Editing SSC/Stability/BiPolytropes/Pt3 (section)

==Splitting Analysis Into Separate Core and Envelope Components==
===Core:===
Given that, <math>~\sqrt{2\pi/3}~r^* = \xi</math>, lets multiply the LAWE through by <math>~3/(2\pi)</math>.  This gives,

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

Specifically for the core, therefore, the finite-difference representation of the LAWE is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta \xi)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ - x_-}{2\delta \xi} \biggr] ~-~ \biggl[ \frac{3\mathcal{K}}{2\pi} \biggr]x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta \xi}{2\xi} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \xi)^2 \biggl[ \frac{3\mathcal{K}}{2\pi} \biggr] x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \xi}{2\xi}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta \xi)^2\biggl( \frac{3\mathcal{K}}{2\pi} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \xi}{2\xi} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>
This also means that, as viewed from the perspective of the core, the slope at the interface is

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{dx}{d\xi}\biggr]_\mathrm{interface}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta \xi} \biggl\{
\biggl[ 2 - (\delta \xi)^2 \biggl( \frac{3\mathcal{K}}{2\pi} \biggr)\biggr] x_i ~-~2x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta \xi}{2\xi}\biggr) \mathcal{H} \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

===Envelope:===
Given that,
<div align="center">
<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 (2\pi)^{1 / 2}~r^* = \eta \, ,</math>
</div>
let's multiply the LAWE through by <math>~(2\pi)^{-1} \theta_i^{-4}( \mu_e/\mu_c)^{-2} </math>.  This gives,

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

Specifically for the envelope, therefore, the finite-difference representation of the LAWE is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta \eta)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2\delta \eta} \biggr] ~-~  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr]x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta \eta}{2\eta} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr] x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \eta}{2\eta}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>

This also means that, once we know the slope at the interface (see immediately below), the amplitude at the first zone outside of the interface will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_{i+1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 - \tfrac{1}{2}(\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{\mathcal{K}}{2\pi \theta_i^4} \biggr)\biggr] x_i 
~+~
\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]
\delta \eta \cdot \biggl[ \frac{dx}{d\eta} \biggr]_\mathrm{interface} \, .
</math>
  </td>
</tr>
</table>

===Interface===
If we consider only cases where <math>~\gamma_e = \gamma_c</math>, then at the interface we expect,

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

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d\ln x}{d\ln \xi} = \frac{d\ln x}{d\ln \eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r^*\frac{dx}{d r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi \frac{dx}{d \xi} = \eta \frac{d x}{d \eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{dx}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\pi}{3} \biggr)^{1 / 2}\frac{dx}{d\xi} = \biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^2 (2\pi)^{1 / 2} \frac{dx}{d\eta} \, .</math>
  </td>
</tr>
</table>
Switching at the interface from <math>~\xi</math> to <math>~\eta</math> therefore means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \biggl[ \frac{dx}{d\eta}\biggr]_\mathrm{interface}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\sqrt{3}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-2} \biggl[ \frac{dx}{d\xi}\biggr]_\mathrm{interface} \, .</math>
  </td>
</tr>
</table>

If, however, we want to consider values for the adiabatic index that are different in the two regions, we have to follow the [[#Interface_Conditions|above-outlined guidelines]], that is,

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

<tr>
  <td align="right">
<math>~ \frac{d\ln x}{d\ln \eta} \biggr|_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x}{d\ln \xi} \biggr)_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \biggl[ \frac{dx}{d\eta} \biggr]_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x_i}{\eta_i} \biggl\{
3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{\xi_i}{x_i}\biggl[ \frac{dx}{d\xi} \biggr]_i 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3x_i}{\eta_i} \biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{1}{\sqrt{3}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-2} \biggl[ \frac{dx}{d\xi} \biggr]_i \, .
</math>
  </td>
</tr>
</table>
</div>