Editing SSC/SynopsisStyleSheet (section)

===Isolated &amp; Pressure-Truncated Configurations===

{| class="Synopsis1E" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
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! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: &nbsp; Applicable to Isolated & Pressure-Truncated Configurations</b></font>
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! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2463;</font></b>&nbsp; &nbsp;<b>Perturbation Theory</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">&#x2466;</font></b>&nbsp; &nbsp;<b>Free-Energy Analysis of Stability</b>

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Given the radial profile of the density and pressure in the equilibrium configuration, solve the [[SSC/VariationalPrinciple#Ledoux_and_Pekeris_.281941.29|eigenvalue problem defined]] by the,
<div align="center">
<font color="#770000">'''LAWE: &nbsp; Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{dr}\biggl[ r^4 \gamma P ~\frac{dx}{dr} \biggr] 
+\biggl[ \omega^2 \rho r^4 + (3\gamma - 4) r^3 \frac{dP}{dr} \biggr] x 
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, &sect;3.7.1, p. 174, Eq. (3.145)
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</table>
</div>
to find one or more radially dependent, radial-displacement eigenvectors, <math>x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>\omega^2</math>.
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The second derivative of the free-energy function is,
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<tr>
  <td align="right">
<math>R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2a\biggl(\frac{R}{R_0}\biggr)^{-3} + (3-3\gamma)(2-3\gamma)b \biggl(\frac{R}{R_0}\biggr)^{1-3\gamma} + 6c\biggl(\frac{R}{R_0}\biggr)
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{R_0}{R} \biggr)^2\biggl[
2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V 
\biggr] \, .
</math>
  </td>
</tr>
</table>
Evaluating this second derivative for an equilibrium configuration &#8212; that is by calling upon the (virial) equilibrium condition to set the value of the internal energy &#8212; we have,
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<tr>
  <td align="right">
<math>3(\gamma-1)U_\mathrm{int}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3P_e V - W_\mathrm{grav} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav}   \biggr] + 6P_e V 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, .
</math>
  </td>
</tr>
</table>
Note the similarity with <b><font color="maroon" size="+1">&#x2465;</font></b>.
----


Alternatively, recalling that,
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<tr>
  <td align="right">
<math>~3(\gamma - 1)U_\mathrm{int}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2S_\mathrm{therm} \, ,
</math>
  </td>
</tr>
</table>
the conditions for virial equilibrium and stability, may be written respectively as,
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<tr>
  <td align="right">
<math>3P_e V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2S_\mathrm{therm}+ W_\mathrm{grav} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>
4W_\mathrm{grav} + 6\gamma S_\mathrm{therm}  \, .
</math>
  </td>
</tr>
</table>


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! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2464;</font></b>&nbsp; &nbsp;<b>Variational Principle</b>

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Multiply the LAWE through by <math>4\pi x dr</math>, and integrate over the volume of the configuration gives the,
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<font color="#770000">'''Governing Variational Relation</font><br />
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<tr>
  <td align="right">
<math>0</math>
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<math>=</math>
  </td>
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<math>
\int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
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<math>
- 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^R x^2 \biggl(\frac{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P  dr
- \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r}  \biggr) 4\pi \rho r^2 dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
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<math>
+ \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
</math>
  </td>
</tr>
</table>
</div>
Now, by setting <math>(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>P = P_e</math> at the surface, in which case this relation becomes,

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<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\omega^2</math> 
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int}
- \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav}
+ 3^2 \gamma  x^2  P_eV}{ \int_0^R  x^2 r^2  dM_r} 
</math>
  </td>
</tr>
</table>
</div>

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! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2465;</font></b>&nbsp; &nbsp;<b>Approximation: &nbsp; Homologous Expansion/Contraction</b>

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! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>x</math> = constant, and the governing variational relation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\omega^2 \int_0^R  r^2  dM_r</math> 
  </td>
  <td align="center">
<math>\leq</math>
  </td>
  <td align="left">
<math>
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma   P_eV \, .
</math>
  </td>
</tr>
</table>
</div>

|}