Editing Appendix/Ramblings/BiPolytrope51AnalyticStability (section)

=====Attempt 1=====
Building on our [[SSC/Stability/InstabilityOnsetOverview#Configurations_Having_an_Index_Less_Than_Three|accompanying discussion of ''Pressure-Truncated Configurations Having a Polytropic Index less than Three'']] &#8212; see, for example, a relevant [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration|''succinct demonstration'']] &#8212; a promising analytic expression is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b}{\eta^2} \biggl[1 - \eta\cot(\eta - C) \biggr] \, ,
</math>
  </td>
</tr>
</table>
where the values of the pair of coefficients, <math>~b</math> and <math>~C</math>, is to be determined.  Most likely, we should set <math>~C = B</math>.

Focusing on the case of <math>~\mu_e/\mu_c = 1</math>, <math>~\gamma_c = 6/5</math>, and <math>~\gamma_e = 2</math>, here are some parameters that we ''think'' we know.
<table border="1" align="center" cellpadding="10">
<tr>
  <th align="center" colspan="5">Relevant Parameters for<br />Marginally Unstable Model <math>~(\Omega^2 = 0)</math> <br />with <math>~\mu_e/\mu_c = 1</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math></th>
</tr>
<tr>
  <td align="right"><math>~\xi_i</math></td>
  <td align="center">&nbsp;</td>
  <td align="center">numerically<br />determined</td>
  <td align="center"><math>~\approx</math></td>
  <td align="left">1.6686460157</td>
</tr>
<tr>
  <td align="right"><math>~x_i</math></td>
  <td align="center"><math>~\equiv</math></td>
  <td align="left"><math>~1 - \frac{\xi_i^2}{15} </math></td>
  <td align="center"><math>~=</math></td>
  <td align="left">0.814374699</td>
</tr>
<tr>
  <td align="right"><math>~\biggl[\frac{d\ln x}{d\ln \xi}\biggr]_i</math></td>
  <td align="center"><math>~\equiv</math></td>
  <td align="left"><math>~- \frac{2\xi_i^2}{(15 - \xi_i^2)}</math></td>
  <td align="center"><math>~=</math></td>
  <td align="left">-0.455871976</td>
</tr>
<tr>
  <td align="right"><math>~\eta_i</math></td>
  <td align="center"><math>~\equiv</math></td>
  <td align="left"><math>~\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3} \xi_i \biggl[1 + \frac{\xi_i^2}{3}\biggr]^{-1 } </math></td>
  <td align="center"><math>~=</math></td>
  <td align="left">1.498957494</td>
</tr>
<tr>
  <td align="right"><math>~B</math></td>
  <td align="center"><math>~\equiv</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="center"><math>~=</math></td>
  <td align="left">- 0.359863579</td>
</tr>
<tr>
  <td align="right"><math>~\eta_s</math></td>
  <td align="center"><math>~\equiv</math></td>
  <td align="left"><math>~\pi + B </math></td>
  <td align="center"><math>~=</math></td>
  <td align="left">2.781729075</td>
</tr>
<tr>
  <td align="right"><math>~\biggl[\frac{d\ln x}{d\ln \eta}\biggr]_i</math></td>
  <td align="center"><math>~\equiv</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>
  <td align="center"><math>~=</math></td>
  <td align="left">-1.473523186</td>
</tr>
<tr>
  <td align="right"><math>~\alpha_e</math></td>
  <td align="center"><math>~\equiv</math></td>
  <td align="left"><math>~3 - \frac{4}{\gamma_e} </math></td>
  <td align="center"><math>~=</math></td>
  <td align="left">+ 1</td>
</tr>
<tr>
  <td align="right"><math>~\biggl[\frac{d\ln x}{d\ln \eta}\biggr]_\mathrm{surf}</math></td>
  <td align="center"><math>~\equiv</math></td>
  <td align="left"><math>~\frac{\cancelto{0}{\Omega^2}}{\gamma_e} - \alpha_e </math></td>
  <td align="center"><math>~=</math></td>
  <td align="left">- 1</td>
</tr>
<tr>
  <td align="right"><math>~b</math></td>
  <td align="center"><math>~\equiv</math></td>
  <td align="left"><math>~\frac{3}{5}\biggl(\frac{\mu_e}{\mu_c}\biggr)  \biggl[\frac{15-\xi_i^2}{3+\xi_i^2}\biggr]</math></td>
  <td align="center"><math>~=</math></td>
  <td align="left">1.26097406</td>
</tr>
</table>

Notice that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cot(\eta_i - B)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan\biggl[ \frac{\pi}{2} -(\eta_i - B ) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \eta_i \cot(\eta_i - B)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~\biggl[ 1 - \frac{\xi_i \eta_i}{\sqrt{3}} \biggr] </math>
  </td>
</tr>

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

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

So if we adopt the expression for <math>~x_P</math> as given above (with C = B), then we can evaluate the leading ''b'' factor by examining the function at the interface, that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_i \eta_i^2 \biggl[1 - \eta_i \cot(\eta_i - B) \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_i \eta_i^2 \biggl[\frac{\xi_i \eta_i }{\sqrt{3}} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\eta_i \biggl[\frac{\sqrt{3}}{\xi_i} \biggr]x_i 
</math>
  </td>
</tr>

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

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

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

<table border="0" align="right">
<tr>
  <th align="center">Figure 7:  Analytic Trial</th>
</tr>
<tr><td align="center">[[File:AnalyticAttempt1.png|400px|First Analytic Trial]]</td></tr>
</table>
An even clearer way of looking at this is to realize that, quite generally,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\eta - B)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta - \eta_i + \frac{\pi}{2} - \tan^{-1} f \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}} \, .</math>
  </td>
</tr>
</table>
Hence, we can write,
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tan(\eta_i - \eta) + f}{1 - f\tan(\eta_i - \eta)} \, .</math>
  </td>
</tr>
</table>
As a result, we can rewrite the expression for our ''guess'' of the envelope segment of the eigenfunction in the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b}{\eta^2} \biggl\{ 1 - \eta \biggl[ \frac{\tan(\eta_i - \eta) + f}{1 - f\tan(\eta_i - \eta)} \biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>
This blows up when <math>\eta \rightarrow \eta_s</math> because, as it turns out, <math>~f = 1/\tan(\eta_i - \eta_s)</math>.  We should point out, as well, that the expression for ''b'' can be rewritten in the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\eta_i^2 (x_P)_i}{(1 - \eta_i f)} \, .
</math>
  </td>
</tr>
</table>

In Figure7 we have reprinted the numerically determined fundamental-mode eigenfunction that was first displayed in [[#Eigenfunction_Details|Figure 5, above]].  We have added to this plot the eigenfunction segments that are defined by our trial analytic functions:  The ''core'' segment, <math>~x_P|_\mathrm{core}</math>, matches the numerically determined segment with sufficient precision that the two curve segments are indiscernible from one another.  However, our analytically defined "env" segment,  <math>~x_P|_\mathrm{env}</math> &#8212; identified in Figure 7 by the solid black, small circular markers &#8212; does not match the numerically determined envelope segment at all.  We therefore have more work to do!