Editing SSC/Structure/Polytropes/VirialSummary (section)

=Virial Equilibrium of Pressure-Truncated Polytropes=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#MoreModels|<b>Virial Equilibrium<br />of<br />Pressure-Truncated<br />Polytropes</b>]]</font>
|}

Here we will draw heavily from: [1] An accompanying [[SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes|''Free Energy Synopsis'']]; and [2] A detailed analysis of the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation|Virial Equilibrium of Adiabatic Spheres]].

&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />

==Groundwork==
===Basic Relation===
In the context of spherically symmetric, pressure-truncated polytropic configurations, the relevant free-energy expression is,

<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>~W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 3\mathcal{A} \biggl[\frac{GM^2}{R} \biggr] + n\mathcal{B} \biggl[ \frac{K_nM^{(n+1)/n}}{R^{3/n}} \biggr] + \frac{4\pi}{3} \cdot P_e R^3 \, ,</math>
  </td>
</tr>
</table>

where, when written in terms of the trio of ''[[SSC/FreeEnergy/Powerpoint#Structural_Form_Factors|structural form factors]]'',  <math>\tilde{\mathfrak{f}}_A,</math> <math>\tilde{\mathfrak{f}}_M,</math> and <math>\tilde{\mathfrak{f}}_W,</math> the pair of constants, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A} \equiv \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>\mathcal{B} \equiv \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, .</math>
  </td>
</tr>
</table>
</div>

===Often-Referenced Dimensionless Expressions===
When rewritten in a suitably dimensionless form &#8212; see two useful alternatives, below &#8212; this expression becomes,
<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>~- a x^{-1} + bx^{-3/n} + c x^3  \, ,</math>
  </td>
</tr>
</table>
where <math>~x</math> is the configuration's dimensionless radius and <math>~a</math>, <math>~b</math>, and  <math>~c</math> are constants.  We therefore have,

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

<tr>
  <td align="right">
<math>~\frac{d\mathfrak{G}^*}{dx}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x^2} \biggl[ a  - \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 3c x^4 \biggr]  \, ,</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right">
<math>~\frac{d^2\mathfrak{G}^*}{dx^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x^3} \biggl[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a  \biggr]  \, .</math>
  </td>
</tr>
</table>

Virial equilibrium is obtained when <math>~d\mathfrak{G}^*/dx = 0</math>, that is, when

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

<tr>
  <td align="right">
<math>~\biggl( \frac{3b}{n} \biggr) x_\mathrm{eq}^{(n-3)/n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ a  + 3c x_\mathrm{eq}^4 \, .</math>
  </td>
</tr>
</table>
And along an equilibrium ''sequence'', the ''specific'' equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations &#8212; henceforth labeled as having the ''critical'' radius, <math>~x_\mathrm{crit}</math> &#8212; is identified by setting  <math>~d^2\mathfrak{G}^*/dx^2 = 0</math>, that is, it is the configuration for which,
<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[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a  \biggr]_{x = x_\mathrm{eq}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
x_\mathrm{crit}^4 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a}{3^2c}\biggl(\frac{n - 3}{n+1}\biggr) \, .
</math>
  </td>
</tr>
</table>

Inserting the adiabatic exponent in place of the polytropic index via the relation, <math>~n = (\gamma - 1)^{-1}</math>, we have equivalently,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
x_\mathrm{crit}^4 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a}{3^2c}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \, .
</math>
  </td>
</tr>
</table>

===Useful Recognition===

By comparing various terms in the first two algebraic ''Setup'' expressions, above, It is clear that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W^*_\mathrm{grav} = -ax^{-1}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~U^*_\mathrm{int} = bx^{-3/n} \, .</math>
  </td>
</tr>
</table>

Notice, then, that in every equilibrium configuration, we should find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{b}{a}\biggr) x_\mathrm{eq}^{(n-3)/n}
=
\frac{n}{3a} \biggl[ a + 3cx^4_\mathrm{eq} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{n}{3} \biggl[ 1 + \biggl(\frac{3c}{a}\biggr) x^4_\mathrm{eq} \biggr] \, .
</math>
  </td>
</tr>
</table>
And, specifically in the ''critical'' configuration we should find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{3(\gamma-1)} \biggl[ 1 + \frac{1}{3}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \biggr] 
=
\frac{4}{3^2\gamma(\gamma-1)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{S^*_\mathrm{therm}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{2}{3\gamma}  \, .
</math>
  </td>
</tr>
</table>
The equivalent of this last expression also appears at the end of subsection <b><font color="maroon" size="+1">&#x2466;</font></b> of an [[SSC/SynopsisStyleSheet#Stability|accompanying ''Tabular Overview'']].

==Equilibrium Sequences==
In all of the polytropic configurations being considered here, {{ Math/MP_PolytropicConstant }} is a constant &#8212; that is, the specific entropy of all fluid elements is assumed to be the same, both spatially and temporally.

===Fix Mass While Varying External Pressure===

In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>R</math> and <math>P_e</math> to vary while holding <math>M</math> fixed.  In our [[SSC/FreeEnergy/PolytropesEmbedded#Case_M|accompanying, more detailed discussion]], this is referred to as '''Case M'''.  Adopting the normalizations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{G}{K_n} \biggr)^n M^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{K_n^{4n}}{G^{3(n+1)} M^{2(n+1)}} \biggr]^{1/(n-3)}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
P_\mathrm{norm} R^3_\mathrm{norm} \, ,
</math>
  </td>
</tr>
</table>
&#8212; which, as is detailed in an [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|accompanying discussion]], are similar but not identical to the normalizations adopted by {{ Horedt70full }} and by {{ Whitworth81full }} &#8212; the coefficients in the above-presented [[#Basic_Relations|Basic Relations]] become,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~3\mathcal{A} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~n\mathcal{B}   
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, .
</math>
  </td>
</tr>
</table>

The relevant dimensionless free-energy surface is, then, given by the expression,
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="1">''Case M'' Free-Energy Surface</th>
</tr>
<tr><td align="center">

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

</td></tr>
</table>

===Fix External Pressure While Varying Mass===

In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>~R</math> and <math>~M</math> to vary while holding <math>~P_e</math> fixed.  In our [[SSC/FreeEnergy/PolytropesEmbedded#Case_P|accompanying, more detailed discussion]], this is referred to as '''Case P'''.  Motivated by the published work of {{ Stahler83full }}, here we adopt the normalizations, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp;&nbsp;</td>
  <td align="right">
<math>~M_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} </math>
  </td>
  <td align="center">&nbsp; &nbsp;&nbsp; and,</td>
</tr>
<tr>
  <td align="right">
<math>~E_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left" colspan="6">
<math>~
\biggl( \frac{n}{n+1} \biggr) \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}}
=
\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .</math>
  </td>
</tr>

</table>
the coefficients in the above-presented [[#Basic_Relations|Basic Relations]] become,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~n\mathcal{B} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n}   
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \, ,
</math>
  </td>
</tr>
</table>

where the pair of constants, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, have the same definitions in terms of the ''[[SSC/FreeEnergy/Powerpoint#Structural_Form_Factors|structural form factors]]'' as provided above. The relevant dimensionless free-energy surface is, then, given by the expression,
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="1">''Case P'': &nbsp; Free-Energy Surfaces</th>
</tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\mathfrak{G}_{K,P_e}^* \equiv \frac{\mathfrak{G}_{K,P_e}}{E_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
+ n\mathcal{B} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} 
+ \frac{4\pi}{3} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \, .
</math>
  </td>
</tr>
</table>

</td></tr>
</table>

Across this '''Case P''' free-energy surface, extrema &#8212; and, hence, equilibrium configurations &#8212; will arise wherever the virial-equilibrium condition is met, that is, wherever

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="1">''Case P'': &nbsp; Virial Equilibrium Sequences</th>
</tr>
<tr><td 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\mathcal{B}}{4\pi} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n}   \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} 
+ \frac{3 \mathcal{A}}{4\pi} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
  </td>
</tr>
</table>
</td></tr>
</table>
An equilibrium model ''sequence'' is thereby defined for each chosen polytropic index in the range, <math>0 \le n < \infty</math>.

Along each equilibrium sequence for which <math>3 \le n < \infty</math>, there is one ''specific'' equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations.  For a given mass, the radius of this ''critical'' configuration is identified by the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mathcal{A}}{4\pi}\biggl(\frac{n - 3}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>

----

Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>  Here, in an effort to provide a broad overview, we set,
<div align="center">
<math>\mathcal{A} = \frac{4\pi n}{3(n+1)}</math> &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;  <math>\mathcal{B} = \biggl( \frac{4\pi}{3} \biggr) \, ,</math>
</div>
and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes.  Specifically, each curve in the left-hand panel of Figure 1 results from the mass-radius expression, 

<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( \frac{M}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}   
+ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
  </td>
</tr>
</table>

<span  id="StahlerSchematic">(As has been demonstrated</span> in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations &#8212; namely, <math>M_\mathrm{mod}</math> and <math>R_\mathrm{mod}</math> &#8212; instead of choosing these specific values of <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>)

<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="2">Figure 1: &nbsp; Virial-Analysis-Based Equilibrium Sequences</th>
</tr>
<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>

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.


----

Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>  Here, in an effort to provide a broad overview, we set all three structural form factors to unity, in which case the pair of coefficients,
<div align="center">
<math>\mathcal{A} = \frac{1}{5}</math> &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;  <math>\mathcal{B} = \biggl( \frac{3}{4\pi} \biggr)^{1/n}</math>,
</div>
and the resulting mass-radius relation of equilibrium sequences is governed by the algebraic mass-radius relation,

<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>
The left-hand panel of Figure 1 displays this behavior for equilibrium sequences having seven different values of the polytropic index, as labeled.  As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation|accompanying discussion]], this governing mass-radius relation can be analytically manipulated into a form that provides either an explicit <math>M(R)</math> or <math>R(M)</math> relation for the cases labeled, <math>n = 1, 3,</math> and <math>\infty</math>; a generic root-finding technique has been used to generate [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#TabulatedValues|points along each of the other depicted equilibrium sequences]].

<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="2">Figure 1: &nbsp; Virial-Analysis-Based Equilibrium Sequences</th>
</tr>
<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>

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.