Editing SSC/Virial/PolytropesSummary (section)

==Mass-Radius Relation==
Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated polytropes.  In effect, we have viewed the problem through the same lens as did [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and, separately, [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)], defining variable normalizations in terms of the polytropic constant, <math>~K</math>, and the configuration mass, <math>~M_\mathrm{tot}</math>, which were both assumed to be held fixed throughout the analysis.  Here we switch to the approach championed by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)], defining variable normalizations in terms of <math>~K</math> and <math>~P_e</math>, and examining the ''mass-radius'' relationship of pressure-truncated polytropes.

===Detailed Force-Balanced Solution===
As has been summarized in our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying review]] of detailed force-balanced models of pressure-truncated polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:
<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} \tilde\theta^{(n-3)/2} (- \tilde\xi^2 \tilde\theta^') \, ,
</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} \tilde\xi \tilde\theta^{(n-1)/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>M_\mathrm{SWS} \equiv 
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
</div>
<div align="center">
<math>
R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .
</math>
</div>

===Mapping from Above Discussion===
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 [[User:Tohline/SSC/Virial/PolytropesSummary#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 [[User:Tohline/SSC/Virial/PolytropesSummary#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 modified length- and mass-normalizations, <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}^{3/2}}{\mathcal{B}^{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}^{1/2}}{\mathcal{B}^{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{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n} 
+ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2 = 0 
\, .
</math>
</div>
This analytic function is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following table.

<!-- COMMENT OUT ORIGINAL FIGURE until reprint permission received

<table border="1" align="center" cellpadding="3">
<tr>
  <td align="center" rowspan="3">
[[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>

-->

<table border="1" align="center" cellpadding="3" width="80%">
<tr>
  <td align="center" rowspan="3">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
  </td>
<td align="center">
Figure 17 extracted<sup>&dagger;</sup>from p. 184 of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S S. W. Stahler (1983)]<p></p>
"''The Equilibria of Rotating Isothermal Clouds. &nbsp; &nbsp; II. Structure and Dynamical Stability''"<p></p>
ApJ, vol. 268, pp. 165-184 &copy; [http://aas.org/ American Astronomical Society]
</td>
</tr>
<tr>
  <td align="center">
<!-- [[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]] -->
[[Image:AAAwaiting01.png|400px|center|Stahler (1983) Figure 17 (edited)]]
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Figure displayed here, as a digital image, has been modified from the original publication only via the addition of the word "<font color="red">SCHEMATIC</font>".</td></tr>
</table>


Now that we have this very general, yet concise, algebraic expression for the mass-radius relationship of all pressure-truncated polytropes, let's replace the new "mod" normalizations with Stahler's original normalizations, <math>~R_\mathrm{SWS}</math> and <math>~M_\mathrm{SWS}</math>, to understand more completely how this general expression should be viewed in relation to the parametric relations provided by solutions of the detailed force-balanced models.  We will henceforth use the notation,
<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{tot}}{M_\mathrm{SWS}} \, .</math>
  </td>
</tr>
</table>
</div>
In the virial theorem expression we will make the following replacements:
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}^{3/2}}{\mathcal{B}^{2n/(n+1)}} \, . </math>
  </td>
</tr>
</table>
</div>
Next, we recognize that, in order to graphically display the mass-radius relation derived from the virial theorem in the <math>~\mathcal{X}-\mathcal{Y}</math> plane, we must write out the expressions for the free-energy coefficients.  After setting <math>~M_\mathrm{limit}/M_\mathrm{tot} = 1</math> in the [[User:Tohline/SSC/Virial/PolytropesSummary#Structural_Form_Factors|above summary expressions]], we obtain,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</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}  = \frac{1}{5-n} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} =
\frac{1}{3(5-n) ( 4\pi )^{1/n}} \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
  </td>
</tr>
</table>
</div>
In an effort not to be caught dividing by zero while investigating the specific case of <math>~n=5</math> polytropes, we will adopt as shorthand notation,
<div align="center">
<math>\mathfrak{b}_n \equiv \biggl[ (4\pi)^{1/n} (5-n)\mathcal{B}\biggr] 
= \biggl[ (n+1) (\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
</div>
Hence, the replacements become,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} </math>
  </td>
  <td align="center">
<math>~~~ \rightarrow ~~~</math>
  </td>
  <td align="left">
<math>~\mathcal{X} \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} 
(5-n)^{(n-1)/[2(n+1)]} (4\pi)^{1/(n+1)} \mathfrak{b}_n^{-n/(n+1)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{X} (5-n)^{(n-1)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{1/2} (4\pi)^{1/2} \cdot 3^{(1-n)/[2(n+1)]} \cdot
\mathfrak{b}_n^{-n/(n+1)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} </math>
  </td>
  <td align="center">
<math>~~~ \rightarrow ~~~</math>
  </td>
  <td align="left">
<math>~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} 
(5-n)^{(n-3)/[2(n+1)]} (4\pi)^{2/(n+1)} \mathfrak{b}_n^{-2n/(n+1)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{Y} (5-n)^{(n-3)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{3/2} (4\pi)^{1/2} \cdot 3^{(3-n)/[2(n+1)]} \cdot \mathfrak{b}_n^{-2n/(n+1)}  \, ,
</math>
  </td>
</tr>
</table>
</div>
and, in particular, the cross term in the virial theorem expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}}  \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl( \frac{n+1}{n} \biggr)^{2} (4\pi)^{(n-1)/n} 
\cdot \mathfrak{b}_n^{(1-3n)/(n+1)}  \, .
</math>
  </td>
</tr>
</table>
</div>

Via these replacements the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialMR|concise and general Virial Theorem expression]] derived above morphs into the,
<div align="center" id="ConciseVirialXY">
<font color="#770000">'''Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\mathfrak{b}_n</math>'''</font><br />

<math>
k_\xi \biggl\{
\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n}  (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr)
\biggr\}  = 0 \, ,
</math>
</div>
where the leading coefficient is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k_\xi </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ 4\pi
\biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl(\frac{n+1}{n} \biggr)^2 \mathfrak{b}_n^{-4n/(n+1)}  \, .
</math>
  </td>
</tr>
</table>
</div>

In carrying out this last derivation we could be accused of reinventing the wheel, as the expression inside the curly braces is simply <math>~(5-n)</math> times the [[User:Tohline/SSC/Virial/PolytropesSummary#CompactStahlerVirial|virial expression presented inside an outlined box, above]], just before we introduced the modified normalization parameters, <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>.

===Relating and Reconciling Two Mass-Radius Relationships for n = 5 Polytropes===

Now, let's examine the case of pressure-truncated, <math>~n=5</math> polytropes.  As we have discussed in the context of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|detailed force-balanced models]], [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] has deduced that all <math>~n=5</math> equilibrium configurations obey the mass-radius relationship,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
where, as [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|reviewed above]], the mass and radius normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, may be treated as constants once the parameters <math>~K</math> and <math>~P_e</math> are specified.  In contrast to this, the mass-radius relationship that we have just derived ''from the virial theorem'' for pressure-truncated, <math>~n=5</math> polytropes is,
<div align="center">
<math>
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{2/5} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{6/5} 
+ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 = 0 \, ,
</math>
</div>
where the mass and radius normalizations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{mod}\biggr|_{n=5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi} \biggr)^{5/3} \biggl[ \frac{2\cdot 5\pi}{3^2 \mathcal{A}} \biggr]^{3/2} 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~R_\mathrm{mod}\biggr|_{n=5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi}\biggr)^{5/6} \biggl[ \frac{2\cdot 5\pi}{3^2\mathcal{A}} \biggr]^{1/2} \, ,</math>
  </td>
</tr>
</table>

depend, not only on <math>~K</math> and <math>~P_e</math> via the definitions of <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, but also on the structural form factors via the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.  While these two separate mass-radius relationships are similar, they are not identical.  In particular, the middle term involving the cross-product of the mass and radius contains different exponents in the two expressions.  It is not immediately obvious how the two different polynomial expressions can be used to describe the same physical sequence.

This apparent discrepancy is reconciled as follows:  The structural form factors &#8212; and, hence, the free-energy coefficients &#8212; vary from equilibrium configuration to equilibrium configuration.  So it does not make sense to discuss ''evolution along the sequence'' that is defined by the second of the two polynomial expressions.  If you want to know how a given system's equilibrium radius will change ''as its mass changes'', the first of the two polynomials will do the trick.  However, the equilibrium radius of ''a given system'' can be found by looking for extrema in the free-energy function while holding the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, constant; more importantly, the relative stability ''of a given equilibrium system'' can be determined by analyzing the behavior of the system's free energy ''while holding the free-energy coefficients constant''.  Dynamically stable versus dynamically unstable configurations can be readily distinguished from one another along the sequence that is defined by the second polynomial expression; they cannot be readily distinguished from one another along the sequence that is defined by the first polynomial expression.  It is useful, therefore, to determine how to map a configuration's position on one of the sequences to the other. 

====Plotting Stahler's Relation====

[[File:CorrectedStahlerN5.png|thumb|300px|Pressure vs. pressure plot]]Switching, again, to the shorthand notation,
<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{tot}}{M_\mathrm{SWS}} \, ,</math>
  </td>
</tr>
</table>
</div>
the equilibrium mass-radius relation defined by the first of the two polynomial expressions can be plotted straightforwardly in either of two ways.  One way is to recognize that the polynomial is a quadratic equation whose solution is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{Y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2} \mathcal{X} \biggl\{ 1 \pm \biggl[ 1 - \biggl( \frac{2^4\cdot \pi}{3\cdot 5} \biggr) \mathcal{X}^2 \biggr]^{1/2} \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>
In the figure shown here on the right &#8212; see also the bottom panel of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler1983Fig17|Figure 2 in our accompanying discussion of detailed force-balance models]] &#8212; Stahler's mass-radius relation has been plotted using the solution to this quadratic equation; the green segment of the displayed curve was derived from the ''positive'' root while the segment derived from the ''negative'' root is shown in orange.  The two curve segments meet at the maximum value of the normalized equilibrium radius, namely, at
<div align="center">
<math>\mathcal{X}_\mathrm{max} \equiv \biggl[ \frac{3\cdot 5}{2^4 \pi} \biggr]^{1/2} \approx 0.54627 \, .</math>
</div>
We note that, when <math>~\mathcal{X} = \mathcal{X}_\mathrm{max}</math>, <math>~\mathcal{Y} = (5\mathcal{X}_\mathrm{max}/2) \approx 1.36569</math>.  Along the entire sequence, the maximum value of <math>~\mathcal{Y}</math> occurs at the location where <math>~d\mathcal{Y}/d\mathcal{X} = 0</math> along the segment of the curve corresponding to the ''positive'' root.  This occurs along the upper segment of the curve where <math>~\mathcal{X}/\mathcal{X}_\mathrm{max} = \sqrt{3}/2</math>, at the location,
<div align="center">
<math>\mathcal{Y}_\mathrm{max} \equiv \biggl[ \frac{3^3 \cdot 5^2}{2^6 } \biggr]^{1/2} \mathcal{X}_\mathrm{max} 
= \biggl[ \frac{3^4 \cdot 5^3}{2^{10} \pi } \biggr]^{1/2}  \approx 1.77408 \, .</math>
</div>


The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relations.  Drawing partly from our [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|above discussion]] and partly from a separate discussion where we provide a [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of the properties of pressure-truncated <math>~n=5</math> polytropes]], these are,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\mathcal{X}\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{2} =
\biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\tilde\xi^2/3}{(1+\tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\mathcal{Y}\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \tilde\theta (- \tilde\xi^2 \tilde\theta^') =
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\tilde\xi^2/3)^3}{(1+\tilde\xi^2/3)^{4}} \biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>
The entire sequence will be traversed by varying the Lane-Emden parameter, <math>~\tilde\xi</math>, from zero to infinity.  Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is,
<div align="center">
<math>
~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = \frac{1}{5^{1/2}} \biggl[ 2^5\pi - 15 + 2^3\pi^{1/2}(2^4\pi-15)^{1/2} \biggr]^{1/2} 
\approx 5.8264 \, .
</math>
</div>
Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,
<div align="center">
<math>
~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3 \, .
</math>
</div>

====Plotting the Virial Theorem Relation====

The relevant relation is obtained by plugging <math>~n = 5</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation derived above]], repeated here for clarity:

<table border="1" cellpadding="8" align="center">
<tr><td>

<div align="center">
<math>
\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n}  (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr) 
= 0 
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_n = \biggl[ (n+1) (-\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} 
</math>
</div>

</td></tr>
</table>
We will begin by plugging <math>~n = 5</math> into these expressions everywhere except for the coefficient <math>~(5-n)</math>, which we will leave unresolved, for the time being, in order to better appreciate the interplay of various terms.  We obtain,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~
\frac{6}{5} \mathcal{Y}^2 - \biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(n-5)\biggl[ \biggl(\frac{4\pi}{3} \biggr) \mathcal{X}^4 -  \biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, if they are to be assigned values that are actually associated with a particular detailed force-balance model having truncation radius, <math>~\tilde\xi</math>,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~
\mathfrak{b}_I^5  
</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ (n+1) (-\tilde\theta^')^2 \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5} 
= 2^5\cdot 3^5 \biggl[ (-\tilde\theta^')^4 {\tilde\xi}^6 \biggr]  
= 2^5 \cdot 3 \cdot \tilde\xi^{10} (1+\tilde\xi^2/3)^{-6}\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\mathfrak{b}_{II}^5
</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{3} ~ \tilde\theta^{n+1}  \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5}
= \frac{1}{3^5} ~ \tilde\theta^{30}  \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{6} 
= 3(1+\tilde\xi^2/3)^{-6} \, .
</math>
  </td>
</tr>
</table>
</div>

Now, if we plug <math>~n=5</math> into the remaining unresolved <math>~(n-5)</math> coefficient, the righthand side goes to zero and the mass-radius relationship provided by the virial theorem becomes,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{6}{5}  \mathcal{Y}^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y}^4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\mathcal{X}^{2} \biggl[ \biggl( \frac{5^5}{2^5\cdot 3^5}\biggr) \frac{\mathfrak{b}^5_{I}}{4\pi} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\mathcal{X}^{1/2} \biggl( \frac{5^5 \mathfrak{b}^5_{I}}{2^7\cdot 3^5 \pi}\biggr)^{1/4} \, .  </math>
  </td>
</tr>
</table>
</div>
Hence, for a given value of the structural form factor(s) &#8212; which implies a specific value of the constant coefficient, <math>~\mathfrak{b}_I</math> &#8212; the scalar virial theorem defines a relationship where the normalized mass <math>~(\mathcal{Y})</math> varies as the square root of the normalized radius <math>~(\mathcal{X})</math>.  On the other hand, if we demand that the expression inside the square brackets on the righthand side of the virial theorem relation go to zero on its own &#8212; without relying on the leading coefficient to knock it zero &#8212; the mass-radius relationship provided by the virial theorem becomes,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{4\pi}{3} ~ \mathcal{X}^4  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y}^6 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\mathcal{X}^{18} \biggl[ \biggl( \frac{4\pi}{3}\biggr)^5 \frac{4\pi}{\mathfrak{b}^5_{II}} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>4\pi \mathcal{X}^{3} ( 3^5\mathfrak{b}^5_{II} )^{-1/6} \, .  </math>
  </td>
</tr>
</table>
</div>
<!-- COMMENT 1 .......

In order to graphically display the mass-radius relation derived from the virial theorem in the <math>~\mathcal{X}-\mathcal{Y}</math> plane, as desired, we must first write out the expressions for the free-energy coefficients.  After setting <math>~M_\mathrm{limit}/M_\mathrm{tot} = 1</math> in the [[User:Tohline/SSC/Virial/PolytropesSummary#Structural_Form_Factors|above summary expressions]], we obtain for all polytropic indexes,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</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}  = \frac{1}{5-n} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} =
\frac{1}{3(5-n) ( 4\pi )^{1/n}} \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
  </td>
</tr>
</table>
</div>
In an effort not to be caught dividing by zero while investigating the specific case of <math>~n=5</math> polytropes, we will use as shorthand notation,
<div align="center">
<math>\mathfrak{b}_5 \equiv \biggl[ (5-n)\mathcal{B}\biggr]_{n=5} 
= \biggl[ \frac{2^3\cdot 3^5}{ \pi} \cdot (-\tilde\theta^')^{4} \tilde\xi^6 \biggr]^{1/5}
= \biggl[ \frac{2^3 \cdot 3}{\pi} \biggl(1 + \frac{\tilde\xi^2}{3} \biggr)^{-6} \tilde\xi^{10} \biggr]^{1/5}
\, ,</math>
</div>
where we have inserted the definition of <math>~\tilde\theta^'</math> as provided for <math>~n=5</math> polytropic structures in another section's [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|summary table]].  For pressure-truncated <math>~n=5</math> polytropes, we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} = \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\mathcal{X} \cdot \biggl( \frac{4\pi}{3\mathcal{B}}\biggr)^{5/6} \biggl[ \frac{3^2\mathcal{A}}{2\cdot 5\pi} \biggr]^{1/2}
= (5-n)^{1/3} \mathcal{X} \cdot \biggl( \frac{4\pi}{3\mathfrak{b}_5}\biggr)^{5/6} \biggl[ \frac{3^2}{2\cdot 5\pi} \biggr]^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
(5-n)^{1/3} \mathcal{X} \cdot \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{1/6} \mathfrak{b}_5^{-5/6} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} = \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~ \mathcal{Y} \cdot \biggl( \frac{4\pi}{3\mathcal{B}} \biggr)^{5/3} \biggl[ \frac{3^2 \mathcal{A}}{2\cdot 5\pi} \biggr]^{3/2}
= (5-n)^{1/6}\mathcal{Y} \cdot \biggl( \frac{4\pi}{3\mathfrak{b}_5} \biggr)^{5/3} \biggl[ \frac{3^2}{2\cdot 5\pi} \biggr]^{3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
(5-n)^{1/6}\mathcal{Y} \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/6}  \mathfrak{b}_5^{-5/3} \, .
</math>
  </td>
</tr>
</table>
</div>

So, the cross term that appears in the mass-radius relation obtained from the virial theorem may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{R_\mathrm{eq}}{R_\mathrm{mod}}\biggr)^{2/5}\biggl(\frac{M_\mathrm{tot}}{M_\mathrm{mod}}\biggr)^{6/5} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\biggl[  
(5-n)^{1/3} \mathcal{X} \cdot \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{1/6} \mathfrak{b}_5^{-5/6} 
\biggr]^2 \times \biggl[
(5-n)^{1/6}\mathcal{Y} \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/6}  \mathfrak{b}_5^{-5/3} 
\biggr]^6 \biggr\}^{1/5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5-n)^{1/3} \mathcal{X}^{2/5} \mathcal{Y}^{6/5}
\biggl[ \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{1/3} \mathfrak{b}_5^{-5/3} \cdot
\biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)  \mathfrak{b}_5^{-10}
\biggr]^{1/5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5-n)^{1/3} \mathcal{X}^{2/5} \mathcal{Y}^{6/5}
\biggl( \frac{2^{8}\cdot 3^{5} \pi}{5^{6}} \biggr)^{1/3} 
\mathfrak{b}_5^{-7/3} \, ,
</math>
  </td>
</tr>
</table>
</div>
and, in entirety, the virial theorem relation becomes,
<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>~
(5-n)^{1/3}\biggl\{ \mathcal{Y}^2 \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/3}  \mathfrak{b}_5^{-10/3}
- \mathcal{X}^{2/5} \mathcal{Y}^{6/5} \biggl( \frac{2^{8}\cdot 3^{5} \pi}{5^{6}} \biggr)^{1/3} \mathfrak{b}_5^{-7/3}
+ (5-n) \mathcal{X}^4 \cdot \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{2/3} \mathfrak{b}_5^{-10/3} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

A nontrivial solution is obtained by requiring that the terms inside the curly braces sum to zero.  Noting that the third term must be set to zero, on its own, because it retains a leading factor of <math>~(5-n)</math>, the virial theorem relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{Y}^2 \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/3}  \mathfrak{b}_5^{-10/3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{X}^{2/5} \mathcal{Y}^{6/5} \biggl( \frac{2^{8}\cdot 3^{5} \pi}{5^{6}} \biggr)^{1/3} \mathfrak{b}_5^{-7/3}
</math>
  </td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\mathcal{Y}^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{X} \biggl( \frac{5\mathfrak{b}_5}{2 \cdot 3}  \biggr)^{5/2} 
= \mathcal{X} \biggl( \frac{5^5}{2^2 \cdot 3^4 \pi} \biggr)^{1/2} \biggl(1 + \frac{\tilde\xi^2}{3} \biggr)^{-3} \tilde\xi^{5}  \, .
</math>
  </td>
</tr>
</table>
</div>

.......   END COMMENT -->
<!-- COMMENT 2 ........
Let's go back to the [[User:Tohline/SSC/Virial/PolytropesSummary#CompactStahlerVirial|earlier virial expression]] that still contains the <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> coefficients and multiply through by <math>~5</math> so that the coefficient of the <math>~\mathcal{X}^4</math> term matches the coefficient found in Stahler's relation.  We have,

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

<tr>
  <td align="right">
<math>~
\biggl(\frac{2^2\cdot 5\pi}{3} \biggr) \mathcal{X}^4  - 
5\mathcal{B} \mathcal{X}^{2/5} \mathcal{Y}^{6/5}  
+ 6\mathcal{A} \mathcal{Y}^{2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .
</math>
  </td>
</tr>
</table>
</div>
Next, let's rewrite the other two terms so that they look more like the terms found in Stahler's expression.

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

<tr>
  <td align="right">
<math>~
\biggl(\frac{2^2\cdot 5\pi}{3} \biggr) \mathcal{X}^4  - 
5\mathcal{X}\mathcal{Y} \biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}
+ 6\mathcal{A} \mathcal{Y}^{2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl(\frac{2^2\cdot 5\pi}{3} \biggr) \mathcal{X}^4  - 
5\mathcal{X}\mathcal{Y} 
+ \mathcal{Y}^{2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5\mathcal{X}\mathcal{Y} \biggl[\biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}-1\biggr]
+(1- 6\mathcal{A} )\mathcal{Y}^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \mathcal{Y} \biggl\{
5\mathcal{X}\biggl[\biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}-1\biggr]
+(1- 6\mathcal{A} )\mathcal{Y} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \mathcal{Y} \biggl\{
5 \mathcal{B} (\mathcal{Y} \mathcal{X}^2 )^{1/5} - 5 \mathcal{X} 
+(1- 6\mathcal{A} )\mathcal{Y} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Now, according to Stahler's relation, the lefthand side of our derived expression should be zero when the chosen <math>~(\mathcal{X}, \mathcal{Y})</math> pair identifies an equilibrium configuration.  Therefore, the terms inside the curly brackets on the righthand side of our derived expression should also sum to zero in equilibrium.  Let's see if, indeed, this is the case; as shorthand, we will use,
<div align="center">
<math>\Lambda \equiv \frac{\tilde{\xi}^2}{3} \, .</math>
</div>
From the [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of detailed force-balanced models]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tilde\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{(1+\Lambda)^{1/2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~-~\tilde\theta'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{3^{1/2}} \cdot
\frac{\Lambda^{1/2}}{(1+\Lambda)^{3/2}}  
~~~~\Rightarrow~~~~ \biggl( \frac{\tilde{\xi}}{-\tilde\theta'} \biggr) = 3(1+\Lambda)^{3/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{X}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \frac{3\cdot 5}{2^2\pi} \biggl[ 
\frac{\Lambda}{(1+\Lambda)^2} \biggr] \biggr\}^{1/2}
=
\biggl( \frac{3\cdot 5}{2^2\pi} \biggr)^{1/2} 
\frac{\Lambda^{1/2}}{(1+\Lambda)}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{Y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \frac{3\cdot 5^3}{2^2\pi} \biggl[ 
\frac{\Lambda^3}{(1+\Lambda)^4} \biggr] \biggr\}^{1/2}
=
5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{1/2} 
\frac{\Lambda^{3/2}}{(1+\Lambda)^2}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \mathcal{Y}\mathcal{X}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{3/2} 
\frac{\Lambda^{5/2}}{(1+\Lambda)^4} \, ;
</math>
  </td>
</tr>
</table>
</div>
and, from the general definitions given, above,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{1}{5-n} \, ,</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
( 4\pi )^{-1/5} \biggl[ \frac{2}{(5-n)} \frac{\Lambda}{(1+\Lambda)^3} + \frac{1}{3}\cdot \frac{1}{(1+\Lambda)^3} \biggr]
3^{6/5} (1+\Lambda)^{9/5} 
</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)^{1/5} \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr]
(1+\Lambda)^{-6/5} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~5\mathcal{B} (\mathcal{Y}\mathcal{X}^2)^{1/5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5
\biggl(\frac{3}{4\pi }\biggr)^{1/5} \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr](1+\Lambda)^{-6/5}
\cdot
\biggl\{ 5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{3/2} 
\frac{\Lambda^{5/2}}{(1+\Lambda)^4} \biggr\}^{1/5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3\cdot 5^3}{4\pi }\biggr)^{1/2}  \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr]\frac{\Lambda^{1/2}}{(1+\Lambda)^2} \, ;
</math>
  </td>
</tr>
</table>
</div>
while,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(1-6\mathcal{A})\mathcal{Y} - 5\mathcal{X}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1- \frac{2\cdot 3}{(5-n)} \biggr]\biggl(\frac{3\cdot 5^3}{2^2\pi}\biggr)^{1/2} 
\frac{\Lambda^{3/2}}{(1+\Lambda)^2} 
- \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} 
\frac{\Lambda^{1/2}}{(1+\Lambda)} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl(\frac{3\cdot 5^3}{4\pi }\biggr)^{1/2}  \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr]\frac{\Lambda^{1/2}}{(1+\Lambda)^2} \, .
</math>
  </td>
</tr>
</table>
</div>
So we have the desired result, namely, that these last two expressions have opposite signs but are otherwise identical and, hence, they sum to zero.
........  END COMMENT 2 -->

===Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes===

For pressure-truncated <math>~n=4</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurations.  However, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\mathcal{X}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{3/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\mathcal{Y}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \tilde\theta^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
</math>
  </td>
</tr>
</table>
</div>
On the other hand, the polynomial that results from plugging <math>~n=4</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,

<div align="center">
<math>
\frac{4\pi}{3} \mathcal{X}^4   - \biggl[ \frac{\mathcal{X} \mathcal{Y}^{5}}{4\pi}\biggr]^{1/4} \mathfrak{b}_{n=4} + \frac{5}{4} \mathcal{Y}^2 
= 0 \, ,
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_{n=4} = \biggl[ 5 (-\tilde\theta^')^2 + \frac{1}{3} \tilde\theta^{5} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{5/4} \, .
</math>
</div>

[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unity.  Its value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>.  A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[User:Tohline/SSC/Virial/PolytropesSummary#Summary|above parameter summary figure]].]

In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations.  The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels.  In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequence.  The "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>. 

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Comparing Two Separate Mass-Radius Relations for Pressure-Truncated ''n = 4'' Polytropes
  </th>
</tr>
<tr><td align="center">
[[File:CompareN4SequencesRevised.png|750px|Comparison of Two Mass-Radius Relations]]
</td></tr>
</table>

According to [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt's (1986)]] tabulated data, the surface of an isolated <math>~(P_e = 0)</math>, spherically symmetric, <math>~n=4</math> polytrope occurs at the dimensionless (Lane-Emden) radius, <math>~\xi_1 = 14.9715463</math>.  In both panels of the above figure, this ''isolated'' configuration is identified by the discrete (blue diamond) point at the origin, that is, at <math>~(\mathcal{X}, \mathcal{Y}) = (0, 0)</math>.   As we begin to examine pressure-truncated models and <math>~\tilde\xi</math> is steadily decreased from <math>~\xi_1</math>, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation.  A maximum mass of <math>~\mathcal{Y} \approx 2.042</math> (corresponding to a radius of <math>~\mathcal{X} \approx 0.4585</math>) is reached ''from the left'' as <math>~\tilde\xi</math> drops to a value of approximately <math>~3.4</math>.  As <math>~\tilde\xi</math> continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554)</math> &#8212; corresponding to <math>~\tilde\xi \approx 2.0</math> &#8212; then decreasing in radius until, once again, the origin is reached, but this time because <math>~\tilde\xi</math> drops to zero.


If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>~\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure.  While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect.  They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>.    In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> &#8212; again, emphasized by a black filled circle &#8212; that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>.  Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.


When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of <math>~\mathfrak{b}_{n=4} = 4.8926</math> (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where <math>~\tilde\xi = 2.8</math>, for the reason just discussed.  After constructing the plot, it became clear that the two curves also intersect at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67)</math> &#8212; also highlighted by a black filled circle &#8212; that corresponds to a detailed force-balanced model having <math>~\tilde\xi \approx 6.0</math>.  This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, <math>~\tilde\xi</math>, that assures precise agreement between the two different mass-radius expressions.


As is detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Stability|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable.  We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs.  As has been detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>\eta_\mathrm{crit}\biggr|_{n=4}~=~\frac{1}{15} \, ;</math>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\Pi_\mathrm{max}\biggr|_{n=4}~=~\frac{15^{15}}{16^{16}} \, ;</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
<math>\Chi_\mathrm{min}\biggr|_{n=4}~=~\biggl( \frac{16}{15} \biggr)^4 \, .</math>
  </td>
</tr>
</table>
</div>
In the context of the above figure, independent of the chosen value of <math>~\mathfrak{b}_{n=4}</math>, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem.  In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of <math>~\mathcal{Y}</math>; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of <math>~\mathcal{X}</math>.  We have deduced (details of the derivation not shown) that, for pressure-truncated <math>~n=4</math> polytropes, the coordinates of this critical point in Stahler's <math>~\mathcal{X}-\mathcal{Y}</math> plane depends on the choice of <math>~\mathfrak{b}_{n=4}</math> as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{X}_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi^{-1/2} 2^{-16/5} (3\mathfrak{b}_{n=4})^{4/5} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{Y}_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi^{-1/2} 2^{-22/5} (3\mathfrak{b}_{n=4})^{8/5} \, .</math>
  </td>
</tr>
</table>
</div>

In practice, for a given plot of the type displayed in the above figure &#8212; that is, for a given choice of the structural parameter, <math>\mathfrak{b}_{n=4}</math> &#8212; it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations.  If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable.  On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable.  We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is ''stable''; however, both configurations identified by filled black circles in the righthand panel are ''unstable''.

It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence.  One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable.  But the righthand panel of our above figure contradicts this expectation.  While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies ''to the right'' of the maximum-mass point along the blue "Stahler" sequence.  This finding is related to [[User:Tohline/SSC/Virial/PolytropesSummary#Curiosity|the curiosity raised earlier]] in our discussion of the structural properties of pressure-truncated, <math>~n=4</math> polytropes.

===Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes===

For pressure-truncated <math>~n=3</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurations.  However, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\mathcal{X}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\mathcal{Y}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3^3}{4\pi} \biggr)^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
</math>
  </td>
</tr>
</table>
</div>
On the other hand, the polynomial that results from plugging <math>~n=3</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,

<div align="center">
<math>
\frac{2^3 \pi}{3} \mathcal{X}^4   - \biggl[ \frac{\mathcal{Y}^{4}}{4\pi}\biggr]^{1/3} \mathfrak{b}_{n=3} + \frac{4}{3} \mathcal{Y}^2 
= 0 \, ,
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_{n=3} = \biggl[ 4 (-\tilde\theta^')^2 + \frac{2}{3} \tilde\theta^{4} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{4/3} \, .
</math>
</div>