Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2 (section)

===Mapping from Above Discussion===
====Deriving Concise Virial Theorem Mass-Radius Relation====
Looking back on the definitions of <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> that we introduced in connection with our initial [[#ConciseVirial|concise algebraic expression of the virial theorem]], we can write,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~P_e </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~P_\mathrm{norm} \biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~R_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~R_\mathrm{norm} \Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} 
\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, .
</math>
  </td>
</tr>

</table>
</div>
The first of these two expressions can be flipped around to give an expression for <math>~M_\mathrm{tot}</math> in terms of <math>~P_e</math> and, then, as normalized to <math>~M_\mathrm{SWS}</math>.  Specifically,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~ M_\mathrm{tot}^{2(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~ \Rightarrow~~~ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{n}{n+1} \biggr)^{3/2} 
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)/[2(n+1)]}  \biggl[ \frac{\mathcal{B}^{2n/(n+1)}}{\mathcal{A}^{3/2}} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

This means, as well, that we can rewrite the equilibrium radius as,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~R_\mathrm{eq}^{n-3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} 
\biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} 
\biggl( \frac{G}{K} \biggr)^n \biggl\{   
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] 
\biggr\}^{(n-1)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} 
\biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{(n-1)/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl( \frac{G}{K} \biggr)^n 
\biggl\{  
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] 
\biggr\}^{(n-1)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl\{ \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{2n(n+1)} 
\biggl[ \frac{\mathcal{B}^{4n(n-1)}}{\mathcal{A}^{3(n+1)(n-1)}} \biggr]\biggr\}^{1/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} 
\biggl\{ \biggl( \frac{G}{K} \biggr)^{2n(n+1)} 
\biggl[ \frac{K^{4n(n-1)}}{G^{3(n+1)(n-1)}P_e^{(n-3)(n-1)} } \biggr] 
\biggr\}^{1/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} 
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
\biggl[ G^{(3-n)(n+1)} K^{2n(n-3)} P_e^{(n-3)(1-n)} \biggr]^{1/[2(n+1)]}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{n}{n+1} \biggr)^{1/2} 
\Chi_\mathrm{ad} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-1)/[2(n+1)]} 
\biggl[ \frac{\mathcal{B}^{n/(n+1)}}{\mathcal{A}^{1/2}} \biggr] \, .
</math>
  </td>
</tr>

</table>
</div>
Flipping both of these expressions around, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}   \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} 
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2} 
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr] 
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(1-n)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2} 
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr] 
\biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}   \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} 
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{(1-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>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)} 
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, our earlier derived [[#ConciseVirial3|compact expression for the virial theorem]] becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

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

Or, rearranged,

<div align="center" id="CompactStahlerVirial">
<table border="1" cellpadding="10" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggr)^4  - 
\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  
+~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2} = 0 \, .
</math> 
  </td>
</tr>
</table>
</div>

After adopting the modified coefficient definitions,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}  =
\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{-(n+1)/n} =
\biggl( \frac{3}{4\pi}\biggr)^{1/n} 
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, ,
</math>
  </td>
</tr>
</table>
</div>

<span id="modNormalizations">as well as the modified length- and mass-normalizations,</span> <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>, such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}} 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}} \, ,</math>
  </td>
</tr>
</table>
</div>
we obtain the
<div align="center" id="ConciseVirialMR">
<font color="#770000">'''Virial Theorem in terms of Mass and Radius'''</font><br />

<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{(n+1)/n} 
+ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2 = 0 
\, .
</math>
</div>


For later use we note as well that, with these modified coefficient definitions, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}^{n-3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl[ \biggl( \frac{4\pi}{3} \biggr)^{n-3} \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}    
\frac{\mathcal{A}_{M_\ell}^{3(n+1)}}{\mathcal{B}_{M_\ell}^{4n}} \biggr] \mathcal{Y}^{2(n+1)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad}^{n-3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~ 
\biggl[ \frac{n}{n+1} \biggl( \frac{\mathcal{B}_{M_\ell}}{\mathcal{A}_{M_\ell}} \biggr)\biggr]^n 
\mathcal{X}^{n-3} \mathcal{Y}^{1-n} \, ,
</math>
  </td>
</tr>
</table>
</div>
where <math>~\mathcal{X}</math> and <math>~\mathcal{Y}</math> are defined [[#Confirmation|immediately below]].

====Corresponding Concise Free-Energy Expression====
Let's also rewrite the [[#FreeEnergyExpression|algebraic free-energy function]] in terms of Stahler's normalized mass and radius variables.  Expressed in terms of the polytropic index, the free-energy function is,
<div align="center">
<math>
\mathfrak{G}^* = 
-3\mathcal{A} \chi^{-1} +~ n\mathcal{B} \chi^{-3/n} +~ \mathcal{D}\chi^3 \, .
</math>
</div>
First, we recognize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi \equiv \frac{R}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{R}{R_\mathrm{SWS}}\biggr) \frac{R_\mathrm{SWS}}{R_\mathrm{norm}} \, .</math>
  </td>
</tr>
</table>
</div>
From the definition of <math>~R_\mathrm{norm}</math> &#8212; reprinted, for example, [[#Detailed_Force-Balanced_Solution|here]] &#8212; we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_\mathrm{SWS}^{n-3} M_\mathrm{SWS}^{1-n} \biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and from the definitions of <math>~R_\mathrm{SWS}</math> and <math>~M_\mathrm{SWS}</math> &#8212; reprinted, for example, [[#Detailed_Force-Balanced_Solution_2|here]] &#8212; we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr]
\biggl( \frac{n+1}{n} \biggr)^{[(n-3) +3(1-n)]/2} G^{[(3-n) + 3(n-1)]/2} K^{n[(n-3)+2(1-n)]/(n+1)} 
</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)^{1-n} \biggl( \frac{n}{n+1} \biggr)^n \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, in each term in the free-energy expression we can make the substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

Next, drawing on the definition of <math>P_\mathrm{norm}</math> &#8212; reprinted, for example, [[#Detailed_Force-Balanced_Solution|here]] &#8212; along with the definition of <math>M_\mathrm{SWS}</math>, we recognize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)} 
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggl( \frac{n+1}{n}\biggr)^3 \biggr\}^{(n+1)/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>

After making these substitutions into the free-energy function, as well as replacing <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> with <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, respectively, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A}_{M_\ell}\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} 
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)}  
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{-1/(n-3)}
</math>
  </td>
</tr>

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A}_{M_\ell} 
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)}  
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)} \biggl( \frac{n+1}{n} \biggr)^n \biggr\}^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~  n\mathcal{B}_{M_\ell}  
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-3(1-n)/[n(n-3)]}  
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)} \biggl( \frac{n+1}{n} \biggr)^{3} \biggr\}^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(5-n)/(n-3)}   
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)}\biggl( \frac{n+1}{n}\biggr)^{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[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)}\biggl( \frac{n+1}{n}\biggr)^{3} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(5-n)} \biggr]^{1/(n-3)} 
\biggl\{
-3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{2} 
+~  n\mathcal{B}_{M_\ell}  
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(n+1)/n}  
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3}   
\biggr\}\, .
</math>
  </td>
</tr>
</table>
</div>

Hence, after defining,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^*_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\mathfrak{G}}{[G^{-3} K^n M_\mathrm{SWS}^{n-5}]^{1/(n-3)}} \biggl( \frac{n}{n+1}\biggr)^{3/(n-3)} 
=
\frac{\mathfrak{G}}{[K^{6n} P_e^{5-n}]^{1/[2(n+1)]}} \biggl( \frac{nG}{n+1}\biggr)^{3/2} \, ,
</math>
  </td>
</tr>
</table>
</div>

we can write,
<div align="center" id="ConciseFreeEnergyExpression">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^*_\mathrm{SWS} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} 
+~ n\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} 
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3}  \, . 
</math>
  </td>
</tr>
</table>
</div>
Setting the first derivative of this function equal to zero should produce the virial theorem expression.  Let's see &hellip;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial\mathfrak{G}^*_\mathrm{SWS}}{\partial \mathcal{X}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-2} 
-~ 3\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-(3+n)/n} 
+~ 4\pi \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 3\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-2}
\biggl[ \mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} 
-~ \mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} 
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{4} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Replacing <math>\mathcal{A}_{M_\ell}</math> and <math>\mathcal{B}_{M_\ell}</math> with <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, as prescribed by their defined relationships, and setting the expression inside the square brackets equal to zero does, indeed, produce the [[#CompactStahlerVirial|above, boxed-in ''viral theorem'' mass-radius relationship]].

====Plotting Concise Mass-Radius Relation====
Our derived, [[#ConciseVirialMR|concise analytic expression for the virial theorem]], namely,
<div align="center">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{(n+1)/n} 
+ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2 = 0 
\, ,
</math>
</div>

is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure.  For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}.  It seems that our derived, analytically prescribable, mass-radius relationship &#8212; which is, in essence, a statement of the scalar virial theorem &#8212; embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983. 
<table border="1" cellpadding="3" align="center" width="70%">
<tr>
  <td align="center" colspan="2">
'''Virial Theorem Mass-Radius Relationships'''
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">
<!-- [[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]] -->
[[File:VirialDeterminedMRsequencesLabeled.png|350px|Virial-Determined MR Sequences]]
  </td>
  <td align="center">
Digital copy of Figure 17 from &hellip;<br />
{{ Stahler83figure }}
  </td>
</tr>
<tr>
  <td align="center">
[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17]]
  </td>
</tr>

<tr>
  <td align="left" colspan="2">
''Left-hand Panel'': &nbsp; As detailed below, the three orange-dashed sequences &#8212; n = 1, n = 3, and isothermal &#8212; are analytically prescribed while the others have been determined via an iterative procedure.  Also as detailed below, a solid-yellow circular marker identifies where along each sequence (n &gt; 1) the model with the largest radius resides; for n = 1, the equilibrium sequence asymptotically approaches the maximum radius, <math>R/R_\mathrm{SWS} = [15/(8\pi)]^{1 / 2}</math>, where the mass climbs to infinity.  The solid-green circular marker identifies where along each sequence (n &ge; 3) the maximum-mass model resides.
  </td>
</tr>
</table>
<!--
<table border="1" align="center" cellpadding="3">
<tr>
  <td align="center" rowspan="2">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
  </td>
  <td align="center">
[[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]]
  </td>
</tr>
<tr>
  <td align="center" bgcolor="white">
[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]]
  </td>
</tr>
</table>
-->

Let's do this again using the mass-radius relation as written explicitly in terms of the normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>.  The relevant, generic nonlinear equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4 
- \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{M}{M_\mathrm{SWS}} \biggr]^{(n+1)/n} 
+ \frac{3}{20\pi} \biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 \, .
</math>
  </td>
</tr>
</table>
</div>

Analytically determined roots:
* <math>~n=1</math>
<div align="center">
<math>~\frac{M}{M_\mathrm{SWS}} = \biggl( \frac{10\pi}{3}\biggr)^{1 / 2} \biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^3 \biggl[ \frac{3\cdot 5}{2^3\pi} - \biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^2
\biggr]^{-1/2} </math> &nbsp; &nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; <math>~0 \le \frac{R}{R_\mathrm{SWS}} \le \biggl(\frac{3\cdot 5}{2^3\pi}\biggr)^{1/2} \, .</math>
</div>
* <math>~n=3</math>
<div align="center">
<math>~\frac{R}{R_\mathrm{SWS}} = \biggl\{ \biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{SWS}}\biggr]^{4/3} 
- \biggl(\frac{1}{5\pi}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggr\}^{1/4} </math> &nbsp; &nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; <math>~0 \le \frac{M}{M_\mathrm{SWS}} \le \biggl(\frac{3^4\cdot 5^3}{2^8\pi}\biggr)^{1/2} \, .</math>
</div>
* <span id="Isothermal">''Isothermal''</span> (explained [[#IsothermalExplained|immediately below]])
<div align="center">
<math>~\frac{M}{M_\mathrm{SWS}} = \frac{5}{2} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr) \biggl\{
1 \pm \biggl[ 1 - \frac{16\pi}{15}\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^2  \biggr]^{1 / 2} \biggr\} </math>
&nbsp; &nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; <math>~0 \le \frac{R}{R_\mathrm{SWS}} \le \biggl(\frac{3\cdot 5}{2^4\pi}\biggr)^{1/2} \, .</math>
</div>


<span id="TabulatedValues">First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.</span>
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr>
  <td align="center" colspan="5">For Various Values of <math>n</math>, Numerically Determined Solutions to the Virial-Equilibrium Relation &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4 
- \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{M}{M_\mathrm{SWS}} \biggr]^{(n+1)/n} 
+ \frac{3}{20\pi} \biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 \, .
</math>
  </td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="center" colspan="1"><math>~n =2</math></td>
  <td align="center" colspan="1"><math>~n =2.8</math></td>
  <td align="center" colspan="1"><math>~n =3.5</math></td>
  <td align="center" colspan="1"><math>~n = 4</math></td>
  <td align="center" colspan="1"><math>~n = 5</math></td>
</tr>
<tr>
<td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr>
  <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.26562</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.477153</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.70919</td>
</tr>
<tr>
  <td align="right">0.5500</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.063602</td>
</tr>
<tr>
  <td align="right">0.5800</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.39755</td>
</tr>
<tr>
  <td align="right">0.5950</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.64662</td>
</tr>
<tr>
  <td align="right">0.6050</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.893915</td>
</tr>
<tr>
  <td align="right">0.6120</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.22372</td>
</tr>
<tr>
  <td align="right"><font color="darkgreen">0.6131721</font></td>
  <td align="center">&nbsp;</td>
  <td align="left"><font color="darkgreen">2.433375</font></td>
</tr>
<tr>
  <td align="right">0.6120</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.64923</td>
</tr>
<tr>
  <td align="right">0.6050</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.01688</td>
</tr>
<tr>
  <td align="right">0.5950</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.32037</td>
</tr>
<tr>
  <td align="right">0.5800</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.658702</td>
</tr>
<tr>
  <td align="right">0.5500</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.19097</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.94599</td>
</tr>
<tr>
  <td align="right">0.4700</td>
  <td align="center">&nbsp;</td>
  <td align="left">5.38791</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">5.69164</td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
</table>
</td>
<td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr>
  <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.266134</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.47971</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.71765</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.881825</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.20977</td>
</tr>
<tr>
  <td align="right">0.5750</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.427183</td>
</tr>
<tr>
  <td align="right">0.5850</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.653232</td>
</tr>
<tr>
  <td align="right">0.5900</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.89304</td>
</tr>
<tr>
  <td align="right"><font color="darkgreen">0.5904492</font></td>
  <td align="center">&nbsp;</td>
  <td align="left"><font color="darkgreen">1.989927</font></td>
</tr>
<tr>
  <td align="right">0.5900</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.086584</td>
</tr>
<tr>
  <td align="right">0.5850</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.32394</td>
</tr>
<tr>
  <td align="right">0.5750</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.54527</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.75612</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.07134</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.460304</td>
</tr>
<tr>
  <td align="right">0.3500</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.75881</td>
</tr>
<tr>
  <td align="right">0.2500</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.97835</td>
</tr>
<tr>
  <td align="right">0.2000</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.09302</td>
</tr>
<tr>
  <td align="right">0.1500</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.232786</td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.430303</td>
</tr>
<tr>
  <td align="right">0.0700</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.60984</td>
</tr>
<tr>
  <td align="right">0.0400</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.9057</td>
</tr>
<tr>
  <td align="right">0.0150</td>
  <td align="center">&nbsp;</td>
  <td align="left">5.47056</td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
</table>
</td>
<td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr>
  <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.26639</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.481072</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.722406</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.89152</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.246123</td>
</tr>
<tr>
  <td align="right">0.5650</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.32113</td>
</tr>
<tr>
  <td align="right">0.5750</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.52651</td>
</tr>
<tr>
  <td align="right">0.5800</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.745165</td>
</tr>
<tr>
  <td align="right"><font color="darkgreen">0.5803836</font></td>
  <td align="center">&nbsp;</td>
  <td align="left"><font color="darkgreen">1.823995</font></td>
</tr>
<tr>
  <td align="right">0.5800</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.90201</td>
</tr>
<tr>
  <td align="right">0.5780</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.01647</td>
</tr>
<tr>
  <td align="right">0.5750</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.11019</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.35906</td>
</tr>
<tr>
  <td align="right">0.5400</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.543602</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.7555746</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.8890287</td>
</tr>
<tr>
  <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.9482952</td>
</tr>
<tr>
  <td align="right" bgcolor="yellow"><font color="black">0.3749583</font></td>
  <td align="center">&nbsp;</td>
  <td align="left" bgcolor="yellow"><font color="black">2.948526</font></td>
</tr>
<tr>
  <td align="right">0.3300</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.93161</td>
</tr>
<tr>
  <td align="right">0.2500</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.829401</td>
</tr>
<tr>
  <td align="right">0.1500</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.578605</td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.380925</td>
</tr>
<tr>
  <td align="right">0.0750</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.2483845</td>
</tr>
<tr>
  <td align="right">0.0400</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.983015</td>
</tr>
<tr>
  <td align="right">0.0200</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.726337</td>
</tr>
<tr>
  <td align="right">0.0100</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.502865</td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
</table>
</td>
<td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.004224</td>
</tr>
<tr>
  <td align="right">0.2000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.034709</td>
</tr>
<tr>
  <td align="right">0.3000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.1230901</td>
</tr>
<tr>
  <td align="right">0.4000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.31735</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.48177</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.72493</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.89686</td>
</tr>
<tr>
  <td align="right">0.5400</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.028495</td>
</tr>
<tr>
  <td align="right">0.5500</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.13574</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.26965</td>
</tr>
<tr>
  <td align="right">0.5730</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.55527</td>
</tr>
<tr>
  <td align="right"><font color="darkgreen">0.5756189</font></td>
  <td align="center">&nbsp;</td>
  <td align="left"><font color="darkgreen">1.750930</font></td>
</tr>
<tr>
  <td align="right">0.5730</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.93949</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.18983</td>
</tr>
<tr>
  <td align="right">0.5400</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.376318</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.46661</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.56895</td>
</tr>
<tr>
  <td align="right">0.4600</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.657809</td>
</tr>
<tr>
  <td align="right" bgcolor="yellow"><font color="black">0.41184646</font></td>
  <td align="center">&nbsp;</td>
  <td align="left" bgcolor="yellow"><font color="black">2.688999</font></td>
</tr>
<tr>
  <td align="right">0.4100</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.68895</td>
</tr>
<tr>
  <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.677703</td>
</tr>
<tr>
  <td align="right">0.3000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.56612</td>
</tr>
<tr>
  <td align="right">0.2500</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.44565</td>
</tr>
<tr>
  <td align="right">0.2000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.28789</td>
</tr>
<tr>
  <td align="right">0.1500</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.08747</td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.82708</td>
</tr>
<tr>
  <td align="right">0.0750</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.660706</td>
</tr>
<tr>
  <td align="right">0.0400</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.3470695</td>
</tr>
<tr>
  <td align="right">0.0200</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.0692</td>
</tr>
<tr>
  <td align="right">0.0100</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.848625</td>
</tr>
</table>
</td>
<td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.004224</td>
</tr>
<tr>
  <td align="right">0.2000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.03471</td>
</tr>
<tr>
  <td align="right">0.3000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.123115</td>
</tr>
<tr>
  <td align="right">0.4000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.31766</td>
</tr>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.48278</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.72866</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.905006</td>
</tr>
<tr>
  <td align="right">0.5400</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.042907</td>
</tr>
<tr>
  <td align="right">0.5500</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.15886</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.313712</td>
</tr>
<tr>
  <td align="right">0.5675</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.511304</td>
</tr>
<tr>
  <td align="right"><font color="darkgreen">0.5692185</font></td>
  <td align="center">&nbsp;</td>
  <td align="left"><font color="darkgreen">1.657839</font></td>
</tr>
<tr>
  <td align="right">0.5675</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.798532</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.97061</td>
</tr>
<tr>
  <td align="right">0.5400</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.17282</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.25888</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.34793</td>
</tr>
<tr>
  <td align="right">0.4600</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.410374</td>
</tr>
<tr>
  <td align="right" bgcolor="yellow"><font color="black">0.4391754</font></td>
  <td align="center">&nbsp;</td>
  <td align="left" bgcolor="yellow"><font color="black">2.417330</font></td>
</tr>
<tr>
  <td align="right">0.4000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.396465</td>
</tr>
<tr>
  <td align="right">0.3000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.19848</td>
</tr>
<tr>
  <td align="right">0.2000</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.84195</td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.31421</td>
</tr>
<tr>
  <td align="right">0.0500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.930314</td>
</tr>
<tr>
  <td align="right">0.0200</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.58847</td>
</tr>
<tr>
  <td align="right">0.0100</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.4161145</td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
</table>
</td>

</tr>

<tr>
  <td align="left" colspan="5">
NOTE: &nbsp; Along each sequence (fixed value of "n"), the coordinates, <math>(R/R_\mathrm{SWS}, M/M_\mathrm{SWS}),</math> of the model with the largest radius are typed in a dark green font; the identified coordinate values not only satisfy the virial-balance equation but also the relation, 
<div align="center">
<math>\biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^4 = \frac{3}{20\pi}\biggl(\frac{n-1}{n}\biggr)\biggl(\frac{M}{M_\mathrm{SWS}} \biggr)^2 \, .</math>
</div>
Similarly, for sequences having n > 3, the coordinates of the model with the maximum mass are highlighted by a yellow background color; the coordinate values not only satisfy the virial-balance equation but also the relation,
<div align="center">
<math>\biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^4 = \frac{1}{20\pi}\biggl(\frac{n-3}{n}\biggr)\biggl(\frac{M}{M_\mathrm{SWS}} \biggr)^2 \, .</math>
</div>
  </td>
</tr>
</table>
</div>

<span id="IsothermalExplained">From a</span> [[SSC/Virial/Isothermal#Bonnor.27s_.281956.29_Equivalent_Relation|free-energy analysis of isothermal spheres]], we have demonstrated that, when the structural form factors are all set to unity, the statement of virial equilibrium is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4 
- \frac{3}{4\pi} \biggr( \frac{M}{M_\mathrm{SWS}} \biggr)  \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)
+ \frac{3}{20\pi} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in order to be consistent with the above polytropic normalizations, we have adopted the ''isothermal'' normalizations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{SWS}\biggr|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1 / 2} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~R_\mathrm{SWS}\biggr|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^4}{G P_e} \biggr)^{1 / 2} \, .</math>
  </td>
</tr>
</table>
</div>
This is a quadratic equation that can be readily solved to provide an analytic expression for the ''isothermal'' mass-radius relation; the relevant expression has already been [[#Isothermal|provided, above]].

====Confirmation====

Rewriting the [[#ConciseVirialMR|just-derived virial theorem expression]] in terms of Stahler's dimensionless radius and mass variables, written in the abbreviated form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{X}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{Y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \, ,</math>
  </td>
</tr>
</table>
</div>
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>~\biggl[ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}} \biggr]^4 
- \biggl[ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}} \biggr]^{(n-3)/n} 
\biggl[ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}} \biggr]^{(n+1)/n} 
+ \biggl[ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}} \biggr]^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{X}^4 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}} \biggr\}^4 
+ \mathcal{Y}^2 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}} \biggr\}^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} 
\biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}}\biggr\}^{(n-3)/n} 
\biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}}  \biggr\}^{(n+1)/n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{X}^4 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} 
\frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\} 
+ \mathcal{Y}^2 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3} \frac{\mathcal{A}_{M_\ell}^{3}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{[(n-3)+2(n+1)]/(n+1)} 
\biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{[(n-3)+3(n+1)]/2n} \frac{\mathcal{A}_{M_\ell}^{[(n-3)+3(n+1)]/2n}}{\mathcal{B}_{M_\ell}^{[(n-3)+2(n+1)]/(n+1)}}\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\mathcal{X}^4 + \biggl[\frac{3(n+1)}{4\pi n}\biggr] \mathcal{A}_{M_\ell} \mathcal{Y}^2 \biggr\} 
\biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{(3n-1)/(n+1)} 
\biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{(3n-1)/(n+1)}}\biggr\} 
</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}\biggr) \mathcal{X}^4 - \mathcal{B}_{M_\ell} \cdot \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} 
+ \biggl(\frac{n+1}{n}\biggr) \mathcal{A}_{M_\ell} \mathcal{Y}^2 \biggr] 
\biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{(3n-1)/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Replacing <math>\mathcal{A}_{M_\ell}</math> and <math>\mathcal{B}_{M_\ell}</math> with <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, as prescribed by their defined relationships, the expression inside the square brackets becomes the [[#CompactStahlerVirial|above, boxed-in mass-radius relationship]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{4\pi}{3} \cdot \mathcal{X}^4  
- \mathcal{B} \cdot \mathcal{X}^{(n-3)/n} \biggl[ \mathcal{Y} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr) \biggr]^{(n+1)/n}  
+~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl[ \mathcal{Y} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr) \biggr]^{2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

====In Terms of Structural Form-Factors====

Alternatively, replacing <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> by their expressions in terms of the structural form factors gives, 

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

<tr>
  <td align="right">
<math>~
\frac{4\pi}{3} \cdot \mathcal{X}^4  
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl( \frac{4\pi}{3} \biggr)^{-1/n}
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} 
+ \mathcal{Y}^2\biggl( \frac{n+1}{5n} \biggr) \frac{\tilde\mathfrak{f}_W  }{\tilde\mathfrak{f}_M^2} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

Finally, inserting into this relation the [[#PTtable|expressions presented above]] for the structural form-factors, <math>\tilde\mathfrak{f}_M</math> and <math>\tilde\mathfrak{f}_A</math>, namely,

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

gives us the desired,
<div align="center" id="ConciseVirialXY">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">
<font color="#770000">'''Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\tilde\mathfrak{f}_W</math>'''</font><br />

<math>~
4\pi \cdot \mathcal{X}^4   ~- ~
\mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} ( 4\pi)^{-1/n}
\biggl[\frac{\tilde\xi}{(-\tilde\theta^')}\biggr]^{(n+1)/n} \biggl[\tilde\theta^{n+1} +  \frac{(n+1)\tilde\xi^2}{3\cdot 5} \cdot \tilde\mathfrak{f}_W \biggr]
~+ ~ \mathcal{Y}^2\biggl( \frac{n+1}{3\cdot 5n} \biggr) \frac{\tilde\xi^2}{(- \tilde\theta^')^2} \cdot \tilde\mathfrak{f}_W 
= 0 \, .
</math>
</td></tr>
</table>
</div>