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=Structural Form Factors=
As has been defined in [[SSCpt1/Virial#Structural_Form_Factors|a companion, introductory discussion]], three key dimensionless structural form factors are:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

As a cross-check, multiplying this expression through by <math>[(R_\mathrm{eq}/R_\mathrm{norm})(M_\mathrm{norm}/M_\mathrm{limit})^2]</math> &#8212; where the expression for <math>R_\mathrm{eq}/R_\mathrm{norm}</math> can be obtained from our [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|discussions of detailed force-balanced models]] &#8212; gives a related result that can be obtained directly from [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|Horedt's expressions]], namely,

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

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

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

[http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala &amp; Horedt (1974)] have provided analytic expressions for the gravitational potential energy and the internal energy &#8212; which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively &#8212; that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres.  [The same expression for <math>~\Omega</math> is also effectively provided in &sect;1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
<div align="center">
<table border="1" align="center" cellpadding="8" width="90%">
<tr><td align="center">

<!-- [[Image:VialaHoredt1974.png|500px|center]] -->
Astronomy &amp; Astrophysics, 33: 195-202, (1974)<br />
POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
Y. P. Viala &amp; Gp. Horedt<br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>\Omega</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- G\int_0^M \frac{MdM}{r} = \frac{16\pi^2 G \rho_0^2 \alpha^5}{(5-n)} \biggl[
\mp \xi^3 \theta^{n+1} - 3\xi^3 (\theta')^2 - 3\xi^2 \theta (\theta')
\biggr] \, ,
</math>
  </td>
</tr>

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

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

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

</td></tr>
<tr>
  <td align="left">
A couple of key equations drawn directly from [http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala &amp; Horedt (1974)] have been shown here.  As its title indicates, the paper includes discussion of &#8212; and accompanying equation derivations for &#8212; equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries:  planar sheets, axisymmetric cylinders, and spheres.  We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only.  These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
  </td>
</tr>
</table>
</div>
Rewriting these two expressions to accommodate our parameter notations &#8212; recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> &#8212; we have from Viala &amp; Horedt's (VH74) work,

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

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

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

===First Reality Check===

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

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

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

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

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

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

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

===Renormalization===
Both of the energy-term expressions derived by Viala &amp; Horedt are written in terms of <math>\rho_c</math> and 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>a_\mathrm{n}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n}\biggr]^{1/2} 
</math>
  </td>
</tr>
</table>
</div>
&#8212; that is, effectively in terms of <math>\rho_c</math> and {{ Template:Math/MP_PolytropicConstant}} &#8212; whereas, in the context of our discussions, we would prefer to express them in terms of [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Adopted_Normalizations|our generally adopted energy normalization]], 
<div align="center">
<table border="0" cellpadding="5">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(This may or may not be useful!)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</td>

<td align="center">

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

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

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

</td>
</tr>
<tr>
  <td align="left" colspan="2">
We should point out that [http://adsabs.harvard.edu/abs/1993ApJS...88..205L Lai, Rasio, &amp; Shapiro (1993b, ApJS, 88, 205)] define a different set of dimensionless structure factors for ''isolated'' polytropic spheres  &#8212; <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10)  is used in the determination of the gravitational potential energy.
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>k_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3}{5-n} \biggl[ \frac{4\pi |\theta^'_1|}{\xi_1} \biggr]^{1 / 3} </math>
  </td>
</tr>
</table>
</div>
Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter.  We note, as well, that for rotating configurations they define two additional dimensionless structure factors &#8212; <math>k_3</math> (their equation 3.17) is used in the determination of the rotational kinetic energy; and <math>\kappa_n</math> (their equation 3.14; also equation 7.4.9 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>])  is used in the determination of the moment of inertia.
  </td>
</tr>
</table>
</div>

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

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

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

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

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

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

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

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

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

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

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

</td>
</tr>

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

As is detailed in our [[SSC/Structure/BiPolytropes/Analytic5_1#Profile|accompanying discussion of bipolytropes]] &#8212; see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope|discussion of the properties of ''isolated'' polytropes]] &#8212; in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,
<div align="center">
<math>
a_{5} =\biggr[ \frac{3K}{2\pi G} \biggr]^{1/2}  \rho_0^{-2/5}  \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

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

<tr>
  <td align="right">
<math>\frac{M_r}{[K^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that &#8212; see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] &#8212; in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2}  \rho_0^{-1/5}  
~~~~\Rightarrow ~~~~
\rho_0^{1/5} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2}  M_\mathrm{tot}^{-1} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles">
<table border="0" cellpadding="5" align="center">

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

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

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

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\xi^2}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
</math>
  </td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above.  Notice that if we decide to truncate an <math>n=5</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> &#8212; as in the discussion that follows &#8212; the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

===Mass2===

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

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

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

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

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

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

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{\tilde\xi^2}{3}\biggr]^{3/2}
\biggl\{ x^3 \biggl[ 1 + \biggl(\frac{\tilde\xi^2}{3}\biggr)x^2 \biggr]^{-3/2}  \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[SSCpt1/Virial/FormFactors#NormalizedProfiles|profile presented just before the "Mass1" subsection]], above.  In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile &#8212; one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

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

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

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

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

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

===Mean-to-Central Density===

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3^{8}}{2^{7}\pi}\biggr)^{1/2} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] \, .
</math>
  </td>
</tr>
</table>
</div>
This exactly matches the normalized thermal energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>S_\mathrm{core}^*</math>.  Its similarity to the expression for the gravitational potential energy &#8212; which is relevant to the virial theorem &#8212; is more apparent if it is rewritten in the following form:
<div align="center">
<table border="0" cellpadding="8" align="center">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\biggl(\frac{2 \cdot 3^{6}}{\pi}\biggr)^{1/2}
\biggl[\frac{\ell}{ (1+\ell^2)} \biggr]^3 \, .
</math>
  </td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]].  Given that, for <math>n=5</math> polytropes &#8212; see the [[#Mass1|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \biggl[ \ell^2(1+\ell^2)^{-1} \biggr]^{3/2}
</math>
  </td>

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

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

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

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

</div>

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

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

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

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

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


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

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

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

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

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

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

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

</td>
</tr>

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


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

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

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

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

<tr>
  <td align="right">
<math>\frac{M_r}{(K/G)^{3/2}\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2}{\pi} \biggr)^{1/2} (\sin\xi - \xi \cos\xi) \, .</math>
  </td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that &#8212; see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] &#8212; in ''isolated'' <math>n=1</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2}  \rho_0  
~~~~\Rightarrow ~~~~
\rho_0 = \biggr[ \frac{G^3}{2\pi K^3} \biggr]^{1/2}   M_\mathrm{tot} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles1">
<table border="0" cellpadding="5" align="center">

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

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

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

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi\cos\xi) \, .
</math>
  </td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above.  Notice that if we decide to truncate an <math>n=1</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> &#8212; as in the discussion that follows &#8212; the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

===Mass2===

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

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

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

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

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

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

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] </math>
  </td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[#NormalizedProfiles1|profile presented just before the "Mass1" subsection]], above.  In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile &#8212; one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

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

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

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

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

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

===Mean-to-Central Density===

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

===Gravitational Potential Energy===
As presented at the [[User:Tohline/SSC/Virial/FormFactors#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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


===Thermal Energy===
As presented at the [[User:Tohline/SSC/Virial/FormFactors#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

===Reality Checks (n=1)===
====Expectation from Stahler's Equilibrium Models====

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} 2\tilde\xi\biggl\{ 1-\cos(2\tilde\xi ) \biggr\}
= \biggl( \frac{1}{2\pi^3} \biggr)^{1/2} \tilde\xi \sin^2(\tilde\xi ) \, .
</math>
  </td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[User:Tohline/SSC/Virial/FormFactors#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]].  Given that, for <math>~n=1</math> polytropes &#8212; see the [[User:Tohline/SSC/Virial/FormFactors#Mass1_2|"Mass1" discussion above]] and our accompanying [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D1.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
</math>
  </td>

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

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

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

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

</div>

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

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

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

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

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

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

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

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

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

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

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

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

which matches the expression presented in the [[User:Tohline/SSC/Virial/FormFactors#Summary_.28n.3D1.29|summary table, above]].  Next,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</table>
</div>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</table>
</div>

=See Also=


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