Editing SSC/FreeEnergy/PolytropesEmbedded/Pt1 (section)

====Case P====

=====First Pass=====
Alternatively, let's examine the [[#Case_P_Free-Energy_Surface|"Case P" free-energy surface]].  Drawing on [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's presentation]], we adopt the following radius and mass normalizations:
<div align="center">
<math>M_\mathrm{SWS} = 
\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
</div>
<div align="center">
<math>
R_\mathrm{SWS} = \biggl( \frac{n+1}{n} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .
</math>
</div>
In terms of these new normalizations, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_\mathrm{norm} \equiv \biggl[\biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{(n-1)} \biggr]^{1/(n-3)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{G}{K} \biggr)^{n/(n-3)} M_\mathrm{tot}^{(n-1)/(n-3)}  
R_\mathrm{SWS} \biggl( \frac{n+1}{n} \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">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+
M_\mathrm{SWS}^{-(n-1)/(n-3)}  \biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^{(n-1)/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_\mathrm{SWS} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)}
\biggl( \frac{n+1}{n} \biggr)^{[3(n-1)-(n-3)]/[2(n-3)]} 
G^{[2n+(n-3)-3(n-1)]/[2(n-3)]}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ K_n^{n[2(n-1) - (n+1) - (n-3)]/[(n+1)(n-3)]} P_\mathrm{e}^{-(n-1)(3-n)/[2(n+1)(n-3)]}
P_\mathrm{e}^{(n-1)(3-n)/[2(n+1)(n-3)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_\mathrm{SWS} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_e 
\biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{-2(n+1)/(n-3)} \biggl( \frac{n+1}{n} \biggr)^{-3(n+1)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

<span id="FirstPassFreeEnergy">Rewriting the expression for the free energy gives,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl(\frac{R_\mathrm{norm}}{R_\mathrm{SWS}} \biggr) 
+~ n\mathcal{B} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}  \biggl(\frac{R_\mathrm{norm}}{R_\mathrm{SWS}} \biggr)^{3/n}
+~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl(\frac{R_\mathrm{norm}}{R_\mathrm{SWS}}\biggr)^{-3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl[ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ n\mathcal{B} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}  \biggl[ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)}\biggr]^{3/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ \biggl( \frac{4\pi}{3} \biggr) \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)} \biggl( \frac{n+1}{n} \biggr)^{3(n+1)/(n-3)} 
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl[ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)}\biggr]^{-3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} 
+~ n\mathcal{B}   \biggl( \frac{n+1}{n} \biggr)^{3/(n-3)}
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{3(n-1)/[n(n-3)]} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ \biggl( \frac{4\pi}{3} \biggr) \biggl( \frac{n+1}{n} \biggr)^{3/(n-3)} 
\biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(5-n)/(n-3)} 
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \, .
</math>
  </td>
</tr>
</table>
</div>


Therefore, in this case, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
n\biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}}  
\biggl( \frac{n+1}{n} \biggr)^{3/(n-3)}
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{3(n-1)/[n(n-3)]} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl( \frac{n+1}{n} \biggr)^{3/(n-3)} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(5-n)/(n-3)}  \, ,
</math>
  </td>
</tr>
</table>
</div>

where the structural form factors for pressure-truncated polytropes are precisely defined [[SSCpt1/Virial/FormFactors#PTtable|here]].  The statement of virial equilibrium is, therefore,

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

<tr>
  <td align="right">
<math>~x^{4}_\mathrm{eq} + \alpha  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\beta  x^{(n-3)/n }_\mathrm{eq} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha \equiv \frac{a}{3c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)} 
\biggl\{ \frac{3}{4\pi} \biggl( \frac{n+1}{n} \biggr)^{-3/(n-3)} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(n-5)/(n-3)} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{20\pi}  \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{4\pi}{3\cdot 5}\biggr) \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr) \mathfrak{m}^{2}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta \equiv \frac{b}{nc}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}}  
\biggl( \frac{n+1}{n} \biggr)^{3/(n-3)}
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{3(n-1)/[n(n-3)]}
\biggl\{ \frac{3}{4\pi} \biggl( \frac{n+1}{n} \biggr)^{-3/(n-3)} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(n-5)/(n-3)} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tilde{\mathfrak{f}}_A \mathfrak{m}^{(n+1)/n}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{m}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{4\pi}\biggr) \frac{1}{\tilde{\mathfrak{f}}_M} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

From a previous derivation, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ b}{nc}\cdot  x^{(n-3)/n }_\mathrm{eq} - \frac{a}{3c}  - x^{4}_\mathrm{eq}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{4\pi} \biggl( \frac{n+1}{n}  \biggr)^{-3/(n-3)} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(n-5)/(n-3)} \biggl\{
\biggl(\frac{3}{4\pi} \biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}}  
\biggl( \frac{n+1}{n} \biggr)^{3/(n-3)}
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{3(n-1)/[n(n-3)]} \biggr\} \cdot  x^{(n-3)/n }_\mathrm{eq} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{3}{4\pi} \biggl( \frac{n+1}{n}  \biggr)^{-3/(n-3)} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(n-5)/(n-3)} \biggl\{
\frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)}  \biggr\}
- x^{4}_\mathrm{eq}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl(\frac{3}{4\pi} \biggr)^{(n+1)/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}}  
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n+1)/n}  x^{(n-3)/n }_\mathrm{eq} 
- \frac{3}{20\pi} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2}  - x^{4}_\mathrm{eq}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \tilde{\mathfrak{f}}_A \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{1}{\tilde{\mathfrak{f}}_M}  
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]^{(n+1)/n}  x^{(n-3)/n }_\mathrm{eq} 
- \frac{1}{5} \biggl(\frac{4\pi}{3}\biggr) \cdot \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr)\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{1}{\tilde{\mathfrak{f}}_M}  
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]^{2}  - x^{4}_\mathrm{eq}
</math>
  </td>
</tr>
</table>
</div>

which, thankfully, matches!  We conclude as well that the transition from stable to dynamically unstable configurations occurs at,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[x_\mathrm{eq}]_\mathrm{crit}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n-3)}{3 (n+1)} \biggr] \alpha \, .
</math>
  </td>
</tr>

<!-- FOR POWERPOINT SYNOPSIS
<tr>
  <td align="right">
<math>~[x_\mathrm{eq}]_\mathrm{crit}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n-3)}{3 (n+1)} \biggr] \frac{a}{3c}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{20\pi n}  \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ [x_\mathrm{eq}]_\mathrm{crit}^{2(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{n}{n-3} \biggr)^{(1-n)} 
\biggl( \frac{3^2\cdot 5}{4\pi \tilde{\mathfrak{f}}_W} \biggr)^{(n+1)} 
\biggl(\frac{\tilde{\mathfrak{f}}_A}{4}  \biggr)^{2n} \, .
</math>
  </td>
</tr>
-->
</table>
</div>

When combined with the statement of virial equilibrium ''at'' this critical point, we find,

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

<tr>
  <td align="right">
<math>~
\biggl\{ \biggl[ \frac{(n-3)}{3 (n+1)} \biggr]  + 1\biggr\}\frac{ \alpha }{\beta} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[x_\mathrm{eq}]^{(n-3)/n }_\mathrm{crit} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[ \frac{4n}{3 (n+1)} \biggr]^{4n} \biggl( \frac{ \alpha }{\beta} \biggr)^{4n}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(n-3)}{3 (n+1)} \biggr]^{(n-3)}  \alpha^{(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[ \frac{3 n}{(n-3)} \biggl( \frac{n+1}{n}\biggr) \biggr]^{(3-n)} \biggl[ \frac{3 }{4} \biggl( \frac{n+1}{n}\biggr) \biggr]^{4n}   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\alpha^{3(n+1)} \beta^{-4n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl( \frac{4\pi}{3\cdot 5}\biggr) \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr) \mathfrak{m}^{2}  \biggr\}^{3(n+1)} 
\biggl\{ \tilde{\mathfrak{f}}_A \mathfrak{m}^{(n+1)/n}  \biggr\}^{-4n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde{\mathfrak{f}}_A^{-4n} 
\biggl[ \biggl( \frac{4\pi}{3\cdot 5}\biggr) \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr)\biggr]^{3(n+1)} \mathfrak{m}^{2(n+1)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\mathfrak{m}^{2(n+1)}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{4\pi}{3\cdot 5}\biggr) \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr)\biggr]^{-3(n+1)} 
\biggl[ \frac{3 n}{(n-3)} \biggl( \frac{n+1}{n}\biggr) \biggr]^{(3-n)} \biggl[ \frac{3\tilde{\mathfrak{f}}_A }{4} \biggl( \frac{n+1}{n}\biggr) \biggr]^{4n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{3\cdot 5}{4\pi}\biggr) \frac{1}{\tilde{\mathfrak{f}}_W} \biggr]^{3(n+1)} 
\biggl[ \frac{3 n}{(n-3)}  \biggr]^{(3-n)} \biggl[ \frac{3\tilde{\mathfrak{f}}_A }{4}  \biggr]^{4n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{3^2 \cdot 5 n}{ 4\pi(n-3)} \cdot \frac{1}{\tilde{\mathfrak{f}}_W}  \biggr]^{(3-n)} 
\biggl[ \biggl( \frac{3^2\cdot 5}{2^4\pi}\biggr) \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_W}  \biggr]^{4n} \, .
</math>
  </td>
</tr>
</table>
</div>

This also means that the critical radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[x_\mathrm{eq}]_\mathrm{crit}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n-3)}{3 (n+1)} \biggr] \biggl( \frac{4\pi}{3\cdot 5}\biggr) \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr) \mathfrak{m}^{2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{3^2\cdot 5 n}{4\pi(n-3)} \cdot \frac{1}{\tilde{\mathfrak{f}}_W}\biggr]^{-1}   \mathfrak{m}^{2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ [x_\mathrm{eq}]_\mathrm{crit}^{4(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3^2\cdot 5 n}{4\pi(n-3)} \cdot \frac{1}{\tilde{\mathfrak{f}}_W} \biggr]^{-(n+1)} 
\biggl[ \frac{3^2 \cdot 5 n}{ 4\pi(n-3)} \cdot \frac{1}{\tilde{\mathfrak{f}}_W}  \biggr]^{(3-n)} 
\biggl[ \biggl( \frac{3^2\cdot 5}{2^4\pi}\biggr) \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_W}  \biggr]^{4n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi(n-3)}{3^2\cdot 5 n} \cdot \tilde{\mathfrak{f}}_W \biggr]^{2(n-1)} 
\biggl[ \biggl( \frac{3^2\cdot 5}{2^4\pi}\biggr) \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_W}  \biggr]^{4n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ [x_\mathrm{eq}]_\mathrm{crit}^{2(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{n}{(n-3)} \biggl( \frac{3^2\cdot 5}{4\pi \tilde{\mathfrak{f}}_W} \biggr) \biggr]^{(1-n)} 
\biggl[ \biggl( \frac{3^2\cdot 5}{4\pi \tilde{\mathfrak{f}}_W}\biggr) \frac{\tilde{\mathfrak{f}}_A}{4}  \biggr]^{2n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{n}{n-3} \biggr)^{(1-n)} 
\biggl( \frac{3^2\cdot 5}{4\pi \tilde{\mathfrak{f}}_W} \biggr)^{(n+1)} 
\biggl(\frac{\tilde{\mathfrak{f}}_A}{4}  \biggr)^{2n} \, .
</math>
  </td>
</tr>
</table>
</div>


<!-- THERE IS A MISTAKE IN THIS OMITTED SUBSECTION

From an earlier derivation, we obtained,


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

<tr>
  <td align="right">
<math>~\biggl[\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr]^4_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n-3)}{3^2 (n+1)} \biggr] \frac{a}{c}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n-3)}{3^2 (n+1)} \biggr] 
\biggl[ \frac{3}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)^{n/(n-3)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/(n-3)}   \biggr]
\biggl[ \frac{3}{4\pi} \biggl( \frac{n+1}{n} \biggr)^{-3/(n-3)} \biggl( \frac{M_\mathrm{tot} }{M_\mathrm{SWS}} \biggr)^{(n-5)/(n-3)}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~[x_\mathrm{eq}]_\mathrm{crit}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{20\pi n}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2}  \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} 
= \tilde{\mathfrak{f}}_W \cdot \frac{(n-3)}{15 n} \biggl(\frac{4\pi}{3}\biggr) \biggl[\biggl(\frac{3}{4\pi}\biggr)\frac{1}{\tilde{\mathfrak{f}}_M}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[\biggl(\frac{3}{4\pi}\biggr)\frac{1}{\tilde{\mathfrak{f}}_M}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{\tilde{\mathfrak{f}}_W} \cdot \frac{15 n}{(n-3)} \biggl(\frac{3}{4\pi}\biggr)\biggr]^{1/2}  [x_\mathrm{eq}]_\mathrm{crit}^2</math>
  </td>
</tr>
</table>
</div>

which, when combined with the statement of virial equilibrium ''at'' this critical point, implies,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde{\mathfrak{f}}_A \biggl\{ \biggl[ \frac{1}{\tilde{\mathfrak{f}}_W} \cdot \frac{15 n}{(n-3)} \biggl(\frac{3}{4\pi}\biggr)\biggr]^{1/2}  [x_\mathrm{eq}]_\mathrm{crit}^2\biggr\}^{(n+1)/n}  [x_\mathrm{eq}]_\mathrm{crit}^{(n-3)/n }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
[x_\mathrm{eq}]_\mathrm{crit}^4
+ \frac{1}{5} \biggl(\frac{4\pi}{3}\biggr) \cdot \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr) \biggl\{ \biggl[ \frac{1}{\tilde{\mathfrak{f}}_W} \cdot \frac{15 n}{(n-3)} \biggl(\frac{3}{4\pi}\biggr)\biggr]^{1/2}  [x_\mathrm{eq}]_\mathrm{crit}^2\biggr\}^{2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \tilde{\mathfrak{f}}_A  \biggl[ \frac{1}{\tilde{\mathfrak{f}}_W} \cdot \frac{15 n}{(n-3)} \biggl(\frac{3}{4\pi}\biggr)\biggr]^{(n+1)/(2n)}  
[x_\mathrm{eq}]_\mathrm{crit}^{(3n-1)/n }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
[x_\mathrm{eq}]_\mathrm{crit}^4 \biggl\{ 1
+ \frac{1}{5} \biggl(\frac{4\pi}{3}\biggr) \cdot \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr) \biggl[ \frac{1}{\tilde{\mathfrak{f}}_W} \cdot \frac{15 n}{(n-3)} \biggl(\frac{3}{4\pi}\biggr)\biggr]^{2} \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ [x_\mathrm{eq}]_\mathrm{crit}^{(n+1)/n}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tilde{\mathfrak{f}}_A  \biggl[ \frac{1}{\tilde{\mathfrak{f}}_W} \cdot \frac{15 n}{(n-3)} \biggl(\frac{3}{4\pi}\biggr)\biggr]^{(n+1)/(2n)}  
\biggl\{ 1+ \frac{1}{5} \biggl(\frac{4\pi}{3}\biggr) \cdot \tilde{\mathfrak{f}}_W \biggl( \frac{n+1}{n} \biggr) \biggl[ \frac{1}{\tilde{\mathfrak{f}}_W} \cdot \frac{15 n}{(n-3)} \biggl(\frac{3}{4\pi}\biggr)\biggr]^{2} \biggr\}^{-1} 
</math>
  </td>
</tr>
</table>
</div>

END OMITTED SUBSECTION -->

The following parallel derivation was done independently.  [<font color="red">Note that a factor of 2n/(n-1) appears to correct a mistake made during the original derivation.</font>]  Beginning with the virial expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta  x^{(n-3)/n }_\mathrm{eq}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\alpha + x^{4}_\mathrm{eq}
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~
\biggl(\frac{3}{4\pi} \biggr)^{(n+1)/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}}  
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n+1)/n}  [x_\mathrm{eq} ]^{(n-3)/n }_\mathrm{crit}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{3}{20\pi} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2}  
+ \frac{(n-3)}{20\pi n}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2}  \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{(n-1)}{10\pi n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2}  \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl[ \frac{2n}{(n-1)}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
[x_\mathrm{eq} ]^{(n-3)/n }_\mathrm{crit}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{2(n-1)}{15 n} \biggl(\frac{4\pi}{3} \biggr)^{1/n}  \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_A\tilde{\mathfrak{f}}_M^{(n-1)/n}} \cdot
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)/n}  \biggl[ \frac{2n}{(n-1)}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
[x_\mathrm{eq} ]^{(n-3) }_\mathrm{crit}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[\frac{2(n-1)}{15 n} \biggr]^n \biggl(\frac{4\pi}{3} \biggr)  \frac{\tilde{\mathfrak{f}}_W^n}{\tilde{\mathfrak{f}}_A^n \tilde{\mathfrak{f}}_M^{(n-1)}} 
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{(n-1)}  \biggl[ \frac{2n}{(n-1)}\biggr]^n
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[\frac{2(n-1)}{15 n} \biggr]^n \biggl(\frac{4\pi}{3} \biggr)  \frac{\tilde{\mathfrak{f}}_W^n}{\tilde{\mathfrak{f}}_A^n \tilde{\mathfrak{f}}_M^{(n-1)}} 
\biggl\{ \biggl[ \frac{20\pi n}{(n-3)} \biggr]^{(n-1)/2}  \biggl( \frac{\tilde{\mathfrak{f}}_M^2}{\tilde{\mathfrak{f}}_W} \biggr)^{(n-1)/2} [x_\mathrm{eq} ]^{2(n-1) }_\mathrm{crit} \biggr\}\biggl[ \frac{2n}{(n-1)}\biggr]^n
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[\frac{2(n-1)}{15 n} \biggr]^n \biggl(\frac{4\pi}{3} \biggr)  \frac{\tilde{\mathfrak{f}}_W^{(n+1)/2}}{\tilde{\mathfrak{f}}_A^n } 
\biggl\{ \biggl[ \frac{20\pi n}{(n-3)} \biggr]^{(n-1)/2}   [x_\mathrm{eq} ]^{2(n-1) }_\mathrm{crit} \biggr\}\biggl[ \frac{2n}{(n-1)}\biggr]^n
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
[x_\mathrm{eq} ]^{(n+1) }_\mathrm{crit}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \biggl(\frac{3}{4\pi} \biggr) 
\biggl[\frac{15 n}{2(n-1)} \biggr]^n
\biggl[ \frac{(n-3)}{20\pi n} \biggr]^{(n-1)/2}   \frac{\tilde{\mathfrak{f}}_A^n }{\tilde{\mathfrak{f}}_W^{(n+1)/2}} \biggl[ \frac{(n-1)}{2n} \biggr]^n
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
[x_\mathrm{eq} ]^{(n+1) }_\mathrm{crit}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \biggl(\frac{3}{4\pi} \biggr) 
\biggl[\frac{15 }{2^2} \biggr]^n
\biggl[ \frac{(n-3)}{20\pi n} \biggr]^{(n-1)/2}   \frac{\tilde{\mathfrak{f}}_A^n }{\tilde{\mathfrak{f}}_W^{(n+1)/2}} 
</math>
  </td>
</tr>
</table>
</div>



Also from [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's work]] we know that the equilibrium mass and radius are,
<div align="center">
<table border="0" cellpadding="3">

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

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl[ \tilde\xi {\tilde\theta}_n^{(n-1)/2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Additional details in support of an associated PowerPoint presentation can be found [[SSC/FreeEnergy/PowerPoint|here]].