View source for SSC/Stability/BiPolytropes

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=Organizational Index=
The [[SSC/Stability/BiPolytropes/Index|accompanying organizational index]] has helped us write this chapter succinctly.

=In Search of Marginally Unstable (n<sub>c</sub>,n<sub>e</sub>) = (5,1) Bipolytropes=

<!-- Our aim is to determine whether or not there is a relationship between (1) equilibrium models at turning points along bipolytrope sequences and (2) bipolytropic models that are marginally (dynamically) unstable toward collapse (or dynamical expansion).-->


==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 = EqSeqCombined]]Figure 1: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
  </th>
</tr>
<tr>
  <td align="center" colspan="1">
[[File:MassVsRadiusCombined02.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
  </td>
</tr>
</table>
-->
<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: &nbsp;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 &#8212; which examines the relative stability of bipolytropes &#8212; 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.

<font color="red">'''Key Realization:'''</font>  ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----


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> &#8212; the ratio of the mean-molecular weight of envelope material to the mean-molecular weight of material in the core &#8212; 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), &frac12; (dark blue), 0.345 (brown), &#8531; (dark green), 0.316943 (purple), 0.309 (orange),  and &frac14; (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 &#8212; both the radius and the mass &#8212; 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 &frac12;.  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.  In Figure 2, a solid green circular marker identifies the location of this ''maximum mass turning point'' along each of these sequences; the analytically determined values of <math>\xi_i, q </math> and <math>\nu</math> that are associated with each of these ''turning points'' are provided in the table adjacent to Figure 2.  (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?''


<table border="1" align="center" cellpadding="3">
<tr>
  <td align="center" rowspan="1">
'''Figure 2:  Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
  </td>
  <td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
  </td>
</tr>
<tr>
  <td align="center" rowspan="8">
[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
  </td>
  <td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
  </td>
  <td align="center">
<math>\xi_i</math>
  </td>
  <td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
  </td>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{3}</math>
  </td>
  <td align="center">
<math>\infty</math> </td>
  <td align="center">0.0 </td>
  <td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
  <td align="center">
0.33
  </td>
  <td align="right">
24.00496  </td>
  <td align="right">
0.038378833  </td>
  <td align="right">
0.52024552  </td>
</tr>

<tr>
  <td align="center">
0.316943
  </td>
  <td align="right">
10.744571  </td>
  <td align="right">
0.068652714  </td>
  <td align="right">
0.382383875  </td>
</tr>

<tr>
  <td align="center">
0.31
  </td>
  <td align="right">
9.014959766  </td>
  <td align="right">
0.0755022550  </td>
  <td align="right">
0.3372170064  </td>
</tr>

<tr>
  <td align="center">
0.3090
  </td>
  <td align="right">
8.8301772  </td>
  <td align="right">
0.076265588  </td>
  <td align="right">
0.331475715  </td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{4}</math>
  </td>
  <td align="right">
4.9379256  </td>
  <td align="right">
0.084824137  </td>
  <td align="right">
0.139370157  </td>
</tr>

<tr>
  <td align="left" colspan="4">
<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
  </td>
</tr>
</table>

==Planned Approach(es)==

In an effort to answer the '''<font color="red">principal question</font>''' posed above, we have pursued each stability-analysis approach described in the introductory section of {{ B-KB74full }}.

<table border="0" align="center" cellpadding="8" width="95%">
<tr><td align="left" rowspan="1">
<font color="darkgreen"><span id="BKB74pt1">"Three different approaches are used in the study of the hydrodynamical stability of stars</span> and other gravitating objects &hellip;" &nbsp; 
<ul><li>"The first approach is based on the use of the equations of small oscillations.  In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations.  The solutions consist of a set of eigenfrequencies and eigenfunctions."</font> </li>

<li>Second, one can derive <font color="darkgreen">"a variational principle from the equations of small oscillations &hellip;</font>  <!-- This principle replaces the straightforward solution of these equations:</font>  In the context of rotating Newtonian systems, see, for example, [http://adsabs.harvard.edu/abs/1964ApJ...140.1045C Clement (1964)], [http://adsabs.harvard.edu/abs/1968ApJ...152..267C Chandrasekhar &amp; Lebovitz (1968)], [http://adsabs.harvard.edu/abs/1967MNRAS.136..293L Lynden-Bell and Ostriker (1967)], or [http://adsabs.harvard.edu/abs/1972ApJS...24..319S Schutz (1972)]. -->  <font color="darkgreen">With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations.  In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated &hellip;"</font></li>

<li>The third approach is what we usually refer to as a free-energy &#8212; and associated virial theorem &#8212; analysis.  <font color="darkgreen">"When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient.  The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability."</font> </li>
</ul>

<span id="BKB74pt2"><font color="darkgreen">"If one wants</span> to know from a stability analysis the answer to only one question &#8212; whether the model is stable or not &#8212; then the most straightforward procedure is to use the third, static method &hellip;  For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation."</font>

<font color="darkgreen">"Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point."</font>
</td></tr>
<tr>
<td align="right">
— Drawn from pp. 391 - 392 of {{ B-KB74 }} 
</td>
</tr>
</table>

==Supplemental Chapters==
<ol>
  <li>Containes [[SSC/Stability/InstabilityOnsetOverview#Displacement_Functions_Summary|Displacement Functions Summary]]</li>
  <li>[[SSC/Stability/BiPolytropes/PlannedApproach|Earlier Planned Approach]]</li>
  <li>[[SSC/Stability/BiPolytropes/HeadScratching|Headscratching]]</li>
  <li>[[SSC/Stability/BiPolytropes/SuccinctDiscussion|Succinct Discussions]]</li>
  <li>[[SSC/Stability/BiPolytropes/51Models|51Models]]</li>
</ol>

=Equilibrium Models=

The original text contained in this section has been commented out.
<!--
The equilibrium sequences displayed in Figure 1 &#8212; for pressure-truncated polytropes &#8212; and in Figure 2 &#8212; for <math>(n_c, n_e) = (5, 1)</math> bipolytropes &#8212; were constructed by solving the second-order ODE that governs detailed force-balance throughout spherically symmetric configurations that obey polytropic equations of state.  In Figure 1, the sequences labeled <math>n = 1</math> and <math>n = 5</math> can be specified entirely through analytical expressions; these are well-known analytic solutions to the Lane-Emden equation.  In Figure 2, ''all seven equilibrium sequences'' are specifiable analytically; the location(s) of ''turning points'' along each sequence (when they exist) are also completely specifiable analytically.

In this chapter we examine the relative stability of various equilibrium <math>(n_c, n_e) = (5, 1)</math> bipolytropes.  A model is uniquely selected once we specify the radial location, <math>\xi_i</math>, of the core/envelope interface, and specify the ratio <math>\mu_e/\mu_c \le 1</math> of the mean-molecular weights of the gas in the envelope to the gas in the core.  Once this pair of parameter values has been specified, the model's radius <math>R_\mathrm{eq}</math> and total mass <math>M_\mathrm{tot}</math> are known analytically, as is the relative size of the core &#8212; both in terms of its fractional radius, <math>q \equiv r_i/R_\mathrm{eq}</math>, and its fractional mass, <math>\nu \equiv m_\mathrm{core}/M_\mathrm{tot}</math>.  Knowledge of the <math>(q, \nu)</math> parameter pair precisely identifies the selected model's location in the Figure 2 diagram.

In line with the '''<font color="red">principal question</font>''' stated above, we are particularly interested in examining the relative stability of models that are associated with the "maximum-mass turning points" along bipolytropic sequences for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math> &#8212; see the solid green circular markers in Figure 2.
-->

=Third Approach: Free-Energy Analysis=

==Old Material==
<table border="0" align="right" cellpadding="5"><tr><td align="center">[[File:VirialStability01.png|350px|Virial Stability01]]</td></tr></table>
The following set of menu tiles include links to chapters where this approach has been applied to: (a) uniform-density configurations, (b) pressure-truncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressure-truncated n = 5 configurations, and (e) bipolytropes having <math>(n_c, n_e) = (1, 5)</math>.

One menu tile, below, links to a chapter in which an analytic (''exact'') demonstration of the variational principle's utility is provided in the context pressure-truncated n = 5 polytropes.

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: &nbsp; [[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 &#8212; at least initially &#8212; 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]] &#8212; including a quantitative comparison against the published work of {{ Schwarzschild41full }};
* [[SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres|Pressure-truncated isothermal spheres]] &#8212; including a quantitative comparison against the published analysis of {{ TVH74full }}; 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 {{ MF85bfull }}.  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 {{ MF85b }} 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 {{ MF85b }}.
** 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 {{ LW58full }}, 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.

==New Material==
Drawing from our [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|accompanying detailed discussion]] &#8212; see also an [[SSC/Stability/BiPolytropes/PlannedApproach#Free_Energy_and_Its_Derivatives|accompanying summary]] &#8212; the normalized free-energy associated with each of our spherically symmetric, bipolytropic configurations is given by the expression,

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

<tr>
  <td align="right">
<math>\mathfrak{g} \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl(\frac{\chi_\mathrm{eq}}{\chi}\biggr)\mathfrak{w} 
+  \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{3 - 3\gamma_c}\mathfrak{s}_\mathrm{core}  
+  \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{3 - 3\gamma_e}\mathfrak{s}_\mathrm{env}  
\, ,
</math>
  </td>
</tr>
</table>

where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively, in the configuration's equilibrium state; and <math>\mathfrak{w}</math> is the absolute value of the normalized total gravitational potential energy of the equilibrium configuration.  For every <math>(n_c, n_e) = (5, 1)</math> bipolytropic model, the values of these four terms can be obtained via [[SSC/Stability/BiPolytropes/PlannedApproach#Free_Energy_and_Its_Derivatives|our derived analytic expressions]]. And the value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>.  

This expression for <math>\mathfrak{g}</math> has been written in such a way that we can readily assess how the free energy varies while the configuration undergoes homologous (<math>\mathfrak{s}_\mathrm{core}, \mathfrak{s}_\mathrm{env}, \mathfrak{w}</math> all held fixed) expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.  Specifically the ''first variation'' is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\mathfrak{g}}{d\chi} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
+\biggl(\frac{\chi_\mathrm{eq}}{\chi^2}\biggr)\mathfrak{w} 
~-~\frac{2}{\chi_\mathrm{eq}} \biggl[\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{2 - 3\gamma_c}\mathfrak{s}_\mathrm{core}  
~+~\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{2 - 3\gamma_e}\mathfrak{s}_\mathrm{env}  \biggr]  \, ;
</math>
  </td>
</tr>
</table>
and the ''second variation'' is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d^2\mathfrak{g}}{d\chi^2} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2\biggl(\frac{\chi_\mathrm{eq}}{\chi^3}\biggr)\mathfrak{w} 
~-~\frac{2}{\chi_\mathrm{eq}^2} \biggl[(2 - 3\gamma_c) \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{1 - 3\gamma_c}\mathfrak{s}_\mathrm{core}  
~+~(2 - 3\gamma_e)\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{1 - 3\gamma_e}\mathfrak{s}_\mathrm{env}  \biggr] \ .
</math>
  </td>
</tr>
</table>
The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability.

=Virial Analysis=

In an [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|accompanying chapter]] we have examined the viral stability of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes &#8212; see also an [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|associated discussion of the free-energy of these configurations]].  We present, here, an overview of this separate discussion in an effort to provide a broader perspective to the focus of this current chapter.

==Various Energy Expressions==
Drawing from our [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|accompanying detailed discussion]], we have,
<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}\biggr|_\mathrm{core} + \mathfrak{S}_A\biggr|_\mathrm{core} + W_\mathrm{grav}\biggr|_\mathrm{env} + \mathfrak{S}_A\biggr|_\mathrm{env} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~W_\mathrm{grav}\biggr|_\mathrm{core} +  \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + W_\mathrm{grav}\biggr|_\mathrm{env} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env}  \, .
</math>
  </td>
</tr>
</table>
</div>

Recognizing that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_i}{R} = \frac{\eta_i}{\eta_s} \, ,</math>
  </td>
</tr>

<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>~\ell</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\xi}{\sqrt{3}}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\pi}{2^3 \cdot 3^6}\biggr)^{1 / 2}
\biggl( \frac{\nu}{q^3}  \biggr)^{2} \biggl( 1 + \ell_i^2 \biggr)^{3}  \biggl( \frac{q}{\ell_i}\biggr)^{5 } \, ,
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~- \chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
\biggl[\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3}  \biggr]_\mathrm{eq}
\biggl[
\ell_i (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>- \frac{\chi_\mathrm{eq}}{\chi} 
\biggl( \frac{3^8}{2^5 \pi}\biggr)^{1 / 2}
\biggl[
\ell_i (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i
\biggr] \, .
</math>
  </td>
</tr>
</table>
Next, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~b_\eta = \eta_s</math>
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
and
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 </td>
  <td align="left">
<math>~b_\eta q = \eta_i = 3\biggl( \frac{\mu_e}{\mu_c} \biggr) \ell_i (1 + \ell_i^2)^{-1}  \, ,</math>
  </td>
</tr>
</table>
</div>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3}  \biggl( \frac{1}{2^3\pi} \biggr)^{1/2} A^2 
\biggl[6 b_\eta x  - 3\sin[2(b_\eta x-B)] - 4b_\eta x \sin^2(b_\eta x - B)  \biggr]_q^1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} 
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3}  \biggl( \frac{1}{2^3\pi} \biggr)^{1/2} A^2 
\biggl\{
\biggl[ 6 \eta_s  - 3\sin[2(\eta_s  -B)] - 4\eta_s   \sin^2(\eta_s   - B)  \biggr]
-
\biggl[ 6 \eta_i    - 3\sin[2(\eta_i -B)] - 4\eta_i  \sin^2(\eta_i  - B)  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~q a_\xi^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\xi_i}{\sqrt{3}} \, ,</math>
  </td>
</tr>
</table>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3({\gamma_c}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_c} 
\biggl\{\biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[ 
\tan^{-1}[a_\xi^{1/2}q] - a_\xi^{1/2}q ~\frac{(1 - a_\xi q^2)}{(1 + a_\xi q^2)^2} \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3({\gamma_c}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_c} 
\biggl\{\biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[ 
\tan^{-1}\biggl( \frac{\xi_i}{\sqrt{3}}\biggr) - \biggl( \frac{\xi_i}{\sqrt{3}}\biggr)\frac{(1 - \xi_i^2/3)}{(1 + \xi_i^2/3)^2} \biggr]
\biggr\} 
</math>
  </td>
</tr>
</table>

Finally, then, we can write,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
  <td align="left">
<math>
\frac{2}{3({\gamma_e}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_e} \biggl\{
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2  \biggl( \frac{3^2}{2^5\pi} \biggr)^{1/2}  \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
  <td align="left">
<math>
\frac{2}{3({\gamma_e}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_e} 
\bigg( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2  \biggl( \frac{3^2}{2^5\pi} \biggr)^{1/2}  \biggl\{ 
\biggl[ 2\eta_s -\sin[2(\eta_s - B)] \biggr]  
-
\biggl[ 2\eta_i -\sin[2(\eta_i - B)] \biggr]  
\biggr\} \ .
</math>
  </td>
</tr>
</table>
</div>

==Free Energy and Its Derivatives==

We can now rewrite the free-energy expression in the form,

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

<tr>
  <td align="right">
<math>~\mathfrak{g} \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~-\biggl(\frac{\chi_\mathrm{eq}}{\chi}\biggr)\mathfrak{w} 
+  \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{3 - 3\gamma_c}\mathfrak{s}_\mathrm{core}  
+  \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{3 - 3\gamma_e}\mathfrak{s}_\mathrm{env}  
\, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[ 
\tan^{-1}\biggl( \frac{\xi_i}{\sqrt{3}}\biggr) - \biggl( \frac{\xi_i}{\sqrt{3}}\biggr)\frac{(1 - \xi_i^2/3)}{(1 + \xi_i^2/3)^2} \biggr]\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\bigg( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2  \biggl( \frac{3^2}{2^5\pi} \biggr)^{1/2}  \biggl\{ 
\biggl[ 2\eta_s -\sin[2(\eta_s - B)] \biggr]  
-
\biggl[ 2\eta_i -\sin[2(\eta_i - B)] \biggr]  
\biggr\} \, ,
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3}  \biggl( \frac{1}{2^3\pi} \biggr)^{1/2} A^2 
\biggl\{
\biggl[ 6 \eta_s  - 3\sin[2(\eta_s  -B)] - 4\eta_s   \sin^2(\eta_s   - B)  \biggr]
-
\biggl[ 6 \eta_i    - 3\sin[2(\eta_i -B)] - 4\eta_i  \sin^2(\eta_i  - B)  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>

The first derivative is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\mathfrak{g}}{d\chi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
+~\biggl(\frac{\chi_\mathrm{eq}}{\chi^2}\biggr)\mathfrak{w} 
~-~\frac{2}{\chi_\mathrm{eq}} \biggl[\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{2 - 3\gamma_c}\mathfrak{s}_\mathrm{core}  
~+~\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{2 - 3\gamma_e}\mathfrak{s}_\mathrm{env}  \biggr]  \, .
</math>
  </td>
</tr>
</table>
And the second derivative is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2\mathfrak{g}}{d\chi^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
-~2\biggl(\frac{\chi_\mathrm{eq}}{\chi^3}\biggr)\mathfrak{w} 
~-~\frac{2}{\chi_\mathrm{eq}^2} \biggl[(2 - 3\gamma_c) \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{1 - 3\gamma_c}\mathfrak{s}_\mathrm{core}  
~+~(2 - 3\gamma_e)\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{1 - 3\gamma_e}\mathfrak{s}_\mathrm{env}  \biggr] \ .
</math>
  </td>
</tr>
</table>

==What to Expect for Equilibrium Configurations==
In equilibrium we should set <math>~d\mathfrak{g}/d\chi = 0</math> and <math>~\chi = \chi_\mathrm{eq}</math>.  In which case we should expect to find that,
<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}{\chi_\mathrm{eq}}\biggl[\mathfrak{w}
~-~2(\mathfrak{s}_\mathrm{core}  
~+~\mathfrak{s}_\mathrm{env})  \biggr]  \, .
</math>
  </td>
</tr>
</table>
In addition, an evaluation of the second derivative should give,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\chi_\mathrm{eq}^2}{2} \cdot \frac{d^2\mathfrak{g}}{d\chi^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
-~\mathfrak{w} 
~-~ \biggl[(2 - 3\gamma_c) \mathfrak{s}_\mathrm{core}  
~+~(2 - 3\gamma_e)\mathfrak{s}_\mathrm{env}  \biggr] \, .
</math>
  </td>
</tr>
</table>
Then, the transition from stable to unstable configurations occurs when <math>~d^2\mathfrak{g}/d\chi^2 = 0</math>, that is, when,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{w} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~-~ \biggl[(2 - 3\gamma_c) \mathfrak{s}_\mathrm{core}  
~+~(2 - 3\gamma_e)\mathfrak{s}_\mathrm{env}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~(4 - 3\gamma_c) \mathfrak{s}_\mathrm{core}  
~+~(4 - 3\gamma_e)\mathfrak{s}_\mathrm{env}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{\mathfrak{s}_\mathrm{core} }{\mathfrak{s}_\mathrm{env}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~-~\frac{(4 - 3\gamma_e) }{(4 - 3\gamma_c) } \, .
</math>
  </td>
</tr>
</table>
For example, if we set <math>~\gamma_c = 6/5</math> and <math>~\gamma_e = 2</math>, this implies,

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

<tr>
  <td align="right">
<math>~\frac{\mathfrak{s}_\mathrm{core} }{\mathfrak{s}_\mathrm{env}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{2 }{(4 - 18/5) } = 5 \, .
</math>
  </td>
</tr>
</table>


==Free-Energy Stability Evaluation==

Here we pull together excerpts from several different [[H_BookTiledMenu#Tiled_Menu|H_Book Chapters]] in which we have presented, from several different perspectives, an analysis of the free-energy of bipolytropes.  
* In [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|one chapter]], using purely analytic techniques, we have derived expressions that detail the structural properties of bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>.  Among these are analytic expressions for various terms that make up the free-energy expression:  <math>~\mathfrak{s}_\mathrm{core}</math>, <math>~\mathfrak{s}_\mathrm{env}</math>, <math>~\mathfrak{w}_\mathrm{core}</math>, <math>~\mathfrak{w}_\mathrm{core}</math>, and <math>~P_iV_\mathrm{core}</math>. Equilibrium model ''sequences'' are defined by fixing the ratio, <math>~\mu_e/\mu_c</math>, then varying the radial location, <math>~r_i</math>, of the core-envelope interface; note that the volume of the core is, then, <math>~V_\mathrm{core} \equiv 4\pi r_i^3/3</math>.  
<div align="left">
* <font color="red">[Virial Equilibrium]</font> In a subsection of this same chapter titled, ''[[SSC/Structure/BiPolytropes/Analytic51#Global|Equilibrium Condition: Global]]'', we have shown that a statement of virial equilibrium &#8212; obtained by setting the first derivative of the free-energy expression to zero &#8212; is,<table border="0" cellpadding="4" align="center">
<tr>
  <td align="right">
<math>~( 2S + W )_\mathrm{tot} ~=~ 2(S_\mathrm{core} + S_\mathrm{env}) + (W_\mathrm{core} + W_\mathrm{env})</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 0 \, .</math>
  </td>
</tr>
</table>In another subsection of this same chapter titled, ''[[SSC/Structure/BiPolytropes/Analytic51#Global|Equilibrium Condition: In Parts]]'', we showed that for each bipolytropic equilibrium structure, the statements
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~2S_\mathrm{core} + W_\mathrm{core} = 3P_i V_\mathrm{core}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~2S_\mathrm{env} + W_\mathrm{env} = - 3P_i V_\mathrm{core}  \, ,</math>
  </td>
</tr>
</table>also hold separately. Therefore, for every equilibrium configuration we should expect the <b>CASE1</b> expression (see Table XXX) to precisely sum to unity.
</div>

<div align="left">
* <font color="red">[Marginally Unstable Model]</font> Near the bottom of this same chapter, in a subsection titled, ''[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Stability Condition]]'', we point out that the model along each sequence that is marginally (dynamically) unstable &#8212; obtained setting the second derivative of the free-energy expression to zero &#8212; is identified by the configuration for which,<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
2(\gamma_e - \gamma_c) \mathfrak{s}_\mathrm{core} + ( \mathfrak{w}_\mathrm{core} +  \mathfrak{w}_\mathrm{env} )(\gamma_e - \tfrac{4}{3})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  \, .</math>
  </td>
</tr>
</table>Therefore, along each equilibrium sequence, the marginally unstable model can be identified by the configuration for which the <b>CASE2</b> expression (see Table XXX) precisely sums to zero.  Immediately above, in a subsection titled, ''[[SSC/Stability/BiPolytropes#What_to_Expect_for_Equilibrium_Configurations|What to Expect for Equilibrium Configurations]]'', we have shown that this same marginally unstable model can be identified by the configuration for which the <b>CASE3</b> energy ratio,  <math>~\mathfrak{s}_\mathrm{core}/\mathfrak{s}_\mathrm{env}</math>, has a value that is precisely 5. And, thirdly, as highlighted in our accompanying [[SSC/SynopsisStyleSheet#Bipolytropes|Tabular Overview]], this same marginally unstable model can be identified by the configuration for which the <b>CASE4</b> expression precisely sums to zero.
</div>


<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="16">
'''Table XXX:'''  Properties of Marginally Unstable Bipolytropes Having<br /><br /><math>~(n_c, n_e) = (5, 1)</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math><br /><br />Determined from Free-Energy Arguments
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~R^*_\mathrm{surf}</math></td>
  <td align="center"><math>~q \equiv \frac{r_\mathrm{core}}{R_\mathrm{surf}}</math></td>
  <td align="center"><math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~P_i</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center"><math>~3P_i V_\mathrm{core}</math></td>
  <td align="center"><b>CASE1</b></td>
  <td align="center"><b>CASE2</b></td>
  <td align="center"><b>CASE3</b></td>
  <td align="center"><b>CASE4</b></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">2.41610822</td>
  <td align="right">2.8049</td>
  <td align="center">0.59520261</td>
  <td align="center">0.68306067</td>
  <td align="center">16.3788</td>
  <td align="center">0.039116848</td>
  <td align="center">4.446748782</td>
  <td align="center">- 6.606135366</td>
  <td align="center">0.889349762</td>
  <td align="center">- 4.066061722</td>
  <td align="center">2.287362198</td>
  <td align="center">1</td>
  <td align="center">0</td>
  <td align="center">5.0000000</td>
  <td align="center">2.1 &times; 10<sup>-8</sup></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">4.1853093</td>
  <td align="right">8.8058</td>
  <td align="center">0.328419479</td>
  <td align="center">0.70131896</td>
  <td align="center">354.089</td>
  <td align="center">0.003126324</td>
  <td align="center">5.76978580</td>
  <td align="center">- 10.58931853</td>
  <td align="center">1.153956968</td>
  <td align="center">- 3.258165567</td>
  <td align="center">0.95025163</td>
  <td align="center">1</td>
  <td align="center">6.3 &times; 10<sup>-8</sup></td>
  <td align="center">5.0000002</td>
  <td align="center">0</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">7.64325</td>
  <td align="right">44.116</td>
  <td align="center">0.119714454</td>
  <td align="center">0.52700045</td>
  <td align="center">2.85 &times; 10<sup>4</sup></td>
  <td align="center">0.000116533</td>
  <td align="center">6.230343527</td>
  <td align="center">- 12.24495934</td>
  <td align="center">1.1246068658</td>
  <td align="center">- 2.707865028</td>
  <td align="center">0.215727713</td>
  <td align="center">1</td>
  <td align="center">6.4 &times; 10<sup>-8</sup></td>
  <td align="center">5.0000002</td>
  <td align="center">0</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">8.548103</td>
  <td align="right">59.643</td>
  <td align="center">0.099032423</td>
  <td align="center">0.47901529</td>
  <td align="center">6.30 &times; 10<sup>4</sup></td>
  <td align="center">6.1337 &times; 10<sup>-5</sup></td>
  <td align="center">6.261548334</td>
  <td align="center">- 12.36425897</td>
  <td align="center">1.252309682</td>
  <td align="center">- 2.663457063</td>
  <td align="center">0.158837699</td>
  <td align="center">1</td>
  <td align="center">0</td>
  <td align="center">4.9999999</td>
  <td align="center">6.0 &times; 10<sup>-8</sup></td>
</tr>
<tr>
  <td align="center">0.316943</td>
  <td align="left">10.7441565</td>
  <td align="right">108.14</td>
  <td align="center">0.068655205</td>
  <td align="center">0.38238387</td>
  <td align="center">2.93 &times; 10<sup>5</sup></td>
  <td align="center">1.6252 &times; 10<sup>-5</sup></td>
  <td align="center">6.301810768</td>
  <td align="center">- 12.52005323</td>
  <td align="center">1.260362204</td>
  <td align="center">- 2.604292714</td>
  <td align="center">0.083568307</td>
  <td align="center">1</td>
  <td align="center">0</td>
  <td align="center">4.9999998</td>
  <td align="center">2.0 &times; 10<sup>-7</sup></td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">12.77156</td>
  <td align="right">166.06</td>
  <td align="center">0.053145011</td>
  <td align="center">0.31696879</td>
  <td align="center">8.70 &times; 10<sup>5</sup></td>
  <td align="center">5.8905 &times; 10<sup>-6</sup></td>
  <td align="center">6.318902171</td>
  <td align="center">- 12.58692884</td>
  <td align="center">1.26378042</td>
  <td align="center">- 2.57843634</td>
  <td align="center">0.050875500</td>
  <td align="center">1</td>
  <td align="center">1.9 &times; 10<sup>-8</sup></td>
  <td align="center">5.0000001</td>
  <td align="center">0</td>
</tr>
<tr>
  <td align="center" colspan="16">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<b>CASE1</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{6P_i V_\mathrm{core}}\biggl[
(2 \mathfrak{s}_\mathrm{core} + \mathfrak{w}_\mathrm{core})  
-(2 \mathfrak{s}_\mathrm{core} + \mathfrak{w}_\mathrm{core})  
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<b>CASE2</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
2(\gamma_e - \gamma_c) \mathfrak{s}_\mathrm{core} + ( \mathfrak{w}_\mathrm{core} +  \mathfrak{w}_\mathrm{env} )(\gamma_e - \tfrac{4}{3})
</math>
  </td>
</tr>

<tr>
  <td align="right">
<b>CASE3</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\mathfrak{s}_\mathrm{core}}{\mathfrak{s}_\mathrm{env}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<b>CASE4</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
(4 - 3\gamma_c)\mathfrak{w}_\mathrm{core} + (4 - 3\gamma_e)\mathfrak{w}_\mathrm{env} + 3^2(\gamma_c - \gamma_e)P_i V_\mathrm{core}
</math>
  </td>
</tr>
</table>

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


The left-hand panel of Figure 3 is identical to Figure 2, above.  It displays in the <math>~q-\nu</math> parameter space, the behavior of <math>~(n_c, n_e) = (5, 1)</math> bipolytropic equilibrium sequences that have, as labeled, seven different values of the ratio of mean-molecular-weights, <math>~\mu_e/\mu_c</math>.  Using a numerical root-finding technique, we have determined where the virial stability condition, <math>~\mathfrak{s}_\mathrm{core}/\mathfrak{s}_\mathrm{env} = 5</math>, is satisfied along each of these sequences &#8212; as well as along a number of additional equilibrium sequences.  Key properties of each of these identified models have been recorded in [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Table 1 of an accompanying discussion]]; see also an [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|associated discussion of the free-energy of these configurations]].  Pulling from this tabulated data, the solid-red circular markers that appear in the right-hand panel of Figure 3 identify where this virial stability condition is satisfied along the separate equilibrium sequences while the accompanying dashed red curve identifies more broadly how the <math>~q-\nu</math> parameter space is divided into stable (below and to the right) versus unstable (above and to the left) regions.  

<table border="0" align="center" cellpadding="8">
<tr>
  <th align="center">Figure 3</th>
</tr>
<tr>
  <td align="center">
[[File:CompositeAlabeled.png|800px|dynamical stability in qNu plane]]
  </td>
</tr>
</table>

In what follows we use a complementary &#8212; and more quantitatively rigorous &#8212; approach to evaluating the stability of equilibrium models, and contrast the results of that analysis with the virial-analysis results presented graphically here in Figure 3.

=Review of the Analysis by Murphy &amp; Fiedler (1985b)=

As we have [[SSC/Stability/Polytropes#Boundary_Conditions|detailed separately]], the boundary condition at the center of a polytropic configuration is,
<div align="center">
<math>\frac{dx}{d\xi} \biggr|_{\xi=0} = 0 \, ;</math>
</div>
and the boundary condition at the surface of an isolated polytropic configuration is,
<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>- \alpha +  \frac{\omega^2}{\gamma_g }  \biggl( \frac{1}{4\pi G \rho_c } \biggr) \frac{\xi}{(-\theta^')} </math>
&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = \xi_s \, .</math>
  </td>
</tr>
</table>
But this surface condition is not applicable to bipolytropes.  Instead, let's return to the [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations|original, more general expression of the surface boundary condition]]:

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln\xi}\biggr|_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \alpha + \frac{\omega^2 R^3}{\gamma_g GM_\mathrm{tot}} \, .</math>
  </td>
</tr>
</table>

<div id="SurfaceCondition">
<table border="1" align="center" width="85%" cellpadding="10"><tr><td align="left">
Utilizing an [[SSC/Stability/Polytropes#Groundwork|accompanying discussion]], let's examine the frequency normalization used by {{ MF85b }} &#8212; see the top of the left-hand column on p. 223:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Omega^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\omega^2 \biggl[ \frac{R^3}{GM_\mathrm{tot}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\omega^2 \biggl[ \frac{3}{4\pi G \bar\rho} \biggr]
=
\omega^2 \biggl[ \frac{3}{4\pi G \rho_c} \biggr] \frac{\rho_c}{\bar\rho}
=
\frac{3\omega^2}{(n_c+1)} \biggl[ \frac{(n_c+1)}{4\pi G \rho_c} \biggr] \frac{\rho_c}{\bar\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3\omega^2}{(n_c+1)} \biggl[ \frac{a_n^2\rho_c}{P_c} \cdot \theta_c \biggr] \frac{\rho_c}{\bar\rho}
=
\frac{3\gamma}{(n_c+1)} \frac{\rho_c}{\bar\rho} \biggl[ \frac{a_n^2\rho_c}{P_c} \cdot \frac{\omega^2 \theta_c}{\gamma} \biggr] \, .
</math>
  </td>
</tr>
</table>

For a given radial quantum number, <math>k</math>, the factor inside the square brackets in this last expression is what {{ MF85b }} refer to as <math>\omega^2_k \theta_c</math>.  Keep in mind, as well, that, in the notation we are using,
<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>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \sigma_c^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\bar\rho}{\rho_c}\biggr) \Omega^2 
=
\frac{6\gamma}{(n_c+1)} \biggl[ \frac{a_n^2\rho_c}{P_c} \cdot \frac{\omega^2 \theta_c}{\gamma} \biggr] 
=
\frac{6\gamma}{(n_c+1)} \biggl[ \omega_k^2 \theta_c \biggr] \, .
</math>
  </td>
</tr>
</table>
This also means that the surface boundary condition may be rewritten as,

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln\xi}\biggr|_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\Omega^2}{\gamma_g } - \alpha  \, .</math>
  </td>
</tr>
</table>

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

Let's apply these relations to the core and envelope, separately.

==Interface Conditions==
Here, we will simply copy the discussion already provided in the context of our attempt to analyze the stability of <math>(n_c, n_e) = (0, 0)</math> bipolytropes; specifically, we will draw from [[SSC/Stability/BiPolytrope00#Piecing_Together|<font color="red">'''STEP 4:'''</font> in the ''Piecing Together'' subsection]].  Following the discussion in &sect;&sect;57 &amp; 58 of {{ LW58 }}, the proper treatment is to ensure that fractional perturbation in the gas pressure (see their equation 57.31),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\delta P}{P}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \gamma x \biggl( 3 + \frac{d\ln x}{d\ln \xi} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
is continuous across the interface.  

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">'''Reaffirmation:'''</font> &nbsp;  In our [[SSC/Perturbations#The_Eigenvalue_Problem|introductory discussion]] of the eigenvalue problem, we adopted the following expression for the time-dependent pressure,

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

<tr>
  <td align="right">
<math>P(m,t)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>P_0(m) + P_1(m,t) = P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \, .</math>
  </td>
</tr>
</table>
In this expression, <math>P_0(m)</math> is the function that details how the unperturbed pressure varies with Lagrangian mass shell <math>(m)</math>, and <math>P_1 = \delta P(m) e^{i\omega t}</math> traces the variation of the pressure away from its equilibrium value at each mass shell. The time-dependent and spatially dependent behavior of <math>P_1</math> has been separated, with <math>\delta P</math> carrying information about the function's spatial dependence.  Furthermore, we have adopted the shorthand notation,
<div align="center"><math>p(m) \equiv \frac{\delta P(m)}{P_0(m)} \, .</math></div>
We can just as well use <math>r_0(m)</math> to tag the (initial, unperturbed location of the) Lagrangian mass shells, in which case we write,
<div align="center"><math>p(r_0) = \frac{\delta P(r_0)}{P_0(r_0)} \, ,</math> &nbsp; &nbsp; and, similarly, &nbsp; &nbsp;<math>d(r_0)= \frac{\delta \rho(r_0)}{\rho_0(r_0)} \, ,</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>x(r_0)= \frac{\delta r(r_0)}{r_0} \, .</math></div>

These three spatially dependent quantities &#8212; <math>p, d,</math> and <math>x</math> &#8212; are related to one another via the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|set of linearized governing relations]], namely,

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
p = \gamma_\mathrm{g} d  \, .
</math>
</td></tr>
</table>
</div>
Using the third of these expressions to replace <math>d</math> in favor of <math>p</math> in the first expression, we find that,
<table border="0" align="center">

<tr>
  <td align="right">
<math>r_0 \frac{dx}{dr_0}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>-3x - \frac{p}{\gamma_g}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~p</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>-\gamma_g x \biggl[3 +  \frac{d \ln x}{d \ln r_0} \biggr] \, ,</math>
  </td>
</tr>
</table>
which is identical to the pressure-perturbation expression used by {{ LW58 }} and referenced above.  As they state, the function, <math>p = \delta P/P_0</math>, should be continuous across the core-envelope interface.
</td></tr></table>
That is to say, at the interface <math>(\xi = \xi_i)</math>, we need to enforce the relation,
<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[ \gamma_c x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \gamma_e x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=\xi_i}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\gamma_e \biggl[ \frac{\gamma_c}{\gamma_e} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=\xi_i}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=\xi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} \, .</math>
  </td>
</tr>
</table>
</div>
In the context of this interface-matching constraint (see their equation 62.1), {{ LW58 }} state the following: &nbsp; <font color="darkgreen"><b>In the static</b></font> (''i.e.,'' unperturbed equilibrium) <font color="darkgreen"><b>model</b></font> &hellip; <font color="darkgreen"><b>discontinuities in <math>\rho</math> or in <math>~\gamma</math> might occur at some [radius]</b></font>.  <font color="darkgreen"><b>In the first case</b></font> &#8212; that is, a discontinuity only in density, while <math>\gamma_e = \gamma_c</math> &#8212; the interface conditions <font color="darkgreen"><b>imply the continuity of <math>\tfrac{1}{x} \cdot \tfrac{dx}{d\xi}</math> at that [radius].  In the second case</b></font> &#8212;  that is, a discontinuity in the adiabatic exponent &#8212; <font color="darkgreen"><b>the dynamical condition may be written</b></font> as above.  <font color="darkgreen"><b>This implies a discontinuity of the first derivative at any discontinuity of <math>~\gamma</math></b></font>.

The algorithm that {{ MF85b }} used to "<font color="#007700">&hellip; [integrate] through each zone &hellip;</font>" was designed "<font color="#007700">&hellip; with continuity in <math>x</math> and <math>dx/d\xi</math> being imposed at the interface &hellip;</font>"  Given that they set <math>\gamma_c = \gamma_e = 5/3</math>, their interface matching condition is consistent with the one prescribed by {{ LW58 }}.

=Radial Oscillations of (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Models=
==Foundation==
In an [[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 />

{{Math/EQ_RadialPulsation01}}
</div>

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.  Assuming that the underlying equilibrium structure is that of a bipolytrope having <math>~(n_c, n_e) = (5, 1)</math>, it makes sense to adopt the normalizations used when defining the equilibrium structure, namely,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\rho_0}{\rho_c}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{[K_c^{1/2}/(G^{1/2}\rho_c^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{K_c\rho_c^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_c^{1/5})]}</math>
  </td>
</tr>

</table>
</div>

We note as well that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM(r_0)}{r_0^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
G \biggl[ M_r^* \rho_c^{-1 / 5} \biggl( \frac{K_c}{G}\biggr)^{3 / 2} \biggr] \biggl[ r^* \rho_c^{-2 / 5}\biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{G M_r^*}{(r^*)^2} \biggl[ \rho_c^{3 / 5} \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] \, .
</math>
  </td>
</tr>
</table>

Hence, multiplying the LAWE through by <math>~(K_c/G)\rho_c^{-4 / 5}</math> gives,

<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>
</table>



<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*^2} + \biggl[\frac{4}{r^*} ~-~ \rho_c^{-2 / 5} \biggl( \frac{K_c}{G} \biggr)^{1 / 2}\biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr*} 
~+~ \rho_c^{-4 / 5}\biggl( \frac{K_c}{G} \biggr)\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">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} ~+~ \biggl\{\frac{4}{r^*} ~-~ \rho_c^{-2 / 5} \biggl( \frac{K_c}{G} \biggr)^{1 / 2}  \frac{G M_r^*}{(r^*)^2} \biggl[ \rho_c^{3 / 5} \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] \biggl[ \frac{\rho_c \rho^*}{P^* K_c \rho_c^{6/5}}\biggr]  \biggr\} \frac{dx}{dr*} 
~+~ \rho_c^{-4 / 5}\biggl( \frac{K_c}{G} \biggr)\biggl[ \frac{\rho_c \rho^*}{\gamma_\mathrm{g}P^* K_c \rho_c^{6/5}}\biggr] 
 \biggl\{\omega^2 ~+~ (4 - 3\gamma_\mathrm{g}) \frac{G M_r^*}{(r^*)^2} \biggl[ \rho_c^{3 / 5} \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] \frac{\rho_c^{2 / 5}}{r^*}\biggl( \frac{G}{K_c}\biggr)^{1 / 2} \biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} ~+~ \biggl\{\frac{4}{r^*} ~-~ \frac{M_r^*}{(r^*)^2} \biggl[ \frac{\rho^*}{P^* }\biggr]  \biggr\} \frac{dx}{dr*} 
~+~ \biggl( \frac{1}{G\rho_c} \biggr)\biggl[ \frac{ \rho^*}{\gamma_\mathrm{g}P^* }\biggr] 
 \biggl\{\omega^2 ~+~ (4 - 3\gamma_\mathrm{g}) \frac{G\rho_c M_r^*}{(r^*)^3} \biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ \frac{4}{r^*} -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)^2}\biggr\} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{\omega^2}{\gamma_\mathrm{g} G\rho_c} + \biggl(\frac{4}{\gamma_\mathrm{g}} - 3\biggr)\frac{ M_r^*}{(r^*)^3}\biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, .
</math>
  </td>
</tr>
</table>

==Profile==

Now, referencing the [[SSC/Structure/BiPolytropes/Analytic51#Profile|derived bipolytropic model profile]], we should incorporate the following relations:


<div align="center">
<b>Table 1:  Radial Profile of Various Physical Variables</b>
<table border="1" cellpadding="6">
<tr>
  <td align="center" rowspan="2">
Variable
  </td>
  <td align="center" rowspan="2">
Throughout the Core<br>
<math>~0 \le \xi \le \xi_i</math>
  </td>
  <td align="center" rowspan="2">
Throughout the Envelope<sup>&dagger;</sup><br>
<math>~\eta_i \le \eta \le \eta_s</math>
  </td>

  <td align="center" colspan="3">
Plotted Profiles
  </td>
</tr>

<tr>
  <td align="center">
<math>~\xi_i = 0.5</math>
  </td>
  <td align="center">
<math>~\xi_i = 1.0</math>
  </td>
  <td align="center">
<math>~\xi_i = 3.0</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~r^*</math>
  </td>
  <td align="center">
<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta</math>
  </td>

  <td align="center" colspan="3">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi(\eta)</math>
  </td>

  <td align="center">
<!--  [[File:PlotDensity_xi_0.5.jpg|thumb|75px]] -->
[[Image:DenXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>~P^*</math>
  </td>
  <td align="center">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}</math>
  </td>
  <td align="center">
<math>\theta^{6}_i [\phi(\eta)]^{2}</math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:PresXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>~M_r^*</math>
  </td>
  <td align="center">
<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>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:MassXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="left" colspan="6">
<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>\phi(\eta)</math> and its first derivative using the information presented in Step 6, above.
  </td>
</tr>
</table>
</div>


Therefore, throughout the core we have,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</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]\biggl( \frac{2\pi}{3}\biggr)^{1 / 2} \frac{1}{\xi} 
= 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} 
\, ;
</math>
  </td>
</tr>
</table>

and, throughout the envelope we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \biggr]^{-1}
=
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{\theta_i}{\eta} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">'''NOTE on 15 May 2019:'''</font>&nbsp; Prior to this date the last RHS expression had an incorrect exponent on <math>~\eta</math>.  It previously (incorrectly) read,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i  \eta \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\, .
</math>
  </td>
</tr>
</table>

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

==Numerical Integration==

===General Approach===
Here, we begin by recognizing that the 2<sup>nd</sup>-order ODE that must be integrated to obtain the desired eigenvectors has the generic form,
<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>~
x'' + \frac{\mathcal{H}}{r^*} x' + \mathcal{K}x \, ,
</math>
  </td>
</tr>
</table>
where,
<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{dx}{dr^*}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~x''</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d(r^*)^2} \, .</math>
  </td>
</tr>
</table>
Adopting the same approach [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|as before when we integrated the LAWE for pressure-truncated polytropes]], we will enlist the finite-difference approximations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{x_+ - x_-}{2\delta r^*}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~x''</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{x_+ -2x_j + x_-}{(\delta r^*)^2} \, .
</math>
  </td>
</tr>
</table>

The finite-difference representation of the LAWE is, therefore,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{r^*} \biggl[ \frac{x_+ - x_-}{2\delta r^*} \biggr] ~-~ \mathcal{K}x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta r^*}{2r^*} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta r^*)^2\mathcal{K}x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>

In what follows we will also find it useful to rewrite <math>~\mathcal{K}</math> in the form,
<div align="center">
<math>~\mathcal{K} ~\rightarrow ~\biggl(\frac{\sigma_c^2}{\gamma_\mathrm{g}}\biggr) \mathcal{K}_1 - \alpha_\mathrm{g} \mathcal{K}_2 \, .</math>
</div>

The relevant coefficient expressions for ''all'' regions of the configuration are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, .
</math>
  </td>
</tr>
</table>

===Special Handling at the Center===
In order to kick-start the integration, we set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \delta\xi</math>, away from the center.  Specifically, we set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
x_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F}  (\delta\xi)^2}{60} \biggr]_{n=5} 
=
x_1 \biggl[ 1 - \frac{\mathfrak{F}  (\delta\xi)^2}{10} \biggr]
\, .</math>
  </td>
</tr>
</table>
</div>

===Special Handling at the Interface===

Integrating outward from the center, the ''general approach'' will work up through the determination of <math>~x_{j+1}</math> when "j+1" refers to the interface location.  In order to properly transition from the core to the envelope, we need to determine the value of the slope at this interface location.  Let's do this by setting j = i, then projecting forward to what <math>~x_+</math> ''would be'' &#8212; that is, to what the amplitude just beyond the interface ''would be'' &#8212; if the core were to be extended one more zone.  Then, the slope at the interface (as viewed from the perspective of the core) will be,

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

<tr>
  <td align="right">
<math>~x'_i\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta r^*} \biggl\{
x_+ - x_{i-1}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{x_{i-1}}{2\delta r^*} 
+
\frac{1}{2\delta r^*} \biggl\{
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta r^*} \biggl\{
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]x_{i-1} ~-~\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta r^*} \biggl\{
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~2x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]^{-1}
</math>
  </td>
</tr>
</table>

Conversely, as viewed from the ''envelope'', if we assume that we know <math>~x_i</math> and <math>~x'_i</math>, we can determine the amplitude, <math>~x_{i+1}</math>, at the first zone beyond the interface as follows:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_-</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
x_{i+1} - 2\delta r^*\cdot x'_i\biggr|_\mathrm{env}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{i+1} \biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
\biggl[ x_{i+1} - 2\delta r^*\cdot x'_i\biggr|_\mathrm{env}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{i+1} \biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]
~+~
\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
x_{i+1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i 
~+~
\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
2\delta r^*\cdot x'_i\biggr|_\mathrm{env} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{i+1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 - \tfrac{1}{2}(\delta r^*)^2\mathcal{K}\biggr] x_i 
~+~
\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
\delta r^*\cdot x'_i\biggr|_\mathrm{env} 
</math>
  </td>
</tr>
</table>

==Splitting Analysis Into Separate Core and Envelope Components==
===Core:===
Given that, <math>~\sqrt{2\pi/3}~r^* = \xi</math>, lets multiply the LAWE through by <math>~3/(2\pi)</math>.  This gives,

<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\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+ \frac{3}{2\pi}\biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, .
</math>
  </td>
</tr>
</table>

Specifically for the core, therefore, the finite-difference representation of the LAWE is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta \xi)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ - x_-}{2\delta \xi} \biggr] ~-~ \biggl[ \frac{3\mathcal{K}}{2\pi} \biggr]x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta \xi}{2\xi} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \xi)^2 \biggl[ \frac{3\mathcal{K}}{2\pi} \biggr] x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \xi}{2\xi}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta \xi)^2\biggl( \frac{3\mathcal{K}}{2\pi} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \xi}{2\xi} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>
This also means that, as viewed from the perspective of the core, the slope at the interface is

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{dx}{d\xi}\biggr]_\mathrm{interface}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta \xi} \biggl\{
\biggl[ 2 - (\delta \xi)^2 \biggl( \frac{3\mathcal{K}}{2\pi} \biggr)\biggr] x_i ~-~2x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta \xi}{2\xi}\biggr) \mathcal{H} \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

===Envelope:===
Given that,
<div align="center">
<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 (2\pi)^{1 / 2}~r^* = \eta \, ,</math>
</div>
let's multiply the LAWE through by <math>~(2\pi)^{-1} \theta_i^{-4}( \mu_e/\mu_c)^{-2} </math>.  This gives,

<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\eta^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{\eta} \cdot \frac{dx}{d\eta} 
+ \frac{1}{2\pi \theta_i^4} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, .
</math>
  </td>
</tr>
</table>

Specifically for the envelope, therefore, the finite-difference representation of the LAWE is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta \eta)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2\delta \eta} \biggr] ~-~  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr]x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta \eta}{2\eta} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr] x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \eta}{2\eta}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>

This also means that, once we know the slope at the interface (see immediately below), the amplitude at the first zone outside of the interface will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_{i+1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 - \tfrac{1}{2}(\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr)\biggr] x_i 
~+~
\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]
\delta \eta \cdot \biggl[ \frac{dx}{d\eta} \biggr]_\mathrm{interface} \, .
</math>
  </td>
</tr>
</table>

===Interface===
If we consider only cases where <math>~\gamma_e = \gamma_c</math>, then at the interface we expect,

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

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d\ln x}{d\ln \xi} = \frac{d\ln x}{d\ln \eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r^*\frac{dx}{d r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi \frac{dx}{d \xi} = \eta \frac{d x}{d \eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{dx}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\pi}{3} \biggr)^{1 / 2}\frac{dx}{d\xi} = \biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^2 (2\pi)^{1 / 2} \frac{dx}{d\eta} \, .</math>
  </td>
</tr>
</table>
Switching at the interface from <math>~\xi</math> to <math>~\eta</math> therefore means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \biggl[ \frac{dx}{d\eta}\biggr]_\mathrm{interface}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\sqrt{3}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-2} \biggl[ \frac{dx}{d\xi}\biggr]_\mathrm{interface} \, .</math>
  </td>
</tr>
</table>

If, however, we want to consider values for the adiabatic index that are different in the two regions, we have to follow the [[#Interface_Conditions|above-outlined guidelines]], that is,

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

<tr>
  <td align="right">
<math>~ \frac{d\ln x}{d\ln \eta} \biggr|_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x}{d\ln \xi} \biggr)_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \biggl[ \frac{dx}{d\eta} \biggr]_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x_i}{\eta_i} \biggl\{
3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{\xi_i}{x_i}\biggl[ \frac{dx}{d\xi} \biggr]_i 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3x_i}{\eta_i} \biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{1}{\sqrt{3}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-2} \biggl[ \frac{dx}{d\xi} \biggr]_i \, .
</math>
  </td>
</tr>
</table>
</div>

==Eigenvectors for Marginally Unstable Models with (&gamma;<sub>c</sub>, &gamma;<sub>e</sub>) = (6/5, 2)==

We now have the tools in hand to identify the eigenvectors &#8212; that is, various radial eigenfunctions and the corresponding eigenfrequency for each &#8212; associated with various modes of oscillation in <math>~(n_c, n_e) = (5,1)</math> bipolytropes.  Which models should we examine?  


In our [[SSC/Stability/MurphyFiedler85#Review_of_the_BiPolytrope_Stability_Analysis_by_Murphy_.26_Fiedler_.281985b.29|accompanying review of the bipolytrope stability analysis presented by Murphy &amp; Fiedler (1983b)]], our primary objective was to show that we were able to match their results quantitatively.  We therefore set <math>~\mu_e/\mu_c</math> = 1 &#8212; the only <math>~\mu</math>-ratio that they considered &#8212; and picked values of the core-envelope interface radius, <math>~\xi_i</math>, that were listed among their set of chosen models.  For a fixed value of <math>~\xi_i</math>, we integrated the relevant LAWE from the center toward the surface for many different eigenfrequency <math>~(\sigma_c^2)</math> ''guesses'' until an eigenfunction was found whose behavior at the surface matched with high precision the physically justified surface boundary condition.


With the [[#Virial_Stability_Evaluation|above virial stability analysis]] in mind (see especially [[#Virial_Stability_Evaluation|Figure 3]]), here we have chosen to focus on models that reside along six of the equilibrium sequences that have already been analytically identified, above &#8212; specifically, the sequences for which <math>~\mu_e/\mu_c</math> = 1, ½, 0.345, ⅓, 0.309, and ¼ &#8212; and to examine oscillation modes under the assumption that,
<div align="center">
<math>~\gamma_c = \frac{(n_c+1)}{n_c} = \frac{6}{5} \, ,</math> &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~\gamma_e = \frac{(n_e+1)}{n_e} =2 \, .</math>
</div>

===Fundamental Modes===

We decided to examine, first, whether any model along each sequence marks a transition from dynamically stable to dynamically unstable configurations.  We accomplished this by setting <math>~\sigma_c^2</math> = 0, then integrating the relevant LAWE from the center toward the surface for many different ''guesses'' of the core-envelope interface radius until an eigenfunction with no radial nodes &#8212; ''i.e.,'' an eigenfunction associated with the fundamental mode of radial oscillation &#8212; was found whose behavior at the surface matched with high precision the physically desired surface boundary condition.  We were successful in this endeavor.  A marginally unstable model was identified on each of the six separate equilibrium sequences.  

====Equilibrium Properties of Marginally Unstable Models====
Table 2 summarizes some of the equilibrium properties of these six models.  For example, the second column of the table gives the value of the core-envelope interface radius, <math>~\xi_i</math>, associated with each marginally unstable model.  The table also lists:  the value of the model's dimensionless radius, <math>~R^*_\mathrm{surf}</math>, the key structural parameters, <math>~q</math> &amp; <math>~\nu</math>, and the central-to-mean density associated with each model; and in each case the dimensionless thermal energy <math>~(\mathfrak{s})</math> and dimensionless gravitational potential energy <math>~(\mathfrak{w})</math> associated, separately, with the core and the envelope.  Note that, once the pair of parameters, <math>~(\mu_e/\mu_c, \xi_i)</math>, has been specified, we can legitimately assign high-precision values to all of the other model parameters because they are [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|analytically prescribed]].

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="10">
'''Table 2:'''  Properties of Marginally Unstable Bipolytropes Having<br /><br /><math>~(n_c, n_e) = (5, 1)</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math><br /><br />Determined from Integration of the LAWE
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~R^*_\mathrm{surf}</math></td>
  <td align="center"><math>~q \equiv \frac{r_\mathrm{core}}{R_\mathrm{surf}}</math></td>
  <td align="center"><math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{env}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">2.139737</td>
  <td align="center">0.53885819</td>
  <td align="center">0.497747626</td>
  <td align="center">8.51704656</td>
  <td align="center">3.021916335</td>
  <td align="center">-3.356583022</td>
  <td align="center">1.47780476</td>
  <td align="center">-5.642859167</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">5.146499</td>
  <td align="center">0.306021732</td>
  <td align="center">0.401776274</td>
  <td align="center">63.29514949</td>
  <td align="center">4.241287117</td>
  <td align="center">-6.074241035</td>
  <td align="center">4.284931508</td>
  <td align="center">-10.97819621</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">9.554041</td>
  <td align="center">0.185160563</td>
  <td align="center">0.234302525</td>
  <td align="center">209.7739052</td>
  <td align="center">4.639705843</td>
  <td align="center">-7.125754184</td>
  <td align="center">11.72861751</td>
  <td align="center">-25.61089252</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">10.120558</td>
  <td align="center">0.176288391</td>
  <td align="center">0.218241608</td>
  <td align="center">230.4125398</td>
  <td align="center">4.667042505</td>
  <td align="center">-7.200966267</td>
  <td align="center">13.15887139</td>
  <td align="center">-28.45086152</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="center">11.464303</td>
  <td align="center">0.158362807</td>
  <td align="center">0.184796947</td>
  <td align="center">279.0788798</td>
  <td align="center">4.722277318</td>
  <td align="center">-7.354156963</td>
  <td align="center">17.1374434</td>
  <td align="center">-36.36528446</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="center">15.895632</td>
  <td align="center">0.118924863</td>
  <td align="center">0.11071211</td>
  <td align="center">430.0444648</td>
  <td align="center">4.84592201</td>
  <td align="center">-7.70305421</td>
  <td align="center">37.84289623</td>
  <td align="center">-77.67458196</td>
</tr>
</table>
</div>

<!--
OLD VERSION; IGNORE!

<div align="center">
<font color="red">'''OLD Table 2'''</font>

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="9">
Properties of Marginally Unstable Bipolytropes Having<br /><br /><math>~(n_c, n_e) = (5, 1)</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math><br /><br />Determined from Integration of the LAWE
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~q \equiv \frac{r_\mathrm{core}}{R_\mathrm{surf}}</math></td>
  <td align="center"><math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{core}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">2.467359668</td>
  <td align="center">0.597684036</td>
  <td align="center">0.692367564</td>
  <td align="center">17.09749847</td>
  <td align="center">4.518031091</td>
  <td align="center">-6.79580606</td>
  <td align="center">0.857904827</td>
  <td align="center">-3.956065776</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.65925887</td>
  <td align="center">0.314759926</td>
  <td align="center">0.481811565</td>
  <td align="center">95.04044773</td>
  <td align="center">4.759771212</td>
  <td align="center">-7.459080087</td>
  <td align="center">3.20977926</td>
  <td align="center">-8.480020858</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.77457272408</td>
  <td align="center">0.183201946</td>
  <td align="center">0.25725514</td>
  <td align="center">281.7482802</td>
  <td align="center">4.887554727</td>
  <td align="center">-7.822410223</td>
  <td align="center">10.22665208</td>
  <td align="center">-22.40600339</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.7843424754</td>
  <td align="center">0.17412463</td>
  <td align="center">0.237677946</td>
  <td align="center">305.4881577</td>
  <td align="center">4.897826446</td>
  <td align="center">-7.852004624</td>
  <td align="center">11.6114224</td>
  <td align="center">-25.16649306</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.8050375512</td>
  <td align="center">0.156021514</td>
  <td align="center">0.197918988</td>
  <td align="center">359.3276918</td>
  <td align="center">4.91930991</td>
  <td align="center">-7.914090174</td>
  <td align="center">15.49005222</td>
  <td align="center">-32.90463409</td>
</tr>
</table>
</div>

OLD VERSION; IGNORE!
-->

As was expected from our [[#What_to_Expect_for_Equilibrium_Configurations|above discussion of virial equilibrium conditions]], we found that to high precision for each of these equilibrium models,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
(\mathfrak{w}_\mathrm{core}  
~+~\mathfrak{w}_\mathrm{env})
~+~2(\mathfrak{s}_\mathrm{core}  
~+~\mathfrak{s}_\mathrm{env})  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
<span id="Figure4">However, contrary to expectations,</span> in no case did we find that <math>~\mathfrak{s}_\mathrm{core}/\mathfrak{s}_\mathrm{env} = 5</math>.  That is to say, we found that ''none'' of the models lies on the (red-dashed) curve in the <math>~q-\nu</math> parameter space that separates stable from unstable models as defined by our [[#What_to_Expect_for_Equilibrium_Configurations|above free-energy-based stability analysis]]. The left-hand panel of Figure 4 shows this (red-dashed) demarcation curve; for all intents and purposes, it is a reproduction of the right-hand panel of [[#Virial_Stability_Evaluation|Figure 3, above]] &#8212; turning-point markers have been removed to minimize clutter, the equilibrium sequences have been labeled, and the horizontal axis has been extended to unity in order to include a longer portion of the <math>~\mu_e/\mu_c = 1</math> sequence.  The orange triangular markers that appear in the right-hand panel of Figure 4 pinpoint where each of the Table 2 "marginally unstable" models resides in this <math>~q-\nu</math> plane.  Clearly, all six of the orange triangles lie well off of &#8212; and to the ''stable'' side of &#8212; the red-dashed demarcation curve.  This discrepancy, which has resulted from our use of two separate approaches to stability analysis, will be discussed further and gratifyingly resolved, below.

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

<tr>
  <th align="center">Figure 4</th>
</tr>
<tr>
  <td align="center" colspan="10">[[File:NEWCompositeDlabeled.png|800px|Marginally unstable models]]</td>
</tr>
</table>

====Eigenfunction Details====

Here we examine some of the properties of the fundamental-mode eigenfunctions that we have found are associated with marginally unstable, <math>~(n_c, n_e) = (5,1)</math> bipolytropes.
<table border="0" align="right" width="40%">
<tr>
  <th align="center">Figure 5</th>
</tr>
<tr><td align="center">
[[File:Mod0MuRatio100.png|450px|Example eigenvector]]
</td></tr>
</table>
Consider the model on the <math>~\mu_e/\mu_c = 1</math> sequence for which <math>~\sigma_c^2=0~</math>; key properties of this specific equilibrium model are enumerated in the first row of numbers provided in [[#Equilibrium_Properties_of_Marginally_Unstable_Models|Table 2, above]].  Figure 5 shows how our numerically derived, fundamental-mode eigenfunction, <math>~x = \delta r/r_0</math>, varies with the fractional radius over the entire range, <math>~0 \le r/R \le 1</math>.  By prescription, the eigenfunction has a value of unity and a slope of zero at the center  <math>~(r/R = 0)</math>.  Integrating the LAWE outward from the center, through the model's core (blue curve segment), <math>~x</math> drops smoothly to the value <math>~x_i = 0.81437</math> at the interface <math>~(\xi_i = 1.6686460157 ~\Rightarrow~ q = r_\mathrm{core}/R_\mathrm{surf} = 0.53885819)</math>.  Our numerical integration of the LAWE showed that, at the interface, the logarithmic slope of the ''core'' (blue) segment of the eigenfunction is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i \biggr\}_\mathrm{core}
=
\biggl\{ \frac{d\ln x}{d\ln \xi}\biggr|_i \biggr\}_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 0.455872 \, .</math>
  </td>
</tr>
</table>
Next, following the [[#Interface|above discussion of matching conditions at the interface]], we determined that, from the perspective of the envelope, the slope of the eigenfunction at the interface must therefore be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i \biggr\}_\mathrm{env}
=
\biggl\{ \frac{d\ln x}{d\ln \eta}\biggr|_i \biggr\}_\mathrm{env}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl\{ \frac{d\ln x}{d\ln \xi}\biggr|_i \biggr\}_\mathrm{core} = -1.47352 \, .</math>
  </td>
</tr>
</table>

Adopting this "env" slope along with the amplitude, <math>~x_i = 0.81437</math>, as the appropriate ''interface'' boundary conditions, we integrated the LAWE from the interface to the surface, obtaining the green-colored segment of the eigenfunction that is shown in Figure 5.  The amplitude continued to steadily decrease, reaching a value of <math>~x_s = 0.38203</math>, at the model's surface <math>~(r/R = 1)</math>.  At the surface, this ''envelope'' (green) segment of the eigenfunction exhibits a logarithmic slope that matches to eight significant digits the value that is [[#SurfaceCondition|expected from astrophysical arguments]] for this marginally unstable <math>~(\sigma_c^2=0)</math> model, namely,
<div align="center">
<math>~
\frac{d\ln x}{d\ln \eta}\biggr|_s = \biggl[ \biggl( \frac{\rho_c}{\bar\rho} \biggr)\frac{\cancelto{0}{\sigma_c^2}}{2\gamma_e} - \biggl(3 - \frac{4}{\gamma_e}\biggr)\biggr] = -1 \, .
</math>
</div>


<table border="1" cellpadding="10" width="80%" align="center"><tr><td align="left">
<font color="red">'''Key Reminder:'''</font>  We were able to find an eigenfunction whose surface boundary condition matched the desired value &#8212; in this particular case, a logarithmic slope of negative one &#8212; to this high level of precision only by iterating many times and, at each step, fine-tuning our choice of the equilibrium model's radial interface location, <math>~\xi_i</math> before performing a numerical integration of the LAWE.
</td></tr></table>


The discontinuous jump that occurs in the ''slope'' of the eigenfunction at the interface results from our assumption that the effective adiabatic index of material in the core <math>~(\gamma_c = 6/5)</math> is different from the effective adiabatic index of the envelope material <math>~(\gamma_e = 2)</math>.   In an effort to emphasize and more clearly illustrate the behavior of this fundamental-mode eigenfunction as it crosses the core/envelope interface, we have added a pair of dashed line segments to the Figure 5 plot.  The red-dashed line segment touches, and is tangent to, the blue segment of the eigenfunction at the location of the core/envelope interface; it has a slope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{dx}{d(r/R)}\biggr|_i
=
\frac{x_i}{(r_i/R)}\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i\biggr\}_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 0.455872 \biggl(\frac{ 0.81437 }{ 0.53885819 }\biggr) = - 0.68895\, .</math>
  </td>
</tr>
</table>
On the other hand, the purple-dashed line segment  touches, and is tangent to, the green segment of the eigenfunction at the location of the core/envelope interface; it has a slope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{dx}{d(r/R)}\biggr|_i
=
\frac{x_i}{(r_i/R)}\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i\biggr\}_\mathrm{env}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 1.47352 \biggl(\frac{ 0.81437 }{ 0.53885819 }\biggr) = - 2.22691\, .</math>
  </td>
</tr>
</table>
For comparison purposes, the eigenfunction shown in Figure 5 has been presented again in Figure 6, along with several other of our numerically derived eigenfunctions, but in Figure 6 the plotted amplitude has been renormalized to give a surface value &#8212; rather than a central value &#8212; of unity.


In Figure 6 we show the behavior of the fundamental-mode eigenfunction for each of the marginally unstable models identified in Table 2.  In the top figure panel, each curve shows &#8212; on a linear-linear plot &#8212; how the amplitude varies with radius; in the bottom figure panel, the amplitude is plotted on a logarithmic scale.  On each curve, the black plus sign marks the radial location of the core-envelope interface; in the bottom panel, these markers are accompanied by the values of <math>~\xi_i</math> that are associated with each corresponding model (see also the second column of Table 2).  Each eigenfunction has been normalized such that the surface amplitude is unity.  In the top panel, the value of the central amplitude of the eigenfunction that results from this normalization is recorded near the point where each eigenfunction touches the vertical axis.   (In each case, the value provided on the plot is simply the inverse of the value of <math>~x_s</math> given in Table 3, below.)

<table border="0" align="center" cellpadding="8">
<tr>
  <th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6:  Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br /> 
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
  </th>
</tr>
<tr>
  <td align="center">
[[File:Mode0EigenfunctionsCombinedSmall.png|800px|Eigenfunctions for Marginally Unstable Models]]
  </td>
</tr>
</table>

Notice that, especially as they approach the surface, the "envelope" segments of these six marginally unstable eigenfunction appear to merge into the same curve, irrespective of their value of the ratio of mean molecular weights.  Note as well that the discontinuous jump that occurs in the ''slope'' of each eigenfunction at the radial location of the core/envelope interface &#8212; resulting from our choice to adopt a different adiabatic index, <math>~\gamma_g</math>, in the core from the one in the envelope &#8212; becomes less and less noticeable for smaller and smaller values of the ratio of mean molecular weights.

====Is There an Analytic Expression for the Eigenfunction?====
After noticing that, in Figure 6,  the ''envelope'' segments of all of the marginally unstable eigenfunctions merge into the same curve, we began to wonder whether a single expression &#8212; and, even better, an ''analytically defined'' expression &#8212; would perfectly describe the eigenfunction.  We had reason to believe that this might actually be possible because, in [[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|pressure-truncated polytropic configurations, we have derived analytic expressions for the marginally unstable, fundamental-mode eigenfunctions]] of both <math>~n = 5</math> and <math>~n=1</math> systems.

Very quickly, we convinced ourselves that a parabolic function does indeed perfectly match the "core" segment of each displayed eigenfunction.  Specifically, throughout the core <math>~(0 \le \xi \le \xi_i)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 - \frac{\xi^2}{15}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{dx_P}{d\xi}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2\xi}{15} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{d\ln x_P}{d\ln \xi}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2\xi^2}{15} \biggl[ \frac{(15 - \xi^2)}{15} \biggr]^{-1} = - \frac{2\xi^2}{(15 - \xi^2)}  \, .</math>
  </td>
</tr>
</table>


The envelope segment posed a much greater challenge.  In the context of our [[SSC/Stability/n1PolytropeLAWE#Radial_Oscillations_of_n_.3D_1_Polytropic_Spheres|discussion of ''Radial Oscillations of n = 1 Polytropic Spheres'']], and in an [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Is_There_an_Analytic_Expression_for_the_Eigenfunction.3F|accompanying ''Ramblings Appendix'' chapter]] we have detailed some trial derivations that are mostly blind alleyways.  Twice &#8212; once in [[SSC/Stability/n1PolytropeLAWE#tagJanuary2019|January, 2019]] and again (independently) in [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B|April 2019]] &#8212; we have analytically demonstrated that the following appears to work for the envelope:

Given that the ''Structural Properties'' of the envelope are described by the Lane-Emden function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_0 \biggl[ \frac{\sin(\eta - b_0)}{\eta} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ Q \equiv - \frac{d\ln \phi}{d\ln\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \eta \cot(\eta - b_0) \biggr] \, ,</math>
  </td>
</tr>
</table>
the relevant LAWE is satisfied by the fractional displacement function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3c_0 Q}{\eta^2} \, ,</math>
  </td>
</tr>
</table>
where, <math>~c_0</math> is an arbitrary scale factor.  

<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left">
Note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3c_0}{\eta^2}\biggl[\eta -\cot(\eta - b_0)
+\eta\cot^2(\eta - b_0) 
\biggr] 
- \frac{6c_0 Q}{\eta^3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{d\ln x_P}{d\ln\eta} = \frac{\eta}{x_P}\cdot \frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\eta^3}{3c_0 Q} \biggl\{
\frac{3c_0}{\eta^2}\biggl[\eta -\cot(\eta - b_0)
+\eta\cot^2(\eta - b_0) 
\biggr] 
- \frac{6c_0 Q}{\eta^3} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{Q} \biggl\{
\biggl[\eta^2 -\eta \cot(\eta - b_0)
+\eta^2\cot^2(\eta - b_0) 
\biggr] 
- 2 Q 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{Q} 
\biggl[\eta^2 -(1-Q)
+(1-Q)^2 
\biggr] 
- 2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{Q} 
\biggl[\eta^2  - Q + Q^2\biggr] 
- 2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\eta^2}{Q} + Q - 3 \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>
But, as far as we have been able to determine (as of 16 April 2019), this analytic displacement function does not match the displacement function that has been generated through numerical integration of the LAWE (see the light-green segment of the eigenfunction displayed [[#Eigenfunction_Details|above in Figure 5]]).  It remains unclear whether (a) the numerical integration is at fault, (b) we are imposing an incorrect slope at the core-envelope interface, or ( c) we are misinterpreting how to compare the two separately derived (one, numerical, and the other, analytic) envelope eigenfunctions.

===Other Modes===
<!--
Keep in mind that, for all models, we ''expect'' that, at the surface, the logarithmic derivative of each proper eigenfunction will be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Omega^2}{\gamma} - \alpha \, .</math>
  </td>
</tr>
</table>

Also, keep in mind that, for Model 10 <math>~(\xi_i = 2.5646)</math>:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r_i}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.39302482</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp;&nbsp;&nbsp;</td>
  <td align="right">
<math>~\frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~34.3460405</math>
  </td>
</tr>
</table>
-->

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="15">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = mu100plot]]'''Our Determinations for Marginally Unstable Model Having <math>~\mu_e/\mu_c = 1</math>'''<br />&nbsp;<br />
<math>~[ ~\xi_i = 1.6686460157 ~]</math>
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" rowspan="2"><math>~\sigma_c^2</math></td>
  <td align="center" rowspan="2"><math>~\Omega^2 \equiv \frac{\sigma_c^2}{2} \biggl( \frac{\rho_c}{\bar\rho}\biggr)</math></td>
  <td align="center" rowspan="2"><math>~x_i</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_i</math></td>
  <td align="center" rowspan="2"><math>~x_\mathrm{surf}</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_3</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_3</math></td>
</tr>
<tr>
  <td align="center">core</td>
  <td align="center">env</td>
  <td align="center">''expected''</td>
  <td align="center">measured</td>
</tr>
<tr>
  <td align="center">1<br /><font size="-1">(Fundamental)</font></td>
  <td align="right">0.00</td>
  <td align="right">0.00</td>
  <td align="right">+0.81437470</td>
  <td align="right">-0.455872</td>
  <td align="right">-1.473523</td>
  <td align="right">+0.3820</td>
  <td align="right">-1</td>
  <td align="right">-0.999999992</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right">2.51513333</td>
  <td align="right">10.7107538</td>
  <td align="right">0.20482050</td>
  <td align="right">-7.09124</td>
  <td align="right">-5.4547441</td>
  <td align="right">- 0.9962</td>
  <td align="right">4.355376917</td>
  <td align="right">4.35537692</td>
  <td align="right">0.64133</td>
  <td align="right">0.3502</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="right">5.72371888</td>
  <td align="right">24.3745901</td>
  <td align="right">-0.14269277</td>
  <td align="right">+8.046019</td>
  <td align="right">+3.627611</td>
  <td align="right">+0.9308</td>
  <td align="right">11.18729505</td>
  <td align="right">11.18729506</td>
  <td align="right">0.4837</td>
  <td align="right">0.5864</td>
  <td align="right">0.842</td>
  <td align="right">0.0854</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="right">10.3458476</td>
  <td align="right">44.0622916</td>
  <td align="right">-0.20845197</td>
  <td align="right">-0.6949966</td>
  <td align="right">-1.61699793</td>
  <td align="right">-1.1443</td>
  <td align="right">21.03114578</td>
  <td align="right">21.03114577</td>
  <td align="right">0.3939</td>
  <td align="right">0.7154</td>
  <td align="right">0.6902</td>
  <td align="right">0.2777</td>
  <td align="center">0.9115</td>
  <td align="center">0.0284</td>
</tr>
<tr>
  <td align="center" colspan="15">
[[File:Mod0MuRatio100.png|550px|Our determination of eigenvector for mu_ratio = 1]]&nbsp;&nbsp;[[File:FourModesMuRatio100.png|550px|Our determination of multiple eigenvectors for mu_ratio = 1]]
  </td>
</tr>
</table>

=Reconciliation=

<table border="0" cellpadding="8" align="right">
<tr>
  <th align="center">Figure 7:  Conflicting Instability Regions</th>
</tr>
<tr>
  <td align="center" colspan="10">[[File:EigenvectorStability.png|400px|Marginally unstable models]]</td>
</tr>
</table>
Figure 7, shown here on the right, is identical to the right-hand panel of [[#Figure4|Figure 4, as displayed above]].  From a standard global, free-energy analysis &#8212; such as the one [[#Virial_Stability_Evaluation|summarized above]] &#8212; we have determined that the red-dashed curve shown in the right panel of Figure 6 divides the <math>~q-\nu</math> plane into dynamically stable (below and to the right) and unstable (above and to the left) regions.


==Variational Principle==

===Setup===

Let's follow the guidelines of the [[SSC/SynopsisStyleSheet|variational principle]].  Instead of starting with the form of the LAWE given above, namely,

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

{{Math/EQ_RadialPulsation01}}
</div>

we will start with a form that is more amenable to the variational principle, namely,
<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_0}\biggl[ r_0^4 \gamma P_0 ~\frac{dx}{dr_0} \biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 + (3\gamma - 4) r_0^3 \frac{dP_0}{dr_0} \biggr] x \, .
</math>
  </td>
</tr>
</table>
</div>

<table border="1" cellpadding="8" align="center" width="85%"><tr><td align="left">
<font color="red">'''ASIDE:'''</font>  Let's show that these two expressions are equivalent.  Remembering that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP_0}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- g_0 \rho_0 = - \biggl(\frac{GM_r}{r_0^2}\biggr)\rho_0 \, ,</math>
  </td>
</tr>
</table>
the second expression 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>~
r_0^4 \gamma P_0 ~\frac{d^2x}{dr_0^2} 
+ ~\gamma \frac{dx}{dr_0}\biggl[ 4r_0^3 P_0 - r_0^4 g_0 \rho_0\biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 - (3\gamma - 4) r_0^3 g_0 \rho_0\biggr] x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_0^4 \gamma P_0 \biggl\{\frac{d^2x}{dr_0^2} 
+ \biggl[ \frac{4}{r_0} - \frac{g_0 \rho_0}{P_0} \biggr] \frac{dx}{dr_0}
+\frac{\rho_0}{\gamma P_0 }\biggl[ \omega^2 - (3\gamma - 4) \frac{g_0}{r_0}  \biggr] x 
\biggr\}\, .
</math>
  </td>
</tr>
</table>
Hence, we must multiply the first expression through by <math>~r_0^4 \gamma P_0</math> in order to obtain the second expression.
</td></tr></table>


From [[#Foundation|above]], we realize that multiplying the second expression through by <math>~(K_c/G)\rho_c^{-4 / 5}</math> gives,

<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_0^4\gamma P_0 \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(r^*)^4 [K_c^{1 / 2}/(G^{1 / 2} \rho_c^{2/5}) ]^4 \gamma P^* [K_c\rho_c^{6/5}] \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{K_c^{3}}{ G^{2} \rho_c^{2/5}} \biggr]  (r^*)^4 \gamma P^* \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

That is, multiplying the second expression through by,  <math>~(K_c/G)\rho_c^{-4 / 5} \cdot G^2 \rho_c^{2 / 5}/K_c^3 = G/(K_c^2 \rho_c^{2 / 5})</math> , should give a desirable, totally dimensionless version of the LAWE. Remembering that,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\rho_0}{\rho_c}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{[K_c^{1/2}/(G^{1/2}\rho_c^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{K_c\rho_c^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_c^{1/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{dP^*}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{M^*_r \rho^*}{(r^*)^2} </math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K_c^5}{G^3}\biggr]^{1 / 2}</math>
  </td>
</tr>

</table>
</div>
let's try it.

<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>~G K_c^{-2} \rho_c^{-2 / 5} \biggl\{
\frac{d}{dr_0}\biggl[ r_0^4 \gamma P_0 ~\frac{dx}{dr_0} \biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 + (3\gamma - 4) r_0^3 \frac{dP_0}{dr_0} \biggr] x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G K_c^{-2} \rho_c^{-2 / 5} \biggl\{ \biggl( \frac{K_c^2 \rho_c^{2/5}}{G} \biggr)
\frac{d}{dr^*}\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+\biggl[ \omega^2 \biggl(\frac{K_c^2}{G^2 \rho_c^{8/5}}\biggr) \rho_c \rho^* (r^*)^4 + (3\gamma - 4)\biggl(\frac{K_c }{G \rho_c^{4/5}}\biggr)K_c\rho_c^{6/5} (r^*)^3 \frac{dP^*}{dr^*} \biggr] x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{dr^*}\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] x \, .
</math>
  </td>
</tr>
</table>

Now, guided by the [[SSC/SynopsisStyleSheet#Stability|accompanying summary]], if we multiply through by <math>~4\pi x dr^*</math> and integrate over the entire volume, we obtain the ''governing variational relation'', namely,
<!-- OLD VERSION; IGNORE

<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  x\cdot
d\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_0^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 4\pi  x(r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr]_0^{R^*} 
- \int_0^{R^*}4\pi  \biggl[ (r^*)^4 \gamma P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
+ \int_0^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4  \biggr] 4\pi x^2 dr^*
- \int_0^{R^*}\biggl[ (3\gamma - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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^*)^2 P^*  dr^*
+ \int_0^{R^*} (3\gamma - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{R^*} 
+ \int_0^{R^*} 4\pi \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4   x^2 dr^* \, .
</math>
  </td>
</tr>
</table>
OLD VERSION; IGNORE-->

<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^*_\mathrm{core}}4\pi  x\cdot
d\biggl[ (r^*)^4 \gamma_c P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_0^{r^*_\mathrm{core}}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma_c - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\int_{r^*_\mathrm{core}}^{R^*}4\pi  x\cdot
d\biggl[ (r^*)^4 \gamma_e P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_{r^*_\mathrm{core}}^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma_e - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 4\pi  x(r^*)^4 \gamma_c P^* ~\frac{dx}{dr^*} \biggr]_0^{r^*_\mathrm{core}} 
- \int_0^{r^*_\mathrm{core}}4\pi  \biggl[ (r^*)^4 \gamma_c P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
- \int_0^{r^*_\mathrm{core}}\biggl[ (3\gamma_c - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ 4\pi  x(r^*)^4 \gamma_e P^* ~\frac{dx}{dr^*} \biggr]_{r^*_\mathrm{core}}^{R^*} 
- \int_{r^*_\mathrm{core}}^{R^*}4\pi  \biggl[ (r^*)^4 \gamma_e P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
- \int_{r^*_\mathrm{core}}^{R^*}\biggl[ (3\gamma_e - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
+ \int_{0}^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4  \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 \gamma_c 4\pi (r^*)^2 P^*  dr^*
+ \int_0^{r^*_\mathrm{core}} (3\gamma_c - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma_c P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{r^*_\mathrm{core}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \int_{r^*_\mathrm{core}}^{R^*}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 \gamma_e 4\pi (r^*)^2 P^*  dr^*
+ \int_{r^*_\mathrm{core}}^{R^*} (3\gamma_e - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma_e P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_{r^*_\mathrm{core}}^{R^*} 
+ \int_{0}^{R^*} 4\pi \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4   x^2 dr^* \, .
</math>
  </td>
</tr>
</table>


Energy Normalization:

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

<tr>
  <td align="right">
<math>~(\gamma - 1)dU_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi r^2 P dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi \biggl[ \frac{K_c^{1 / 2}}{G^{1 / 2} \rho_c^{2 / 5}} \biggr]^3 K_c \rho_c^{6 / 5} (r^*)^2 P^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi \biggl[ \frac{K_c^{5}}{G^{3}} \biggr]^{1 / 2} (r^*)^2 P^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{K_c^{5}}{G^{3}} \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dW_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{GM^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^* \biggl[ \frac{K_c}{G\rho_c^{4 / 5}} \biggr]\rho_c \biggl[ \frac{ K_c^{3 / 2} }{G^{3 / 2}\rho_c^{1 / 5}  } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{M^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^*  E_\mathrm{norm}
</math>
  </td>
</tr>
</table>

<span id="VariationPrincipleRelation">Hence, the ''dimensionless'' governing variational relation becomes,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_0^{R^*} \biggl( \frac{2\pi}{3}\biggr)\sigma_c^2(r^*)^2   x^2 dM_r^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \int_0^{r^*_\mathrm{core}}  \gamma_c(\gamma_c - 1) x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- \int_0^{r^*_\mathrm{core}} (3\gamma_c - 4)  x^2 dW^*_\mathrm{grav}
+ \biggl[ 4\pi  x^2 (r^*)^3 \gamma_c P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{r^*_\mathrm{core}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \int_{r^*_\mathrm{core}}^{R^*}  \gamma_e(\gamma_e - 1) x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- \int_{r^*_\mathrm{core}}^{R^*} (3\gamma_e - 4)  x^2 dW^*_\mathrm{grav}
+ \biggl[ 4\pi  x^2 (r^*)^3 \gamma_e P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_{r^*_\mathrm{core}}^{R^*} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
+ 4\pi  x_i^2 (r_\mathrm{core}^*)^3 \gamma_c P_i^* \biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 \gamma_e P_i^* \biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^*\biggl[ \gamma_c \biggl\{ \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}  - \gamma_e \biggl\{ \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^*\biggl[ 3(\gamma_e - \gamma_c) \biggr] \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dM^*_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi (r^*)^2 \rho^* dr^*  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dU^*_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\gamma-1)} \biggl[4\pi (r^*)^2 P^*  dr^* \biggr] = \biggl[ \frac{2}{3(\gamma - 1)} \biggr]dS^*_\mathrm{therm}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( \frac{M^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^*  \, .</math>
  </td>
</tr>
</table>

Or, for inclusion in our accompanying ''[[SSC/SynopsisStyleSheet#Bipolytropes|Tabular Overview]]'',
<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">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </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">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^*  \, .
</math>
  </td>
</tr>
</table>

===Implementation===

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="15">
Table 3  
  </th>
</tr>
<tr>
  <td rowspan="3" align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td rowspan="3" align="center"><math>~\xi_i</math></td>
  <td colspan="6" align="center" bgcolor="lightblue">Core</td>
  <td colspan="6" align="center" bgcolor="lightgreen">Envelope</td>
  <td align="center">Virial</td>
</tr>
<tr>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dS^*_\mathrm{therm}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dW^*_\mathrm{grav}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dS^*_\mathrm{therm}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dW^*_\mathrm{grav}</math></td>
  <td rowspan="2" align="center">Numerical<p></p><math>~\biggl[\frac{2\mathfrak{s}_\mathrm{tot}}{|\mathfrak{w}_\mathrm{tot}|}-1\biggr] </math></td>
</tr>
<tr>
  <td align="center">Analytic<br /><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center">'''TERM1'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center">'''TERM2'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center">'''TERM3'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center">'''TERM4'''</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">3.021916335</td>
  <td align="center">3.021921</td>
  <td align="center">0.116389175</td>
  <td align="center">-3.356583022</td>
  <td align="center">-3.35666</td>
  <td align="center">-2.649752079</td>
  <td align="center">1.47780476</td>
  <td align="center">1.47791</td>
  <td align="center">1.0720821</td>
  <td align="center">-5.642859167</td>
  <td align="center">-5.642820</td>
  <td align="center">-1.91142893</td>
  <td align="center">0.000020</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">4.241287117</td>
  <td align="center">4.241410819</td>
  <td align="center">0.440878529</td>
  <td align="center">-6.074241035</td>
  <td align="center">-6.074317546</td>
  <td align="center">-4.150731169</td>
  <td align="center">4.284931508</td>
  <td align="center">4.28547195</td>
  <td align="center">1.44651932</td>
  <td align="center">-10.97819621</td>
  <td align="center">-10.97847622</td>
  <td align="center">-0.92598634</td>
  <td align="center">0.000057</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">4.639705843</td>
  <td align="center">4.6399114</td>
  <td align="center">0.6794857</td>
  <td align="center">-7.125754184</td>
  <td align="center">-7.125854025</td>
  <td align="center">-4.5487829</td>
  <td align="center">11.72861751</td>
  <td align="center">11.730381</td>
  <td align="center">1.51410084</td>
  <td align="center">-25.61089252</td>
  <td align="center">-25.6115597</td>
  <td align="center">-0.4496513</td>
  <td align="center">0.000097</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">4.667042505</td>
  <td align="center">4.667254935</td>
  <td align="center">0.700414598</td>
  <td align="center">-7.200966267</td>
  <td align="center">-7.201068684</td>
  <td align="center">-4.57274936</td>
  <td align="center">13.15887139</td>
  <td align="center">13.1608467</td>
  <td align="center">1.51408246</td>
  <td align="center">-28.45086152</td>
  <td align="center">-28.45153761</td>
  <td align="center">-0.4170461</td>
  <td align="center">0.000101</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="right">4.722277318</td>
  <td align="center">4.722504339</td>
  <td align="center">0.744964507</td>
  <td align="center">-7.354156963</td>
  <td align="center">-7.3542507</td>
  <td align="center">-4.61961058</td>
  <td align="center">17.1374434</td>
  <td align="center">17.1399773</td>
  <td align="center">1.51055838</td>
  <td align="center">-36.36528446</td>
  <td align="center">-36.36591543</td>
  <td align="center">-0.3524855</td>
  <td align="center">0.000110</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="right">4.84592201</td>
  <td align="center">4.846185027</td>
  <td align="center">0.857001395</td>
  <td align="center">-7.70305421</td>
  <td align="center">-7.703178009</td>
  <td align="center">-4.7163542</td>
  <td align="center">37.84289623</td>
  <td align="center">37.8479208</td>
  <td align="center">1.47966673</td>
  <td align="center">-77.67458196</td>
  <td align="center">-77.67408155</td>
  <td align="center">-0.2194152</td>
  <td align="center">0.000128</td>
</tr>
<tr>
  <td colspan="15" align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{r^*_\mathrm{core}}  dS^*_\mathrm{therm}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM1'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{w}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^{r^*_\mathrm{core}}  dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM2'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_{r^*_\mathrm{core}}^{R^*} dS^*_\mathrm{therm}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM3'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{w}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_{r^*_\mathrm{core}}^{R^*} dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM4'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}</math>
  </td>
</tr>
<tr>
  <td colspan="7" align="center">NOTE:  In all integrals, the fractional radial-displacement function, <math>~x</math>, has been normalized<br />to unity at the center of the spherical model.</td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{2\pi \cdot \mathbf{TERM5} } \biggl\{
\frac{2\gamma_c}{3} [\mathbf{TERM1}]
- (3\gamma_c - 4) [\mathbf{TERM2}]
+ \frac{2 \gamma_e}{3} [\mathbf{TERM3}]
- (3\gamma_e - 4)   [\mathbf{TERM4}]
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^* [3(\gamma_e - \gamma_c)] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.036312577\times \biggl[0.093111340 - 1.059900832 + 1.4294428 - (-3.82285786) - 1.782200484\times (2.4) \biggr] = 0.036312577\times [0.00819] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0002973 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.020704186\times \biggl[0.352702823 - 1.660292468 + 1.928692426 - (-1.85197268) - 1.029029184\times (2.4) \biggr] = 0.020704186\times [0.003405] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0000705 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 0.345}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019377703\times \biggl[0.54358856 - 1.81951316 + 2.01880112 - (-0.8993026) - 0.68747441\times (2.4) \biggr] = 0.019377703\times [0.003585] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0000695 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019467503\times \biggl[0.560331678 - 1.81099744 + 2.018776613 - (-0.8340922) - 0.658354814\times (2.4) \biggr] = 0.019467503\times [1.602203052 - 1.580051555] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.00043123 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 0.309}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019781992\times \biggl[0.595971606 - 1.847844232 + 2.01407784 - (-0.704971) - 0.60905486\times (2.4) \biggr] = 0.019781992\times [1.467176214 - 1.461731665] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0001077 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.021344909\times \biggl[0.685601116 - 1.88654168 + 1.972888973 - (-0.4388304) - 0.499262049\times (2.4) \biggr] = 0.021344909\times [1.210778809 - 1.198228918] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.000267876 \, .</math>
  </td>
</tr>
</table>


<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="11">
Table 4
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center">Analytic<br /><math>~\mathfrak{m}_\mathrm{tot}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{m}_\mathrm{tot}</math></td>
  <td align="center">'''TERM5'''</td>
  <td align="center"><math>~x_i</math></td>
  <td align="center"><math>~r^*_\mathrm{core}</math></td>
  <td align="center"><math>~P^*_i</math></td>
  <td align="center"><math>~\biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}</math></td>
  <td align="center"><math>~\biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env}</math></td>
  <td align="center"><math>~\biggl[\sigma_c^2\biggr]^\mathrm{VP}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">4.818155928</td>
  <td align="center">4.818145</td>
  <td align="center">13.14874521</td>
  <td align="center">0.814374698</td>
  <td align="center">1.153014872</td>
  <td align="center">0.139506172</td>
  <td align="center">+0.455871977</td>
  <td align="center">+1.47352</td>
  <td align="center">0.0002973</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">9.020985415</td>
  <td align="center">9.021084268</td>
  <td align="center">23.0612702</td>
  <td align="center">0.653665497</td>
  <td align="center">1.574940686</td>
  <td align="center">0.049058481</td>
  <td align="center">+1.059668912</td>
  <td align="center">+1.835801347</td>
  <td align="center">0.0000705</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">17.41399388</td>
  <td align="center">17.4141672</td>
  <td align="center">24.63990825</td>
  <td align="center">0.563043202</td>
  <td align="center">1.769031527</td>
  <td align="center">0.030957085</td>
  <td align="center">+1.552125296</td>
  <td align="center">+2.131275177</td>
  <td align="center">0.0000695</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">18.8449906</td>
  <td align="center">18.8451614</td>
  <td align="center">24.52624951</td>
  <td align="center">0.555549156</td>
  <td align="center">1.78413696</td>
  <td align="center">0.029889634</td>
  <td align="center">+1.600041467</td>
  <td align="center">+2.16002488</td>
  <td align="center">0.0004312</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="right">22.61541791</td>
  <td align="center">22.6155686</td>
  <td align="center">24.13633675</td>
  <td align="center">0.539776219</td>
  <td align="center">1.815519219</td>
  <td align="center">0.027798189</td>
  <td align="center">+1.705239188</td>
  <td align="center">+2.223143513</td>
  <td align="center">0.0001077</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="right">39.12088278</td>
  <td align="center">39.1208075</td>
  <td align="center">22.36902623</td>
  <td align="center">0.501037082</td>
  <td align="center">1.890385851</td>
  <td align="center">0.023427588</td>
  <td align="center">+1.991720516</td>
  <td align="center">+2.39503231</td>
  <td align="center">0.0002679</td>
</tr>
<tr>
  <td colspan="11" align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{m}_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}dM_r^*</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~I_\mathrm{sphere}^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}(r^*)^2   dM_r^*</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM5'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}(r^*)^2   x^2 dM_r^*</math>
  </td>
</tr>
<tr>
  <td colspan="11" align="center">NOTE:  In the '''TERM5''' integral (as elsewhere), the fractional radial-displacement function, <math>~x</math>, <br />has been normalized to unity at the center of the spherical model.</td>
</tr>
</table>
  </td>

</tr>
</table>

==Revised Free-Energy Analysis==
If we set <math>~x </math> = constant in the [[#VariationPrincipleRelation|variational principle relation]], we have,
<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^*)^2dM_r^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  dW^*_\mathrm{grav}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} dW^*_\mathrm{grav}
- 4\pi  (r_\mathrm{core}^*)^3 P_i^*\biggl[ 3(\gamma_e - \gamma_c) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~~~ \biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 I_\mathrm{sphere}^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(4 - 3\gamma_c )\mathfrak{w}_\mathrm{core}
+ (4 - 3\gamma_e )  \mathfrak{w}_\mathrm{env}
+ P_i^*V^*_\mathrm{core}\biggl[ 3^2(\gamma_c - \gamma_e) \biggr] \, ,
</math>
  </td>
</tr>
</table>
Notice the similarity between this last expression and the pair of expressions &#8212; numbered <b><font color="maroon" size="+1">&#x2465;</font></b> and <b><font color="maroon" size="+1">&#x2466;</font></b> &#8212; that arise in the context of [[SSC/SynopsisStyleSheet#Stability|pressure-truncated polytropes]].

=See Also=

* [http://adsabs.harvard.edu/abs/2018Sci...362..201D K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206)], ''A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.''


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