Editing SSC/Structure/Polytropes/VirialSummary (section)

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

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  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
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<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, 
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<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;
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<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,
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  <td align="right">
<math>~\mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<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,

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<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,

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<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

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<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,
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  <td align="right">
<math>~0</math>
  </td>
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<math>~=</math>
  </td>
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<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>

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  <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,
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<math>~ 
x_\mathrm{crit}^4 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<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,
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<math>~W^*_\mathrm{grav} = -ax^{-1}</math>
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&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
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<math>~U^*_\mathrm{int} = bx^{-3/n} \, .</math>
  </td>
</tr>
</table>

Notice, then, that in every equilibrium configuration, we should find,
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  <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>
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<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,
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<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>

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  <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'']].