Editing SSC/Virial/PolytropesSummary (section)

===Determining Expressions for Free-Energy Coefficients===
We should be able to convert the separately derived expression for <math>~W_\mathrm{grav}^*</math> into an expression for the free-energy coefficient, <math>~A</math>, in equilibrium configurations.  As [[User:Tohline/SSC/Virial/PolytropesSummary#Virial_Theorem|noted above]], for a fixed value of <math>~A</math>,
<div align="center">
<math>~W_\mathrm{grav}^* ~~\rightarrow ~~ -3A\chi^{-1} \, .</math>
</div>
Therefore, in an equilibrium configuration, we can write,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~W_\mathrm{grav}^*
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-3A\chi_\mathrm{eq}^{-1} = -3A \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)
= -3A \biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{eq}}\biggr)\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\Rightarrow ~~~ A
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{1}{3} W_\mathrm{grav}^* \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .</math>
  </td>
</tr>
</table>
</div>
Now, from immediately above, we know that,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}  \, ;
</math>
  </td>
</tr>
</table>
</div>
and, from our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|accompanying discussion of the free-energy of bipolytropic configurations]], we know that,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~W^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} 
\biggl[ x_i\biggl(x_i^4 - \frac{8}{3}x_i^2 - 1\biggr) (1 + x_i^2)^{-3}  + \tan^{-1}(x_i) \biggr] \, . </math>
  </td>
</tr>
</table>
</div>
So, again realizing that <math>~x_i</math> and <math>~\ell</math> are interchangeable, we have,
<div align="center">
<table border="0" cellpadding="4">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .
</math>
  </td>
</tr>
</table>
</div>
Finally, we need to determine an expression for the ratio, <math>~R_\mathrm{SWS}/R_\mathrm{norm}</math>.  Drawing the definition of <math>~R_\mathrm{norm}</math> from [[User:Tohline/SphericallySymmetricConfigurations/Virial#Normalizations|our introductory chapter on the virial equilibrium]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_\mathrm{norm}\biggr|_{\gamma = 6/5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{2-6/5} \biggr]^{1/(4-18/5)}
= \biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{4/5} \biggr]^{5/2}
= \biggl( \frac{G}{K}\biggr)^{5/2} M_\mathrm{tot}^{2} \, .
</math>
  </td>
</tr>
</table>
</div>
From our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of Stahler's derived mass &amp; radius relationships for truncated, <math>~n=5</math> polytropes]] we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{limit}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
M_\mathrm{SWS}^2 \biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
In addition, from our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|review of Stahler's defined normalizations]], we see that,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~M_\mathrm{SWS}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3 \cdot 3^3}{5^3} \biggr) G^{-3} K^{10/3} P_e^{-1/3} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
and, &nbsp;&nbsp;&nbsp;<math>~R_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2 \cdot 3}{5} \biggr)^{1/2} G^{-1/2} K^{5/6} P_e^{-1/3} \, ,
</math>
  </td>
</tr>
</table>
</div>
which, when combined to cancel <math>~P_e</math> gives,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~M_\mathrm{SWS}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^6 \cdot 3^6}{5^6} \biggr)^{1/2} G^{-3} K^{10/3} 
\biggl( \frac{5}{2 \cdot 3} \biggr)^{1/2} G^{1/2} K^{-5/6} R_\mathrm{SWS} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} R_\mathrm{SWS} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}  
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \frac{1}{M_\mathrm{limit}^2} \, .
</math>
  </td>
</tr>
</table>
</div>
In combination with the expression for <math>~R_\mathrm{norm}</math>, then, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{norm}}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2} 
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \biggl(\frac{M_\mathrm{tot}}{M_\mathrm{limit}}\biggr)^{2} \, ,
</math>
  </td>
</tr>
</table>
</div>
which means that, for truncated <math>~n=5</math> polytropes, the expression for the free-energy coefficient is,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \biggl( \frac{\pi^2}{2\cdot 3^7 \cdot 5} \biggr)^{1/2}
\ell (1 + \ell^2)^{-1} \cdot \ell^{-6} (1+\ell^2)^{4}  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
2^{-4} \ell^{-5} (1+\ell^2)^{3}  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, .
</math>
  </td>
</tr>
</table>
</div>
Finally, drawing from our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|accompanying derivation of expressions for the structural form factors in this case]], we know that,
<div align="center">
<table border="0" cellpadding="3">

<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>
which gives,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~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>
</table>
</div>

This exactly matches the expression for the free-energy coefficient, <math>~\mathcal{A}</math>, that we derived separately in conjunction with our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|derivation of expressions for the structural form factors]].