Editing Appendix/Ramblings/BiPolytropeStability (section)

=Begin Our Analysis=
==Relevant LAWEs==

The LAWE that is relevant to polytropic spheres may be written as,
<div align="center">
{{ Math/EQ_RadialPulsation02 }}
</div>

===Core Layers With n = 5===
The LAWE for n = 5 structures is, 
<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 - 6Q_5 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ 6 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr)  \frac{\xi^2}{\theta_5} 
- \alpha_\mathrm{core} Q_5\biggr]  \frac{x}{\xi^2}
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>Q_5</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>- \frac{d\ln\theta_5}{d\ln\xi} \, .</math>
  </td>
</tr>
</table>

From our study of the [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|equilibrium structure of <math>(n_c, n_e) = (5, 1)</math> bipolytropes]], we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\theta_5 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1 / 2} \,  ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{d\theta_5}{d\xi} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\xi}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
Q_5 = - \frac{\xi}{\theta_5} \cdot \frac{d\theta_5}{d\xi}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{1 / 2} \frac{\xi^2}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\xi^2}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

Hence, for the core the governing LAWE is,
<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 -  2\xi^2\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \biggr\} \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ 6 \biggl\{ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr)  \xi^2\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{1 / 2} 
- ~\alpha_\mathrm{core} \cdot \frac{\xi^2}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \biggr\}  \frac{x}{\xi^2}
</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\{ 2 -  \xi^2\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \biggr\} \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{\gamma_\mathrm{core} } \biggr)  \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{1 / 2} 
- ~2\alpha_\mathrm{core} \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \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\{ 2 -  3\xi^2\biggl[ 3 + \xi^2 \biggr]^{-1} \biggr\} \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  \biggl[ 3 + \xi^2 \biggr]^{1 / 2} 
- ~6\alpha_\mathrm{core} \biggl[ 3 + \xi^2 \biggr]^{-1} \biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(3 + \xi^2) \frac{d^2x}{d\xi^2} + \biggl\{ 2(3 + \xi^2) -  3\xi^2 \biggr\} \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(3 + \xi^2) \frac{d^2x}{d\xi^2} + ( 6 - \xi^2 ) \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  x \, .
</math>
  </td>
</tr>
</table>
This exactly matches our [[SSC/Stability/n5PolytropeLAWE#Specifically_for_n.3D5_Configurations|derivation performed in the context of pressure-truncated polytropes]].  When we insert the eigenfunction obtained via a ''[[SSC/Stability/n5PolytropeLAWE#Eureka_Moment|Eureka Moment]]'' on 3/6/2017,
<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>x_0\biggl[1 - \frac{\xi^2}{15}\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{2x_0 \xi}{15} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{d^2x}{d\xi^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{2x_0 }{15} \, ,</math>
  </td>
</tr>
</table>
we obtain,

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

<tr>
  <td align="right">
<math>
(3 + \xi^2) \frac{d^2x}{d\xi^2} + ( 6 - \xi^2 ) \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  x 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~(3 + \xi^2) \frac{2x_0 }{15} - 2( 6 - \xi^2 ) \frac{2x_0 }{15}  
+ \frac{x_0}{15}\biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  (15 - \xi^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{x_0}{15} \biggl\{
2(3 + \xi^2)  + 4( 6 - \xi^2 )   
- \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  (15 - \xi^2) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{x_0}{15} \biggl\{
30 - 2\xi^2 ~+ ~6\alpha_\mathrm{core}   (15 - \xi^2) 
~- \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
\biggr]  (15 - \xi^2) 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

The right-hand-side goes to zero if <math>\alpha_\mathrm{core} = - 1/3</math> and <math>\sigma_c = 0</math>.  Also, notice that,

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

<tr>
  <td align="right">
<math>~ \frac{d\ln x}{d\ln \xi}\biggr|_\mathrm{core} = \frac{\xi}{x}\cdot \frac{dx}{d\xi} \biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2x_0 \xi^2}{15} \biggl\{ x_0\biggl[1 - \frac{\xi^2}{15}\biggr] \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl\{ \frac{15}{2\xi^2}\biggl[\frac{15 -\xi^2}{15}\biggr] \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl[\frac{15 -\xi^2}{2\xi^2}\biggr]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \biggl[1 - \frac{15}{\xi^2} \biggr]^{-1} \, .</math>
  </td>
</tr>
</table>

So, with <math>~\gamma_c = 6/5</math> and <math>~\gamma_e = 2</math> we need,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=\xi_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_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\biggl(\frac{3}{5}  -1\biggr) + \frac{6}{5}  \biggl[1 - \frac{15}{\xi_i^2} \biggr]^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{6}{5} \biggl\{  \biggl[\frac{\xi_i^2 - 15}{\xi_i^2} \biggr]^{-1} - 1 \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{6}{5} \biggl[\frac{15}{\xi_i^2 - 15} \biggr] \, .</math>
  </td>
</tr>
</table>

===Envelope Layers With n = 1===
And for n = 1 structures the LAWE is,

<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 - 2 Q_1 \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{env} } \biggr)  \frac{\eta^2}{\theta_1} 
- \alpha_\mathrm{env} Q_1\biggr]  \frac{x}{\eta^2} 
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{d\ln\theta_1}{d\ln\eta} \, .</math>
  </td>
</tr>
</table>

As has already been pointed out, above, for n = 1 polytropic spheres, this LAWE becomes,

<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} +  \frac{2}{\eta} \biggl[ 1 +  \eta\cot\eta  \biggr]\frac{dx}{d\eta}  
+ \biggl[ \frac{\gamma_g}{\gamma_\mathrm{env}}\biggl( \omega_k^2 \theta_c \biggr)  \frac{\eta}{\sin\eta} 
+ \frac{2 \alpha_\mathrm{env} ( \eta\cos\eta - \sin\eta) }{\eta^2 \sin\eta} \biggr]  x \, .
</math>
  </td>
</tr>
</table>

In a [[SSC/Stability/InstabilityOnsetOverview#Configurations_Having_an_Index_Less_Than_Three|separate chapter]], we explain that the analytically defined eigenfunction that satisfies this LAWE &#8212; when <math>\omega_k^2 = 0</math> and <math>\alpha_\mathrm{env} = +1</math> &#8212; is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_P\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3b_e}{\eta^2}\biggl[ 1- \eta \cot\eta  \biggr]   \, .
</math>
  </td>
</tr>
</table>
</div>

===Summary===

For a given choice of the ''equilibrium'' model parameters, <math>\xi_i</math> and <math>\mu_e/\mu_c</math>, we can pull the parameters and profiles of the base equilibrium model from our accompanying chapter on <math>(n_c, n_e) = (5, 1)</math> bipolytropes.  [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|Note, in particular, that]]:  
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi \, ,</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>0 \le \xi \le \xi_i \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{1}{\sqrt{2\pi}} \biggl(1 + \frac{\xi_i^2}{3}  \biggr) \biggr] \eta \, ,</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>\eta_i \le \eta \le \eta_s \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i + \frac{\pi}{2} + \tan^{-1} \biggl[\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}   \biggr]
\, .
</math>
  </td>
  <td align="left" colspan="2">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  
  </td>
</tr>
</table>


Then, for a choice of the pair of exponents, <math>\gamma_\mathrm{core}</math> and <math>\gamma_\mathrm{env}</math>, that govern the behavior of adiabatic oscillations, the LAWE to be numerically integrated is:
<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>
(3 + \xi^2) \frac{d^2x_\mathrm{core}}{d\xi^2} + ( 6 - \xi^2 ) \frac{2}{\xi} \cdot \frac{dx_\mathrm{core}}{d\xi}  
+ \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  x_\mathrm{core} \, ,
</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>~0 \le \xi \le \xi_i \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x_\mathrm{env}}{d\eta^2} + \biggl[ 1 +  \eta\cot\eta  \biggr] \frac{2}{\eta} \cdot \frac{dx_\mathrm{env}}{d\eta}  
+ 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{env} } \biggr)  \frac{\eta^3}{\sin\eta} 
- \alpha_\mathrm{env} ( 1 -  \eta\cot\eta )\biggr]  \frac{x_\mathrm{env}}{\eta^2} \, ,
</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>~\eta_i \le \eta \le \eta_s \, .</math>
  </td>
</tr>
</table>

The boundary conditions at the center of the configuration are, <math>x_\mathrm{env} = 1</math>, while <math>dx_\mathrm{env}/d\xi = 0</math>.

As [[#Interface_Conditon|described above]], the two interface boundary conditions are, <math>x_\mathrm{env} = x_\mathrm{core}</math>, and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=\xi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3\biggl(\frac{\gamma_\mathrm{core}}{\gamma_\mathrm{env}}  -1\biggr) + \frac{\gamma_\mathrm{core}}{\gamma_\mathrm{env}} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} \, .</math>
  </td>
</tr>
</table>

Finally, drawing from a [[SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29|related discussion of the surface boundary condition for isolated n = 3 polytropes]], at the surface we need,

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

<tr>
  <td align="right">
<math>\frac{d\ln x_\mathrm{env}}{d\ln \eta}\biggr|_\mathrm{surface}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\gamma_\mathrm{env}} \biggl( 4 - 3\gamma_\mathrm{env} + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{3\omega^2 R^3}{4\pi G \gamma_\mathrm{env} \bar\rho} - \alpha_\mathrm{env} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \biggl[ \frac{\sigma_c^2}{\gamma_\mathrm{env} } \biggl(\frac{\rho_c}{\bar\rho}\biggr) -2 \alpha_\mathrm{env} \biggr] \, ,</math>
  </td>
</tr>
</table>

where, for an <math>(n_c, n_e) = (5, 1)</math> bipolytrope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s^2}{3A\theta_i^5} \, .
</math>
  </td>
</tr>
</table>