Editing SSC/Virial/PolytropesEmbeddedOutline (section)

===Whitworth's (1981) Case P Analysis of Uniform-Density Configurations===

====Coefficient Definitions====

The {{ Whitworth81 }} <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function &#8212; referred to by Whitworth as the "global potential function":

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation copied without modification from p. 973 of<br />
{{ Whitworth81figure }}<br />
&copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
<!-- [[File:Whitworth1981Eq10.jpg|750px|center|Whitworth (1981, MNRAS, 195, 967)]] -->

<table border="0" align="center" width="100%" cellpadding="5">
<tr>
  <td align="center">
<math>
\frac{2\mathcal{U}}{3M_0 K_1} = 
-\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} 
+ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 
+ (1 - \delta_{1\eta})\frac{2}{3(\eta - 1)}\biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{3(1-\eta)}
- \delta_{1\eta} 2 \ln \biggl(\frac{R}{R_\mathrm{rf}}\biggr) 
</math>
  </td>
  <td align="right" width="8%">(10)</td>
</tr>
</table>

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

After setting <math>\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>~\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>~(\eta)</math> in terms of the corresponding polytropic index, Whitworth's free-energy expression becomes,  
<div align="center" id="WhitworthFreeEnergyExpression">
<math>
\frac{2\mathcal{U}}{3M_0 K_1} = 
-\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} +~ \frac{2n}{3}\biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-3/n} 
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 \, .
</math>
</div>
This expression is identical to the [[SSC/Virial/PolytropesEmbeddedOutline#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made: 

<div align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
{{ Whitworth81 }} <font color="red">Case P</font> Analysis
</td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{3}{2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{2n}{3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{R}{R_\mathrm{rf}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{G}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{2}{3} \biggl( \frac{\mathcal{U}}{\mathcal{U}_\mathrm{rf}} \biggr) </math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>
where (see an [[SSC/Structure/PolytropesASIDE1#ASIDE:_Whitworth.27s_Scaling|accompanying ASIDE]]),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_\mathrm{rf}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{rf}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} 
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{U}_\mathrm{rf} \equiv (M_0K_1)_\mathrm{Whitworth}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr) K^n G^{-3} M_\mathrm{limit}^{n-5}\biggr]^{1/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>

====Virial Equilibrium====
Plugging these coefficient assignments into the [[SSC/Virial/PolytropesEmbeddedOutline#Equilibrium_Configurations|above mathematical prescription of the virial theorem]] gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressure-truncated polytropic configurations,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{2}{3}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} + \frac{1}{6} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr) 
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 \, .
</math>
  </td>
</tr>

</table>
</div>
A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} -3\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-3(n+1)/n} -3\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4}  \, ,</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~P_e </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4P_\mathrm{rf} R_\mathrm{rf}^{3(n+1)/n} R_\mathrm{eq}^{-3(n+1)/n} -3P_\mathrm{rf} R_\mathrm{rf}^4 R_\mathrm{eq}^{-4} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} 
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{3(n+1)/[n(n-3)]}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} 
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{4/(n-3)}  
</math>
  </td>
</tr>


<tr>
  <td align="right">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2^2 R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2n(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n(n+1)} 
K^{4n^2} G^{-3n(n+1)} M_\mathrm{limit}^{-2n(n+1)}\cdot
\frac{\pi^{3(n+1)}}{5^{3n(n+1)}} \biggl( \frac{2^2}{3}\biggr)^{3(n+1)(n+1)} K^{-3n(n+1)} G^{3n(n+1)} M_\mathrm{limit}^{3(n-1)(n+1)} \biggr]^{1/[n(n-3)]}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} 
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\cdot
\frac{\pi^4}{5^{4n}} \biggl( \frac{2^2}{3}\biggr)^{4(n+1)} K^{-4n} G^{4n} M_\mathrm{limit}^{4(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>~
R_\mathrm{eq}^{-3(n+1)/n} \biggl( \frac{3}{2^2\pi}\biggr)^{(n+1)/n} K M_\mathrm{limit}^{(n+1)/n}  
- R_\mathrm{eq}^{-4} \biggl(\frac{3}{2^2\cdot 5\pi}\biggr) G M_\mathrm{limit}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n}  
- \frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4}  \, .</math>
  </td>
</tr>
</table>
</div>

Recalling that <math>~\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>~P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of {{ Whitworth81 }}, which reads:

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation and accompanying sentence drawn directly from p. 970 of<br />
{{ Whitworth81figure }}<br />
&copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="left">
<!--
[[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
[[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]]
-->
The general equilibrium condition, <math>(d\mathcal{U}/dR)_{R_0} = 0</math>, reduces to
<table border="0" align="center" width="100%" cellpadding="5">
<tr>
  <td align="right" width="25%"><math>R_0 \rightarrow R_\mathrm{eq}\, ,</math></td>
  <td align="center" width="5%">&nbsp;</td>
  <td align="left">
<math>P_\mathrm{ex} = K(3M_0/4\pi R^3_\mathrm{eq})^\eta - 3GM_0^2/20\pi R^4_\mathrm{eq}</math>
  </td>
  <td align="right" width="8%">(5)</td>
</tr>
</table>
(subscript 'eq' for equilibrium).
  </td>
</tr>
</table>
</div>

====Stability====
Similarly, according to the [[SSC/Virial/PolytropesEmbeddedOutline#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 </math>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>\frac{(n-3)}{(n+1)} 
\biggl( \frac{P_e}{P_\mathrm{rf}} \biggr)^{-1} \, .
</math>
  </td>
</tr>

</table>
</div>
Or, given that <math>~P_\mathrm{rf}R_\mathrm{rf}^4 = (G M_\mathrm{limit}^2)/(20\pi)</math>, the criterion for stability may be written as,
<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>~
\frac{(n-3)}{20\pi(n+1)} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr) 
\, .
</math>
  </td>
</tr>

</table>
</div>

====ASIDE:  Isothermal Configurations====
While our focus in this chapter is on polytropic systems, it is advantageous to review the {{ Whitworth81 }} discussion of pressure-truncated isothermal configurations because that discussion includes presentation of a free-energy surface &#8212; see, specifically, Whitworth's Figure 2, which is reproduced in the lower-right quadrant of our [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|composite Figure 2, below]].   Setting <math>\delta_{1\eta} = 1</math> in Whitworth's free-energy expression (his equation 10, [[SSC/Virial/PolytropesEmbeddedOutline#Coefficient_Definitions|copied above]]) gives,  
<div align="center" id="WhitworthFreeEnergyExpression">
<math>
\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}} = -\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} 
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 
-2\ln\biggl( \frac{R}{R_\mathrm{rf}}\biggr) - \mathcal{G}_0 \, ,
</math>
</div>
where, as earlier, we have inserted the additional constant, <math>\mathcal{G}_0</math>, to accommodate normalization. A segment of the free-energy surface defined by this function is displayed in the righthand panel of our [[SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|Figure 1]].  In constructing this figure, <math>\mathcal{G}_0</math> has been set to a value that ensures that <math>\mathcal{U}</math> is everywhere positive over the displayed domain: <math>0.1 \le P_e/P_\mathrm{rf} \le 1.1</math> and <math>0.3 \le R/R_\mathrm{rf} \le 3.0</math>.     

<span id="ScalarVirialTheorem">
For a given choice of <math>P_e</math>, equilibrium radii are identified by setting <math>d\mathcal{U}/dR = 0</math>, that is, they are defined by the (scalar virial theorem) relation,</span>
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{P_e}{P_\mathrm{rf}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr) - 3  \biggr] 
\, ;
</math>
  </td>
</tr>

</table>
</div>

and equilibria are stable &#8212; that is, <math>[d^2\mathcal{U}/dR^2]_\mathrm{eq} > 0</math> &#8212; if <math>R_\mathrm{eq}/R_\mathrm{rf} > 1</math>.  For physically realistic systems, of course, <math>P_e</math> must be positive &#8212; which means that all equilibria have <math>R_\mathrm{eq}/R_\mathrm{rf} > 3/4</math>.  But, from this algebraic virial theorem expression, it is also clear that physically realistic equilibrium configurations only exist when <math>P_e/P_\mathrm{rf} \le 1</math>.  The sequence of small, colored spherical dots in the righthand panel of our [[SSC/Virial/PolytropesEmbeddedOutline#IntroFigures|Figure 1]] identify parameter-value pairs, <math>(R_\mathrm{eq}, P_e)</math>, associated with fourteen different equilibrium configurations:  Blue dots &#8212; tracing out the valley of the free-energy surface &#8212; identify stable configurations; white dots &#8212; balanced along the crest of the surface ridge &#8212; identify dynamically unstable configurations; and the lone red dot identifies the critical neutral equilibrium state, which is also associated with the maximum allowable value of <math>P_e</math> along the equilibrium model sequence.


<div align="center" id="3DIsothermalSurface">
<table border="1" cellpadding="5" align="center" width="500px">
<tr><td align="center" colspan="2">
Figure 2:<br />
The black &amp; white line-drawing in lower-right quadrant<br />of this composite image is Figure 2 extracted<sup>&dagger;</sup> from p. 973 of<br />
{{ Whitworth81figure }}<br />
&copy; Royal Astronomical Society
</td></tr>
<tr>
<td align="center" colspan="1" rowspan="1" bgcolor="#C0FFFF">
[[File:EnergyRadiusViewTop5.png|250px|Whitworth's (1981) Isothermal Free-Energy Surface]]
</td>
<td align="center" colspan="1" rowspan="1" bgcolor="#D0FFFF">
[[File:FEmovie02.gif|250px|Animated Isothermal Free-Energy Surface]]
</td>
</tr>
<tr>
<td align="center" colspan="2" rowspan="1" bgcolor="#D0FFFF">
[[File:EnergyRadiusViewBottom6.png|500px|Whitworth's (1981) Isothermal Free-Energy Surface]]
<!-- [[Image:AAAwaiting01.png|440px|Whitworth's (1981) Isothermal Free-Energy Surface]]  -->
</td>
</tr>
</table>
</div>

Our Figure 2 provides graphical depictions of the free-energy surface, <math>\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of {{ Whitworth81 }} or our restatement of this equation, [[SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression|above]]:
<ul>  
<li>''Lower-right quadrant'':  <sup>&dagger;</sup>A reproduction of the black &amp; white Figure 2 from {{ Whitworth81 }} ; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this figure.  Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled.  Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>. </li> 
<li>''Upper-right quadrant'':  The undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy.  The three coordinate axes are labeled and colored as follows:  Radius (red), External Pressure (green), and Free Energy (blue).  The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots:  Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>.</li>
<li>''Upper-left quadrant'':  The two-dimensional projected image that results from viewing the free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>~(Z)</math> axis and looking directly down onto the radius-pressure <math>~(X-Y)</math> plane.  From this vantage point, the sixteen small colored dots cleanly trace out the <math>~P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[SSC/Virial/PolytropesEmbeddedOutline#ScalarVirialTheorem|algebraic expression of the scalar virial theorem]]. </li> 
<li>''Lower-left quadrant'':  The two-dimensional projected image that results from viewing the free-energy surface "from underneath," along a light of sight that is parallel to the external-pressure <math>~(Y)</math> axis and looking directly up at the radius-free-energy <math>~(X-Z)</math> plane.  This image can be directly compared with Figure 2 from {{ Whitworth81 }}.  Seven of the nine equilibrium configurations that are marked in Whitworth's diagram also appear among the fifteen equilibria that are identified (as small colored dots) in our projected image.  For example, the red dot in our image corresponds to the marginally stable configuration that Whitworth marks with a cross; and the white dot that is peeking from behind the "Radius" axis &#8212; that is, the unstable equilibrium configuration that has a free-energy value of zero &#8212; corresponds to the open circle in Whitworth's plot that is labeled as <math>P_e = 0.632</math> (see also the coordinate values given in our accompanying table).</li>
<li>''Table Identifying Properties of Virial Equilibria'' (see immediately below):  Coordinates <math>(X, Y, Z)</math> = <math>(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}, \mathcal{G})</math> and, hence, also the physical properties are provided for each of the sixteen equilibria that are marked by small colored spherical dots in our attending color plots of the free-energy surface. [Actually, two of the three attending color plots display only fifteen dots because the position of the (unstable) equilibrium configuration of highest energy falls outside the boundaries of the plot.]  Seven of the nine equilibrium configurations that are identified in Figure 2 of {{ Whitworth81 }} are among the sixteen equilibria that are identified here, including the critical neutral equilibrium state, which is highlighted in red.  For completeness, the value of the corresponding normalization energy, <math>~\mathcal{G}_0(P_e)</math>, is also tabulated.</li>
</ul>



<div align="center" id="3DIsothermalSurfaceTable">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="4">
Properties of Virial Equilibria
  </th>
</tr>
<tr>
  <td align="center">
<math>~\frac{P_e}{P_\mathrm{rf}}</math>
  </td>
  <td align="center">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{rf}}</math>
  </td>
  <td align="center">
<math>~\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}}</math>
  </td>
  <td align="center">
<math>~\mathcal{G}_0</math>
  </td>
</tr>

<tr>
  <td align="right">
0.200
  </td>
  <td align="right">
2.395
  </td>
  <td align="right">
-0.03205
  </td>
  <td align="right">
-1.8831
  </td>
</tr>

<tr>
  <td align="right">
0.30946
  </td>
  <td align="right">
2.008
  </td>
  <td align="right">
-0.04507
  </td>
  <td align="right">
-1.6786
  </td>
</tr>

<tr>
  <td align="right">
0.400
  </td>
  <td align="right">
1.800
  </td>
  <td align="right">
-0.05548
  </td>
  <td align="right">
-1.5646
  </td>
</tr>

<tr>
  <td align="right">
0.50364
  </td>
  <td align="right">
1.622
  </td>
  <td align="right">
-0.06730
  </td>
  <td align="right">
-1.4666
  </td>
</tr>

<tr>
  <td align="right">
0.632
  </td>
  <td align="right">
1.452
  </td>
  <td align="right">
-0.08213
  </td>
  <td align="right">
-1.3743
  </td>
</tr>

<tr>
  <td align="right">
0.72543
  </td>
  <td align="right">
1.347
  </td>
  <td align="right">
-0.09327
  </td>
  <td align="right">
-1.3205
  </td>
</tr>

<tr>
  <td align="right">
0.800
  </td>
  <td align="right">
1.270
  </td>
  <td align="right">
-0.10252
  </td>
  <td align="right">
-1.2835
  </td>
</tr>

<tr>
  <td align="right">
0.93516
  </td>
  <td align="right">
1.127
  </td>
  <td align="right">
-0.12070
  </td>
  <td align="right">
-1.2263
  </td>
</tr>

<tr>
  <td align="right">
0.97564
  </td>
  <td align="right">
1.071
  </td>
  <td align="right">
-0.12681
  </td>
  <td align="right">
-1.2111
  </td>
</tr>

<tr>
  <td align="right" bgcolor="red">
1.000
  </td>
  <td align="right" bgcolor="red">
1.000
  </td>
  <td align="right" bgcolor="red">
-0.13086
  </td>
  <td align="right" bgcolor="red">
-1.2025
  </td>
</tr>

<tr>
  <td align="right">
0.9897
  </td>
  <td align="right">
0.9612
  </td>
  <td align="right">
-0.12880
  </td>
  <td align="right">
-1.2061
  </td>
</tr>

<tr>
  <td align="right">
0.92636
  </td>
  <td align="right">
0.9061
  </td>
  <td align="right">
-0.11370
  </td>
  <td align="right">
-1.2297
  </td>
</tr>

<tr>
  <td align="right">
0.800
  </td>
  <td align="right">
0.859
  </td>
  <td align="right">
-0.07423
  </td>
  <td align="right">
-1.2835
  </td>
</tr>

<tr>
  <td align="right">
0.632
  </td>
  <td align="right">
0.822
  </td>
  <td align="right">
0.000
  </td>
  <td align="right">
-1.3743
  </td>
</tr>

<tr>
  <td align="right">
0.400
  </td>
  <td align="right">
0.789
  </td>
  <td align="right">
+0.17021
  </td>
  <td align="right">
-1.5646
  </td>
</tr>

<tr>
  <td align="right">
0.200
  </td>
  <td align="right">
0.767
  </td>
  <td align="right">
+0.47300
  </td>
  <td align="right">
-1.8831
  </td>
</tr>
</table>
</div>

Figure 2 from {{ Whitworth81 }} &#8212; reproduced in the lower-right quadrant of [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|our Figure 2 composite image]] &#8212; does an excellent job of conveying many of the essential elements of this isothermal free-energy surface within the constraints imposed by a two-dimensional black &amp; white line plot.  In order to construct this compact plot, Whitworth employed a different free-energy normalization parameter for each selected value of the external pressure.  Specifically, he used,
<div align="center">
<math>~\mathcal{G}_0(P_e) = \frac{2}{3}\biggl[ 1 + \ln\biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)\biggr] 
- \frac{3}{2} \biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)^{1/3} \, .</math>
</div>
In an effort to quantitatively compare (and check for accuracy) our results with Whitworth's, we have adopted the same <math>~\mathcal{G}_0(P_e)</math> normalization function when generating the multicolored, three-dimensional free-energy surface that is displayed (with three different projections) in [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|our Figure 2 composite image]], along with the reproduction of Whitworth's Figure 2.  Aside from this pressure-dependent normalization parameter, the surface drawn for comparison with Whitworth's Figure 2 is identical to the one displayed in the righthand panel of [[SSC/Virial/PolytropesEmbeddedOutline#IntroFigures|our Figure]], at the top of this page.