Editing SSC/Virial/Polytropes/Pt2 (section)

====More Precise Form Factors====
Here we attempt to determine proper expressions for several form factors such that the equilibrium configurations determined from virial analysis will precisely match the [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|pressure-truncated polytropic configurations]] that have been determined from detailed force-balanced models that have been derived and published separately by {{ Horedt70full }}, by {{ Whitworth81full }} and by {{ Stahler83full }}.  It seems simplest to begin with the [[SSC/BipolytropeGeneralizationVersion2#Generalized_Free-Energy_Expression|free-energy expressions that we have already generalized in the context of bipolytropic configurations]], properly modified to embed the "core" in an external medium of pressure, <math>~P_e</math>, rather then inside an envelope that has a different polytropic index.  Specifically,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core}
+ \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}
+ \biggl( \frac{P_e V}{E_\mathrm{norm}} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A} \chi^{-1} - \frac{\mathcal{B}_\mathrm{core}}{(1-\gamma_c)} \chi^{3-3\gamma_c}  + \mathcal{D} \chi^3 \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\nu}{q^3} \biggr) \int_0^{q} \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core}  \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core}  x dx  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_0^q  3\biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx 
=
\frac{4\pi}{3} \biggl[ \frac{P_e \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ q^3 s_\mathrm{core} \biggr] 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\chi^{-3} \biggl[ \frac{P_e V}{P_\mathrm{norm} R_\mathrm{norm}^3} \biggr] = \biggl(\frac{P_e}{P_\mathrm{norm}} \biggr) \int_0^q  4\pi  x^2 dx 
= \frac{4\pi q^3}{3} \biggl(\frac{P_e}{P_\mathrm{norm}} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

Virial equilibrium occurs where <math>~\partial\mathfrak{G}/\partial \chi = 0</math>, that is, when,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{A} \chi_\mathrm{eq}^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{B} \chi_\mathrm{eq}^{3-3\gamma_c} - \mathcal{D} \chi_\mathrm{eq}^{3} \, .</math>
  </td>
</tr>
</table>
</div>

=====n = 5 Polytropic=====

From our [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|analysis of the free energy of <math>(n_c, n_e) = (5, 0)</math> bipolytropes]], we deduce that the coefficient, <math>\mathcal{A}</math>, that quantifies the [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_2|gravitational potential energy of a pressure-truncated <math>n = 5</math> polytrope]] is,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A} = - \frac{\chi}{3} \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
\chi_\mathrm{eq} \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2}  
\biggl[
a_\xi^{1/2} q(a_\xi^2 q^4 - \frac{8}{3}a_\xi q^2 - 1) (a_\xi q^2 +1)^{-3} + \tan^{-1}(a_\xi^{1/2} q)
\biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
the coefficient, <math>\mathcal{B}_\mathrm{core}</math>, that quantifies the [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_3|thermal energy content of a pressure-truncated <math>n = 5</math> polytrope]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{B} = (\gamma_c-1) \chi^{3\gamma_c-3}\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\chi_\mathrm{eq}^{3/5} 
\biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \biggl[ 
\tan^{-1}[a_\xi^{1/2}q] - a_\xi^{1/2}q ~\frac{(1 - a_\xi q^2)}{(1 + a_\xi q^2)^2} \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and the coefficient, <math>\mathcal{D}</math> &#8212; which can be obtained by setting <math>P_{ic} \rightarrow P_e</math> in expressions drawn from out analysis of the [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_3|thermal energy content of an <math>~(5, 0)</math>bipolytrope]] &#8212; is,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{D} = \biggl(\frac{2q^3}{3 \chi_\mathrm{eq}^3} \biggr) \biggl[ \frac{2\pi P_{ic} \chi_\mathrm{eq}^3}{P_\mathrm{norm}} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\chi_\mathrm{eq}^{-3} \biggl( \frac{2\cdot 3^4}{\pi} \biggr)^{1/2} \biggl(1 + a_\xi q^2 \biggr)^{-3} a_\xi^{3/2}q^3 \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, virial equilibrium occurs when,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{A} \chi_\mathrm{eq}^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{B} \chi_\mathrm{eq}^{-3/5} - \mathcal{D} \chi_\mathrm{eq}^{3} \, ,</math>
  </td>
</tr>
</table>
</div>
that is, when,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2}  
\biggl[ a_\xi^{1/2} q(a_\xi^2 q^4 - \frac{8}{3}a_\xi q^2 - 1) (a_\xi q^2 +1)^{-3} + \tan^{-1}(a_\xi^{1/2} q)\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \biggl[ 
\tan^{-1}[a_\xi^{1/2}q] - a_\xi^{1/2}q ~\frac{(1 - a_\xi q^2)}{(1 + a_\xi q^2)^2} \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\biggl[ a_\xi^{1/2} q(a_\xi^2 q^4 - \frac{8}{3}a_\xi q^2 - 1) (a_\xi q^2 +1)^{-3} + \tan^{-1}(a_\xi^{1/2} q)\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \tan^{-1}[a_\xi^{1/2}q] - a_\xi^{1/2}q ~\frac{(1 - a_\xi q^2)}{(1 + a_\xi q^2)^2} \biggr] 
-
\biggl( \frac{2^3}{3} \biggr) \biggl(1 + a_\xi q^2 \biggr)^{-3} a_\xi^{3/2}q^3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
a_\xi^{1/2} q(a_\xi^2 q^4 - \frac{8}{3}a_\xi q^2 - 1) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 - a_\xi^{1/2}q ~(1 - a_\xi q^2)(1 + a_\xi q^2) 
-
\biggl( \frac{2^3}{3} \biggr) a_\xi^{3/2}q^3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
(a_\xi^2 q^4 - \frac{8}{3}a_\xi q^2 - 1) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (1 - a_\xi^2 q^4)  - \biggl( \frac{2^3}{3} \biggr) a_\xi q^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>
</div>
So this will always be true!
 


The [[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|detailed force-balanced analysis]] of <math>~n=5</math> polytropes shows that the pair of equations defining the equilibrium (truncated) radius for a specified external pressure are,
<div align="center">
<math>
R_\mathrm{eq} = \biggl[ \frac{\pi M^4 G^5}{2^3 \cdot 3^7 K^5} \biggr]^{1/2} \frac{(3+\xi_e^2)^3}{\xi_e^5} \, ,
</math>

<math>P_e= \biggr( \frac{2^3\cdot 3^{15} K^{10}}{\pi^3 M^{6} G^9} \biggr) \frac{\xi_e^{18}}{(3 + \xi_e^2)^{12}} </math> .
</div>
Applying our chosen normalizations, this pair of defining equations becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\frac{P_e}{P_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggr( \frac{2^3\cdot 3^{15} K^{10}}{\pi^3 M^{6} G^9} \biggr) \frac{\xi_e^{18}}{(3 + \xi_e^2)^{12}} 
\cdot \biggl[\frac{G^9 M_\mathrm{tot}^6}{K^{10}} \biggr]
= \biggr( \frac{2^3\cdot 3^{3} }{\pi^3} \biggr) \xi_e^{18} \biggl(1 + \frac{1}{3}\xi_e^2 \biggr)^{-12} \, .
</math>
  </td>
</tr>
</table>
</div>