to Heat," [http://www.tandfonline.com/doi/abs/10.1080/.U35XrK1dUys ''Philosophical Magazine, Ser. 4, vol. 40, issue 265], pp. 122-127.</li>
<ul>
<li>[http://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA122&lpg=PA122&dq=Mechanical+Theorem+Applicable+to+Heat&source=bl&ots=dhToAIe8k9&sig=0TaKVTmMnZ5qWvxLkdzuExMoU-U&hl=en&sa=X&ei=YFd-U5qiGdizyASS74CIAw&ved=0CDYQ6AEwAQ#v=onepage&q=Mechanical%20Theorem%20Applicable%20to%20Heat&f=false Google Book Reference]
</li>
<li>
[http://www.elastic-plastic.de/Clausius1870.pdf Alternative, stand-alone document]
</li>
</ul>
<li>[http://en.wikipedia.org/wiki/Virial_theorem Wikipedia discussion of the virial theorem]</li>
<li>[http://ads.harvard.edu/books/1978vtsa.book/ The Virial Theorem in Stellar Astrophysics (1978)] by George W. Collins, II</li>
<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
</ul>

{{ SGFfooter }}

SSCpt2/SolutionStrategies

2021-07-28T16:45:46Z

174.64.8.247: /* Spherically Symmetric Configurations (Part II) */

__FORCETOC__ 

=Spherically Symmetric Configurations (Part II)=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Solution<br />Strategies</b>]]</font>
|}


Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|identified set of simplified governing equations]]. The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is,
 <br />
 <br />
<div align="center">
<math>~\vec{v} = \hat{e}_r v_r = 0 \, .</math>
</div>

After setting the radial velocity, <math>~v_r</math>, and all time-derivatives to zero, we see that the 1<sup>st</sup> (continuity) and 3<sup>rd</sup> (first law of thermodynamics) equations are trivially satisfied while the 2<sup>nd</sup> (Euler) and 4<sup>th</sup> give, respectively,

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,<br />
</div>
and,
<div align="center">
<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .<br />
</div>

(We recognize the first of these expressions as being the statement of [[PGE/ConservingMomentum#Time-independent_Behavior|hydrostatic balance]] appropriate for spherically symmetric configurations.)

We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions — {{Math/VAR_Pressure01}}<math>(r)</math>, {{Math/VAR_Density01}}<math>(r)</math>, and {{Math/VAR_NewtonianPotential01}}<math>(r)</math>. As has been outlined in our discussion of [[SR#Time-Independent_Problems|supplemental relations for time-independent problems]] — and as is discussed further, below — in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{Math/VAR_Pressure01}} and {{Math/VAR_Density01}}.

==Solution Strategies==
When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where {{Math/VAR_Density01}} is not a function of <math>~\theta</math> or <math>~\varphi</math>), the differential mass <math>~dm_r</math> that is enclosed within a spherical shell of thickness <math>~dr</math> is,

<div align="center">
<math>~dm_r = \rho dr \oint dS = r^2 \rho dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\varphi = 4\pi r^2 \rho dr</math> ,
</div>

where we have pulled from the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates] to define the spherical surface element, <math>~dS</math>. Integrating from the center of the spherical configuration <math>~(r=0)</math> out to some finite radius, <math>~r</math>, that is still inside the configuration gives the mass enclosed within that radius, <math>~M_r</math>; specifically,

<div align="center">
<math>~M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
</div>

We can also state that,

<div align="center">
{{Math/EQ_SSmassConservation01}}
</div>

This differential relation is often identified as a statement of mass conservation that replaces the equation of continuity for spherically symmetric, static equilibrium structures.

===Technique 1===
Integrating the Poisson equation once, from the center of the configuration <math>~(r=0)</math> out to some finite radius, <math>~r</math>, that is still inside the configuration, gives,

<div align="center">
<math>
~\int_0^r d\biggl( r^2 \frac{d \Phi}{dr} \biggr) = \int_0^r 4\pi G r^2 \rho dr
</math><br />

<math>
\Rightarrow ~~~~~ r^2 \frac{d \Phi}{dr} \biggr|_0^r = GM_r \, .
</math>
</div>

Now, as long as <math>~d\Phi/dr</math> increases less steeply than <math>~r^{-2}</math> as we move toward the center of the configuration — indeed, we will find that <math>~d\Phi/dr</math> usually goes smoothly to zero at the center — the term on the left-hand-side of this last expression will go to zero at <math>~r=0</math>. Hence, this first integration of the Poisson equation gives,

<div align="center">
<math>
~\frac{d \Phi}{dr} = \frac{G M_r}{r^2} \, .
</math>
</div>

Substituting this expression into the hydrostatic balance equation gives,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>

that is, a single governing integro-differential equation which depends only on the two unknown functions, {{Math/VAR_Pressure01}} and {{Math/VAR_Density01}} .

===Technique 2===
As long as we are examining only barotropic structures, we can replace <math>~dP/\rho</math> by <math>~dH</math> in the hydrostatic balance relation to obtain,

<div align="center">
<math>~\frac{dH}{dr} =- \frac{d\Phi}{dr} \, .</math>
</div>

If we multiply this expression through by <math>~r^2</math> then differentiate it with respect to <math>~r</math>, we obtain,
<div align="center">
<math>\frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) \, ,</math>
</div>

which can be used to replace the left-hand-side of the Poisson equation and give,

<div align="center">
<math>~\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \rho \, ,</math>
</div>

that is, a single second-order governing differential equation which depends only on the two unknown functions, {{Math/VAR_Enthalpy01}} and {{Math/VAR_Density01}}.

<b>Numerical integration examples:</b>
* Isothermal sphere —
** [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden's (1907) tabulation]].
** Tabulation by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar & Wares (1949, ApJ, 109, 551)].
** Outline of [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|our numerical integration scheme]].
* Spherical polytropes —
** [[SSC/Structure/Polytropes#Tabulated_Properties|Some tabulated global properties]].
** Outline of [[SSC/Structure/Polytropes#Straight_Numerical_Integration|our numerical integration scheme]].

===Technique 3===
As in Technique #2, we replace <math>~dP/\rho</math> by <math>~dH</math> in the hydrostatic balance relation, but this time we realize that the resulting expression can be written in the form,

<div align="center">
<math>~\frac{d}{dr}(H+\Phi) = 0 \, .</math>
</div>

This means that, throughout our configuration, the functions {{Math/VAR_Enthalpy01}}<math>(\rho)~</math> and {{Math/VAR_NewtonianPotential01}}<math>(\rho)~</math> must sum to a constant value, call it <math>~C_\mathrm{B}</math>. That is to say, the statement of hydrostatic balance reduces to the ''algebraic'' expression,

<div align="center">
<math>H + \Phi = C_\mathrm{B} \, .</math>
</div>

This relation must be solved in conjunction with the Poisson equation,

<div align="center">
<math>~\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho \, ,</math>
</div>

giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, {{Math/VAR_Enthalpy01}}, {{Math/VAR_Density01}}, and {{Math/VAR_NewtonianPotential01}} to one another.

<b>Self-Consistent Field (SCF) Technique:</b>
* Spherical polytropes —
** [[SSC/Structure/Polytropes#Tabulated_Properties|Some tabulated global properties]].
** Outline of [[SSC/Structure/Polytropes#HSCF_Technique|our implementation of the HSCF scheme]].

=See Also=

* Part I of ''Spherically Symmetric Configurations'': [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|Simplified Governing Equations]]

{{ SGFfooter }}

SSCpt2/IntroductorySummary

2021-07-28T16:44:36Z

174.64.8.247: /* Applications */

__FORCETOC__


=Applications=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:lightgreen;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|Hydrostatic<br />Balance<br />Equations]]</font>
|}
==Spherically Symmetric Configurations==

===Structure:===

Here we show how the set of [[PGE#Principal_Governing_Equations|principal governing equations]] (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds.

====Isolated Configurations====
After supplementing the PGEs by specifying an [[SR#Supplemental_Relations|equation of state of the fluid]], the system of equations is usually solved by employing one of [[SSCpt2/SolutionStrategies#Solution_Strategies|three techniques]] to obtain a "detailed force-balanced" model that provides the radius, <math>R_\mathrm{eq}</math>, of the equilibrium configuration — given its mass, <math>M</math>, and central pressure, <math>P_c</math>, for example — as well as details regarding the internal radial profiles of the fluid density and fluid pressure. If a ''polytropic'' equation of state,
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:FreeNRGpressureRadiusIsothermal.png|250px|right|border|Whitworth's (1981) Isothermal Free-Energy Surface]]
</td>
</tr>
<tr>
<td align="center">
[[SSCpt1/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|Free-Energy Surface]]
</td>
</tr>
</table>

<div align="center">
{{ Math/EQ_Polytrope01 }}
</div>
is adopted, for example, a detailed force-balanced model is fully described by the radially dependent function, <math>\Theta_H(\xi) = [\rho(\xi)/\rho_c]^{1/n}</math>, which is obtained by solving the
<div align="center">
Lane-Emden Equation<br />

{{ Math/EQ_SSLaneEmden01 }}
</div>
As our various discussions illustrate (see the table of contents, below), simply varying the power-law index, {{ Math/MP_PolytropicIndex }}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.

If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression|global free energy]]. An illustrative, undulating free-energy surface is displayed here, on the right; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.

In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem|''scalar virial theorem'']]. Specifically, for [[SSC/Virial/Polytropes#TwoPointsOfView|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W}
\, ,</math>
</td>
</tr>
</table>
</div>
where all three of the dimensionless ''structural form factors'', <math>\mathfrak{f}_M</math>, <math>\mathfrak{f}_W</math>, and <math>\mathfrak{f}_A</math>, are unity, under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, and are otherwise generically ''of order unity'' for detailed force-balanced models having a wide range of internal structures. Alternatively, if the parameter, {{ Math/MP_PolytropicConstant }} (which defines the specific entropy of fluid elements throughout the configuration), rather than the central pressure, is held fixed while searching for extrema in the free-energy, the [[SSC/Virial/Polytropes#TwoPointsOfView|virial equilibrium relation for isolated polytropes]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M^{(n-1)/n} R_\mathrm{eq}^{(3-n)/n} \biggl( \frac{G}{K_\mathrm{n}} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggl(\frac{3}{4\pi \mathfrak{f}_M}\biggr)^{1/n} \, .
</math>
</td>
</tr>
</table>
</div>

====Pressure-Truncated Configurations====
If the physical system under consideration — such as a protostellar gas cloud — is not isolated but is, instead, surrounded and ''truncated'' by a hot, tenuous medium that exerts on the system a confining external pressure, <math>P_e</math>, the configuration's equilibrium parameters will be related via the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W} \biggl[ \mathfrak{f}_A - \frac{P_e}{P_c} \biggr]
\, ;</math>
</td>
</tr>
</table>
</div>
or, fixing {{ Math/MP_PolytropicConstant }} instead of <math>P_c</math>, the [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_2|relevant virial equilibrium expression]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>P_e </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_\mathrm{n} \mathfrak{f}_A \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \cdot \frac{1}{\mathfrak{f}_M} \biggr)^{1 + 1/n}
- \frac{\mathfrak{f}_W}{5} \biggl(\frac{3GM^2}{4\pi R_\mathrm{eq}^4} \cdot \frac{1}{\mathfrak{f}_M^2} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
It is a virial expression specifically of this form <math>(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbow-colored free-energy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, [[SSCpt1/Virial/PolytropesEmbeddedOutline#ASIDE:__Isothermal_Configurations|displayed above]]. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>P_e</math> is greater than some limiting value, <math>P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>P_e = P_\mathrm{crit}</math>, is marked by a red dot on the above-displayed free-energy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]].

After rearranging terms, for any specified values of the parameters <math>P_e</math> and {{ Math/MP_PolytropicConstant }}, this virial equilibrium expression can also be viewed as a [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_3|mass-radius relation of the form]],
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:MassRadiusVirialLabeled.png|250px|right|border|Virial Mass-Radius Relation]]
</td>
</tr>
</table>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>a R_\mathrm{eq}^4 - b M^{(n+1)/n} R_\mathrm{eq}^{(n-3)/n} + c M^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>0\, ,</math>
</td>
</tr>
</table>
</div>
where the constants,
<div align="center">
<math>a \equiv \frac{4\pi}{3} \cdot P_e \, ,</math>      <math>b \equiv \biggl( \frac{3}{4\pi} \biggr)^{1/n} \cdot K_\mathrm{n} \mathfrak{f}_A \mathfrak{f}_M^{-(n+1)/n} \, ,</math>       and      <math>c \equiv \frac{G\mathfrak{f}_W}{5\mathfrak{f}_M^2} \, .</math>
</div>

Using this virial equilibrium relation (and, for illustration purposes, assuming a = b = c = 1), the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation|curves drawn in the figure that is displayed here, on the right]], show how the equilibrium radius of an embedded, pressure-truncated polytropic sphere varies with mass for seven different adopted polytropic indexes. In direct analogy with the critical pressure that is associated with Bonnor-Ebert spheres, for systems having <math>~n \ge 3</math>, there is a mass, <math>M_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, when <math>n > 3</math>, multiple equilibrium configurations having different radii can be constructed for any system having a mass, <math>M < M_\mathrm{max}</math>.

A [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|detailed force-balance analysis of the structure of embedded, pressure-truncated polytropic configurations]] reveals, for each choice of {{ Math/MP_PolytropicIndex }}, a mass-radius relationship that is qualitatively similar to the one deduced from a virial equilibrium analysis. However, the resulting mass-radius relationship is invariably different in detail and quantitatively more correct than the one prescribed by the virial theorem. At its foundation are models whose internal structural profile — of, for example, the fluid pressure and fluid density — is not uniform but, rather, is precisely that which is required to achieve detailed force balance throughout. Hence, we appreciate that even as <math>P_e</math> and {{ Math/MP_PolytropicConstant }} are held fixed, in essence the structural form factors must vary from model to model along the more precise "detailed force-balanced" equilibrium ''sequence.''

====Bipolytropic Configurations====
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:PlotSequencesBest02.png|300px|right|border|Virial Mass-Radius Relation]]
</td>
</tr>
</table>
Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' — also sometimes referred to as ''composite polytropes.'' These are models in which the configuration's "core" is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the configuration's "envelope" is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.

Following the lead of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] — see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon (1998)] — we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface. As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>. Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>. As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme — specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here — there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>. The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch. It is usually referred to as the Schönberg-Chandrasekhar mass limit.

====Details====
In the following table, each green check mark identifies and provides a link to an H_Book chapter that presents a detailed discussion of the topic that is identified on the left — for example, the equilibirum structure of "isolated polytropes" or an "isothermal sphere embedded in an external medium." Mathematical models that provide full solutions to the PGEs, including details regarding the internal structural profiles of equilibrium configurations, are derived in chapters whose check marks fall under the column labeled "Detailed Force-Balance." Insight into the properties of equilibrium systems that is revealed via an analysis of a system's free-energy and the corresponding scalar virial theorem is presented in chapters whose check marks fall under the column labeled "Virial Equilibrium." Motivated by a striking similarity between the sets of model sequences displayed in the two "mass-radius" diagrams shown, above, a link also is provided to a chapter that discusses the relationship between the limiting mass associated with Bonnor-Ebert spheres and the Schönberg-Chandrasekhar mass limit.

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<ul>
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174.64.8.247:

<table border="1" width="100%" cellspacing="2" cellpadding="8">
<tr>
<th align="right" width="50%">
Solution Strategies:    
</th>
<th align="center" width="25%">
<font color="black">Detailed Force-Balance</font><br>([[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Introduction]])
</th>
<th align="center">
<font color="black">Virial Equilibrium</font><br>([[SSCpt1/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|Introduction]])
</th>
</tr>
<tr>
<td align="left" colspan="3">
<table border="0" cellpadding="0" width="100%">
<tr>
<td align="left" colspan="4">Example Solutions: </td>
</tr>
<tr>
<td align="left" colspan="4">
* Uniform-density sphere
</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/VirialEquilibrium/UniformDensity#Uniform-Density_Sphere|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium …</td>
<td align="center" width="25%">[[SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/VirialEquilibrium/UniformDensity#Adiabatic_Evolution_of_Pressure-truncated_Sphere|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Polytropes
</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[SSC/Structure/Polytropes#Polytropic_Spheres|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium …</td>
<td align="center" width="25%">[[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|<font size="+2" color="green">✓</font>]]   [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|<font size="+2">🎦</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Isothermal sphere
</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[SSC/Structure/IsothermalSphere#Isothermal_Sphere|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/Virial/Isothermal#Isothermal_Evolutions|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium (Bonnor-Ebert Sphere) …</td>
<td align="center" width="25%">[[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|<font size="+2" color="green">✓</font>]]<td align="center" width="25%">[[SSC/Virial/Isothermal#Bounded_Isothermal|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Zero-temperature White Dwarf — [[SSC/Structure/WhiteDwarfs#White_Dwarfs|Overview]]
* Power-law density distribution — [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|Overview]]
</td>
</tr>
<tr>
<td align="left" colspan="2">
* Other Analytically Definable Models
</td>
<td align="center" width="25%">[[SSC/Structure/Other_Analytic_Models|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>
<tr>
<td align="left" colspan="2">
* BiPolytropes (also referred to as ''Composite Polytropes'')
</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes#BiPolytropes|Overview]]</td>
<td align="center" width="25%">[[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalization|Overview]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (0,0)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/FreeEnergy0_0#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (5,1)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/FreeEnergy5_1#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (1,5)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (\tfrac{3}{2},3)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic1.5_3|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>

<tr>
<td align="left" colspan="4">
* Limiting Masses — [[SSC/Structure/LimitingMasses|Summary]]
* [[SSC/FreeEnergy/Equilibrium_Sequence_Instabilities|Instabilities Associated with Equilibrium Sequence Turning Points]]
** [[SSC/FreeEnergy/PowerPoint#General_Free-Energy_Expression|Material Underpinning Summary PowerPoint Slides]]
</td>
</tr>
</table>
</td>
</tr>
</table>

Template:SSCstructure

2021-07-28T16:25:47Z

174.64.8.247:

<table border="1" width="100%" cellspacing="2" cellpadding="8">
<tr>
<th align="right" width="50%">
Solution Strategies:    
</th>
<th align="center" width="25%">
<font color="black">Detailed Force-Balance</font><br>([[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Introduction]])
</th>
<th align="center">
<font color="black">Virial Equilibrium</font><br>([[SSCpt1/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|Introduction]])
</th>
</tr>
<tr>
<td align="left" colspan="3">
<table border="0" cellpadding="0" width="100%">
<tr>
<td align="left" colspan="4">Example Solutions: </td>
</tr>
<tr>
<td align="left" colspan="4">
* Uniform-density sphere
</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/VirialEquilibrium/UniformDensity#Uniform-Density_Sphere|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium …</td>
<td align="center" width="25%">[[SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/VirialEquilibrium/UniformDensity#Adiabatic_Evolution_of_Pressure-truncated_Sphere|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Polytropes
</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[SSC/Structure/Polytropes#Polytropic_Spheres|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium …</td>
<td align="center" width="25%">[[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|<font size="+2" color="green">✓</font>]]   [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|<font size="+2">🎦</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Isothermal sphere
</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[SSC/Structure/IsothermalSphere#Isothermal_Sphere|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[SSC/Virial/Isothermal#Isothermal_Evolutions|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium (Bonnor-Ebert Sphere) …</td>
<td align="center" width="25%">[[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|<font size="+2" color="green">✓</font>]]<td align="center" width="25%">[[SSC/Virial/Isothermal#Bounded_Isothermal|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Zero-temperature White Dwarf — [[SSC/Structure/WhiteDwarfs#White_Dwarfs|Overview]]
* Power-law density distribution — [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|Overview]]
</td>
</tr>
<tr>
<td align="left" colspan="2">
* Other Analytically Definable Models
</td>
<td align="center" width="25%">[[SSC/Structure/Other_Analytic_Models|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>
<tr>
<td align="left" colspan="2">
* BiPolytropes (also referred to as ''Composite Polytropes'')
</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes#BiPolytropes|Overview]]</td>
<td align="center" width="25%">[[SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalization|Overview]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (0,0)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/FreeEnergy0_0#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (5,1)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/FreeEnergy5_1#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (1,5)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (\tfrac{3}{2},3)</math> …</td>
<td align="center" width="25%">[[SSC/Structure/BiPolytropes/Analytic1.5_3|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>

<tr>
<td align="left" colspan="4">
* Limiting Masses — [[SSC/Structure/LimitingMasses#Mass_Upper_Limits|Summary]]
* [[SSC/FreeEnergy/Equilibrium_Sequence_Instabilities|Instabilities Associated with Equilibrium Sequence Turning Points]]
** [[SSC/FreeEnergy/PowerPoint#General_Free-Energy_Expression|Material Underpinning Summary PowerPoint Slides]]
</td>
</tr>
</table>
</td>
</tr>
</table>

Template:SSCstructure

2021-07-28T16:19:47Z

174.64.8.247: Created page with "<table border="1" width="100%" cellspacing="2" cellpadding="8"> <tr> <th align="right" width="50%"> Solution Strategies:     </th> <th align="center" width="25..."

<table border="1" width="100%" cellspacing="2" cellpadding="8">
<tr>
<th align="right" width="50%">
Solution Strategies:    
</th>
<th align="center" width="25%">
<font color="black">Detailed Force-Balance</font><br>([[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Introduction]])
</th>
<th align="center">
<font color="black">Virial Equilibrium</font><br>([[SSCpt1/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|Introduction]])
</th>
</tr>
<tr>
<td align="left" colspan="3">
<table border="0" cellpadding="0" width="100%">
<tr>
<td align="left" colspan="4">Example Solutions: </td>
</tr>
<tr>
<td align="left" colspan="4">
* Uniform-density sphere
</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/VirialEquilibrium/UniformDensity#Uniform-Density_Sphere|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/VirialEquilibrium/UniformDensity#Adiabatic_Evolution_of_Pressure-truncated_Sphere|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Polytropes
</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|<font size="+2" color="green">✓</font>]]   [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|<font size="+2">🎦</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Isothermal sphere
</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Isolated …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere|<font size="+2" color="green">✓</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/Virial/Isothermal#Isothermal_Evolutions|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">    </td><td align="left">Embedded in an External Medium (Bonnor-Ebert Sphere) …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|<font size="+2" color="green">✓</font>]]<td align="center" width="25%">[[User:Tohline/SSC/Virial/Isothermal#Bounded_Isothermal|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left" colspan="4">
* Zero-temperature White Dwarf — [[User:Tohline/SSC/Structure/WhiteDwarfs#White_Dwarfs|Overview]]
* Power-law density distribution — [[User:Tohline/SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|Overview]]
</td>
</tr>
<tr>
<td align="left" colspan="2">
* Other Analytically Definable Models
</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/Other_Analytic_Models|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>
<tr>
<td align="left" colspan="2">
* BiPolytropes (also referred to as ''Composite Polytropes'')
</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes#BiPolytropes|Overview]]</td>
<td align="center" width="25%">[[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalization|Overview]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (0,0)</math> …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy0_0#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (5,1)</math> …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy5_1#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">✓</font>]]</td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (1,5)</math> …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>
<tr>
<td align="left">        </td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (\tfrac{3}{2},3)</math> …</td>
<td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/Analytic1.5_3|<font size="+2" color="green">✓</font>]]</td>
<td align="center" width="25%">  …  </td>
</tr>

<tr>
<td align="left" colspan="4">
* Limiting Masses — [[User:Tohline/SSC/Structure/LimitingMasses#Mass_Upper_Limits|Summary]]
* [[User:Tohline/SSC/FreeEnergy/Equilibrium_Sequence_Instabilities|Instabilities Associated with Equilibrium Sequence Turning Points]]
** [[User:Tohline/SSC/FreeEnergy/PowerPoint#General_Free-Energy_Expression|Material Underpinning Summary PowerPoint Slides]]
</td>
</tr>
</table>
</td>
</tr>
</table>

SSCpt2/IntroductorySummary

2021-07-28T16:15:33Z

174.64.8.247: /* Applications */

__FORCETOC__


=Applications=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:lightgreen;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|Hydrostatic<br />Balance<br />Equations]]</font>
|}
==Spherically Symmetric Configurations==

===Structure:===

Here we show how the set of [[PGE#Principal_Governing_Equations|principal governing equations]] (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds.

====Isolated Configurations====
After supplementing the PGEs by specifying an [[SR#Supplemental_Relations|equation of state of the fluid]], the system of equations is usually solved by employing one of [[SSCpt2/SolutionStrategies#Solution_Strategies|three techniques]] to obtain a "detailed force-balanced" model that provides the radius, <math>R_\mathrm{eq}</math>, of the equilibrium configuration — given its mass, <math>M</math>, and central pressure, <math>P_c</math>, for example — as well as details regarding the internal radial profiles of the fluid density and fluid pressure. If a ''polytropic'' equation of state,
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:FreeNRGpressureRadiusIsothermal.png|250px|right|border|Whitworth's (1981) Isothermal Free-Energy Surface]]
</td>
</tr>
<tr>
<td align="center">
[[SSCpt1/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|Free-Energy Surface]]
</td>
</tr>
</table>

<div align="center">
{{ Math/EQ_Polytrope01 }}
</div>
is adopted, for example, a detailed force-balanced model is fully described by the radially dependent function, <math>\Theta_H(\xi) = [\rho(\xi)/\rho_c]^{1/n}</math>, which is obtained by solving the
<div align="center">
Lane-Emden Equation<br />

{{ Math/EQ_SSLaneEmden01 }}
</div>
As our various discussions illustrate (see the table of contents, below), simply varying the power-law index, {{ Math/MP_PolytropicIndex }}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.

If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression|global free energy]]. An illustrative, undulating free-energy surface is displayed here, on the right; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.

In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem|''scalar virial theorem'']]. Specifically, for [[SSC/Virial/Polytropes#TwoPointsOfView|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W}
\, ,</math>
</td>
</tr>
</table>
</div>
where all three of the dimensionless ''structural form factors'', <math>\mathfrak{f}_M</math>, <math>\mathfrak{f}_W</math>, and <math>\mathfrak{f}_A</math>, are unity, under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, and are otherwise generically ''of order unity'' for detailed force-balanced models having a wide range of internal structures. Alternatively, if the parameter, {{ Math/MP_PolytropicConstant }} (which defines the specific entropy of fluid elements throughout the configuration), rather than the central pressure, is held fixed while searching for extrema in the free-energy, the [[SSC/Virial/Polytropes#TwoPointsOfView|virial equilibrium relation for isolated polytropes]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M^{(n-1)/n} R_\mathrm{eq}^{(3-n)/n} \biggl( \frac{G}{K_\mathrm{n}} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggl(\frac{3}{4\pi \mathfrak{f}_M}\biggr)^{1/n} \, .
</math>
</td>
</tr>
</table>
</div>

====Pressure-Truncated Configurations====
If the physical system under consideration — such as a protostellar gas cloud — is not isolated but is, instead, surrounded and ''truncated'' by a hot, tenuous medium that exerts on the system a confining external pressure, <math>P_e</math>, the configuration's equilibrium parameters will be related via the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W} \biggl[ \mathfrak{f}_A - \frac{P_e}{P_c} \biggr]
\, ;</math>
</td>
</tr>
</table>
</div>
or, fixing {{ Math/MP_PolytropicConstant }} instead of <math>P_c</math>, the [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_2|relevant virial equilibrium expression]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>P_e </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_\mathrm{n} \mathfrak{f}_A \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \cdot \frac{1}{\mathfrak{f}_M} \biggr)^{1 + 1/n}
- \frac{\mathfrak{f}_W}{5} \biggl(\frac{3GM^2}{4\pi R_\mathrm{eq}^4} \cdot \frac{1}{\mathfrak{f}_M^2} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
It is a virial expression specifically of this form <math>(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbow-colored free-energy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, [[SSCpt1/Virial/PolytropesEmbeddedOutline#ASIDE:__Isothermal_Configurations|displayed above]]. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>P_e</math> is greater than some limiting value, <math>P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>P_e = P_\mathrm{crit}</math>, is marked by a red dot on the above-displayed free-energy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]].

After rearranging terms, for any specified values of the parameters <math>P_e</math> and {{ Math/MP_PolytropicConstant }}, this virial equilibrium expression can also be viewed as a [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_3|mass-radius relation of the form]],
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:MassRadiusVirialLabeled.png|250px|right|border|Virial Mass-Radius Relation]]
</td>
</tr>
</table>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>a R_\mathrm{eq}^4 - b M^{(n+1)/n} R_\mathrm{eq}^{(n-3)/n} + c M^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>0\, ,</math>
</td>
</tr>
</table>
</div>
where the constants,
<div align="center">
<math>a \equiv \frac{4\pi}{3} \cdot P_e \, ,</math>      <math>b \equiv \biggl( \frac{3}{4\pi} \biggr)^{1/n} \cdot K_\mathrm{n} \mathfrak{f}_A \mathfrak{f}_M^{-(n+1)/n} \, ,</math>       and      <math>c \equiv \frac{G\mathfrak{f}_W}{5\mathfrak{f}_M^2} \, .</math>
</div>

Using this virial equilibrium relation (and, for illustration purposes, assuming a = b = c = 1), the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation|curves drawn in the figure that is displayed here, on the right]], show how the equilibrium radius of an embedded, pressure-truncated polytropic sphere varies with mass for seven different adopted polytropic indexes. In direct analogy with the critical pressure that is associated with Bonnor-Ebert spheres, for systems having <math>~n \ge 3</math>, there is a mass, <math>M_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, when <math>n > 3</math>, multiple equilibrium configurations having different radii can be constructed for any system having a mass, <math>M < M_\mathrm{max}</math>.

A [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|detailed force-balance analysis of the structure of embedded, pressure-truncated polytropic configurations]] reveals, for each choice of {{ Math/MP_PolytropicIndex }}, a mass-radius relationship that is qualitatively similar to the one deduced from a virial equilibrium analysis. However, the resulting mass-radius relationship is invariably different in detail and quantitatively more correct than the one prescribed by the virial theorem. At its foundation are models whose internal structural profile — of, for example, the fluid pressure and fluid density — is not uniform but, rather, is precisely that which is required to achieve detailed force balance throughout. Hence, we appreciate that even as <math>P_e</math> and {{ Math/MP_PolytropicConstant }} are held fixed, in essence the structural form factors must vary from model to model along the more precise "detailed force-balanced" equilibrium ''sequence.''

====Bipolytropic Configurations====
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:PlotSequencesBest02.png|300px|right|border|Virial Mass-Radius Relation]]
</td>
</tr>
</table>
Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' — also sometimes referred to as ''composite polytropes.'' These are models in which the configuration's "core" is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the configuration's "envelope" is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.

Following the lead of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] — see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon (1998)] — we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{User:Tohline/Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface. As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>. Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>. As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme — specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here — there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>. The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch. It is usually referred to as the Schönberg-Chandrasekhar mass limit.

====Details====
In the following table, each green check mark identifies and provides a link to an H_Book chapter that presents a detailed discussion of the topic that is identified on the left — for example, the equilibirum structure of "isolated polytropes" or an "isothermal sphere embedded in an external medium." Mathematical models that provide full solutions to the PGEs, including details regarding the internal structural profiles of equilibrium configurations, are derived in chapters whose check marks fall under the column labeled "Detailed Force-Balance." Insight into the properties of equilibrium systems that is revealed via an analysis of a system's free-energy and the corresponding scalar virial theorem is presented in chapters whose check marks fall under the column labeled "Virial Equilibrium." Motivated by a striking similarity between the sets of model sequences displayed in the two "mass-radius" diagrams shown, above, a link also is provided to a chapter that discusses the relationship between the limiting mass associated with Bonnor-Ebert spheres and the Schönberg-Chandrasekhar mass limit.

{{ SSCstructure }}

{{ SGFfooter }}

SSCpt1/Virial

2021-07-28T16:03:53Z

174.64.8.247: /* Virial Equilibrium of Spherically Symmetric Configurations */

__FORCETOC__


=Virial Equilibrium of Spherically Symmetric Configurations=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|<b>Free-Energy<br />of<br />Spherical<br />Systems</b>]]</font>
|}
==Free Energy Expression==
===Review===
As has been [[VE#Free_Energy_Expression|introduced elsewhere in a more general context]], associated with any isolated, self-gravitating, gaseous configuration we can identify a total Gibbs-like free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots
</math>
</div>
Here, we have explicitly included the gravitational potential energy, <math>W_\mathrm{grav}</math>, the ordered kinetic energy, <math>T_\mathrm{kin}</math>, a term that accounts for surface effects if the configuration of volume <math>V</math> is embedded in an external medium of pressure <math>P_e,</math> and <math>\mathfrak{S}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts. A mathematical expression encapsulating the physical definition of each of these energy terms, in full three-dimensional generality, [[VE#Free_Energy_Expression|can be found in our introductory discussion]] of the scalar virial theorem and the free-energy function.

===Expressions for Various Energy Terms===
We begin, here, by deriving an expression for each of the terms in the free-energy function as appropriate for spherically symmetric systems. In deriving each expression, we keep in mind two issues: First, for a given size system a determination of each term's total contribution to the free energy generally will involve integration through the entire volume of the configuration, effectively "summing up" the differential mass in each radial shell,
<div align="center">
<math>
dm = \rho(\vec{x}) d^3x = 4\pi \rho(r) r^2 dr \, ,
</math>
</div>
weighted by some specific energy expression. Second, each term must be formulated in such a way that it is clear how the energy contribution depends on the overall system size.

====Volume Integrals====
We note, first, that the mass enclosed within each interior radius, <math>r</math>, is
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>M_r(r) = \int\limits_V dm</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_0^r 4\pi r^2 \rho dr \, .</math>
</td>
</tr>
</table>
</div>
Hence, if the volume of the configuration extends out to a radius denoted by <math>R_\mathrm{limit}</math>, the configuration mass is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>M_\mathrm{limit}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_0^{R_\mathrm{limit}} 4\pi r^2 \rho dr \, .</math>
</td>
</tr>
</table>
</div>

<table align="center" border="1" width="80%" cellpadding="8">
<tr><td align="left">
NOTE: The following considerations have led us to formally draw a distinction between <math>M_\mathrm{limit}</math> and the "total" mass, <math>M_\mathrm{tot}</math>, that we use (see below) for normalization.

<font color="maroon"><b>Isolated Polytropes</b></font>: For [[SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|isolated polytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will be the natural edge of the configuration, where the pressure and mass-density drop to zero. In this case, <math>M_\mathrm{limit}</math> quite naturally corresponds to the total mass of the configuration.

<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|configuration embedded in an external medium]] of pressure, <math>P_e</math>, will have a (pressure-truncated) surface whose radius, <math>R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>P_e</math>. In this case as well, one might choose to refer to <math>M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>M_\mathrm{limit}</math> from <math>M_\mathrm{tot}</math>, continuing to rely on <math>M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.

<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalization|bipolytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.
</td></tr>
</table>

<font color="red">Confinement by External Pressure:</font> For spherically symmetric configurations, the energy term due to confinement by an external pressure can be expressed, simply, in terms of the configuration's radius, <math>R_\mathrm{limit}</math>, as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>P_e V</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>P_e \int_0^{R_\mathrm{limit}} 4\pi r^2 dr = \frac{4\pi}{3} P_e R_\mathrm{limit}^3 \, .</math>
</td>
</tr>
</table>
</div>

<font color="red">Gravitational Potential Energy:</font> From our discussion of the [[VE#Scalar_Virial_Theorem|scalar virial theorem]] — see, specifically, the reference to Equation (18), on p. 18 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] — the gravitational potential energy is given by the expression,
<div align="center">
<math>
W_\mathrm{grav} = - \int\limits_V \rho x_i \frac{\partial\Phi}{\partial x_i} d^3 x
= - \int\limits_V \vec{r} \cdot \nabla\Phi dm = - \int_0^{R_\mathrm{limit}} \biggl( r \frac{d\Phi}{dr} \biggr) dm \, .
</math>
</div>
For spherically symmetric systems, the

<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{ Template:Math/EQ_Poisson01 }}
</div>
becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\frac{1}{r^2} \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi G \rho(r) \, , </math>
</td>
</tr>
</table>
</div>
which implies,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>r^2 \frac{d\Phi}{dr} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_0^r 4\pi G \rho(r) r^2 dr = GM_r(r) \, .</math>
</td>
</tr>
</table>
</div>
<span id="Wgrav">Hence</span> — see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] — the desired expression for the gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>W_\mathrm{grav}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) dm = - \int_0^{R_\mathrm{limit}} \frac{G}{r}\biggl[\int_0^r 4\pi r^2 \rho dr \biggr] 4\pi r^2 \rho dr \, .</math>
</td>
</tr>
</table>
</div>

<div id="AlternateGravPotEnergy">
<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
Also, as pointed out by [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] — see p. 64, Equation (16) — it may sometimes prove advantageous to recognize that, if a spherically symmetric system is in hydrostatic balance, an alternate expression for the total gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>W_\mathrm{grav}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>+ \frac{1}{2} \int_0^{R_\mathrm{limit}} \Phi(r) dm \, .</math>
</td>
</tr>
</table>
</div>
</td></tr>
</table>
</div>

<div>
<font color="red">Rotational Kinetic Energy:</font> We will also consider a system that is rotating with a specified [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple'' angular velocity profile]], <math>\dot\varphi(\varpi)</math>, in which case, from our discussion of the [[VE#Scalar_Virial_Theorem|scalar virial theorem]] — see, specifically, the reference to Equation (8), on p. 16 of [[Appendix/References#EFE|EFE]] — the (ordered) kinetic energy,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>T_\mathrm{kin}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \int\limits_V \rho |\vec{v} |^2 d^3x = \frac{1}{2} \int\limits_V |\vec{v} |^2 dm \, ,</math>
</td>
</tr>
</table>
</div>
is entirely rotational kinetic energy, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>T_\mathrm{kin} = T_\mathrm{rot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \int\int\int \dot\varphi^2 \varpi^2 dm
= \frac{1}{2} \int_0^{R_\mathrm{limit}} \dot\varphi^2 \varpi^2 \int_{-\sqrt{{R_\mathrm{limit}}^2 - \varpi^2}}^{\sqrt{{R_\mathrm{limit}}^2 - \varpi^2}} \rho(r(\varpi,z)) 2\pi \varpi d\varpi dz\, .</math>
</td>
</tr>
</table>
</div>

<font color="red">Reservoir of Thermodynamic Energy:</font> As has been explained in [[VE#Reservoir_of_Thermodynamic_Energy|our introductory discussion of the Gibbs-like free energy]], formulation of an expression for the reservoir of thermodynamic energy, <math>\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally. For [[VE#Isothermal_Systems|isothermal systems]],
<div align="center" id="Reservoir">
<math>
\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~\mathfrak{S}_I
= + \int\limits_V c_s^2 \ln \biggl(\frac{\rho}{\rho_\mathrm{norm}}\biggr) dm
= c_s^2 \int_0^{R_\mathrm{limit}} \ln \biggl(\frac{\rho}{\rho_\mathrm{norm}}\biggr) 4\pi r^2 \rho dr \, ,
</math>
</div>
where, <math>c_s</math> is the isothermal sound speed and <math>\rho_\mathrm{norm}</math> is a (as yet unspecified) reference mass density; while, for [[VE#Adiabatic_Systems|adiabatic systems]],
<div align="center">
<math>
\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~ \mathfrak{S}_A
= + \int\limits_V \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) dm
= \frac{1}{({\gamma_g}-1)} \int_0^{R_\mathrm{limit}} 4\pi r^2 P dr
\, ,</math>
</div>
where, <math>P(r)</math> is the system's pressure distribution and <math>\gamma_g</math> is the specified adiabatic index.

====Normalizations====

=====Our Choices=====
It is appropriate for us to define some characteristic scales against which various physical parameters can be normalized — and, hence, their relative significance can be specified or measured — as the free energy of various systems is examined. As the system size is varied in search of extrema in the free energy, we generally will hold constant the total system mass and the specific entropy of each fluid element. (When isothermal rather than adiabatic variations are considered, the sound speed rather than the specific entropy will be held constant.) Hence, following the lead of both [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970, MNRAS, 151, 81)] and [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981, MNRAS, 195, 967)], we will express the various characteristic scales in terms of the constants, <math>G, M_\mathrm{tot},</math> and the polytropic constant, <math>K.</math> Specifically, we will normalize all length scales, pressures, energies, mass densities, and the square of the speed of sound by, respectively,
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="2">
Adopted Normalizations
</th></tr>
<tr>
<td align="center">
Adiabatic Cases
</td>
<td align="center">
Isothermal Case
<math>~(\gamma = 1; K = c_s^2)</math>
</td>
</tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr]^{1/(4-3\gamma)} </math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[ \frac{K^4}{G^{3\gamma} M_\mathrm{tot}^{2\gamma}} \biggr]^{1/(4-3\gamma)} </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl[ KG^{3(1-\gamma)}M_\mathrm{tot}^{6-5\gamma} \biggr]^{1/(4-3\gamma)} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(4-3\gamma )} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl[ \frac{K}{(G^3 M_\mathrm{tot}^2)^{\gamma-1}} \biggr]^{1/(4-3\gamma )} </math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{G M_\mathrm{tot}}{c_s^2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{c_s^8}{G^{3} M_\mathrm{tot}^{2}} </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>M_\mathrm{tot} c_s^2 </math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{3}{4\pi} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) c_s^2 </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><td align="left" colspan="2">
Note that, given the above definitions, the following relations hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</td></tr>
</table>
</div>

It should be emphasized that, as we discuss how a configuration's free energy varies with its size, the variable <math>R_\mathrm{limit}</math> will be used to identify the configuration's size ''whether or not the system is in equilibrium,'' and the parameter,
<div align="center">
<math>\chi \equiv \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \, ,</math>
</div>
will be used to identify the size as referenced to <math>R_\mathrm{norm}</math>. When an equilibrium configuration is identified <math>(R_\mathrm{limit} \rightarrow R_\mathrm{eq})</math>, we will affix the subscript "eq," specifically,
<div align="center">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, .</math>
</div>

=====Choices Made by Other Researchers=====

As is detailed in a [[SSC/Structure/PolytropesEmbedded#General_Properties|related discussion]], our definitions of <math>R_\mathrm{norm}</math> and <math>P_\mathrm{norm}</math> are close, but not identical, to the scalings adopted by [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970, MNRAS, 151, 81)] and by [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981, MNRAS, 195, 967)]. The following relations can be used to switch from our normalizations to theirs:
<div align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">[[SSC/Structure/PolytropesEmbedded|Hoerdt's (1970)]] Normalization</th><tr>

<tr>
<td align="right">
<math>\biggl( \frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl( \frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3\gamma} \biggl( \frac{1}{4\pi} \biggr)^{\gamma}</math>
</td>
</tr>
</table>

</td>
<td align="center">
     
</td>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">[[SSC/Structure/PolytropesEmbedded|Whitworth's (1981)]] Normalization</th><tr>

<tr>
<td align="right">
<math>\biggl( \frac{R_\mathrm{rf}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{5\pi} \biggl( \frac{4\pi}{3} \biggr)^\gamma</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl( \frac{P_\mathrm{rf}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2^{-2(4+\gamma)} \biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr)^\gamma</math>
</td>
</tr>
</table>

</td>
</tr>
</table>
</div>

It is also worth noting how the length-scale normalization that we are adopting here relates to the characteristic length scale,
<div align="center">
<math>a_n \equiv \biggl[ \frac{1}{4\pi G} \biggl( \frac{H_c}{\rho_c} \biggr) \biggr]^{1/2} \, ,</math>
</div>
that has classically been adopted in the context of the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden equation]], the solution of which provides a detailed description of the internal structure of spherical polytropes for a wide range of values of the polytropic index, {{ Template:Math/MP_PolytropicIndex }}.
Recognizing that, via the [[SR#Barotropic_Structure|polytropic equation of state]], the pressure, density, and enthalpy of every element of fluid are related to one another via the expressions,
<div align="center">
<math>H\rho = (n+1)P</math>     … and
…      <math>P = K_n\rho^{1+1/n} \, ,</math>
</div>
the specific enthalpy at the center of a polytropic sphere, <math>H_c/\rho_c</math>, can be rewritten in terms of {{ Template:Math/MP_PolytropicConstant }} and <math>\rho_c</math> to give,
<div align="center">
<math>~a_n = \biggl[ \frac{(n+1)K_n}{4\pi G} \rho_c^{(1/n) -1} \biggr]^{1/2} \, ,</math>
</div>
which is the definition of this classical length scale introduced by [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] (see, specifically, his equation 10 on p. 87). Switching from {{ Template:Math/MP_PolytropicIndex }} to the associated adiabatic exponent via the relation, <math>\gamma = 1+1/n ~~~\Rightarrow~~~ n = 1/(\gamma-1)</math>, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K_n \rho_c^{(\gamma-2)}}{4\pi G} \cdot \frac{1}{R_\mathrm{norm}^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{4\pi}\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K_n }{G} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{(\gamma-2)}
\biggl( \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr)^{(\gamma-2)}
\cdot \frac{1}{R_\mathrm{norm}^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{3}{4\pi } \cdot \frac{\rho_c}{\bar\rho} \biggr)^{\gamma-2}
\biggl[ \frac{K_n M_\mathrm{tot}^{\gamma-2} }{G} \biggr]
\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}} \biggr)^{3{(\gamma-2)}}
\cdot \frac{1}{R_\mathrm{norm}^{3\gamma-4}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{3}{4\pi } \cdot \frac{\rho_c}{\bar\rho} \biggr)^{2-\gamma}
\chi_\mathrm{eq}^{6-3\gamma}
\biggl[ \frac{K_n M_\mathrm{tot}^{\gamma-2} }{G} \biggr]
\cdot \biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{3}{4\pi } \cdot \frac{\rho_c}{\bar\rho} \biggr)^{2-\gamma}
\chi_\mathrm{eq}^{6-3\gamma} \, .
</math>
</td>
</tr>
</table>
</div>

Notice that, written in this manner, the scale length, <math>a_n</math>, cannot actually be determined unless the normalized equilibrium radius, <math>\chi_\mathrm{eq}</math>, is known. We will encounter analogous situations whenever the free energy function is used to identify the physical parameters that define equilibrium configurations — key attributes of a system that should be held fixed as the system size (or some other order parameter) is varied cannot actually be evaluated until an extremum in the free energy is identified and the corresponding value of <math>\chi_\mathrm{eq}</math> is known. Because solutions of the Lane-Emden equation directly provide detailed force-balance models of polytropic spheres, [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] did not encounter this issue. As we have [[SSC/Structure/Polytropes#Known_Analytic_Solutions|discussed elsewhere]], the equilibrium radius of a polytropic sphere is identified as the radial location,
<div align="center">
<math>\xi_1 = \frac{R_\mathrm{eq}}{a_n} \, ,</math>
</div>
at which the Lane-Emden function, <math>\Theta_H(\xi)</math>, first goes to zero. Bypassing the free-energy analysis and using knowledge of <math>\xi_1</math> to identify the equilibrium radius — specifically, setting,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\chi_\mathrm{eq}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \xi_1 \biggl(\frac{a_n}{R_\mathrm{norm}} \biggr) \, ,</math>
</td>
</tr>
</table>
</div>

we can extend the above analysis to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{4\pi }{3} \cdot \frac{\rho_c}{\bar\rho} \biggr)^{2-\gamma}
\biggl[ \xi_1 \biggl(\frac{a_n}{R_\mathrm{norm}} \biggr) \biggr]^{6-3\gamma}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~~\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl( \frac{\gamma-1}{\gamma} \biggr) \biggl( \frac{4\pi }{3} \cdot \frac{\rho_c}{\bar\rho} \cdot \xi_1^3\biggr)^{\gamma-2}
\, .
</math>
</td>
</tr>
</table>
</div>

====Implementation====
=====Normalize=====
We will now judiciously introduce our adopted normalizations into the [[#Expressions_for_Various_Energy_Terms|above-defined free-energy term expressions]], using asterisks to denote dimensionless variables that have been accordingly normalized; for example,
<div align="center">
<math>
r^* \equiv \frac{r}{R_\mathrm{norm}} \, , ~~~~~~ P^* \equiv \frac{P}{P_\mathrm{norm}} \, , ~~~~~~ </math>
        and       <math>\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}} \, .
</math>
</div>

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^*) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R_\mathrm{norm}^3 \rho_\mathrm{norm} \int_0^{r^*} 4\pi (r^*)^2 \rho^* dr^*
= M_\mathrm{tot} \int_0^{r^*} 3(r^*)^2 \rho^* dr^* \, .
</math>
</td>
</tr>
</table>
</div>

<font color="red">Confinement by External Pressure (Normalized Volume):</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>P_e V</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} \biggl[ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)
\biggl(\frac{R_\mathrm{limit}}{R_\mathrm{norm}}\biggr)^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>

<font color="red">Normalized Gravitational Potential Energy:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>W_\mathrm{grav}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 4\pi GM_\mathrm{tot} R_\mathrm{norm}^2 \rho_\mathrm{norm} \int_0^{\chi=R_\mathrm{limit}^*} \biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^*
</math>
</td>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- E_\mathrm{norm} \int_0^{\chi = R_\mathrm{limit}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^* \, .
</math>
</td>
</tr>
</table>
</div>

<font color="red">Normalized Reservoir of Thermodynamic Energy:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{S}_I</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} \int_0^{\chi=R_\mathrm{limit}^*} 3 \ln (\rho^*) (r^*)^2 \rho^* dr^* \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{S}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{E_\mathrm{norm}}{({\gamma_g}-1)} \int_0^{\chi=R_\mathrm{limit}^*} 4\pi (r^*)^2 P^* dr^* \, .</math>
</td>
</tr>
</table>
</div>

<font color="red">Normalized Rotational Kinetic Energy:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>T_\mathrm{rot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\pi \dot\varphi_c^2 R_\mathrm{norm}^5 \rho_\mathrm{norm}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3 d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}} (\rho^*) dz^*
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{5^2\pi}{2^2} \biggr) \biggl[ \frac{J^2 R_\mathrm{norm} \rho_\mathrm{norm}}{M_\mathrm{tot}^2} \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3 d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}} (\rho^*) dz^*
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3\cdot 5^2}{2^4} \biggr) \biggl[ \frac{J^2}{M_\mathrm{tot}} \biggl(\frac{E_\mathrm{norm} }{G M_\mathrm{tot}^2 }\biggr)^2 \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3 d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}} (\rho^*) dz^*
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm}
\biggl( \frac{3^2\cdot 5^2}{2^6 \pi} \biggr) \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3 d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}} (\rho^*) dz^* \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\dot\varphi_c \equiv \frac{5J}{2M_\mathrm{tot} R_\mathrm{eq}^2} =
\frac{5}{2} \biggl[ \frac{J}{M_\mathrm{tot} R_\mathrm{norm}^2} \biggr] \chi_\mathrm{eq}^{-2} \, ,</math>
</div>
is a characteristic rotation frequency in the equilibrium configuration whose value is set once the system's total angular momentum, <math>J</math>, is specified.

=====Separate Time & Space=====
Our intent is to vary the size of the configuration <math>(R_\mathrm{limit})</math> while holding the (properly normalized) internal structural profile fixed, so let's separate the spatial integral over the (fixed) structural profile from the time-varying configuration size. Making use of the dimensionless ''internal'' coordinates,
<div align="center">
<math>x \equiv \frac{r}{R_\mathrm{limit}} \, ,~~~~w \equiv \frac{\varpi}{R_\mathrm{limit}} \, ,
~~~~\zeta \equiv \frac{z}{R_\mathrm{limit}} \, ,
</math>
</div>
that always run from zero to one, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^*</math>
</td>
<td align="center">
<math>\rightarrow~</math>
</td>
<td align="left">
<math>
x \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \biggr) = x \chi \, ;
</math>
   and, likewise,    
<math>
~~~~\varpi^* ~\rightarrow~ w \chi \, ;
~~~~z^* ~\rightarrow~ \zeta \chi \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^*</math>
</td>
<td align="center">
<math>\rightarrow~</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl( \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr)
= \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl( \frac{M_\mathrm{limit}/R_\mathrm{limit}^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr)
= \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \chi^{-3}
= \frac{\rho_c}{\bar\rho} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \chi^{-3} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^*</math>
</td>
<td align="center">
<math>\rightarrow~</math>
</td>
<td align="left">
<math>
\biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{K\rho_c^\gamma}{P_\mathrm{norm}} \biggr)
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{\rho_c}{\bar\rho} \biggr)^\gamma
\biggl[ \frac{(3M_\mathrm{limit}/4\pi R_\mathrm{limit}^3)^\gamma}{K^{-1}P_\mathrm{norm}} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
    
</td>
<td align="left">
<math>
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^\gamma
\biggl[ \frac{K M_\mathrm{tot}^\gamma}{P_\mathrm{norm} R_\mathrm{norm}^{3\gamma}} \biggr] \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \biggr)^{-3\gamma}
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^\gamma \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\chi^{-3\gamma} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\dot\varphi}{\dot\varphi_c}</math>
</td>
<td align="center">
<math>\rightarrow~</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \biggl( \frac{\dot\varphi_\mathrm{limit}}{\dot\varphi_c}\biggr)
= \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{eq}}\biggr)^{-2}
= \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \chi_\mathrm{eq}^{2} \chi^{-2} \, .
</math>
</td>
</tr>
</table>
</div>

=====Summary of Normalized Expressions=====
Hence, our normalized expressions become,
<div align="center">
<table border="1" cellpadding="8">
<tr><th align="center">
Normalized Expressions
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P_e V}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi^3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr] \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx\biggr\} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1} 3x^2 \biggl[ \frac{P(x)}{P_c} \biggr] dx \biggr\} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_0^{1}
\biggl\{ \ln \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] -3\ln \biggl[ \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr] \biggr\}
3 x^2 \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-3 \ln \chi + \mathrm{constant} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^6 \pi} \biggr) \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq}
\int_0^{1} \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{edge}} \biggr]^2 w^3 dw
\int_{-\sqrt{1 - w^2}}^{\sqrt{1 - w^2}} \biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr] d\zeta \, .
</math>
</td>
</tr>
</table>

</td></tr>
<tr><td align="left>
<font color="red">NOTE to self (21 September 2014)<b></b></font>: The expressions for <math>\mathfrak{S}_I</math> and <math>T_\mathrm{rot}</math> may not properly account for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>.
</td></tr>
</table>
</div>

It should be emphasized that the coefficient involving the density ratio, <math>(\rho_c/\bar\rho)</math>, that lies outside of the integral in most of these expressions depends only on the internal structure, and not the overall size, of the configuration. It can therefore be evaluated at any time. We usually will choose to evaluate this coefficient in an equilibrium state, that is, when <math>R_\mathrm{limit} \rightarrow R_\mathrm{eq}</math>. Accordingly, the subscript "eq" has been attached to this coefficient. The inverse of this density ratio can be obtained from the integral expression for <math>M_r</math> by recognizing that <math>M_r \rightarrow M_\mathrm{limit}</math> when the upper limit on the integral <math>x \rightarrow 1</math>. Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl(\frac{\rho_c}{\bar\rho} \biggr)^{-1}_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]_\mathrm{eq} dx \, .</math>
</td>
</tr>
</table>
</div>
This coefficient also may be rewritten in terms of the central pressure in the equilibrium state; specifically, using a sequence of steps similar to the ones that were used, above, in rewriting <math>P^*</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>
\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi^{3\gamma} \biggr]_\mathrm{eq} \, .</math>
</td>
</tr>
</table>
</div>

=====Looking Ahead to Bipolytropes=====
<div id="BiPolytrope">
<table border="1" align="center" width="90%" cellpadding="20">
<tr><td align="left">
<b><font color="purple">ASIDE:</font></b> When we discuss the free energy of bipolytropic configurations, we will need to divide the expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math> into two parts — one accounting for the reservoir of thermodynamic energy in the bipolytrope's "core" and one accounting for the reservoir of thermodynamic energy in the bipolytrope's "envelope." It is useful to develop this two-part expression here, while the definition of <math>\mathfrak{S}_A</math> is fresh in our minds and to show how the two-part expression reduces to the simpler expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math>, just derived, when there is no distinction drawn between the properties of the core and the envelope.

In what follows, we will use the subscript ''core'' (or "c") when referencing physical properties of the bipolytrope's core and the subscript ''env'' (or "e") for the envelope; and, as above, we will use <math>x \equiv r/R_\mathrm{edge}</math> to denote the dimensionless radial location within a configuration of radius, <math>R_\mathrm{edge}</math>. The dimensionless radial coordinate, <math>q \equiv x_i = r_i/R_\mathrm{edge}</math>, will identify the radial ''interface'' where the core meets the envelope; that is, <math>q</math> will identify both the outer edge of the core and the inner edge of the envelope. In general, separate expressions will define the run of pressure through the core and through the envelope. We can assume that, for the core, the pressure drops monotonically from a value of <math>P_0</math> at the center of the configuration according to an expression of the form,
<div align="center">
<math>P_\mathrm{core}(x) = P_0 [1 - p_c(x)]</math>      for      <math>0 \leq x \leq q \, ,</math>
</div>
and that, for the envelope, the pressure drops monotonically from a value of <math>P_{ie}</math> at the interface according to an expression of the form,
<div align="center">
<math>P_\mathrm{env}(x) = P_{ie} [1 - p_e(x)]</math>      for      <math>q \leq x \leq 1 \, ,</math>
</div>
where <math>p_c(x)</math> and <math>p_e(x)</math> are both dimensionless functions that will depend on the equations of state that are chosen for the core and envelope, respectively. By prescription, the pressure in the envelope must drop to zero at the surface of the bipolytropic configuration, hence, we should expect that <math>p_e(1) = 1</math>. Furthermore, by prescription, the pressure in the core will drop to a value, <math>P_{ic}</math>, at the interface, so we can write,
<div align="center">
<math>P_{ic} = P_0 [1 - p_c(q)] \, .</math>
</div>

In equilibrium — that is, when <math>R_\mathrm{edge} = R_\mathrm{eq}</math> — we will demand that the pressure at the interface be the same, whether it is referenced in the core or in the envelope, that is, we will demand that <math>P_{ic} = P_{ie} \, .</math> It will therefore prove to be strategically advantageous to rewrite the expression for the run of pressure through the core in terms of the pressure at the interface rather than in terms of the central pressure; specifically,
<div align="center">
<math>P_\mathrm{core}(x) = P_{ic} \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] \, .</math>
</div>
Referencing these prescriptions for <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math>, the two-part expression for the reservoir of thermodynamic energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{({\gamma_c}-1)} \int_0^{r_i/R_\mathrm{norm}} 4\pi (r^*)^2 P^*_\mathrm{core} dr^*
+ \frac{1}{({\gamma_e}-1)} \int_{r_i/R_\mathrm{norm}}^\chi 4\pi (r^*)^2 P^*_\mathrm{env} dr^*
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi \chi^3 }{({\gamma_c}-1)} \biggl[ \frac{P_{ic}}{P_\mathrm{norm}} \biggr] \int_0^q \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx
+ \frac{4\pi \chi^3 }{({\gamma_e}-1)} \biggl[ \frac{P_{ie}}{P_\mathrm{norm}} \biggr] \int_q^1 \biggl[1 - p_e(x) \biggr] x^2 dx \, .
</math>
</td>
</tr>
</table>
</div>
As is implied by the subscripts on the adiabatic exponents that appear in the leading factor of each of the two terms, we are assuming that, as the bipolytropic system expands or contracts, the thermodynamic properties of the material in the envelope will vary as prescribed by an adiabat of index, <math>\gamma_e</math>, while the thermodynamic properties of material in the core will vary as prescribed by a, generally different, adiabat of index, <math>\gamma_c</math>. Therefore, as the radius of the bipolytropic configuration, <math>R_\mathrm{edge}</math>, is varied, the density of each fluid element will vary and, in the core, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_c}</math> while, in the envelope, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_e}</math>. If we furthermore assume that the mass in the core and the mass in the envelope remain constant during a phase of contraction or expansion, the density of each fluid element will vary as <math>R_\mathrm{edge}^{-3}</math>, whether the material is associated with the core or with the envelope. Therefore, using the subscript, "eq," to identify the value of thermodynamic quantities when the system is in an equilibrium state and, accordingly, <math>R_\mathrm{edge} = R_\mathrm{eq}</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{core}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_c} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{env}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_e} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} \, .</math>
</td>
</tr>
</table>
</div>
In particular, for any <math>R_\mathrm{edge}</math>, material associated with the core that lies at the interface will have a pressure given by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>P_{ic}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c}
= (P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_c}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_c}
= (P_{ic})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_c} \chi^{-3\gamma_c}
\, ,</math>
</td>
</tr>
</table>
</div>
while material associated with the envelope that lies at the interface will have a pressure given by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>P_{ie}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e}
= (P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_e}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_e}
= (P_{ie})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_e} \chi^{-3\gamma_e}
\, .</math>
</td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c}
\int_0^q \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ ~\frac{4\pi }{({\gamma_e}-1)} \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_e}
\int_q^1 \biggl[1 - p_e(x) \biggr] x^2 dx \, .
</math>
</td>
</tr>
</table>
</div>
----

Now, let's see how this expression simplifies if <math>P_{ie} = P_{ic}</math> and <math>\gamma_e = \gamma_c</math> and, hence, the properties of the envelope are indistinguishable from the properties of the core. We note, first, that in this limit, <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math> must be identical functions of <math>x</math>, that is, it must be the case that <math>p_e(x)</math> is related to <math>p_c(x)</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>1 - p_e(x) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1 - p_c(x)}{1-p_c(q)} \, .</math>
</td>
</tr>
</table>
</div>

We therefore obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c}
\biggl\{ \int_0^q \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx + \int_q^1 \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] x^2 dx \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)} \biggl[ \frac{P_0 \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c}
\biggl\{ \int_0^1 \biggl[1 - p_c(x)\biggr] x^2 dx \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi }{({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma}
\int_0^{1} \biggl[ \frac{P(x)}{P_c} \biggr] x^2 dx \biggr\} \, ,
</math>
</td>
</tr>
</table>
</div>
as desired.

</td></tr>
</table>
</div>

====Idealized Configuration====
(For simplicity throughout this subsection, we will assume that the mass enclosed within the configuration's limiting radius, <math>M_\mathrm{limit}</math>, equals the normalization mass, <math>M_\mathrm{tot}</math>.) In the idealized situation of a configuration that has uniform density, <math>\rho(x) = \rho_c</math> — and, hence, the density ratio <math>\rho_c/\bar\rho = 1</math> — the mass interior to each radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_0^{x} 3x^2 dx = x^3 \, ,</math>
</td>
</tr>
</table>
</div>
and the normalized gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \int_0^{1} 5x \biggl\{ x^3\biggr\} dx = -\frac{3}{5} \chi^{-1} \, .
</math>
</td>
</tr>
</table>
</div>

If, in addition, the configuration is uniformly rotating with angular velocity, <math>\dot\varphi = \dot\varphi_\mathrm{edge}</math>, and has uniform pressure, <math>P_c</math>, evaluation of the ordered kinetic energy and thermodynamic energy integrals yields,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm} }</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^6\pi} \biggr) \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
\int_0^{1} w^3 dw
\int_{0}^{\sqrt{1 - w^2}} d\zeta
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^5\pi} \biggr) \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
\int_0^1 w^3 (1-w^2)^{1/2} dw
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^5\pi} \biggr)\biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
\biggl[ -\frac{1}{15} (1-w^2)^{3/2} (3w^2 +2) \biggr]_0^1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\chi^{-2}
\biggl( \frac{3\cdot 5}{2^4 \pi} \biggr) \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi }{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl(\frac{3}{4\pi} \biggr)^{\gamma}\int_0^{1} 3x^2 dx \biggr\}
= \frac{1}{({\gamma_g}-1)} \biggl(\frac{3}{4\pi} \biggr)^{\gamma-1} \chi^{3-3\gamma}
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-3 \ln \chi + \mathrm{constant} \, ,
</math>
</td>
</tr>
</table>
</div>
where the various dimensionless integration variables are, <math>x \equiv (r/R)</math>, <math>\zeta \equiv (z/R)</math>, and <math>w \equiv (\varpi/R)</math>.

====Structural Form Factors====
Keeping in mind the expressions that arise in the case of our just-defined, idealized configuration, in more realistic cases we generally will write each energy term as follows:
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3\cdot 5}{2^4 \pi} \biggr)\chi^{-2} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \cdot \frac{\mathfrak{f}_T}{\mathfrak{f}_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \biggr]_\mathrm{eq}^{\gamma}
\cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma}} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq} \cdot \mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>

<span id="FormFactors">where the dimensionless form factors, <math>\mathfrak{f}_i</math>, which are assumed to be independent of the overall configuration size and will each usually of order unity, are</span>,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_T</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{15}{2} \int_0^1 \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{edge}} \biggr]^2 w^3 dw \int_0^{\sqrt{1 - w^2}}
\biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr] d\zeta\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>
</td>
</tr>

</table>
</div>
In each case, the "idealized" energy expression is retrieved if/when the relevant form factor, <math>\mathfrak{f}_i</math>, is set to unity.

====Some Detailed Examples====

In an [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|accompanying discussion]], we derive detailed expressions for a selected subset of the above structural form factors and corresponding energy terms in the case of spherically symmetric configurations that obey an {{ Template:Math/MP_PolytropicIndex }} = 5 or an {{ Template:Math/MP_PolytropicIndex }} = 1 polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should help with the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases, which cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|cores of bipolytropes]].

===Gathering it All Together===
Gathering all of the terms together we find that, to within an additive constant, the expression for the normalized free energy is,
<div align="center">
<math>
\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}} =
-3A\chi^{-1} -~ \frac{(1-\delta_{1\gamma_g})}{(1-\gamma_g)} B \chi^{3-3\gamma_g} -~ \delta_{1\gamma_g} 3\ln \chi
+~ C \chi^{-2} +~ D\chi^3 \, ,
</math>
</div>
where,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>B</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{\gamma}
\cdot \mathfrak{f}_A
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3}
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq}
\cdot \mathfrak{f}_A \, ,
</math>
</td>
</tr>
<tr>
<td align="right">
<math>C</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \cdot \frac{\mathfrak{f}_T}{\mathfrak{f}_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>D</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .
</math>
</td>
</tr>
</table>
</div>

Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M_\mathrm{tot}</math>), angular momentum (<math>J</math>), and specific entropy (via {{ Template:Math/MP_PolytropicConstant }}) — or, in the isothermal case, sound speed (<math>c_s</math>) — have been specified, the values of all of the coefficients are known and the above algebraic expression for <math>\mathfrak{G}^*</math> describes how the free energy of the configuration will vary with the configuration's size (<math>\chi</math>) for a given choice of <math>\gamma_g</math>.

==Visual Representation==
<div align="center">
<table border="2" cellpadding="8">
<tr>
<td align="center" colspan="2">
'''Figure 1:''' <font color="darkblue">Free Energy Surface </font>
</td>
</tr>
<tr>
<td valign="top" width=450>
This segment of the free energy "surface" shows how the free energy varies as the size of the configuration and the applied external pressure are varied, while all other relevant physical attributes are held fixed.

The plotted function — derived from the above expression for <math>\mathfrak{G}^*</math>, with <math>\gamma_\mathrm{g} = 1</math> and <math>C=0</math> (see [[SSCpt1/Virial#Examples|further discussion]], below) — is, specifically,
<div align="center">
<font size="-1">
<math>
\mathfrak{G}^* = 3000\biggl[ - \frac{1}{\chi} - \ln\chi + \frac{\Pi}{3}\chi^3 + 0.9558 \biggr] \, .
</math>
</font>
</div>
As shown, the size of the configuration <math>(\chi)</math> increases to the right from <math>1.2</math> to <math>1.51</math>; the dimensionless external pressure <math>(\Pi)</math> increases into the screen from <math>0.103</math> to <math>0.104</math>; and the dimensionless free energy, <math>\mathfrak{G}^*</math>, increases upward.
</td>
<td align="center" bgcolor="black">
[[File:3DFreeEnergy.jpg|350px|center|Free Energy Surface]]
</td>
</tr>
</table>
</div>

==Energy Extrema==
As is illustrated in [[#Visual_Representation|Figure 1]], the free energy surface generally will exhibit multiple local minima and local maxima, and may also possess one or more points of inflection. The locations along the energy surface where these special points arise identify equilibrium states, and the associated values of <math>(R/R_0)_\mathrm{eq}</math> give the radii of the equilibrium configurations.

For a given choice of the set of physical parameters <math>M</math>, {{ Template:Math/MP_PolytropicConstant }}, <math>J</math>, <math>P_e</math>, and <math>\gamma_g</math>, extrema occur wherever,
<div align="center">
<math>
\frac{d\mathfrak{G^*}}{d\chi} = 0 \, .
</math>
</div>
For the free energy function identified above,
<div align="center">
<math>
\frac{d\mathfrak{G^*}}{d\chi} =
3A\chi^{-2} -~ (1-\delta_{1\gamma_g})~3 B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} 3\chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2 \, ,
</math>
</div>
<span id="GeneralVirial">so <math>\chi_\mathrm{eq} \equiv R_\mathrm{eq}/R_\mathrm{norm}</math> is obtained from the real root(s) of the equation,</span>
<div align="center">
<math>
2C \chi_\mathrm{eq}^{-2} + ~ (1-\delta_{1\gamma_g})~3 B\chi_\mathrm{eq}^{3 -3\gamma_g} +~ \delta_{1\gamma_g} 3 ~
-~3A\chi_\mathrm{eq}^{-1} -~ 3D\chi_\mathrm{eq}^3 = 0 \, .
</math>
</div>

<div id="Tohline85">
<table border="1" align="center" width="90%" cellpadding="20">
<tr><td align="left">
<b><font color="purple">ASIDE:</font></b> When we discuss the equilibrium of isothermal, rotating configurations that are immersed in an external medium, we will draw on the work of [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber (1976, ApJ, 208, 113)] — ''Oscillation and Collapse of Interstellar Clouds'' — and the work of [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985, ApJ, 292, 181)] — ''Star Formation: Phase Transition, not Jeans Instability'' — which, in turn draws upon [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981, ApJ, 248, 717)]. In preparation for that discussion, we will go ahead and show how [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline's (1985)] statement of virial equilibrium — his equation (9) — is the same as the equation that defines free energy extrema that has been derived here; and we will show how [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)] "energy integral" — his equation (B3) — relates to our dimensionless free-energy function.

----

<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">

The Astrophysical Journal, 292: 181-187, 1985 May 1<br />
STAR FORMATION: PHASE TRANSITION, NOT JEANS INSTABILITY<br />
Joel E. Tohline<br />
<math>\beta(\sin^{-1}e/e)^2 [\tfrac{1}{2} - \beta ] + kV^* - F_s^* = 0 \, .</math>       (9)
</td></tr>
</table>

First, in order to match sign conventions, we need to multiply our "free energy extrema" equation through by minus one; second, we should set <math>\delta_{1\gamma_g} = 1</math> because [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] was only concerned with isothermal systems; then, because [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] normalizes each energy term by
<div align="center">
<math>E^* \equiv \biggl( \frac{2^2 \cdot 3^2}{5^3} \biggr) \frac{G^2 M_\mathrm{tot}^5}{J^2} \, ,</math>
</div>
instead of by our <math>E_\mathrm{norm}</math>, we need to multiply our equation through by the ratio,
<div align="center">
<math>\frac{E_\mathrm{norm}}{E^*} = \biggl( \frac{5^3}{2^4 \cdot 3\pi} \biggr) \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \, .</math>
</div>
With these three modifications, our "free energy extrema" relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3E_\mathrm{norm}}{E^*}\biggl[~A\chi_\mathrm{eq}^{-1} ~- \biggl( \frac{2C}{3}\biggr) \chi_\mathrm{eq}^{-2}
~ +~ D\chi_\mathrm{eq}^3 - ~ B_I \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Next, because [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] considered only uniform-density configurations, all of the dimensionless filling factors can be set to unity in the definitions of the leading coefficients of all of our energy terms; but, following [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981)], the leading coefficients of two of our energy terms should be modified to include a factor involving the configuration's eccentricity,
<div align="center">
<math>e \equiv \biggl( 1 - \frac{Z_\mathrm{eq}^2}{R_\mathrm{eq}^2} \biggr)^{1/2} \, ,</math>
</div>
in order to account for rotational flattening. Properly adjusted, the four coefficients are,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>B_I</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
1 \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>C</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
= \biggl( \frac{3^2}{5^2} \biggr) \frac{E_\mathrm{norm}}{E^*} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>D</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \, .
</math>
</td>
</tr>
</table>
</div>
Inserting these coefficient definitions, our "free energy extrema" relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3E_\mathrm{norm}}{E^*}
\biggl[~\frac{1}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \chi_\mathrm{eq}^{-1}
~- \frac{E_\mathrm{norm}}{E^*} \biggl( \frac{2\cdot 3}{5^2} \biggr) \chi_\mathrm{eq}^{-2}
~ +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \chi_\mathrm{eq}^3 - ~ 1 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Next we need to appreciate that [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] adopted the dimensionless parameter, <math>\beta \equiv T_\mathrm{rot}/|W_\mathrm{grav}|</math>, instead of the normalized radius, <math>\chi</math>, as the order parameter that is varied when searching for extrema in the free-energy function. So, in our equation that defines "free energy extrema" we need to replace <math>\chi_\mathrm{eq}</math> with <math>\beta_\mathrm{eq}</math>, using the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\beta \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{C\chi^{-2}}{3A \chi^{-1}} = \biggl( \frac{3}{5} \biggr) \frac{E_\mathrm{norm}}{E^*}
\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-1} \chi^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~\chi_\mathrm{eq}^{-1} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{5}{3} \biggr) \frac{E^*}{E_\mathrm{norm}}
\biggl( \frac{\sin^{-1}e}{e} \biggr)\beta_\mathrm{eq} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, our expression for the "free energy extrema" becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \beta_\mathrm{eq} ~- 2\biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \beta_\mathrm{eq}^{2}
~ +~ \frac{4\pi P_e}{P_\mathrm{norm}} (1-e^2)^{1/2}
\biggl[ \biggl( \frac{3^3}{5^3} \biggr) \biggl( \frac{E_\mathrm{norm}}{E^*} \biggr)^4
\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-3}\biggr] \beta_\mathrm{eq}^{-3}
- ~ \frac{3E_\mathrm{norm}}{E^*} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2 \biggl\{
\beta_\mathrm{eq} \biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \biggl( \frac{1}{2} - \beta_\mathrm{eq} \biggr)
~ + \frac{2\pi \cdot 3^3}{5^3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggl( \frac{E_\mathrm{norm}}{E^*} \biggr)^4
\biggl[
\beta_\mathrm{eq}^{-3}\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-3} (1-e^2)^{1/2}\biggr]
- ~ \biggl( \frac{3}{2} \biggr) \frac{5^3}{2^4 \cdot 3 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
\biggr\} \, .</math>
</td>
</tr>
</table>
</div>
Now,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{2\pi \cdot 3^3}{5^3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggl( \frac{E_\mathrm{norm}}{E^*} \biggr)^4</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{2\pi \cdot 3^3}{5^3} \biggl[ \frac{P_e}{(E^*)^4} \biggr] ( GM_\mathrm{tot}^2)^3</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{2\pi \cdot 3^3}{5^3} \biggl[\biggl( \frac{2^2 \cdot 3^2}{5^3} \biggr) \frac{G^2 M_\mathrm{tot}^5}{J^2} \biggr]^{-4} ( P_e G^3 M_\mathrm{tot}^6)</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\pi\biggl( \frac{5^{9}}{2^7 \cdot 3^5} \biggr) \frac{J^8 P_e }{G^5 M_\mathrm{tot}^{14}}
= \frac{10 \pi}{3} \biggl( \frac{5^{2}}{2^2 \cdot 3} \biggr)^4 \frac{J^8 P_e }{G^5 M_\mathrm{tot}^{14}} \, ,
</math>
</td>
</tr>
</table>
</div>
which is the definition of the coefficient "<math>k</math>" that is provided as equation (7) of [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)]. Hence, dropping the factor of two out front, our expression for "free energy extrema" becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\beta_\mathrm{eq} \biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \biggl( \frac{1}{2} - \beta_\mathrm{eq} \biggr)
~ + \frac{10 \pi}{3} \biggl( \frac{5^{2}}{2^2 \cdot 3} \biggr)^4 \frac{J^8 P_e }{G^5 M_\mathrm{tot}^{14}}
\biggl[
\beta_\mathrm{eq}^{-3}\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-3} (1-e^2)^{1/2}\biggr]
- ~ \frac{3}{4\pi} \biggr( \frac{5^3}{2^3 \cdot 3} \biggr) \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>0 \, .</math>
</td>
</tr>
</table>
</div>
Finally, realizing that the square of the sound speed is related to our <math>c_\mathrm{norm}^2</math> via the relation [note that [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] uses <math>a^2</math> in place of <math>c_s^2</math>],
<div align="center">
<math>c_s^2 = \biggl( \frac{3}{4\pi} \biggr) c_\mathrm{norm}^2 \, ,</math>
</div>
it is clear that this last form of our "free energy extrema" expression is identical to [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline's (1985)] virial equilibrium equation (9), which appears in print in a simpler but also more cryptic form as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\beta_\mathrm{eq} \biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \biggl( \frac{1}{2} - \beta_\mathrm{eq} \biggr) + kV^* - F_s^*
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>0 \, .</math>
</td>
</tr>
</table>
</div>

----

<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">

The Astrophysical Journal, 208: 113-126, 1976 August 15<br />
OSCILLATION AND COLLAPSE OF INTERSTELLAR CLOUDS<br />
Stephen V. Weber<br />
<math>E = \dot{a}^2 + \dot\gamma^2/2 + J^2/\alpha^2
- \tfrac{4}{3} \log \alpha^2 \gamma + \tfrac{1}{3}\alpha^2 \gamma P_\mathrm{ext} -
\begin{cases}\biggl(\frac{3}{\alpha e}\biggr) \sin^{-1}e, & e > 0, \\ \biggl( \frac{3}{2\alpha e} \biggr) \log Q , & e<0 \, . \end{cases}
</math>       (B3)
</td></tr>
</table>

Plugging the same set of modified leading coefficients into our derived expression for the free energy gives,
<div align="center">
<math>
\mathfrak{G}^* = \frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \chi^{-2}
-~ 3 \ln \chi +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \chi^3
- \frac{3}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \chi^{-1}\, .
</math>
</div>
Now, recognize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\alpha \biggl( \frac{R_0}{R_\mathrm{norm}} \biggr) = \biggl( \frac{2^2}{3\cdot 5} \biggr) \alpha \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>(1 - e^2)^{1/2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{Z}{R} = \frac{\gamma}{\alpha} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P_e}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{P_e}{P_0} \cdot \frac{P_0}{P_\mathrm{norm}} = \frac{3^4 \cdot 5^3}{2^{10} \pi} \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber}
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl( \frac{2}{5} J_\mathrm{Weber} \biggr)^2
\, ,</math>
</td>
</tr>
</table>
</div>
where, for axisymmetric configurations (set <math>\beta=\alpha</math> in [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)] equation 12),
<div align="center">
<math>J_\mathrm{Weber} \equiv \alpha^2 \Omega = \biggl( \frac{R}{R_0} \biggr)^2 (\dot\varphi_c t_0)^2 \, .</math>
</div>
Hence, our expression for the free energy may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{G}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{3} \biggl( \frac{2}{5} J_\mathrm{Weber}\biggr)^2 \biggl( \frac{3\cdot 5}{2^2} \biggr)^2 \alpha^{-2}
-~ 3 \ln \chi +~ \biggl( \frac{4\pi}{3} \biggr) \frac{3^4 \cdot 5^3}{2^{10} \pi} \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber} \biggl( \frac{2^2}{3\cdot 5} \biggr)^3 \alpha^2 \gamma
- \frac{3}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \biggl( \frac{3\cdot 5}{2^2} \biggr) \alpha^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3}{2^2} \biggr) J^2_\mathrm{Weber} \alpha^{-2}
-~ \ln \chi^3 +~ \frac{1}{2^{2} } \alpha^2 \gamma \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber}
- \frac{3^2}{2^2} \biggl( \frac{\sin^{-1}e}{e} \biggr) \alpha^{-1} \, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~\frac{4}{3} \mathfrak{G}^*</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
J^2_\mathrm{Weber} \alpha^{-2}
-~ \frac{4}{3} \ln \chi^3 +~ \frac{1}{3} \alpha^2 \gamma \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber}
- 3 \biggl( \frac{\sin^{-1}e}{e} \biggr) \alpha^{-1} \, .
</math>
</td>
</tr>
</table>
</div>
The right-hand-side of this expression exactly matches [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)] "energy integral" for oblate-spheroidal configurations — see his equation (B3) for the case, <math>e > 0</math> — except that Weber's energy integral includes an additional pair of terms (<math>{\dot\alpha}^2 + {\dot\gamma}^2/2</math>) to account for kinetic energy associated with the overall collapse or expansion of the configuration. [NOTE: The logarithmic term ultimately needs to be <math>\ln \alpha^2\gamma</math> instead of <math>\ln\chi^3</math> in order to reflect an oblate-spheroidal, rather than spherical, volume. This term also needs to be fixed in the above discussion of Tohline's work.]
</td></tr>
</table>
</div>

=Examples=
* Polytropic Spheres
** [[SSC/Virial/Polytropes#Isolated.2C_Nonrotating_Configuration|Isolated, Nonrotating Configuration]]
** [[SSC/Virial/Polytropes#Nonrotating_Configuration_Embedded_in_an_External_Medium|Nonrotating Configuration Embedded in an External Medium]]

* Isothermal Spheres
** [[SSC/Virial/Isothermal#Isolated.2C_Nonrotating_Configuration|Isolated, Nonrotating Configuration]]
** [[SSC/Virial/Isothermal#Nonrotating_Configuration_Embedded_in_an_External_Medium|Nonrotating Configuration Embedded in an External Medium]]

{{ SGFworkInProgress }}

=BiPolytrope=
[Following a discussion that Tohline had with Kundan Kadam on <font color="red">3 July 2013</font>, we have decided to carry out a virial equilibrium and stability analysis of nonrotating bipolytropes.]

We will adopt the following approach:
* Properties of the core <math>\cdots</math>
** Uniform density, <math>\rho_c</math>;
** Polytropic constant, <math>K_c</math>, and polytropic index, <math>n_c</math>;
** Surface of the core at <math>r_i</math>;
* Properties of the envelope <math>\cdots</math>
** Uniform density, <math>\rho_e</math>;
** Polytropic constant, <math>K_e</math>, and polytropic index, <math>n_e</math>;
** Base of the core at <math>r_i</math> and surface at <math>R</math>.

Use the dimensionless radius,
<div align="center">
<math>\xi \equiv \frac{r}{r_i}</math>.
</div>
Then, <math>\xi_i = 1</math> and <math>\xi_s \equiv R/r_i</math>.

==Expressions for Mass==
Inside the core, the expression for the mass interior to any radius, <math>0 \le \xi \le 1</math>, is,
<div align="center">
<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .
</div>
The expression for the mass interior to any position within the envelope, <math>1 \le \xi \le \xi_s</math>, is,
<div align="center">
<math>M_\xi = \frac{4\pi}{3} r_i^3 \biggl[\rho_c + \rho_e(\xi^3 - 1) \biggr]</math> .
</div>
Hence, in terms of a reference mass, <math>~M_0 \equiv 4\pi \rho_0 R_0^3/3</math>, the mass of the core, the mass of the envelope, and the total mass are, respectively,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_\mathrm{core}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} \rho_c r_i^3
= M_0 \biggl[ \frac{\rho_c}{\rho_0} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr]
~~~~~\Rightarrow~~~~~
\frac{\rho_c}{\rho_0} = \frac{M_\mathrm{core}}{M_0} \biggl( \frac{r_i}{R_0}\biggr)^{-3} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{env}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} r_i^3 \biggl[\rho_e (\xi_s^3 - 1) \biggr] =
M_0 (\xi_s^3 - 1) \biggl[ \frac{\rho_e}{\rho_0} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr]
~~~~~\Rightarrow~~~~~
\frac{\rho_e}{\rho_0} = \frac{M_\mathrm{env}}{M_0} \biggl( \frac{r_i}{R_0}\biggr)^{-3} (\xi_s^3 - 1)^{-1}\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{4\pi}{3} r_i^3 \biggl[\rho_c + \rho_e(\xi_s^3 - 1) \biggr]
= M_0 \biggl( \frac{\rho_c}{\rho_0} \biggr) \biggl( \frac{r_i}{R_0}\biggr)^3 \biggl[ 1 + \frac{\rho_e}{\rho_c} (\xi_s^3 - 1) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Following the work of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942, ApJ, 96, 1615)] — see [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|our accompanying discussion]] — we will discuss bipolytropic equilibrium configurations in the context of a <math>\nu - q</math> plane where,
<table align="center" border="0" cellpadding="10">
<tr>
<td align="right">
<math>\nu</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{M_\mathrm{core}}{M_\mathrm{tot}} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>q</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{r_i}{R} = \frac{1}{\xi_s} \, .</math>
</td>
</tr>
</table>

With this in mind we can write,
<div align="center">
<math>\frac{\rho_e}{\rho_c} = \frac{M_\mathrm{env}}{M_\mathrm{core}} (\xi_s^3 - 1)^{-1}
= \frac{q^3 (1-\nu)}{\nu (1-q^3)} </math> ,
</div>
and,
<div align="center">
<math>\nu \biggl(\frac{1-q^3}{q^3}\biggr) \biggl( \frac{\rho_e}{\rho_c} \biggr) = (1-\nu)
~~~~~\Rightarrow~~~~~
\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1-q^3}{q^3}\biggr) \biggr]^{-1} \, .</math>
</div>

==Energy Expressions==

The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,
<div align="center">
<math>dW_\mathrm{grav} = - \biggl( \frac{GM_r}{r} \biggr) dm</math> .
</div>

Hence,
<table border="0" align="center">

<tr>
<td align="right">
<math>W_\mathrm{grav} = W_\mathrm{core} + W_\mathrm{env}</math>
</td>
<td align="left">
<math>
= - G \biggl\{ \int_0^{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_c dr + \int^R_{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_e dr \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="left">
<math>
= - G \biggl\{ \int_0^1 \biggl( \frac{4\pi }{3} \rho_c r_i^3 \xi^3 \biggr) 4\pi r_i^2 \rho_c \xi d\xi
+ \int_1^{\xi_s} \frac{4\pi}{3} \rho_c r_i^3 \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] 4\pi r_i^2 \rho_e \xi d\xi \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="left">
<math>
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \int_0^1 \xi^4 d\xi
+ \int_1^{\xi_s} \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] \biggl( \frac{\rho_e}{\rho_c} \biggr) \xi d\xi \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="left">
<math>
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \frac{1}{5}
+ \biggl( \frac{\rho_e}{\rho_c} \biggr) \int_1^{\xi_s} \xi d\xi
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \int_1^{\xi_s} (\xi^3 - 1) \xi d\xi \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="left">
<math>
= - \frac{3GM^2_\mathrm{tot}}{R} \biggl( \frac{M_\mathrm{core}}{M_\mathrm{tot}} \biggr)^2 \xi_s \biggl\{ \frac{1}{5}
+ \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{5}(\xi_s^5 - 1) - \frac{1}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
</td>
</tr>

</table>

I like the form of this expression. The leading term, which scales as <math>R^{-1}</math>, encapsulates the behavior of the gravitational potential energy for a given choice of the internal structure, namely, a given choice of <math>\xi_s</math>, <math>\nu</math>, and density ratio <math>(\rho_e/\rho_c)</math>. Actually, only two internal structural parameters need to be specified — <math>\xi_s</math> and <math>f_c</math>; from these two, the expressions shown above allow the determination of both <math>(\rho_e/\rho_c)</math> and <math>\nu</math>. Keeping in mind our desire to discuss the properties of bipolytropes in the context of the <math>\nu - q</math> plane introduced by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942, ApJ, 96, 1615)], we will rewrite this expression for the gravitational potential energy as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>W_\mathrm{grav}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \frac{\nu^2}{q} \cdot f\biggl(q, \frac{\rho_e}{\rho_c} \biggr) \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>f\biggl(q, \frac{\rho_e}{\rho_c} \biggr)</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2} - 1 \biggr)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \biggl( \frac{1}{q^5} - 1 \biggr) - \frac{5}{2}\biggl(\frac{1}{q^2} - 1 \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (q^3- q^5 )
+ \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{2}{5} -q^3 + \frac{3}{5}q^5\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

=See Also=
<ul>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>

{{ SGFfooter }}

SSCpt1/Virial/FormFactors

2021-07-28T15:58:31Z

174.64.8.247:

__NOTOC__ 
=Structural Form Factors=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|<b>Structural<br />Form<br />Factors</b>]]</font>
|}
As has been defined in [[SSCpt1/Virial#Structural_Form_Factors|a companion, introductory discussion]], three key dimensionless structural form factors are:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>x \equiv r/R_\mathrm{limit}</math>, and the subscript "0" denotes central values. The principal purpose of this chapter is to carry out the integrations that are required to obtain expressions for these structural form factors, at least in the few cases where they can be determined analytically. These form-factor expressions will then be used to provide expressions for the two constants, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, that appear in the free-energy function and in the virial theorem, and to provide corresponding expressions for the normalized energies, <math>W_\mathrm{grav}/E_\mathrm{norm}</math> and <math>S_\mathrm{therm}/E_\mathrm{norm}</math>.

==Expectation in Context of Pressure-Truncated Polytropes==
For pressure-truncated polytropic configurations, the normalized virial theorem states that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>2 \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr) + \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} \, .</math>
</td>
</tr>
</table>
</div>
This provides one mechanism by which the correctness of our form-factor expressions can be checked. Specifically, having determined <math>S_\mathrm{therm}</math> and <math>W_\mathrm{grav}</math> from the derived form factors, we can see whether the sum of these energies as specified on the lefthand-side of this virial theorem expression indeed match the normalized energy term involving the external pressure, as specified on the righthand side. In order to facilitate this "reality check" at the end of each example, below, we will use [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's detailed force-balanced solution of the equilibrium structure of embedded polytropes]] to provide an expression for the term on the righthand side of the virial theorem expression.

We begin by plugging our [[SSCpt1/Virial#Normalizations|general expression for <math>E_\mathrm{norm}</math>]] into this righthand-side term and grouping factors to facilitate insertion of Stahler's expressions.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{tot}^{(5-n)} \biggr]^{1/(n-3)} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)/(n-3)} P_e R_\mathrm{eq}^3
\biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)} \, . </math>
</td>
</tr>
</table>
</div>
From [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's equilibrium solution]], we have,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
R_\mathrm{eq}
</math>
</td>
<td align="center">
<math>=~</math>
</td>
<td align="left">
<math>
R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
~P_e R_\mathrm{eq}^3
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{1 + 3(1-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
M_\mathrm{limit}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
K^{-n} G^3 M_\mathrm{limit}^{(5-n)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3-3(5-n)/2} K_n^{-n +2n(5-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3(n-3)/2} K_n^{3n(3-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~~
P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} \biggr\}^{(5-n)/(n-3)} G^{3/2} K_n^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>\times ~\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(5-n)/2}
\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^{3(n-3)}_{\tilde\xi} \biggl( \frac{n+1}{4\pi} \biggr)^{3(n-3)/2} \biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl\{
(n+1)^{3[(5-n)+(n-3)]/2} (4\pi)^{[(n-5)+(9-3n)]/2}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{[(n-3)(5-n) + 3(n-1)(n-3)]/2} \tilde\xi^{[2(5-n) + 3(n-3)]}
\biggr\}^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{ (n+1)^{3} (4\pi)^{(2-n)}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{(n+1)(n-3)} \tilde\xi^{(n+1)}
\biggr\}^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, the expectation based on Stahler's equilibrium models is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

As a cross-check, multiplying this expression through by <math>[(R_\mathrm{eq}/R_\mathrm{norm})(M_\mathrm{norm}/M_\mathrm{limit})^2]</math> — where the expression for <math>R_\mathrm{eq}/R_\mathrm{norm}</math> can be obtained from our [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|discussions of detailed force-balanced models]] — gives a related result that can be obtained directly from [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|Horedt's expressions]], namely,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{4\pi P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(n+1)( -\tilde\theta' )^{2}} \, .
</math>
</td>
</tr>
</table>
</div>

==Viala and Horedt (1974) Expressions==
===Presentation===

[http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have provided analytic expressions for the gravitational potential energy and the internal energy — which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively — that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres. [The same expression for <math>~\Omega</math> is also effectively provided in §1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
<div align="center">
<table border="1" align="center" cellpadding="8" width="90%">
<tr><td align="center">


Astronomy & Astrophysics, 33: 195-202, (1974)<br />
POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
Y. P. Viala & Gp. Horedt<br />
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>\Omega</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
- G\int_0^M \frac{MdM}{r} = \frac{16\pi^2 G \rho_0^2 \alpha^5}{(5-n)} \biggl[
\mp \xi^3 \theta^{n+1} - 3\xi^3 (\theta')^2 - 3\xi^2 \theta (\theta')
\biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>U</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{\gamma - 1}\int_V pdV = \frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \int_0^\xi \theta^{n+1} 4\pi \alpha^2 \xi^2 d\xi
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \cdot
\frac{4\pi \alpha^2(n+1)}{(5-n)}
\biggl[
\frac{2\xi^3 \theta^{n+1}}{n+1} \pm \xi^3 (\theta')^2 \pm \xi^2 \theta (\theta')
\biggr]_0^\xi \, .
</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
(the superior sign holds if <math>-1 < n < \infty</math>, the inferior if <math>-\infty < n < -1</math>)
</td>
</tr>
</table>

</td></tr>
<tr>
<td align="left">
A couple of key equations drawn directly from [http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)] have been shown here. As its title indicates, the paper includes discussion of — and accompanying equation derivations for — equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries: planar sheets, axisymmetric cylinders, and spheres. We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only. These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
</td>
</tr>
</table>
</div>
Rewriting these two expressions to accommodate our parameter notations — recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> — we have from Viala & Horedt's (VH74) work,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \mathfrak{S}_\mathrm{A} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n(4\pi)^2}{3(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

===First Reality Check===

As a quick reality check, let's see whether, when appropriately added together, these two energies satisfy the scalar virial theorem for isolated polytropes.

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
W_\mathrm{grav} + \frac{3}{n} \mathfrak{S}_A</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>+
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)} - 1 \biggr] \tilde\xi^3 \tilde\theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(4\pi)^2}{(n+1)} \cdot G \rho_c^2 a_n^5 \tilde\xi^3 \tilde\theta^{n+1} \, .
</math>
</td>
</tr>
</table>
</div>
For ''isolated polytropes'', <math>\tilde\theta \rightarrow 0</math>, so this sum of terms goes to zero, as it should if the system is in virial equilibrium.

===Renormalization===
Both of the energy-term expressions derived by Viala & Horedt are written in terms of <math>\rho_c</math> and
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>a_\mathrm{n}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n}\biggr]^{1/2}
</math>
</td>
</tr>
</table>
</div>
— that is, effectively in terms of <math>\rho_c</math> and {{ Template:Math/MP_PolytropicConstant}} — whereas, in the context of our discussions, we would prefer to express them in terms of [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Adopted_Normalizations|our generally adopted energy normalization]],
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K_n^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>
In order to accomplish this, we need to replace the central density with the total mass of an ''isolated polytrope'', <math>M_\mathrm{tot}</math>, whose generic expression is (see, for example, equation 69 of Chandrasekhar),
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>E_\mathrm{norm}^{n-3}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
K_n^n G^{-3}\biggl\{ (4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr\}^{n-5} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
K_n^{[2n + 3(n-5)]/2} G^{[-6-3(n-5)]/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5}
\rho_c^{(n-3)(5-n)/2n} K_n^{5(n-3)/2} G^{-3(n-3)/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)}
\rho_c^{(5-n)/2n} K_n^{5/2} G^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\rho_c^{[ - 4n +(5-n)]/2n} \biggl( \frac{K_n}{G}\biggr)^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{K_n}{G} \cdot \rho_c^{(1-n)/n}\biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{4\pi}{(n+1)} \cdot a_n^2 \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~(4\pi)^2 G\rho_c^2 a_n^5</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (4\pi)^2
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(5-n)/(n-3)}
\biggl[ \frac{(n+1)}{4\pi} \biggr]^{5/2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)/(n-3)}
(4\pi)^{[-(n-3)-(5-n)]/2(n-3)} (n+1)^{[3(5-n)+5(n-3)]/2(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>E_\mathrm{norm} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
So, employing our preferred normalization, the VH74 expressions become,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{\mathfrak{S}_\mathrm{A}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{n}{3(5-n)}
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

===Second Reality Check===
If we now renormalize the sum of energy terms discussed in our [[SSCpt1/Virial/FormFactors#First_Reality_Check|first reality check, above]], we have,

<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>
\frac{1}{E_\mathrm{norm}} \biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}
= \frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(n+1)^{-1} \tilde\xi^3 \tilde\theta^{n+1} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>

(This may or may not be useful!)

===Implication for Structural Form Factors===
On the other hand, our expressions for these two [[SSCpt1/Virial#Structural_Form_Factors|normalized energy components written in terms of the structural form factors]] are,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
</td>
</tr>
</table>
</div>
where, in equilibrium (see [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|here]] and [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|here]] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl\{ \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl\{ \biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-1)/(n-3)}
\biggr\} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
Hence, we deduce that,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> - \frac{5}{3} \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]
\chi_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \cdot \tilde\mathfrak{f}^2_M
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{5}{3} \biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggr\}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(n-1)-2(n-3)]/(n-3)}
\cdot \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr)^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{[(1-n)+2(n-3)]/(n-3)} \tilde\xi^{[-(n-3)+2(1-n)]/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{(n-5)/(n-3)} \tilde\xi^{(5-3n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
If we now adopt the VH74 expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\biggl\{~~~\biggr\}_\mathrm{VH74}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)} \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)}
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)}
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{(5-n)}
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
(-\tilde\xi^2 \tilde\theta^')^{(5-n)/(n-3)} \, .
</math>
</td>
</tr>
</table>
</div>
Therefore,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_W \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)} \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\tilde\xi^{-5}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
\, .
</math>
</td>
</tr>
</table>
</div>

Now, from [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|our earlier work]] we deduced that <math>\tilde\mathfrak{f}_A</math> is related to <math>\tilde\mathfrak{f}_W</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \tilde\mathfrak{f}_W\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \, .</math>
</td>
</tr>
</table>
</div>
Hence, we now have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl[ \tilde\mathfrak{f}_A \biggr]_\mathrm{VH74}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\tilde\theta^{n+1} + \frac{(n+1)}{(5-n)}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
\, .
</math>
</td>
</tr>
</table>
</div>

Building on the work of VH74, we have, quite generally,

<div align="center" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1}
</math>
</td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2}
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
</td>
</tr>
</table>

</td>
</tr>
<tr>
<td align="left" colspan="2">
We should point out that [http://adsabs.harvard.edu/abs/1993ApJS...88..205L Lai, Rasio, & Shapiro (1993b, ApJS, 88, 205)] define a different set of dimensionless structure factors for ''isolated'' polytropic spheres — <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10) is used in the determination of the gravitational potential energy.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>k_1</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[ \frac{n(n+1)}{5-n} \biggr] \xi_1|\theta^'_1|</math>
</td>
</tr>

<tr>
<td align="right">
<math>k_2</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{5-n} \biggl[ \frac{4\pi |\theta^'_1|}{\xi_1} \biggr]^{1 / 3} </math>
</td>
</tr>
</table>
</div>
Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter. We note, as well, that for rotating configurations they define two additional dimensionless structure factors — <math>k_3</math> (their equation 3.17) is used in the determination of the rotational kinetic energy; and <math>\kappa_n</math> (their equation 3.14; also equation 7.4.9 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]) is used in the determination of the moment of inertia.
</td>
</tr>
</table>
</div>

The singularity that arises when <math>n = 5</math> leads us to suspect that these general expressions fail in that one specific case. Fortunately, as [[#Summary_.28n.3D5.29|we have shown in an accompanying discussion]], <math>\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, as well as <math>\mathfrak{f}_M</math>, can be determined by direct integration in this single case.

===Related Discussions===
* See [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Model_Sequences|our plot of, what Kimura (1981b) would refer to as, several <math>M_1</math> sequences]]

==First Detailed Example (n = 5)==

Here we complete these integrals to derive detailed expressions for the above subset of structural form factors in the case of spherically symmetric configurations that obey an <math>~n=5</math> polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should simplify the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases that cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|cores of bipolytropes]].

===Foundation (n = 5)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=5; ~\gamma=6/5)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{K^{10}}{G^{9} M_\mathrm{tot}^{6}} \biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl( \frac{K}{G} \biggr)^{15/2} M_\mathrm{tot}^{-5} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{K^5}{G^3} \biggr)^{1/2} M_\mathrm{tot}^{-1} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/BiPolytropes/Analytic5_1#Profile|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,
<div align="center">
<math>
a_{5} =\biggr[ \frac{3K}{2\pi G} \biggr]^{1/2} \rho_0^{-2/5} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{[K^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^{6/5}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{[K^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5}
~~~~\Rightarrow ~~~~
\rho_0^{1/5} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} M_\mathrm{tot}^{-1} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr) \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi
= \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2\cdot 3^4}{\pi} \biggr)^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\pi}{2\cdot 3^4} \biggr)^{1/2}
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
= \biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2}
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 5)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math>, and recognizing that,
<div align="center">
<math>dr^\dagger = \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} d\xi \, ,</math>
</div>
gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], [[Image:OnlineIntegral01.png|250px|right|Mathematica Integral]]
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{3/2} \biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2}
\int_0^{\xi} \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \biggl( \frac{1}{3} \biggr)^{3/2}
\biggl[ \frac{\xi^3}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-3/2} \biggr]_0^{\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\xi^2}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=5</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 5)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_0^{-1/5} \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=5</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_5 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_5 \xi}{a_5 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=5</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{3/2} \biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[SSCpt1/Virial/FormFactors#NormalizedProfiles|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2} \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 5)===

From the above line of reasoning we appreciate that, for any spherically symmetric configuration, the ratio of the configuration's mean density to its central density can be obtained by setting the upper limit of our just-completed "Mass2" integration to <math>x=1</math>. That is to say, quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx </math>
</td>
</tr>
</table>
</div>
But the integral expression on the righthand side of this relation is also the definition of the structural form factor, <math>~\mathfrak{f}_M</math>, given at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]]. Hence, we can say, quite generally, that,
<div align="center">
<math>\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} \, .</math>
</div>
And, given that we have just completed this integral for the case of truncated <math>n=5</math> polytropic structures, we can state, specifically, that,
<div align="center">
<math>\mathfrak{f}_M\biggr|_{n=5} = \biggl[ 1 + \frac{\tilde\xi^2}{3} \biggr]^{-3/2} \, .</math>
</div>

===Gravitational Potential Energy (n = 5)===
As presented at the [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral02.png|225px|right|Mathematica Integral]]Given that an expression for the normalized density profile, <math>\rho(x)/\rho_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been determined, above]], we can carry out the nested pair of integrals immediately. Indeed, the integral contained inside of the curly braces has already been completed [[SSCpt1/Virial/FormFactors#Mass2|in the "Mass2" subsection, above]], in order to determine the radial mass profile. Specifically, we have already determined that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ \int_0^{x} 3x^2 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} dx\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{3} \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2} \biggr\}
\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-5/2} x dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
5 \int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-4} x^4 dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \biggl( \frac{\tilde\xi^2}{3}\biggr)^{-5/2} \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-3}
\biggl\{
\biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggl[ 3\biggl( \frac{\tilde\xi^2}{3}\biggr)^2 - 8\biggl( \frac{\tilde\xi^2}{3}\biggr) - 3 \biggr]
+ 3\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr)^3\tan^{-1}\biggl[ \biggl( \frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>\mathfrak{f}_M</math> and <math>\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>n=5</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{\pi}{2^3\cdot 3^7}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
In order to simplify typing, we will switch to the variable,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
in which case a summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>( 1 + \ell^2 )^{-3/2} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4\cdot 3} \cdot \ell^{-5} (1 + \ell^2)^{-3}
\biggl\{ \ell [ 3\ell^4 - 8\ell^2 - 3 ] + 3( 1 + \ell^2 )^3\tan^{-1}(\ell ) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{3}{5} \biggl(\frac{2^3\cdot 3^6}{\pi}\biggr)^{1/2} \frac{1}{\ell} \cdot (1 + \ell^2)^3
\mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5 \pi}\biggr)^{1/2} \cdot \ell^{-6} (1 + \ell^2)^3
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized gravitational potential energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>W_\mathrm{core}^*</math>.

Hence, also, as defined in the [[SSCpt1/Virial#Gathering_it_All_Together|accompanying introductory discussion]], the constant, <math>\mathcal{A}</math>, that appears in our general free-energy equation is (for <math>n=5</math> polytropic configurations),
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 5)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
[[File:OnlineIntegral03.png|225px|right|Mathematica Integral]]Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[SSCpt1/Virial/FormFactors#rhoofx|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>3\int_0^1 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x \biggr]^{-3} x^2 dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3}{2^3}
\biggl\{\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-3/2} \tan^{-1}\biggl[ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{1/2} \biggr]
+ \biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-1}
- 2\biggl(\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggl[1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)\biggr]^{-2}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=5 (\gamma=6/5)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)^{1/5}
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{6/5}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>

If, as above, we adopt the simplifying variable notation,
<div align="center">
<math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~\Rightarrow~~~~ \frac{\tilde\xi^2}{3} = \ell^2 \, ,</math>
</div>
the various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell^3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} [ \ell^{-3} \tan^{-1}(\ell ) + \ell^{-2}(1+\ell^2)^{-1} - 2\ell^{-2}(1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, ;
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3/5} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2} \cdot \biggl[ \biggl(\frac{\pi}{2^3\cdot 3^6}\biggr)^{1/2} \ell \biggr]^{-3/5}
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5} \ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{3^{10}}{2^{10}} \biggl(\frac{2^9\cdot 3^{18}}{\pi^3}\biggr)
\biggl(\frac{3^{12}}{2^{34} \pi^2} \biggr) \biggr]^{1/10} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^{8}}{2^{7}\pi}\biggr)^{1/2} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
</td>
</tr>
</table>
</div>
This exactly matches the normalized thermal energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>S_\mathrm{core}^*</math>. Its similarity to the expression for the gravitational potential energy — which is relevant to the virial theorem — is more apparent if it is rewritten in the following form:
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2}
\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 5)===
In summary, for <math>n=5</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
( 1 + \ell^2 )^{-3/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{2^4} \cdot \ell^{-5}
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 5)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\ell}{2^4} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3^6}{2^{17} \pi} \biggr)^{1/5}~\ell^{3/5} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 5)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2} \biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2} [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{3^8}{2^5\pi}\biggr)^{1/2} \cdot
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Check (n = 5)===

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}\biggl\{ [ \ell (\ell^4-1) (1+\ell^2)^{-3} + \tan^{-1}(\ell )]
-
\biggl[\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1+\ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{3^{8}}{2^{5}\pi}\biggr)^{1/2}
\biggl[\frac{8}{3}\ell^3 (1+\ell^2)^{-3}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2 \cdot 3^{6}}{\pi}\biggr)^{1/2}
\biggl[\frac{\ell}{ (1+\ell^2)} \biggr]^3 \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=5</math> polytropes — see the [[#Mass1|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \biggl[ \ell^2(1+\ell^2)^{-1} \biggr]^{3/2}
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_5 = ( 1 + \ell^2 )^{-1/2}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_5}{d\xi} \biggr|_{\xi_e} = 3^{1/2} \ell ( 1 + \ell^2 )^{-3/2} \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2\cdot 3^3}{\pi}\biggr]^{1/2} ( 1 + \ell^2 )^{-3} (3^{1/2}\ell)^{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{2\cdot 3^6}{\pi}\biggr)^{1/2} \biggl[ \frac{\ell}{( 1 + \ell^2 )} \biggr]^{3} \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors. This gives us confidence that our form-factor expressions are correct, at least in the case of embedded <math>n=5</math> polytropic structures.

==Second Detailed Example (n = 1)==

===Foundation (n = 1)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=1; ~\gamma=2)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl(\frac{K}{G}\biggr)^{1/2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{G^3 M_\mathrm{tot}^2}{K^2}\biggr) </math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
----
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 =
\biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot}^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi}\biggl( \frac{G}{K} \biggr)^{3/2} M_\mathrm{tot} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c^2_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot} </math>
</td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}}
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>

As is detailed in our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of the properties of ''isolated'' polytropes]], in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{1}</math>, where,
<div align="center">
<math>
a_{1} = \biggl( \frac{K}{2\pi G} \biggr)^{1/2} \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{r}{(K/G)^{1/2}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{P}{K\rho_0^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^{2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{(K/G)^{3/2}\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl(\frac{2}{\pi} \biggr)^{1/2} (\sin\xi - \xi \cos\xi) \, .</math>
</td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] — in ''isolated'' <math>n=1</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0
~~~~\Rightarrow ~~~~
\rho_0 = \biggr[ \frac{G^3}{2\pi K^3} \biggr]^{1/2} M_\mathrm{tot} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles1">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{2^3 \pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\pi} \biggl( \frac{\sin\xi}{\xi}\biggr)^2 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi \cos\xi)
\, .</math>
</td>
</tr>
</table>
</div>

===Mass1 (n = 1)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile. After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the,

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_r(r^\dagger) </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, .
</math>
</td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math> gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3 \int_0^{\xi} \frac{\xi^2}{2\pi} \biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \cdot \frac{d\xi }{(2\pi)^{1/2}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
3( 2\pi)^{-3/2}\biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \int_0^{\xi} \xi \sin\xi d\xi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi\cos\xi) \, .
</math>
</td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=1</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

===Mass2 (n = 1)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x} 3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr] dx \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2} \rho_0 \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=1</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1 = \pi</math>. In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_1 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_1 \xi}{a_1 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx1">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin(\tilde\xi x)}{\tilde\xi x} \, ,</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi} \int_0^{x} x \sin(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{3}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, . </math>
</td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>. Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=1</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
\biggl\{ \frac{[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ]}{( \sin \tilde\xi - \tilde\xi \cos \tilde\xi )} \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] </math>
</td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[#NormalizedProfiles1|profile presented just before the "Mass1" subsection]], above. In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile — one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} (\sin\xi - \xi \cos\xi ) \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Mean-to-Central Density (n = 1)===

[[SSCpt1/Virial/FormFactors#Mean-to-Central_Density|Following the line of reasoning provided above]], we can use the just-derived central-to-mean density ratio to specify one of the structural form factors. Specifically,
<div align="center">
<math>~\mathfrak{f}_M\biggr|_{n=1} = \frac{\bar\rho}{\rho_c} = \frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, .</math>
</div>

===Gravitational Potential Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
</td>
</tr>
</table>
</div>
From the derivations already presented, above, for <math>~n=1</math> polytropic configurations, we know all of the functions under this integral. We know, for example, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\tilde\xi^3}
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \int_0^1 \biggl\{ [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \biggr\}
\sin(\tilde\xi x) dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{\tilde\xi^4} \biggl\{ \frac{x}{2} - \frac{\sin(2\tilde\xi x)}{4\tilde\xi} - \tilde\xi\biggl[
\frac{\sin(2\tilde\xi x) - 2\tilde\xi x\cos(2\tilde\xi x)}{8\tilde\xi^2}
\biggr] \biggr\}_0^1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>~n=1</math> polytropes. As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- 3\mathcal{A} \chi^{-1} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\chi</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
A summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_M</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>~\mathcal{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{5} \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^{2} \mathfrak{f}_W
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Thermal Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr] x^2 dx \, ,</math>
</td>
</tr>
</table>
</div>
Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[#Foundation_2|been provided, above]], we can carry out the integral immediately. Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{P(\xi)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\sin\xi}{\xi}\biggr)^{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{\sin(\tilde\xi x)}{(\tilde\xi x)}\biggr]^{2} \, .</math>
</td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2} \int_0^1 \sin^2(\tilde\xi x) dx </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{3}{\tilde\xi^2}
\biggl[\frac{x}{2}- \frac{\sin(2\tilde\xi x)}{4\tilde\xi} \biggr]_0^1
= \frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=1 (\gamma=2)</math> polytropic configurations. From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2}
\cdot \mathfrak{f}_A \, .
</math>
</td>
</tr>
</table>
</div>
The various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\chi</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\tilde\xi^3}{3\pi} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr) \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^2
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr]
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3} \mathcal{B}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3 \tilde\xi^3}{2^5\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
\biggl[ (2\pi)^{3/2} \tilde\xi^{-3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

===Summary (n = 1)===
In summary, for <math>~n=1</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{\tilde\xi^3} [\sin \tilde\xi - \tilde\xi \cos \tilde\xi ]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2^2\tilde\xi^3} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 1)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi - \sin(2\tilde\xi ) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]
</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Reality Checks (n = 1)===
====Expectation from Stahler's Equilibrium Models====
If we add twice the thermal energy to the gravitational potential energy, we obtain,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} \biggl\{ \biggl[6\tilde\xi - 3\sin(2\tilde\xi ) \biggr]
- \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} 2\tilde\xi\biggl\{ 1-\cos(2\tilde\xi ) \biggr\}
= \biggl( \frac{1}{2\pi^3} \biggr)^{1/2} \tilde\xi \sin^2(\tilde\xi ) \, .
</math>
</td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]]. Given that, for <math>n=1</math> polytropes — see the [[#Mass1_.28n_.3D_1.29|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D1.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
</math>
</td>

<td align="center">
  ;        
</td>

<td align="right">
<math>
\theta_1 = \frac{\sin\tilde\xi}{\tilde\xi}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
-\frac{d\theta_1}{d\xi} \biggr|_{\tilde\xi} = \frac{1}{\tilde\xi^2}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, ,
</math>
</td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)}
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2}{\pi} \biggr]^{-1/2} \biggl[ \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \tilde\xi^2(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)^{-1}\biggr]^{2}
\biggl( \frac{\sin\tilde\xi}{\tilde\xi}\biggr)^2 \tilde\xi^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{2\pi^3}\biggr)^{1/2} \tilde\xi \sin^2\tilde\xi \, .
</math>
</td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors, giving us additional confidence that our form-factor expressions are correct.

====Compare With General Expressions Based on VH74 Work====
Based on the general expressions [[#PTtable|derived above]] in the context of VH74's work, for the specific case of <math>n=1</math> polytropic configurations, the three structural form factor should be,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\tilde\xi^2}
\biggl[\tilde\theta^{2} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ 3\tilde\theta^{2} +
3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, .
</math>
</td>
</tr>
</table>
Also, remember that,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\theta</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sin\xi}{\xi}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~\theta^' \equiv \frac{d\theta}{d\xi}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\cos\xi}{\xi} - \frac{\sin\xi}{\xi^2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~(\theta^' )^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\xi^4} \biggl[ \xi\cos\xi - \sin\xi \biggr]^2
=\frac{1}{\xi^4} \biggl[ \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr] \, .
</math>
</td>
</tr>
</table>
Now, let's look at the structural form factors, one at a time. First, we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{\xi^3} \biggl[\sin\xi - \xi\cos\xi \biggr]</math>
</td>
</tr>
</table>

which matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Next,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~ \frac{3\cdot 5}{4\xi^2} \biggl[ \frac{\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 + 2\xi^2\cos^2\xi - 3 \xi\sin\xi \cos\xi \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{4\xi^6} \biggl\{ \xi^2 + \xi^2[1+\cos(2\xi)] - \frac{3}{2} \xi\sin(2\xi) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3\cdot 5}{8\xi^6} \biggl[ 4\xi^2 + 2\xi^2 \cos(2\xi) - 3 \xi\sin(2\xi) \biggr] \, ,
</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. Finally,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\mathfrak{f}_A</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} \biggl[ \frac{3\sin^2\xi}{\xi^2}
+ \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi - \xi\cos\xi \biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+ 3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi \biggr)
+ 3\xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2\xi^4} \biggl[ \xi^2 - \xi \sin\xi \cos\xi \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{2^2\xi^3} \biggl[ 2\xi - \sin(2\xi) \biggr] \, ,</math>
</td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]]. So this adds support to the deduction, above, that VH74 have provided us with the information necessary to develop general expressions for the three structural form factors.

==Fiddling Around==
NOTE (from Tohline on 17 March 2015): Chronologically, this "Fiddling Around" subsection was developed before our discovery of the VH74 derivations. It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres. But this subsection's conclusions are superseded by the VH74 work.

In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>. In the case of <math>n=1</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{\sin\xi}{\xi}\biggr)^2
= \frac{1}{2\xi^2} \biggl[ 1 - \cos(2\xi) \biggr]
= \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\mathfrak{f}_A - \theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[6\xi - 3\sin(2\xi ) \biggr]
- \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2^2\xi^3} \biggl[4\xi - 3\sin(2\xi ) + 2\xi\cos(2\xi) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^3 \xi^5}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ 4\xi - 3\sin(2\xi) + 2\xi \cos(2\xi ) \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2 \xi^2}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

Similarly, in the case of <math>n = 5</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\theta^{n+1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~\biggl( \frac{2^3}{3} \ell^{3} \biggr) \biggl[ \mathfrak{f}_A - \theta^{n+1} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
[ \tan^{-1}(\ell ) + \ell (\ell^4-1) (1+\ell^2)^{-3} ] - \biggl( \frac{2^3}{3} \ell^{3} \biggr) (1 + \ell^2)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\tan^{-1}(\ell ) + \ell \biggl(\ell^4-\frac{8}{3}\ell^2 - 1 \biggr) (1+\ell^2)^{-3} \, .
</math>
</td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2^4}{5} \cdot \ell^{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \, .
</math>
</td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl( \frac{2\cdot 3}{5} \cdot \ell^{2} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{2\xi^2}{5} \biggr) \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>
This is pretty amazing! Both examples produce almost exactly the same relationship between the two structural form factors, <math>\mathfrak{f}_A</math> and <math>\mathfrak{f}_W</math>. I think that we are well on our way toward nailing down the generic, analytic relationship and, in turn, a generally applicable mass-radius relationship for pressure-truncated polytropic configurations.

Okay … here is the final piece of information. In the case of isolated polytropes, we know that the correct expressions for the structural form factors are as summarized in the following table:
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\tilde\mathfrak{f}_M</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_W </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\tilde\mathfrak{f}_A </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi}
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>
</div>
We notice, from this, that the ratio,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} \cdot \frac{5-n}{3^2\cdot 5} \biggl[ \frac{\xi}{\Theta^'} \biggr]^{2}_{\tilde\xi}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(n+1)\tilde\xi^2 }{3\cdot 5} \, .
</math>
</td>
</tr>
</table>
Even in the case of the two pressure-truncated polytropes, analyzed above, this ratio proves to give the correct prefactor on <math>\mathfrak{f}_W</math>. So we ''suspect'' that the universal relationship between the two form factors is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{(n+1) \xi^2 }{3\cdot 5} \biggr] \mathfrak{f}_W</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
</td>
</tr>

</table>
</div>

=See Also=
<ul>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>

{{ SGFfooter }}

SSCpt1/Virial/PolytropesEmbeddedOutline

2021-07-16T23:13:26Z

174.64.8.247: /* See Also */

__FORCETOC__ 

=Virial Equilibrium of Embedded Polytropic Spheres=

<div align="center" id="IntroFigures">
<table border="1" cellpadding="8" align="center">
<tr>
<th align="center" colspan="2">
Figure 1:   Free-Energy Surfaces for Pressure-Truncated Structures
</th>
<tr>
<td align="center" colspan="1" bgcolor="#CCFFFF">
[[File:FreeNRGmassRadiusN5.png|300px|n = 5 Free-Energy Surface]]
</td>
<td align="center" colspan="1" bgcolor="CCFFFF">
[[File:FreeNRGpressureRadiusIsothermal.png|300px|Whitworth's (1981) Isothermal Free-Energy Surface]]
</td>
</tr>
<tr>
<td align="center">
<math>\mathfrak{G}(M,R)</math> surface for <math>n = 5</math> polytropic configurations<br />
[[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_the_Virial_Theorem_Relation|(click here for a technical discussion)]]
</td>
<td align="center">
<math>\mathfrak{G}(P_e,R)</math> surface for isothermal configurations<br />
(see further elaboration [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_the_Virial_Theorem_Relation|below]])
</td>
</tr>
</table>
</div>

==Overview==
The free-energy function that is relevant to a discussion of the structure and stability of a pressure-truncated configuration having polytropic index, {{ Template:Math/MP_PolytropicIndex }}, has the form,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\mathfrak{G}(x)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-ax^{-1} +b x^{-3/n} + c x^3 + \mathfrak{G}_0
\, ,
</math>
</td>
</tr>

</table>
</div>
where <math>x</math> identifies the size of the configuration and <math>\mathfrak{G}_0</math> is an arbitrary constant. (As is explained more fully, below, the left-hand panel of Figure 1 displays a free-energy surface of this form for the case, <math>n=5</math>.) If the coefficients, <math>a, b</math>, and <math>c</math>, are held constant while varying the configuration's size, we see that,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\frac{d\mathfrak{G}}{dx}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
ax^{-2} - \frac{3b}{n}\cdot x^{-(3+n)/n} + 3c x^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
x^{-2} \biggl[ a - \frac{3b}{n}\cdot x^{(n-3)/n} + 3c x^4 \biggr]
\, ,
</math>
</td>
</tr>

</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\frac{d^2\mathfrak{G}}{dx^2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
x^{-3} \biggl[ -2a + \frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} + 6c x^4 \biggr]
\, .
</math>
</td>
</tr>

</table>
</div>

===Equilibrium Configurations===
The size, <math>x_\mathrm{eq}</math>, of each equilibrium configuration is determined by setting, <math>d\mathfrak{G}/dx = 0</math>. Hence, <math>x_\mathrm{eq}</math> is given by the root(s) of the polynomial expression that is often referred to as the,
<div align="center" id="ScalarVT">
<font color="#770000">'''Scalar Virial Theorem'''</font><br />
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>x^{(n-3)/n}_\mathrm{eq} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{n}{b} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
(The equilibrium radii of <math>n = 5</math> polytropic configurations having a variety of different masses are identified by the sequence of a dozen, small colored spherical dots shown in the [[#IntroFigures|left-hand panel of Figure 1]].)

===Stability===
The relative stability of each equilibrium configuration is determined by the sign of the second derivative of the free-energy function, evaluated at the specified equilibrium radius. Specifically, the systems being considered here are stable if the second derivative is positive, but they are unstable if the second derivative is negative. Evaluating the second derivative in this manner gives,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\biggl[ x^{3} \cdot \frac{d^2\mathfrak{G}}{dx^2}\biggr]_\mathrm{eq}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-2a + \frac{3(3+n)}{n} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] + 6c x^4_\mathrm{eq}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-2a + \frac{(3+n)a}{n} + \frac{3(3+n)c}{n} \cdot x^4_\mathrm{eq} + 6c x^4_\mathrm{eq}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{9(n+1)c}{n}\cdot x^4_\mathrm{eq} - \frac{a(n-3)}{n} \, .
</math>
</td>
</tr>

</table>
</div>
Defining <math>x_\mathrm{crit}</math> as the equilibrium radius at which this function goes to zero gives,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>x_\mathrm{crit} </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ \frac{a(n-3)}{3^2c(n+1)} \biggr]^{1/4} \, .
</math>
</td>
</tr>

</table>
</div>
(The small red spherical dot in the [[#IntroFigures|left-hand panel of Figure 1]] identifies the equilibrium configuration at <math>x_\mathrm{crit} </math>.) We conclude, therefore, that pressure-truncated, equilibrium polytropic configurations having <math>n > 3</math> are stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>x_\mathrm{eq}</math>
</td>
<td align="center">
<math>></math>
</td>
<td align="left">
<math>
x_\mathrm{crit} \, ,
</math>
</td>
</tr>

</table>
</div>
while they are unstable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>x_\mathrm{eq}</math>
</td>
<td align="center">
<math><</math>
</td>
<td align="left">
<math>
x_\mathrm{crit} \, .
</math>
</td>
</tr>

</table>
</div>

==The Physics==
The above mathematical statements — ostensibly defining the free-energy function, the scalar virial theorem, and stability — cannot be interpreted in physical terms until the definitions of the various coefficients have been provided. In the discussion that follows, we will focus on sequences of equilibrium configurations that have a polytropic index <math>n > 3</math> because, as has been foreshadowed in [[#Overview|the above overview]], such sequences include both stable and unstable equilbria and are therefore of considerable interest in an astrophysical context. Isothermal sequences — corresponding to <math>n = \infty</math> — are of particular astrophysical interest; however, we will devote a great deal of attention to <math>n=5</math> configurations because their structures can be defined entirely in terms of analytic expressions.


In order to determine the equilibrium radius, <math>R_\mathrm{eq}</math>, of any pressure-truncated polytropic configuration, we must specify the configuration's mass, <math>M_\mathrm{limit}</math>, its polytropic constant, {{ Template:Math/MP_PolytropicConstant }}, and the pressure, <math>P_e</math>, of the external medium in which the configuration is embedded, then we must locate extrema in the resulting <math>\mathfrak{G}(R)</math> function. A ''sequence'' of equilibria can be identified if, for example:
* <font color="red">'''Case P'''</font>: <math>P_e</math> is varied while holding {{ Template:Math/MP_PolytropicConstant }} and <math>M_\mathrm{limit}</math> fixed; or
* <font color="red">'''Case M'''</font>: <math>M_\mathrm{limit}</math> is varied while holding {{ Template:Math/MP_PolytropicConstant }} and <math>P_e</math> fixed.
In the first case, the analysis reveals how <math>R_\mathrm{eq}</math> varies with the applied external pressure and usually is displayed as a <math>P_e(R_\mathrm{eq})</math> function. The second case identifies a mass-radius relationship for the polytropic sequence under consideration and is usually displayed as a <math>M_\mathrm{limit}(R_\mathrm{eq})</math> function. (In the image presented in the [[#IntroFigures|left-hand panel of Figure 1]], a "Case M" mass-radius relation for pressure-truncated, <math>n = 5</math> polytropic configurations is traced by the sequence of a dozen, small colored spherical dots that each reside at an extremum in the displayed free-energy function.)

===Whitworth's (1981) Case P Analysis of Uniform-Density Configurations===

====Coefficient Definitions====

[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function — referred to by Whitworth as the "global potential function":

<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">

MNRAS, 195: 967-977, 1981 June<br />
GLOBAL GRAVITATIONAL STABILITY FOR ONE-DIMENSIONAL POLYTROPES<br />
Ant. Whitworth<br />
<math>
\frac{2\mathcal{U}}{3M_0 K_1} = -\frac{3}{2}\biggl(\frac{R}{R_\mathrm{rf}}\biggr)^{-1}
+ \frac{1}{6} \biggl(\frac{P_\mathrm{ex}}{P_\mathrm{rf}}\biggr)\biggl(\frac{R}{R_\mathrm{rf}}\biggr)^3
+ (1-\delta_{1\eta})\frac{2}{3(\eta-1)} \biggl(\frac{R}{R_\mathrm{rf}}\biggr)^{3(1-\eta)}
- \delta_{1\eta} 2\ln\biggl(\frac{R}{R_\mathrm{rf}}\biggr)
\, .</math>       (10)
</td></tr>
</table>

After setting <math>\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>(\eta)</math> in terms of the corresponding polytropic index, Whitworth's free-energy expression becomes,
<div align="center" id="WhitworthFreeEnergyExpression">
<math>
\frac{2\mathcal{U}}{3M_0 K_1} =
-\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} +~ \frac{2n}{3}\biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-3/n}
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 \, .
</math>
</div>
This expression is identical to the [[#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:

<div align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] <font color="red">Case P</font> Analysis
</td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>a</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\frac{3}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>b</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\frac{2n}{3}</math>
</td>
</tr>

<tr>
<td align="right">
<math>c</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr)</math>
</td>
</tr>

<tr>
<td align="right">
<math>x</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\frac{R}{R_\mathrm{rf}}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{G}</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\frac{2}{3} \biggl( \frac{\mathcal{U}}{\mathcal{U}_\mathrm{rf}} \biggr) </math>
</td>
</tr>
</table>
</td></tr>
</table>
</div>
where (see an [[SSC/Structure/PolytropesASIDE1#ASIDE:_Whitworth.27s_Scaling|accompanying ASIDE]]),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>R_\mathrm{rf}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{rf}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{U}_\mathrm{rf} \equiv (M_0K_1)_\mathrm{Whitworth}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr) K^n G^{-3} M_\mathrm{limit}^{n-5}\biggr]^{1/(n-3)} \, .</math>
</td>
</tr>
</table>
</div>

====Virial Equilibrium====
Plugging these coefficient assignments into the [[#Equilibrium_Configurations|above mathematical prescription of the virial theorem]] gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressure-truncated polytropic configurations,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\frac{2}{3}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{2} + \frac{1}{6} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr)
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 \, .
</math>
</td>
</tr>

</table>
</div>
A rearrangement of terms explicitly provides the desired <math>P_e(R_\mathrm{eq})</math> function, namely,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} -3\biggr] \, ,</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-3(n+1)/n} -3\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4} \, ,</math>
</td>
</tr>
</table>
or,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>P_e </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
4P_\mathrm{rf} R_\mathrm{rf}^{3(n+1)/n} R_\mathrm{eq}^{-3(n+1)/n} -3P_\mathrm{rf} R_\mathrm{rf}^4 R_\mathrm{eq}^{-4}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
4R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{3(n+1)/[n(n-3)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{4/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2^2 R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2n(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n(n+1)}
K^{4n^2} G^{-3n(n+1)} M_\mathrm{limit}^{-2n(n+1)}\cdot
\frac{\pi^{3(n+1)}}{5^{3n(n+1)}} \biggl( \frac{2^2}{3}\biggr)^{3(n+1)(n+1)} K^{-3n(n+1)} G^{3n(n+1)} M_\mathrm{limit}^{3(n-1)(n+1)} \biggr]^{1/[n(n-3)]}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\cdot
\frac{\pi^4}{5^{4n}} \biggl( \frac{2^2}{3}\biggr)^{4(n+1)} K^{-4n} G^{4n} M_\mathrm{limit}^{4(n-1)} \biggr]^{1/(n-3)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
R_\mathrm{eq}^{-3(n+1)/n} \biggl( \frac{3}{2^2\pi}\biggr)^{(n+1)/n} K M_\mathrm{limit}^{(n+1)/n}
- R_\mathrm{eq}^{-4} \biggl(\frac{3}{2^2\cdot 5\pi}\biggr) G M_\mathrm{limit}^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n}
- \frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \, .</math>
</td>
</tr>
</table>

Recalling that <math>\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:

<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">

MNRAS, 195: 967-977, 1981 June<br />
GLOBAL GRAVITATIONAL STABILITY FOR ONE-DIMENSIONAL POLYTROPES<br />
Ant. Whitworth<br />
<math>
R_0 \rightarrow R_\mathrm{eq}
\, ,</math>         
<math>
P_\mathrm{ex} = K(3M_0/4\pi R_\mathrm{eq}^3)^\eta - 3GM_0^2/20\pi R_\mathrm{eq}^4
</math>       (5)
</td></tr>
</table>

====Stability====
Similarly, according to the [[#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 </math>
</td>
<td align="center">
<math>></math>
</td>
<td align="left">
<math>\frac{(n-3)}{(n+1)}
\biggl( \frac{P_e}{P_\mathrm{rf}} \biggr)^{-1} \, .
</math>
</td>
</tr>

</table>
</div>
Or, given that <math>P_\mathrm{rf}R_\mathrm{rf}^4 = (G M_\mathrm{limit}^2)/(20\pi)</math>, the criterion for stability may be written as,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>P_e </math>
</td>
<td align="center">
<math>></math>
</td>
<td align="left">
<math>
\frac{(n-3)}{20\pi(n+1)} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr)
\, .
</math>
</td>
</tr>
</table>

====ASIDE: Isothermal Configurations====
While our focus in this chapter is on polytropic systems, it is advantageous to review [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] discussion of pressure-truncated isothermal configurations because that discussion includes presentation of a free-energy surface — see, specifically, Whitworth's Figure 2.

Setting <math>\delta_{1\eta} = 1</math> in Whitworth's free-energy expression (his equation 10, [[#Coefficient_Definitions|reprinted above]]) gives,
<div align="center" id="WhitworthFreeEnergyExpression">
<math>
\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}} = -\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1}
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3
-2\ln\biggl( \frac{R}{R_\mathrm{rf}}\biggr) - \mathfrak{G}_0 \, ,
</math>
</div>
where, as earlier, we have inserted the additional constant, <math>\mathfrak{G}_0</math>, to accommodate normalization. A segment of the free-energy surface defined by this function is displayed in the right-hand panel of our [[#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|Figure 1, at the top of this page]]. In constructing this figure, <math>\mathfrak{G}_0</math> has been set to a value that ensures that <math>\mathcal{U}</math> is everywhere positive over the displayed domain: <math>0.1 \le P_e/P_\mathrm{rf} \le 1.1</math> and <math>0.3 \le R/R_\mathrm{rf} \le 3.0</math>.

<span id="ScalarVirialTheorem">
For a given choice of <math>P_e</math>, equilibrium radii are identified by setting <math>d\mathcal{U}/dR = 0</math>, that is, they are defined by the (scalar virial theorem) relation,</span>
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>\frac{P_e}{P_\mathrm{rf}} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr) - 3 \biggr]
\, ;
</math>
</td>
</tr>
</table>

and equilibria are stable — that is, <math>[d^2\mathcal{U}/dR^2]_\mathrm{eq} > 0</math> — if <math>R_\mathrm{eq}/R_\mathrm{rf} > 1</math>. For physically realistic systems, of course, <math>P_e</math> must be positive — which means that all equilibria have <math>R_\mathrm{eq}/R_\mathrm{rf} > 3/4</math>. But, from this algebraic virial theorem expression, it is also clear that physically realistic equilibrium configurations only exist when <math>P_e/P_\mathrm{rf} \le 1</math>. The sequence of small, colored spherical dots in the right-hand panel of our [[#IntroFigures|Figure 1]] identify parameter-value pairs, <math>(R_\mathrm{eq}, P_e)</math>, associated with fourteen different equilibrium configurations: Blue dots — lying along the valley of the free-energy surface — identify stable configurations; white dots — balanced along the crest of the surface ridge — identify dynamically unstable configurations; and the lone red dot identifies the critical neutral equilibrium state, which is also associated with the maximum allowable value of <math>P_e</math> along the equilibrium model sequence.

<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">

Figure 2 extracted without modification from §6 (p. 973) of …<br />
MNRAS, 195: 967-977, 1981 June<br />
GLOBAL GRAVITATIONAL STABILITY FOR ONE-DIMENSIONAL POLYTROPES<br />
Ant. Whitworth<br />
</td></tr>
<tr>
<td align="center">[[File:Whitworth81Figure2.png|600px|Figure 2 from Whitworth (1981)]]
</tr>
<tr>
<td align="left">
Caption (''verbatim''):   The continuous lines are potentials controlling radial motions of a uniform-density spherical cloud with isothermal equation of state (i.e., <math>\eta = 1</math>), for several different values of the external pressure <math>P_\mathrm{ex}</math> (as labelled). The filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state. The dotted lines are freefall collapse potentials for comparison.
</td>
</tr>
</table>

Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)] — reproduced immediately above — does an excellent job of conveying many of the essential elements of this isothermal free-energy surface within the constraints imposed by a two-dimensional black & white line plot. In order to construct this compact plot, Whitworth employed a different free-energy normalization parameter for each selected value of the external pressure. Specifically, he used,
<div align="center">
<math>\mathfrak{G}_0(P_e) = \frac{2}{3}\biggl[ 1 + \ln\biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)\biggr]
- \frac{3}{2} \biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)^{1/3} \, .</math>
</div>
In an effort to quantitatively compare (and check for accuracy) our results with Whitworth's, we have adopted the same <math>\mathfrak{G}_0(P_e)</math> normalization function when generating the multicolored, three-dimensional free-energy surface that is displayed (with three different projections) immediately below. Aside from this pressure-dependent normalization parameter, the multi-colored free-energy surface that has been drawn, here, for comparison with Whitworth's Figure 2 is identical to the one displayed in the right-hand panel of our Figure 1 (see [[#IntroFigures|the top of this page]]).

<div align="center" id="3DIsothermalSurface">
<table border="1" cellpadding="5" align="center" width="80%">
<tr>
<td align="center" colspan="2">
Figure 3:   Our Depiction of Whitworth's (1981) 3D Isothermal Free-Energy Surface<br />As Seen From Different Lines of Sight
</td>
<td align="center" rowspan="5">
<table border="0" cellpadding="5" align="center">
<tr>
<th align="center" colspan="4">
Table 1:Properties of<br />Selected Virial Equilibria
</th>
</tr>
<tr>
<td align="center">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{rf}}</math>
</td>
<td align="center">
<math>\frac{P_e}{P_\mathrm{rf}}</math>
</td>
<td align="center">
<math>\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}}</math>
</td>
<td align="center">
<math>\mathfrak{G}_0</math>
</td>
</tr>

<tr>
<td align="right">
2.395
</td>
<td align="right">
0.200
</td>
<td align="right">
-0.03205
</td>
<td align="right">
-1.8831
</td>
</tr>

<tr>
<td align="right">
2.008
</td>
<td align="right">
0.30946
</td>
<td align="right">
-0.04507
</td>
<td align="right">
-1.6786
</td>
</tr>

<tr>
<td align="right">
1.800
</td>
<td align="right">
0.400
</td>
<td align="right">
-0.05548
</td>
<td align="right">
-1.5646
</td>
</tr>

<tr>
<td align="right">
1.622
</td>
<td align="right">
0.50364
</td>
<td align="right">
-0.06730
</td>
<td align="right">
-1.4666
</td>
</tr>

<tr>
<td align="right">
1.452
</td>
<td align="right">
0.632
</td>
<td align="right">
-0.08213
</td>
<td align="right">
-1.3743
</td>
</tr>

<tr>
<td align="right">
1.347
</td>
<td align="right">
0.72543
</td>
<td align="right">
-0.09327
</td>
<td align="right">
-1.3205
</td>
</tr>

<tr>
<td align="right">
1.270
</td>
<td align="right">
0.800
</td>
<td align="right">
-0.10252
</td>
<td align="right">
-1.2835
</td>
</tr>

<tr>
<td align="right">
1.127
</td>
<td align="right">
0.93516
</td>
<td align="right">
-0.12070
</td>
<td align="right">
-1.2263
</td>
</tr>

<tr>
<td align="right">
1.071
</td>
<td align="right">
0.97564
</td>
<td align="right">
-0.12681
</td>
<td align="right">
-1.2111
</td>
</tr>

<tr>
<td align="right" bgcolor="red">
<font color="white">1.000</font>
</td>
<td align="right" bgcolor="red">
<font color="white">1.000</font>
</td>
<td align="right" bgcolor="red">
<font color="white">-0.13086</font>
</td>
<td align="right" bgcolor="red">
<font color="white">-1.2025</font>
</td>
</tr>

<tr>
<td align="right">
0.9612
</td>
<td align="right">
0.9897
</td>
<td align="right">
-0.12880
</td>
<td align="right">
-1.2061
</td>
</tr>

<tr>
<td align="right">
0.9061
</td>
<td align="right">
0.92636
</td>
<td align="right">
-0.11370
</td>
<td align="right">
-1.2297
</td>
</tr>

<tr>
<td align="right">
0.859
</td>
<td align="right">
0.800
</td>
<td align="right">
-0.07423
</td>
<td align="right">
-1.2835
</td>
</tr>

<tr>
<td align="right">
0.822
</td>
<td align="right">
0.632
</td>
<td align="right">
0.000
</td>
<td align="right">
-1.3743
</td>
</tr>

<tr>
<td align="right">
0.789
</td>
<td align="right">
0.400
</td>
<td align="right">
+0.17021
</td>
<td align="right">
-1.5646
</td>
</tr>

<tr>
<td align="right">
0.767
</td>
<td align="right">
0.200
</td>
<td align="right">
+0.47300
</td>
<td align="right">
-1.8831
</td>
</tr>
</table>
</td>
</tr>
<tr>
<td align="center" colspan="1" rowspan="1" bgcolor="#C0FFFF">
[[File:EnergyRadiusViewTop5.png|350px|Whitworth's (1981) Isothermal Free-Energy Surface]]
</td>
<td align="center" colspan="1" rowspan="1" bgcolor="#CCFFFF">
[[File:FEmovie02.gif|350px|Animated Isothermal Free-Energy Surface]]
</td>
</tr>
<tr>
<td align="center" colspan="2" rowspan="1" bgcolor="#C0FFFF">
[[File:EnergyRadiusViewBottom6.png|600px|Whitworth's (1981) Isothermal Free-Energy Surface]]

</td>
</tr>
<tr>
<td align="left" colspan="2">
Graphical depictions of the free-energy surface, <math>~\mathfrak{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations — see equation (10) of Whitworth (1981) or our restatement of this equation, [[#WhitworthFreeEnergyExpression|above]].
*''Upper-right quadrant'': The undulating free-energy surface is drawn in 3D and viewed from a (time-varying) vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. The three coordinate axes are labeled and colored as follows: Radius (red; hereafter, <math>X</math>), External Pressure (green; hereafter, <math>Y</math>), and Free Energy (blue; hereafter, <math>Z</math>). The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots: Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>.
*''Upper-left quadrant'': The two-dimensional projected image that results from viewing the free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>(Z)</math> axis and looking directly down onto the radius-pressure <math>(X-Y)</math> plane. From this vantage point, the sixteen small colored dots cleanly trace out the <math>P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[#ScalarVirialTheorem|algebraic expression of the scalar virial theorem]].
*''Bottom'': The two-dimensional projected image that results from viewing the free-energy surface "from underneath," along a line of sight that is parallel to the external-pressure <math>(Y)</math> axis and looking directly up at the radius-free-energy <math>(X-Z)</math> plane. This image can be directly compared with [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] Figure 2. (Whitworth's black & white plot has been reprinted next to our multi-colored image, and the aspect-ratio of the plot has been modified slightly, in order to facilitate comparison.) Seven of the nine equilibrium configurations that are marked in Whitworth's diagram also appear among the fifteen equilibria that are identified (as small colored dots) in our projected image. For example, the red dot in our image corresponds to the marginally stable configuration that Whitworth marks with a cross; and the white dot that is peeking from behind the "Radius" axis — that is, the unstable equilibrium configuration that has a free-energy value of zero — corresponds to the open circle in Whitworth's plot that is labeled as <math>P_e = 0.632</math> (see also the coordinate values given in our accompanying, Table 1).
</td>
</tr>
</table>
</div>

Caption to Table 1, which accompanies our Figure 3:   Coordinates <math>(X, Y, Z)</math> = <math>(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}, \mathfrak{G})</math> and, hence, also the physical properties are provided for each of the sixteen equilibria that are marked by small colored spherical dots in our attending color plots of the free-energy surface. [Actually, two of the three attending color plots display only fifteen dots because the position of the (unstable) equilibrium configuration of highest energy falls outside the boundaries of the plot.] Seven of the nine equilibrium configurations that are identified in Whitworth's Figure 2 are among the sixteen equilibria that are identified here, including the critical neutral equilibrium state, which is highlighted in red. For completeness, the value of the corresponding normalization energy, <math>\mathfrak{G}_0(P_e)</math>, is also tabulated.

===Our Case P Analysis Allowing for Nonuniform Density Structures===

====Coefficient Definitions (P)====
As has been both summarized and detailed in [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Free_Energy_Function_and_Virial_Theorem|an accompanying discussion]], our <font color="red">'''Case P'''</font> analysis has demonstrated that the following,
<div align="center" id="FreeEnergyExpression">
<font color="#770000">'''Algebraic Free-Energy Function'''</font><br />

<math>
\mathfrak{G}^* =
-3\mathcal{A} \chi^{-1} +~ n\mathcal{B} \chi^{-3/n} +~ \mathcal{D}\chi^3 \, ,
</math>
</div>
properly governs the equilibrium structure and stability of pressure-truncated polytropic configurations. This algebraic expression is identical to the [[#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:
<div align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
Our <font color="red">Case P</font> Analysis
</td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>a</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>3\mathcal{A}</math>
</td>
</tr>

<tr>
<td align="right">
<math>b</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>n\mathcal{B} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\mathcal{D} \equiv \frac{4\pi}{3} \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr)</math>
</td>
</tr>

<tr>
<td align="right">
<math>x</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\chi \equiv \frac{R}{R_\mathrm{norm}}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{G}</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\mathfrak{G}^{*} \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
</td>
</tr>
</table>
</td></tr>
</table>
</div>

where — see, for example, our [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review|accompanying review]],
<div align="center">
<table border="0" cellpadding="5">

<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, ,</math>
</td>
</tr>

</table>
</div>
and, in terms of the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Structural_Form_Factors|structural form factors]], <math>\tilde\mathfrak{f}_M</math>, <math>\tilde\mathfrak{f}_A</math>, and <math>\tilde\mathfrak{f}_W</math>,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\mathcal{A}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{4\pi}\biggr)^{1/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n+1)/n}
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, .
</math>
</td>
</tr>

</table>
</div>

====Virial Equilibrium (P)====
Plugging these coefficient assignments into the [[#Equilibrium_Configurations|above mathematical prescription of the virial theorem]] gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressure-truncated polytropic configurations,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\mathcal{A} + \frac{4\pi}{3} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{norm}} \biggr)
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \, ,
</math>
</td>
</tr>

</table>
</div>
which matches the [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|statement of virial equilibrium presented in our accompanying, more detailed analysis]]. A rearrangement of terms explicitly provides the desired <math>P_e(R_\mathrm{eq})</math> function, namely,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{P_\mathrm{e}}{P_\mathrm{norm}}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3}{4\pi}
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{-4}
\biggl[ \mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} -\mathcal{A} \biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
or (see the [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.29|accompanying derivation]] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>P_e </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}
- \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr)\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>
</td>
</tr>
</table>
</div>

Notice that this equilibrium relation exactly matches the one derived by Whitworth — and [[#Virial_Equilibrium|rederived above]] — when all three structural form factors are set to unity. This is as it should be because all of Whitworth's results were derived assuming uniform-density configurations. Also notice that, when <math>P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>K</math> and <math>M_\mathrm{limit}</math> (see the left-hand column of our [[SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]]).

====Stability (P)====
Similarly, according to the [[#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math>
</td>
<td align="center">
<math>></math>
</td>
<td align="left">
<math>\frac{(n-3)\mathcal{A}}{4\pi(n+1)}
\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \, .
</math>
</td>
</tr>

</table>
</div>
Or, given that <math>P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>P_e </math>
</td>
<td align="center">
<math>></math>
</td>
<td align="left">
<math>
\frac{(n-3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
=\frac{(n-3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr)
\, .
</math>
</td>
</tr>

</table>
</div>

===Our Case M Analysis===

====Coefficient Definitions (M)====
As has been detailed in [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseFreeEnergyExpression|an accompanying discussion]], our <font color="red">'''Case M'''</font> analysis has demonstrated that the free-energy expression governing the equilibrium structure and stability of pressure-truncated polytropic configurations can also be written as,
<div align="center" id="CaseMFreeEnergyExpression">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathfrak{G}^*_\mathrm{SWS}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
+~ n\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} \, .
</math>
</td>
</tr>
</table>
</div>
Written in this form, the expression highlights the functional dependence on the configuration's mass while assuming that the external pressure is held fixed. Still, the expression is identical to the [[#Overview|free-energy function given above]], but viewed in this manner the appropriate coefficient and variable substitutions are:
<div align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
Our <font color="red">Case M</font> Analysis
</td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>a</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>b</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>n\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} </math>
</td>
</tr>

<tr>
<td align="right">
<math>c</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\frac{4\pi}{3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>x</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\frac{R}{R_\mathrm{SWS}}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathfrak{G}</math>
</td>
<td align="center">
<math>\rightarrow</math>
</td>
<td align="left">
<math>\mathfrak{G}^{*}_\mathrm{SWS} \equiv \frac{\mathfrak{G}}{E_\mathrm{SWS}} </math>
</td>
</tr>
</table>
</td></tr>
</table>
</div>

where, drawing from [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Steven W. Stahler's (1983)] work — see also our [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>M_\mathrm{SWS}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>R_\mathrm{SWS}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>E_\mathrm{SWS} \equiv \biggl( \frac{n}{n+1}\biggr)\frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}} </math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, ,</math>
</td>
</tr>
</table>
</div>

and, in terms of the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Structural_Form_Factors|structural form factors]], <math>\tilde\mathfrak{f}_M</math>, <math>\tilde\mathfrak{f}_A</math>, and <math>\tilde\mathfrak{f}_W</math>,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\mathcal{A}_{M_\ell}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} =
\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\mathcal{B}_{M_\ell}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{-(n+1)/n} =
\biggl( \frac{3}{4\pi}\biggr)^{1/n}
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, .
</math>
</td>
</tr>
</table>
</div>

====Virial Equilibrium (M)====
Plugging these coefficient assignments into the [[#Equilibrium_Configurations|above mathematical prescription of the virial theorem]] gives the mass-radius relationship for pressure-truncated, polytropic equilibrium configurations, namely,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2}
+ \frac{4\pi}{3} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4 \, ,
</math>
</td>
</tr>

</table>
</div>
which matches the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseFreeEnergyExpression|statement of virial equilibrium presented in our accompanying, more detailed analysis]].

====Stability (M)====
Similarly, according to the [[#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} </math>
</td>
<td align="center">
<math>></math>
</td>
<td align="left">
<math>\biggl[ \frac{\mathcal{A}_{M_\ell}(n-3)}{4\pi n}\biggr]^{1/4} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{1/2}
\, .
</math>
</td>
</tr>

</table>
</div>
Or, flipped around, the criterion for stability may be written as,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} </math>
</td>
<td align="center">
<math><</math>
</td>
<td align="left">
<math>
\biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n-3)}\biggr]^{1/2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^2
\, .
</math>
</td>
</tr>

</table>
</div>

{{ SGFworkInProgress }}

==Outline of Detailed Investigations Leading to Above Summary==
===First Effort===
My [[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|first attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres]] was developed as a direct extension of my description of the virial equilibrium of ''isolated'' polytropes. An important outcome of this "first effort" was the unveiling of analytic expressions for the key structural form factors, both [[SSC/Virial/Polytropes#Summary|for ''isolated'' polytropes]] and, separately, [[SSC/Virial/Polytropes#PTtable|for ''pressure-truncated'' polytropic structures]].

I am very confident that the form-factor expressions presented for isolated polytropes are all correct because they have been cross-checked with expressions for closely related "integral" parameters discussed by Chandrasekhar [C67]. Although the form-factor expressions derived for pressure-truncated polytropes make some sense — they look very similar to the ones presented for isolated polytropes and seem to behave properly for models which, based on detailed force-balanced analysis, are known to be in equilibrium — I have much less confidence that they are correct. A couple of strategies were developed in an effort to demonstrate the validity and utility of these more general form-factor expressions, resulting in the derivation of a ''concise virial equilibrium'' relation,
<div align="center">
<math>\Pi_\mathrm{ad} = \chi_\mathrm{ad}^{-3\gamma} - \chi_\mathrm{ad}^{-4} \, ,</math>
</div>
that incorporates the newly defined normalization parameters, <math>~R_\mathrm{ad}</math> and <math>~P_\mathrm{ad}</math>. But subsequent derivations aimed at more conclusively demonstrating the correctness of the more general form-factor expressions were messy and got bogged down.

===Second Effort===
My [[SSC/Virial/PolytropesSummary#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|second attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres]] built upon my ''first effort'' and, for a couple of different polytropic indexes, focused on comparing the mass-radius relationship embodied in detailed force-balanced models against the mass-radius relationship implied by the virial theorem. A lot of reasonable results seem to have arisen from a discussion of [[SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_4_Polytropes|models (done numerically using Excel) with <math>~n=4</math> polytropic index]]. And there are some nice aspects of [[SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_5_Polytropes|models with an <math>~n=5</math> index]], but these models raise some [[SSC/Virial/PolytropesSummary#Serious_Concern|serious concerns]] related to the fact that two of our "derived" form-factor expressions involve division by the factor, <math>~(5-n)</math>, that is, division by zero.

===Third Effort===
In an attempt to answer the serious concern(s) raised during our first two efforts, we finally buckled down and [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|performed the integrals necessary to determine expressions for key structural form factors in the cases where the internal structure is known analytically]], specifically, for indexes <math>~n=5</math> and <math>~n=1</math>. The result is that the individual expressions derived by direct integration for <math>~\mathfrak{f}_W</math> and for <math>~\mathfrak{f}_A</math> ''do not match'' the general form-factor expressions that were rather cavalierly "derived" during our first effort. Oddly enough, as we discovered while [[SSCpt1/Virial/FormFactors#Fiddling_Around|fiddling around with the new results]], the ''ratio'' of these form factors appears to be the same as before, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_A - \tilde\theta^{n+1}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{(n+1) \tilde\xi^2 } \biggr] \, .
</math>
</td>
</tr>

</table>
</div>
It is worth noting that, as a result of this more thorough "third effort" examination, we have confirmed that the third key form factor,
<div align="center">
<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} = \biggl[- \frac{3\tilde\theta^'}{\tilde\xi}\biggr] \, ,</math>
</div>
which is the same as before and the same as for isolated polytropes. We also have determined that,
<div align="center">
<math>~\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} =
\biggl(\frac{\tilde\xi^2 \tilde\theta^'}{\xi_1 \theta^'_1} \biggr)\biggl[- \frac{\tilde\xi}{3\tilde\theta^'}\biggr]
= - \frac{\tilde\xi^3 }{3\xi_1 \theta^'_1} \, ,
</math>
</div>
except in the case of <math>~n=5</math> structures, for which we have determined,
<div align="center">
<math>~\biggl[\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_{n=5} =
\ell^3 = \biggl( \frac{\tilde\xi^2}{3} \biggr)^{3/2} \, .
</math>
</div>

===First Effort, Second Time Around===
[[SSC/Virial/PolytropesEmbedded/FirstEffortAgain|In an accompanying chapter]], we reproduce the discussion associated with our [[SSCpt1/Virial/PolytropesEmbeddedOutline#First_Effort|"First Effort", as referenced above]], but correct expressions for <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math>, as identified in our "Third Effort" and, accordingly, re-derive various affected expressions that follow.

===Second Effort, Second Time Around===
[[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|In an accompanying chapter]], we reproduce the discussion associated with our [[SSCpt1/Virial/PolytropesEmbeddedOutline#Second_Effort|"Second Effort", as referenced above]], but revise key sections to incorporate corrected expressions for the structural form factors.

=See Also=
<ul>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>

{{ SGFfooter }}