Editing Appendix/Ramblings/51BiPolytropeStability/BetterInterface (section)

==Solid Foundation==

Here we pull primarily from the chapters labeled II and III, above.

===Entire Configuration===
Beginning with the familiar,
<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

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

if we adopt the variable normalizations,
<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>

the LAWE takes 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>~
\frac{d^2x}{dr*^2} + \frac{\mathcal{H}}{r^*} \frac{dx}{dr*} 
+ 
\biggl[\biggl(\frac{\sigma_c^2}{\gamma_g}\biggr) \mathcal{K}_1 - \alpha_g \mathcal{K}_2\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
where,
<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>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\sigma_c^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3\omega^2}{2\pi G\rho_c}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl(3 - \frac{4}{\gamma_g}\biggr) \, .
</math>
  </td>
<td align="center" colspan="4">&nbsp; </td>
</tr>
</table>

===Core===
Given that, in the core, <math>\gamma_g = 6/5</math> and,

<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>
</tr>
</table>
we can rewrite the LAWE to read,

<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} + \frac{\mathcal{H}}{\xi} \frac{dx}{d\xi} 
+ 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi }{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 - 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{4\pi}{3}\biggr) \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, .
</math>
  </td>
</tr>
</table>

===Structure at the Interface===

Once <math>\mu_e/\mu_c</math> and <math>\xi_i</math> have been specified, other [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values at the interface]] are:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( 1 + \frac{1}{3}\xi^2_i \biggr)^{-1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3}~\theta_i^2 \xi_i \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta_i}  - \frac{\xi_i}{\sqrt{3}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i(1+\Lambda_i^2)^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i - \frac{\pi}{2} + \tan^{-1}(\Lambda_i) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
B + \pi \, .
</math>
  </td>
</tr>
</table>

===Linearized Perturbation at the Interface===

At all radial locations throughout the equilibrium configuration, the three spatially dependent quantities &#8212; <math>p \equiv \delta p/P^*, d \equiv \delta\rho/\rho^*,</math> and <math>x \equiv \delta r/r^*</math> &#8212; are related to one another via the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|set of linearized governing relations]], namely,

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
p = \gamma_\mathrm{g} d  \, .
</math>
</td></tr>
</table>
</div>

Combining the 2<sup>nd</sup> and 3<sup>rd</sup> equations, we find,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>(4x + \gamma_g d)g_0 + \omega^2 r_0 x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{P_0}{\rho_0} \frac{d (\gamma_g d)}{dr_0}</math></td>
</tr>
</table>


----

At the interface, presumably the ''dimensional'' structural variables, <math>P^*</math> and <math>r^*</math> have the same values, whether viewed from the perspective of the core or from the perspective of the envelope. But <math>\rho^*</math> has a different value, depending on the point of view.  Specifically, 
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\rho^*\biggr|_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)\rho^*\biggr|_\mathrm{core}\, .</math></td>
</tr>
</table>

Hence, from the perspective of the core, the linearized equation of continuity may be written as,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[ r_0 \cdot \frac{dx}{dr_0} + 3x \biggr]_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- \frac{\delta\rho}{[\rho^*]_\mathrm{core}} \, ;</math></td>
</tr>
</table>

while, from the perspective of the envelope, the linearized equation of continuity may be written as,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[ r_0 \cdot \frac{dx}{dr_0} + 3x \biggr]_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- \frac{\delta\rho}{[\rho^*]_\mathrm{core}} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}</math></td>
</tr>
</table>

----

<font color="red">Try again</font>

From [[SSC/Perturbations#Entropy_Conservation|here]], we know &hellip;

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

<tr>
  <td align="right">
<math>
\frac{P}{\rho^{\gamma_g}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]\, .
</math>
  </td>
</tr>
</table>
And, from my [[Appendix/Ramblings/PatrickMotl#Understanding_the_Step_Function_at_the_Core-Envelope_Interface|discussions with Patrick Motl]], we find &hellip;

<font color="red"><b>CORE:</b></font> &nbsp; Throughout the core,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-3}</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-5/2}</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{6}{5} \, .</math>
  </td>
</tr>
</table>
Hence, independent of the radial location, <math>~\xi</math>, throughout the core,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5\ln (5) \, . 
</math>
  </td>
</tr>
</table>

<font color="red"><b>ENVELOPE:</b></font> &nbsp; Throughout the envelope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\theta_i^6 [\phi(\eta)]^2</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)]</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \, .</math>
  </td>
</tr>
</table>
Hence, independent of the radial location, <math>~\eta</math>, throughout the envelope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{-4} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr] \, .
</math>
  </td>
</tr>
</table>

===Envelope===
Given that, throughout the envelope <math>\gamma_g = 2</math> and,

<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{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \, ,</math>
  </td>
</tr>
</table>
we can rewrite the LAWE to read,

<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{\mathcal{H}}{\eta} \frac{dx}{d\eta} 
+ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}  \biggl[\biggl(\frac{\sigma_c^2}{2}\biggr) \mathcal{K}_1 -  \mathcal{K}_2\biggr]  x\, ,
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
2\biggl[1 + \frac{\eta}{\tan(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2} 
\, .
</math>
  </td>
</tr>
</table>