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

====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]].