Editing SSC/Virial/Polytropes (section)

=====Compare With Detailed Force Balanced Solution=====
In a [[SSC/Structure/PolytropesEmbedded#.3D_1_Polytrope|separate discussion]], we presented the detailed force-balanced model of an <math>~n=1</math> polytrope that is embedded in an external medium.  We showed that, for an applied external pressure given by,
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<math>\frac{P_e}{P_\mathrm{norm}} = \frac{\pi}{2} \biggl[ \frac{\sin\xi_e}{\xi_e(\sin\xi_e - \xi_e \cos\xi_e )} \biggr]^2 \, ,</math> 
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the associated equilibrium radius of the pressure-confined configuration is,
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<math>
R_\mathrm{eq} = \xi_e a_\mathrm{n=1} = \biggl[ \frac{K}{2\pi G} \biggr]^{1/2} \xi_e \, .
</math>
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Flipping this around, after we use a plot of the free-energy expression to identify the equilibrium radius, <math>~\chi_\mathrm{eq}</math>, the corresponding dimensionless radius as used in the Lane-Emden equation should be,
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<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~\xi_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{2\pi G}{K} \biggr]^{1/2} R_\mathrm{norm} \cdot \chi_\mathrm{eq} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2\pi)^{1/2} \chi_\mathrm{eq} \, .
</math>
  </td>
</tr>

</table>
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Keep in mind that, for an [[SSC/Structure/Polytropes#.3D_1_Polytrope|isolated <math>~n=1</math> polytrope]], the (zero pressure) surface is identified by <math>~\xi_1 = \pi</math>. Hence we should expect free-energy extrema to occur at values of <math>~\chi_\mathrm{eq} \le \pi/2</math>.