Editing SSC/Virial/PolytropesSummary (section)

===Stability===

Analysis of the free-energy function allows us to not only ascertain the equilibrium radius of isolated polytropes and pressure-truncated polytropic configurations, but also the relative stability of these configurations.  We begin by repeating 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>
The first and second derivatives of <math>~\mathfrak{G}^{**}</math>, with respect to the dimensionless radius, <math>~\Chi</math>, are, respectively,
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<math>~\frac{\partial\mathfrak{G}^{**}}{\partial\Chi}</math>
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<math>~=</math>
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<math>~3 \Chi^{-2} -3\Chi^{-(n+3)/n} + 3\Pi_\mathrm{ad} \Chi^2 \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}</math>
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<math>~=</math>
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<math>~-6 \Chi^{-3} + \frac{3(n+3)}{n} \Chi^{-(2n+3)/n} + 6\Pi_\mathrm{ad} \Chi \, .</math>
  </td>
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</table>
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As alluded to, above, equilibrium radii are identified by values of <math>~\Chi</math> that satisfy the equation, <math>\partial\mathfrak{G}^{**}/\partial\Chi = 0</math>.  Specifically, marking equilibrium radii with the subscript "ad", they will 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>
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Dynamical stability then depends on the sign of the second derivative of <math>~\mathfrak{G}^{**}</math>, evaluated at the equilibrium radius; specifically, configurations will be stable if,
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<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}\biggr|_{\Chi_\mathrm{ad}}</math>
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<math>~></math>
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<math>~0 \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
  </td>
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</table>
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and they will be unstable if, upon evaluation at the equilibrium radius, the sign of the second derivative is less than zero.  Hence, isolated polytropes as well as pressure-truncated polytropic configurations will be stable if,
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<math>~0</math>
  </td>
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<math>~< </math>
  </td>
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<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ - 2 + \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 \biggr]</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~< </math>
  </td>
  <td align="left">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl\{ \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2[\Chi_\mathrm{ad}^{(n-3)/n} -1] - 2\biggr\}</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~< </math>
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<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ \frac{3(n+1)}{n} \Chi_\mathrm{ad}^{(n-3)/n}  - 4\biggr]</math>
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<math>\Rightarrow~~~~\Chi_\mathrm{ad}</math>
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<math>~> </math>
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<math>~\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, .</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
  </td>
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</table>
</div>
Reference to this stability condition proves to be simpler if we define the limiting configuration size as,
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<math>~\Chi_\mathrm{min} \equiv \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, ,</math>
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and write the stability condition as,
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<math>~\Chi_\mathrm{ad} > \Chi_\mathrm{min} \, .</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
</div>

When examining the equilibrium sequences found in the upper-righthand quadrant of the figure at the top of this page &#8212; each corresponding to a different value of the polytropic index, <math>~n > 3</math> or <math>~n < 0</math> &#8212; we find that <math>~\Chi_\mathrm{min}</math> corresponds to the location along each sequence where the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, reaches a maximum.  (Keeping in mind that the virial theorem defines each of these sequences, this statement of fact can be checked by identifying where the condition, <math>~\partial\Pi_\mathrm{ad}/\partial\Chi_\mathrm{ad} = 0</math>, occurs according to the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial2|algebraic expression of the virial theorem]].)  Hence, we conclude that, along each sequence, no equilibrium configurations exist for values of the dimensionless external pressure that are greater than,
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<math>~\Pi_\mathrm{max}</math>
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<math>~\equiv</math>
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<math>~\Chi_\mathrm{min}^{-4}  \biggl[ \Chi_\mathrm{min}^{(n-3)/n} - 1 \biggr] </math>
  </td>
</tr>

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&nbsp;
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<math>~=</math>
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<math>~\biggl[ \frac{3(n+1)}{4n} \biggr]^{4n/(n-3)} \biggl[\frac{4n}{3(n+1)}  - 1 \biggr]</math>
  </td>
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&nbsp;
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<math>~=</math>
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<math>~\biggl\{ \biggl[ \frac{3(n+1)}{4n} \biggr]^{4n} \biggl[\frac{n-3}{3(n+1)} \biggr]^{n-3} \biggr\}^{1/(n-3)}</math>
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<math>~\Rightarrow~~~~\Pi_\mathrm{max}^{n-3}</math>
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<math>~=</math>
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<math>~(4n)^{-4n}~[3(n+1)]^{3(n+1)} ~(n-3)^{n-3} \, .</math>
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</table>
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In the context of a general examination of the free-energy of pressure-truncated polytropes, it is worth noting that this limit on the external pressure also establishes a limit on the coefficient, <math>~\mathcal{D}</math>, that appears in the free energy function.  Specifically, we will not expect to find any extrema in the free energy if,
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<math>~\mathcal{D} > \mathcal{D}_\mathrm{max}</math>
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<math>~\equiv</math>
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<math>~(n-3) \biggl\{ \biggl[ \frac{\mathcal{B}}{4n} \biggr]^{4n}~\biggl[ \frac{3(n+1)}{\mathcal{A}} \biggr]^{3(n+1)} ~\biggr\}^{1/(n-3)} \, .</math>
  </td>
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</table>
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Finally, it is worth noting that the point along each equilibrium sequence that is identified by the coordinates, <math>~(\Chi_\mathrm{min}, \Pi_\mathrm{max})</math> always corresponds to,
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<math>~\eta_\mathrm{ad} = \eta_\mathrm{crit} \equiv \frac{n-3}{3(n+1)} \, .</math>
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Summary
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<math>~\eta_\mathrm{crit}</math>
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<math>~\equiv</math>
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<math>~ \frac{n-3}{3(n+1)}  </math>
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  <td align="right">
<math>~\Pi_\mathrm{max}</math>
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  <td align="center">
<math>~\equiv</math>
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<math>~(n-3) \biggl\{~\frac{ [3(n+1)]^{3(n+1)} }{(4n)^{4n}} \biggr\}^{1/(n-3)} </math>
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<math>~\Chi_\mathrm{min} </math>
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  <td align="center">
<math>~\equiv</math>
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<math>
\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}  
</math>
  </td>
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</table>

</td>
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</table>
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