Editing SSC/Stability/BiPolytropes/RedGiantToPN/Pt4 (section)

==Succinct==

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

{{Math/EQ_RadialPulsation01}}
</div>

may also be written as &hellip;

<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\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ 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\}  x \, .
</math>
  </td>
</tr>
</table>

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>

===Specific Polytropes===

In a [[SSC/Stability/Polytropes#Adiabatic_(Polytropic)_Wave_Equation|separate discussion]], we have shown that configurations with a polytropic equation of state exhibit a characteristic length scale that is given by the expression,

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

<tr>
  <td align="right">
<math>a_n</math>
  </td>
  <td align="center">
<math>\equiv</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>
and, once the dimensionless polytropic temperature, <math>\theta(\xi)</math>, is known, the radial dependence of key physical variables is given by the expressions,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">&nbsp;</td>
  <td align="center" bgcolor="lightgray" rowspan="8">&nbsp;&nbsp;</td>
  <td align="center" colspan="3">
if, as [[SSC/Stability/BiPolytropes/Pt3#Foundation|in a separate discussion]], <math>n=5</math> and <math>\theta = (1+\xi^2/3)^{-1 / 2}</math> &hellip;
  </td>
</tr>

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

  <td align="right">
&nbsp; &nbsp; &nbsp;<math>r_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[\frac{K_5}{G} \cdot \rho_c^{-4/5} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \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>

  <td align="right">
&nbsp; &nbsp; &nbsp;<math>\rho_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c \theta^{5} \, ,</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>

  <td align="right">
&nbsp; &nbsp; &nbsp;<math>P_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>K_5\rho_c^{6/5} \theta^{6} \, ,</math>
  </td>

</tr>

<tr>
  <td align="right">
<math>M(r_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>4\pi \rho_c a_n^3 \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr)
=
\rho_c^{(3-n)/(2n)} \biggl[\frac{(n+1)^3 K^3}{4\pi G^3}  \biggr]^{1/2} \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr)
\, ,</math>
  </td>

  <td align="right">
&nbsp; &nbsp; &nbsp;<math>M(r_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{ K_5^3}{ G^3}\cdot \rho_c^{-2/5}  \biggr]^{1/2}
\biggl(\frac{2\cdot 3^3 }{\pi }  \biggr)^{1/2} 
\biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr)
\, ,</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>

  <td align="right">
&nbsp; &nbsp; &nbsp;<math>g_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{~\biggl[\frac{K_5}{G} \cdot \rho_c^{-4/5} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \xi ~\biggr\}^{-2}
\biggl[\frac{6K_5}{4\pi G} \cdot \rho_c^{-4/5} \biggr]^{3/2}\biggl[4\pi  G \rho_c \biggr]
\biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]
</math>
  </td>

</tr>

<tr>
  <td align="center" colspan="3">
&nbsp;
  </td>

  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2^3\pi^2}{3}\biggr) 
\biggl[\frac{6}{4\pi} \biggr]^{3/2}
\biggl[\frac{G}{K_5} \cdot \rho_c^{4/5} \biggr] \biggl[\frac{K_5}{G} \cdot \rho_c^{-4/5} \biggr]^{3/2}
\biggl[ G \rho_c \biggr]
\biggl(-~ \frac{d\theta}{d\xi}\biggr) \biggr]
</math>
  </td>

</tr>

<tr>
  <td align="center" colspan="3">
&nbsp;
  </td>

  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
(2^3\cdot 3 \pi)^{1 / 2}
\biggl[K_5 G \cdot \rho_c^{6/5} \biggr]^{1/2}
\biggl(-~ \frac{d\theta}{d\xi}\biggr) \biggr]
\, .</math>
  </td>

</tr>
</table>

Combining variable expressions from the above right-hand column, we find that for <math>n=5</math> polytropes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{g_0 \rho_0 r_0}{P_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(2^3\cdot 3 \pi)^{1 / 2}\biggl[K_5 G \cdot \rho_c^{6/5} \biggr]^{1/2}\biggl(-~ \frac{d\theta}{d\xi}\biggr) \biggr]
\cdot \rho_c \theta^5
\cdot \biggl[\frac{K_5}{G} \cdot \rho_c^{-4/5} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \xi
~\biggl[K_5\rho_c^{6/5} \theta^{6}  \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
6~\biggl(-~\frac{\xi}{\theta} \frac{d\theta}{d\xi}\biggr) \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
More generally, combining variable  expressions from the above left-hand column, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{g_0 \rho_0 r_0}{P_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{G}{a_n^2 \xi^2} \biggl[ 4\pi a_n^3 \rho_c \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]
\cdot \rho_c \theta^{n} \cdot a_n \xi \cdot \biggl[ K\rho_c^{(n+1)/n} \theta^{n+1} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi G }{K} \biggl[ \rho_c^{1- 1/n} \biggr] \biggl(-\xi \frac{d\theta}{d\xi}\biggr) 
\cdot \theta^{-1} \cdot a_n^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(n+1)\biggl(- \frac{\xi}{\theta} \cdot \frac{d\theta}{d\xi}\biggr) 
\, ;  
</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_c \theta^{n} \cdot \biggl(a_n \xi\biggr)^2 \cdot \biggl[ K\rho_c^{(n+1)/n} \theta^{n+1} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
K^{-1}\rho_c^{-1/n}  \cdot a_n^2 \cdot \frac{\xi^2}{\theta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{(n+1)}{4\pi G \rho_c} \biggr] \cdot \frac{\xi^2}{\theta}
\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

As a result, for polytropes 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}{dr_0^2} + \biggl[4 - \biggl(\frac{g_0 \rho_0 r_0}{P_0}\biggr) \biggr] \frac{1}{r_0} \cdot\frac{dx}{dr_0} 
+ \biggl(\frac{\rho_0 r_0^2}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  \frac{x}{r_0^2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x}{dr_0^2} + \biggl[4 - \biggl(\frac{g_0 \rho_0 r_0}{P_0}\biggr) \biggr] \frac{1}{r_0} \cdot\frac{dx}{dr_0} 
+ \biggl[\frac{\omega^2}{\gamma_\mathrm{g}}\biggl(\frac{\rho_0 r_0^2}{ P_0} \biggr) 
- \biggl(3 - \frac{4}{\gamma_\mathrm{g}} \biggr)\biggl(\frac{g_0 \rho_0 r_0}{P_0}\biggr)  \biggr]  \frac{x}{r_0^2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x}{dr_0^2} + \biggl[4 - (n+1) Q \biggr] \frac{1}{r_0} \cdot\frac{dx}{dr_0} 
+ (n+1)\biggl[\frac{\omega^2}{\gamma_\mathrm{g}}\biggl[\frac{1}{4\pi G \rho_c} \biggr] \cdot \frac{\xi^2}{\theta} 
- \biggl(3 - \frac{4}{\gamma_\mathrm{g}} \biggr)Q \biggr]  \frac{x}{r_0^2} 
\, . 
</math>
  </td>
</tr>
</table>
Finally, multiplying through by <math>a_n^2</math> &#8212; which everywhere converts <math>r_0</math> to <math>\xi</math> &#8212; gives, what we will refer to as the,

<div align="center">
<font color="maroon"><b>Polytropic LAWE (linear adiabatic wave equation)</b></font><br />
{{ Math/EQ_RadialPulsation02 }}
</div>