Editing Appendix/Ramblings/BiPolytrope51ContinueSearch

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=Continue Search for Marginally Unstable (5,1) Bipolytropes=

This ''Ramblings Appendix'' chapter &#8212; see also, [[Appendix/Ramblings/BiPolytrope51AnalyticStability|various trials]] &#8212; provides some detailed trial derivations in support of the [[SSC/Stability/BiPolytropes#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|accompanying, thorough discussion of this topic]].

==Key Differential Equation==
In an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Linear 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.  After adopting an appropriate set of variable normalizations &#8212; as detailed [[SSC/Stability/BiPolytropes#Foundation|here]] &#8212; this 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}{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}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, ,
</math>
  </td>
</tr>
</table>
where, <math>~\alpha_g \equiv (3 - 4/\gamma_g)</math>.  Alternatively &#8212; see, for example, our [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|introductory discussion]] &#8212; for polytropic configurations we may 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}{d\xi^2} + \biggl[4 - (n+1) \biggl(- \frac{d\ln \theta}{d\ln \xi} \biggr)\biggr] \frac{1}{\xi} \frac{dx}{d\xi} + 
\biggl\{ \frac{(n+1)}{\theta} \biggl[ \frac{\sigma_c^2}{6\gamma_g}\biggr]  
- 
(n+1) \biggl(- \frac{d\ln \theta}{d\ln \xi} \biggr) \frac{\alpha_g}{\xi^2 } \biggr\}  x \, .</math>
  </td>
</tr>
</table>

===Applied to the Core===
As we have already summarized in an [[SSC/Stability/BiPolytropes#Profile|accompanying discussion]], throughout the core we have,
<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="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}  \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} 
\, .
</math>
  </td>
</tr>
</table>
So the relevant ''core'' 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>~
\biggl( \frac{2\pi}{3} \biggr) \frac{d^2x}{d\xi^2} 
+ \biggl( \frac{2\pi}{3} \biggr) \biggl\{ 4 -  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \biggl[ 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2} \biggr]\biggr\}\frac{1}{\xi} \frac{dx}{d\xi} 
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\biggl( \frac{2\pi}{3} \biggr)\frac{\alpha_\mathrm{g} }{\xi^2} \biggl[ 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2} \biggr] \biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{3}{4\pi} \biggr) \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\cdot \frac{d^2x}{d\xi^2} 
+ \biggl[ 2 -  \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr] \frac{1}{\xi} \frac{dx}{d\xi} 
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl[ \frac{\sigma_c^2}{2\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2}  \biggr]  x \, .
</math>
  </td>
</tr>
</table>
Now, following our [[SSC/Stability/BiPolytropes#Is_There_an_Analytic_Expression_for_the_Eigenfunction.3F|separate discussion of an analytic solution to this LAWE]], we try,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 - \frac{\xi^2}{15}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{dx_P}{d\xi}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2\xi}{15} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{d\ln x_P}{d\ln \xi}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2\xi^2}{15} \biggl[ \frac{(15 - \xi^2)}{15} \biggr]^{-1} = - \frac{2\xi^2}{(15 - \xi^2)}  \, .</math>
  </td>
</tr>
</table>
Plugging this trial function into the relevant LAWE gives,

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

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \biggl( -\frac{2}{3\cdot 5}\biggr)
+ \biggl( -\frac{2}{3\cdot 5}\biggr)\biggl[ 2 -  \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr]
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl[ \frac{\sigma_c^2}{2\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2}  \biggr]  \biggl[1 - \frac{\xi^2}{15}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{3}
+ \biggl( \frac{2}{3\cdot 5}\biggr)\biggl[ \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr]
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl[ \frac{\sigma_c^2}{2\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2}  \biggr]  \biggl[1 - \frac{\xi^2}{15}\biggr]
</math>
  </td>
</tr>
</table>
Now, if we set <math>~\sigma_c^2 = 0</math> and <math>~\gamma_g = \gamma_c = \tfrac{6}{5} ~~\Rightarrow ~~ \alpha_g = -1/3</math>, we find that the terms on the RHS sum to zero.  It therefore appears that we have identified a dimensionless displacement function that satisfies the ''core'' LAWE.
<!--  SHOW THAT LAWE IS SATISFIED IN THE CORE
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{3}
+ \biggl( \frac{2}{3\cdot 5}\biggr)\biggl[ \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr]
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl[ \cancelto{0}{\frac{\sigma_c^2}{2\gamma_\mathrm{g}}}  ~+~\frac{1}{3} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2}  \biggr]  \biggl[1 - \frac{\xi^2}{15}\biggr]
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .
</math>
  </td>
</tr>
</table>

END SHOWING THAT LAWE SUMS TO ZERO -->

===Applied to the Envelope===
And as we have also summarized in the same [[SSC/Stability/BiPolytropes#Profile|accompanying discussion]], throughout the envelope we have,
<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>
<td align="center">&nbsp; &nbsp; </td>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1}
\, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; </td>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr)
\, .
</math>
  </td>
</tr>
</table>
So the relevant ''envelope'' LAWE becomes,

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

<tr>
  <td align="right">
<math>~\mathrm{LAWE}</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}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^{-2}\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr] \biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i  \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr]
\biggr\}\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^{-2}\frac{1}{\eta} \frac{dx}{d\eta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr]\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g}}{\eta^2} 
\biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i  \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^{-2}\biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{2} \theta^{4}_i (2\pi) \biggr]^{-1} \cdot~ \mathrm{LAWE}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr] \biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i  \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr]
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr]\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{2} \theta^{4}_i (2\pi) \biggr]^{-1}  
~-~\frac{\alpha_\mathrm{g}}{\eta^2} 
\biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr]  \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\eta^2} 
+ \biggl\{ 4 - \biggl[ 2  \biggl(-\frac{d\ln \phi}{d\ln \eta} \biggr) \biggr]
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta} 
+ \biggl\{ \frac{\sigma_c^2}{3\gamma_\mathrm{g}}  \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-5}_i \phi^{-1}\biggr]
~-~\frac{\alpha_\mathrm{g}}{\eta^2} \biggl[ 2 \biggl(- \frac{d\ln \phi}{d\ln \eta} \biggr) \biggr]  \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\eta^2} 
+ \biggl\{ 4 - 2Q_\eta
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta} 
+ \biggl\{ \frac{\sigma_c^2}{3\gamma_\mathrm{g}}  \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-5}_i \phi^{-1}\biggr]
~-~(2Q_\eta)\frac{\alpha_\mathrm{g}}{\eta^2}  \biggr\}  x 
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi(\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{A\sin(\eta - B)}{\eta}</math>
  </td>
<td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~Q_\eta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{d\ln \phi}{d\ln\eta} = \biggl[1 - \eta\cot(\eta-B) \biggr] \, .</math>
  </td>
</tr>
</table>
If we set <math>~\sigma_c^2 = 0</math> and <math>~\gamma_g = \gamma_e = 2 ~~\Rightarrow ~~ \alpha_g = +1</math>, the envelope LAWE simplifies to the form,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~ \biggl(\frac{r^*}{\eta}\biggr)^2 \cdot~ \mathrm{LAWE}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - 2Q_\eta
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta} 
- \biggl\{ \frac{2Q_\eta}{\eta^2}  \biggr\}  x \, .
</math>
  </td>
</tr>
</table>


In [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B|yet another ''Ramblings Appendix'' derivation]] we have explored a trial dimensionless displacement for the envelope of the form,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x_P\biggr|_\mathrm{env} </math>
  </td>
  <td align="left">
<math>~= \frac{3c_0}{\eta^2} \cdot Q_\eta \, .</math>
  </td>
</tr>
</table>
In this case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{3c_0}\cdot \frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\eta^2} \frac{dQ_\eta}{d\eta} - \frac{2Q_\eta}{\eta^3} </math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\eta\sin^2(\eta - B)} + \frac{\cot(\eta-B)}{\eta^2} - \frac{2}{\eta^3}
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{3c_0}\cdot \frac{d^2x_P}{d\eta^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\eta}\biggl[\frac{1}{\eta\sin^2(\eta - B)} + \frac{\cot(\eta-B)}{\eta^2} - \frac{2}{\eta^3} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6}{\eta^4} - \frac{2\cot(\eta-B)}{\eta^3} - \frac{2}{\eta^2\sin^2(\eta-B)} - \frac{2\cos(\eta-B)}{\eta \sin^3(\eta-B)}
\, ,
</math>
  </td>
</tr>
</table>
and [[Appendix/Ramblings/BiPolytrope51AnalyticStability#proof|it can be shown]] that the simplified ''envelope'' LAWE is perfectly satisfied.  Notice that, with this adopted segment of the eigenfunction for the envelope, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_P}{d\ln\eta}\biggr|_\mathrm{env} = \frac{\eta^3}{3c_0 Q_\eta}\cdot \frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\eta^3}{Q_\eta} \biggl\{ \frac{1}{\eta^2}\biggl[\eta -\cot(\eta - B)
+\eta\cot^2(\eta - B) 
\biggr] 
- \frac{2Q_\eta}{\eta^3} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{Q_\eta} \biggl[\eta^2 - \eta \cot(\eta - B)
+\eta^2 \cot^2(\eta - B) 
\biggr] 
- 2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{[\eta^2 - 2 + \eta \cot(\eta - B)+\eta^2 \cot^2(\eta - B) ] }{[1 - \eta\cot(\eta-B)]}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{[\eta^2  - 2\sin^2(\eta-B) + \eta \sin(\eta-B) \cos(\eta - B) ] }{[\sin^2(\eta-B) - \eta \sin(\eta-B) \cos(\eta-B)]}  \, .
</math>
  </td>
</tr>
</table>

==Interface Matching==
According to our [[SSC/Stability/BiPolytropes#Interface_Conditions|accompanying discussion]] of the interface matching condition &#8212; as we presently understand it &#8212; the proper eigenfunction will exhibit a discontinuity in the slope of the dimensionless displacement function such that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_\mathrm{env}}{d\ln \eta} \biggr|_{\eta=\eta_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>~\frac{3}{5}\biggl[ \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} -2 \biggr] \, .</math>
  </td>
</tr>
</table>

=See Also=

* [http://adsabs.harvard.edu/abs/2018Sci...362..201D K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206)], ''A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.''

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