Editing SSC/Stability/NeutralMode (section)

=LAWE=
==Most General Form==
In an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

<!--
<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave Equation'''</font><br />

<math>
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x = 0 \, ,
</math>
</div>
-->
where the [[SSC/Perturbations#g0|gravitational acceleration]],
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>g_0</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\frac{GM_r}{r_0^2} = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} 
~~~\Rightarrow ~~~ 
\frac{g_0\rho_0 r_0}{P_0} = - \frac{d\ln P_0}{d\ln r_0}
\, .
</math>
  </td>
</tr>
</table>
The solution to this equation gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.  The boundary condition conventionally used in connection with the adiabatic wave equation is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r_0 \frac{d\ln x}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>r_0 = R \, .</math>
  </td>
</tr>
</table>

==Polytropic Configurations==
===Part 1===
If the initial, unperturbed equilibrium configuration is a [[SSC/Structure/Polytropes#Polytropic_Spheres|polytropic sphere]] whose internal structure is defined by the function, <math>\theta(\xi)</math>, that provides a solution to the,
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
then, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a_n \xi \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_c \theta^{n} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K\rho_0^{(n+1)/n} = K\rho_c^{(n+1)/n} \theta^{n+1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>g_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{GM(r_0)}{r_0^2} = \frac{G}{r_0^2} \biggl[ 4\pi a_n^3 \rho_c \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]
\, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a_n</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[\frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, after multiplying through by <math>~a_n^2</math>, the above adiabatic wave equation can be rewritten in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{g_0}{a_n}\biggl(\frac{a_n^2 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{d\xi} + \biggl(\frac{a_n^2\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{a_n\xi} \biggr]  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
In addition, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{g_0}{a_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho_c \biggl(-\frac{d \theta}{d\xi} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>

and, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{a_n^2 \rho_0}{P_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(n+1)}{(4\pi G\rho_c)\theta} = \frac{a_n^2 \rho_c}{P_c} \cdot \frac{\theta_c}{\theta}\, ,</math>
  </td>
</tr>
</table>
</div>
we can write,

<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^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + 
\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta} - 
\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot \frac{(n+1)V(x)}{\xi^2} \biggr]  x </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + 
(n+1)\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\xi^2 \theta_c}{(n+1)\theta} - 
\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot V(x) \biggr]  \frac{x}{\xi^2} </math>
  </td>
</tr>
</table>

where we have adopted the function notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>V(\xi)</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>- \frac{\xi}{\theta} \frac{d \theta}{d\xi} \, .</math>
  </td>
</tr>
</table>
</div>

===Part 2===
Drawing from  an [[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|accompanying discussion]], we have the following:

<div align="center">
<font color="maroon"><b>Polytropic LAWE (linear adiabatic wave equation)</b></font><br />
{{ Math/EQ_RadialPulsation02 }}
</div>
<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In order to reconcile with the "Part 1" expression, we note first that <math>V(\xi) \leftrightarrow Q(\xi)</math>.  We note as well that since,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl(\frac{a_n^2 \rho_c }{P_c} \biggr)\theta_c</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{(n+1)}{4\pi G\rho_c}\, , </math>
  </td>
</tr>
</table>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\xi^2 \theta_c}{(n+1)\theta}</math>
  </td>
  <td align="center">
<math>\leftrightarrow</math>
  </td>
  <td align="left">
<math>
\frac{\omega^2}{\gamma_g} \biggl[\frac{(n+1)}{4\pi G\rho_c} \biggr] \frac{\xi^2 }{(n+1)\theta}
=
\frac{1}{6\gamma_g} \biggl[\frac{3\omega^2}{2\pi G\rho_c} \biggr] \frac{\xi^2 }{\theta}
=
\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \frac{\xi^2 }{\theta} 
\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>
All physically reasonable solutions are subject to the inner boundary condition,
<div align="center">
<math>\frac{dx}{d\xi} = 0</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = 0 \, ,</math>
</div>
but the relevant outer boundary condition depends on whether the underlying equilibrium configuration is isolated (surface pressure is zero), or whether it is a "pressure-truncated" configuration.  As is the case with the pressure-truncated isothermal spheres, discussed above, if the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,
<div align="center">
<math>-\frac{d\ln x}{d\ln\xi} = 3</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = \tilde\xi \, .</math>
</div>
But, for ''isolated'' polytropes, the sought-after solution is subject to the more conventional boundary condition,
<div align="center">
<math>- \frac{d\ln x}{d\ln \xi} = \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr] </math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = \xi_\mathrm{surf} \, .</math><br />
</div>