Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

====Is There an Analytic Expression for the Eigenfunction?====
After noticing that, in Figure 6,  the ''envelope'' segments of all of the marginally unstable eigenfunctions merge into the same curve, we began to wonder whether a single expression &#8212; and, even better, an ''analytically defined'' expression &#8212; would perfectly describe the eigenfunction.  We had reason to believe that this might actually be possible because, in [[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|pressure-truncated polytropic configurations, we have derived analytic expressions for the marginally unstable, fundamental-mode eigenfunctions]] of both <math>~n = 5</math> and <math>~n=1</math> systems.

Very quickly, we convinced ourselves that a parabolic function does indeed perfectly match the "core" segment of each displayed eigenfunction.  Specifically, throughout the core <math>~(0 \le \xi \le \xi_i)</math>,
<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>


The envelope segment posed a much greater challenge.  In the context of our [[SSC/Stability/n1PolytropeLAWE#Radial_Oscillations_of_n_.3D_1_Polytropic_Spheres|discussion of ''Radial Oscillations of n = 1 Polytropic Spheres'']], and in an [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Is_There_an_Analytic_Expression_for_the_Eigenfunction.3F|accompanying ''Ramblings Appendix'' chapter]] we have detailed some trial derivations that are mostly blind alleyways.  Twice &#8212; once in [[SSC/Stability/n1PolytropeLAWE#tagJanuary2019|January, 2019]] and again (independently) in [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B|April 2019]] &#8212; we have analytically demonstrated that the following appears to work for the envelope:

Given that the ''Structural Properties'' of the envelope are described by the Lane-Emden function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_0 \biggl[ \frac{\sin(\eta - b_0)}{\eta} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ Q \equiv - \frac{d\ln \phi}{d\ln\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \eta \cot(\eta - b_0) \biggr] \, ,</math>
  </td>
</tr>
</table>
the relevant LAWE is satisfied by the fractional displacement function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3c_0 Q}{\eta^2} \, ,</math>
  </td>
</tr>
</table>
where, <math>~c_0</math> is an arbitrary scale factor.  

<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left">
Note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3c_0}{\eta^2}\biggl[\eta -\cot(\eta - b_0)
+\eta\cot^2(\eta - b_0) 
\biggr] 
- \frac{6c_0 Q}{\eta^3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{d\ln x_P}{d\ln\eta} = \frac{\eta}{x_P}\cdot \frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\eta^3}{3c_0 Q} \biggl\{
\frac{3c_0}{\eta^2}\biggl[\eta -\cot(\eta - b_0)
+\eta\cot^2(\eta - b_0) 
\biggr] 
- \frac{6c_0 Q}{\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} \biggl\{
\biggl[\eta^2 -\eta \cot(\eta - b_0)
+\eta^2\cot^2(\eta - b_0) 
\biggr] 
- 2 Q 
\biggr\}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\eta^2}{Q} + Q - 3 \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>
But, as far as we have been able to determine (as of 16 April 2019), this analytic displacement function does not match the displacement function that has been generated through numerical integration of the LAWE (see the light-green segment of the eigenfunction displayed [[#Eigenfunction_Details|above in Figure 5]]).  It remains unclear whether (a) the numerical integration is at fault, (b) we are imposing an incorrect slope at the core-envelope interface, or ( c) we are misinterpreting how to compare the two separately derived (one, numerical, and the other, analytic) envelope eigenfunctions.