Editing SSC/Stability/BiPolytropes/RedGiantToPN/Pt2 (section)

==Foundation==
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>

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.  
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">

Introducing the dimensionless frequency-squared, <math>\sigma_c^2 \equiv 3\omega^2/(2\pi G\rho_c)</math>, we can rewrite this LAWE as,

<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}{dr_0^2} + \biggl[4 - \biggl(\frac{g_0}{r_0} \cdot \frac{\rho_0 r_0^2}{P_0}\biggr) \biggr] \frac{1}{r_0}\cdot \frac{dx}{dr_0} 
+ \biggl(\frac{\rho_0 r_0^2}{ P_0} \biggr)\biggl[\frac{2\pi G\rho_c \sigma_c^2}{3\gamma_\mathrm{g}} 
- \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)\frac{g_0}{r_0} \biggr]  \frac{x}{r_0^2} 
\, ,
</math>
  </td>
</tr>
</table>
where, as a reminder, <math>g_0 \equiv GM(r_0)/r_0^2</math>.  Now, for our <math>(n_c, n_e) = (5, 1)</math> bipolytrope, we have found it useful to adopt the following four dimensionless variables:

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\rho_0}{\rho_c}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{[K_c^{1/2}/(G^{1/2}\rho_c^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{K_c\rho_c^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_c^{1/5})]}</math>
  </td>
</tr>

</table>
</div>

This means that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{g_0}{r_0} = \frac{G M(r_0)}{r_0^3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
G M_r^* \biggl[K_c^{3/2} G^{-3/2} \rho_c^{-1/5} \biggr]
r_*^{-3} \biggl[K_c^{-3 / 2} G^{3 / 2} \rho_c^{6/5}  \biggr]
= \biggl[G \rho_c \biggr] M_r^* r_*^{-3}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\rho_0 r_0^2}{P_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\rho^* \rho_c(r^*)^2 \biggl[K_c G^{-1} \rho_c^{-4 / 5} \biggr] (P^*)^{-1} \biggl[K_c^{-1} \rho_c^{-6 / 5}  \biggr]
=
\biggl[G^{-1} \rho_c^{-1}\biggr]\rho^* (r^*)^2 (P^*)^{-1}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{g_0}{r_0}  \cdot \frac{\rho_0 r_0^2}{P_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[G \rho_c \biggr] M_r^* r_*^{-3}
\cdot
\biggl[G^{-1} \rho_c^{-1}\biggr]\rho^* (r^*)^2 (P^*)^{-1}
=
\frac{M_r^* \rho^*}{P^* r^*}
\, .
</math>
  </td>
</tr>
</table>

Making these substitutions, the LAWE can be rewritten as,

<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}{dr_0^2} + \biggl[4 - \frac{M_r^* \rho^*}{P^* r^*} \biggr] \frac{1}{r_0}\cdot \frac{dx}{dr_0} 
+ \frac{1}{G\rho_c}\biggl[\frac{\rho^* (r^*)^2}{ P^*} \biggr]\biggl[\frac{2\pi G\rho_c \sigma_c^2}{3\gamma_\mathrm{g}} 
- \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)\frac{G \rho_c M_r^*}{(r^*)^3} \biggr]  \frac{x}{r_0^2} 
\, ;
</math>
  </td>
</tr>
</table>
then, multiplying through by <math>[K G^{-1}\rho_c^{-4/5}]</math> allows us to everywhere switch from <math>(r_0)^2</math> to <math>(r^*)^2</math>, namely,

<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(r^*)^2} + \biggl[4 - \frac{M_r^* \rho^*}{P^* r^*} \biggr] \frac{1}{r^*}\cdot \frac{dx}{d(r^*)} 
+ \biggl[\frac{\rho^* (r^*)^2}{ P^*} \biggr]\biggl[\frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} 
- \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)\frac{ M_r^*}{(r^*)^3} \biggr]  \frac{x}{(r^*)^2} 
\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>
<!--  DELETE
Multiplying this LAWE through by <math>(K_c/G)\rho_c^{-4 / 5}</math> and recognizing that,
<table border="0" cellpadding="5" align="center">

<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 M_r^*}{(r^*)^2} \biggl[ \rho_c^{3 / 5} \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] 
</math>
  </td>
</tr>
</table>

we have,
<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}{dr*^2} + \biggl\{ \frac{4}{r^*} -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)^2}\biggr\} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{\omega^2}{\gamma_\mathrm{g} G\rho_c} + \biggl(\frac{4}{\gamma_\mathrm{g}} - 3\biggr)\frac{ M_r^*}{(r^*)^3}\biggr\}  x 
</math>
  </td>
</tr>
</table>
DELETE -->

In shorthand, we can rewrite this equation in the form,

<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>~
x'' + \frac{\mathcal{H}}{r^*} x' + \mathcal{K}x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{dx}{dr^*}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~x''</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d(r^*)^2} \, ;</math>
  </td>
</tr>
</table>
and,
<div align="center">
<math>~\mathcal{K} \equiv ~\biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \biggl(\frac{\sigma_c^2}{\gamma_\mathrm{g}}\biggr)\frac{2\pi }{3} 
- \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr) \frac{M_r^*}{(r^*)^3} \biggr] \, ;</math>
</div>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\, .
</math>
  </td>
</tr>
</table>