Editing SSC/Virial/PolytropesSummary (section)

====Groundwork====
In an effort to more fully understand what can be learned from an examination of the free-energy, let's play with <math>~n=4</math> polytropic models.  First, let's plot <math>~\mathfrak{G}^{**}(\Chi)</math> using a specific, trial value of the coefficient, <math>~\Pi_\mathrm{ad}</math>, keeping in mind that,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\eta_\mathrm{crit}\biggr|_{n=4}</math>
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  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~\frac{1}{15} = 0.066667 \, ;</math>
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  <td align="right">
<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
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<math>~=</math>
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<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ;</math>
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  <td align="right">
<math>~\Chi_\mathrm{min}\biggr|_{n=4}</math>
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<math>~=</math>
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<math>~\biggl( \frac{16}{15} \biggr)^4 = 1.294538 \, .</math>
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</table>
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At the top of the table, shown below, we display a plot of the,
<div align="center" id="RenormalizedFreeEnergyExpression2">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />

<math>
\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, ,
</math>
</div>
where we have set <math>~n = 4</math>, and <math>~\Pi_\mathrm{ad} = 0.01</math>.  Reading quantities off of the plot, the left and right extrema identify equilibria having the following approximate dimensionless radii:  <math>~\Chi_\mathrm{left} \approx 1.03</math> and <math>~\Chi_\mathrm{right} \approx 2.13</math>.  Upon closer examination (plots not shown), we have determined that, <math>~\Chi_\mathrm{left} \approx 1.0494</math> and <math>~\Chi_\mathrm{right} \approx 2.13905</math>.  In accordance with our stability analysis, these values of <math>~\Chi_\mathrm{ad}</math> fall on either side of the demarcation value, <math>~\Chi_\mathrm{min} = (16/15)^4</math>, with the one on the left being a local maximum in the free energy &#8212; indicating an unstable equilibrium &#8212; while the one on the right is a local minimum &#8212; indicating a stable equilibrium.  Next, let's check to see if both extrema satisfy the,
<div align="center" id="ConciseVirial2">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
</math>
</div>
For the unstable equilibrium configuration, we calculate,
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<math>\Pi_\mathrm{ad} \approx [(1.0494)^{1/4} - 1]/(1.0494)^4 = 1.000024 \times 10^{-2}</math>;
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while, for the stable equilibrium we calculate,
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<math>\Pi_\mathrm{ad} \approx [(2.13905)^{1/4} - 1]/(2.13905)^4 = 1.000018 \times 10^{-2}</math>.
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Because we inserted a value of <math>~\Pi_\mathrm{ad} = 0.01</math> into the free-energy expression, we conclude that, as desired, both identified extrema satisfy the virial relation to the measured accuracy.  These parameter values, and the corresponding values of many other related physical parameters are summarized in the following table, along with the algebraic relations that were used to calculate them.