Editing Appendix/Ramblings/BiPolytrope51AnalyticStability (section)

=====Attempt 2=====

Using an Excel spreadsheet as a sandbox, we employed crude, brute force iterations in an effort to fit the numerically constructed envelope eigenfunction.  Here is a trial function that works pretty well.  Using <math>~\eta_\mathrm{F}</math> to represent the envelope's dimensionless radial coordinate, over the range, 
<div align="center">
<math>~\eta_i \le \eta_\mathrm{F} \le \eta_s \, ,</math>
</div>
and defining the parameter,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_\mathrm{F}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\pi}{8(\eta_s - \eta_i)}
\, ,</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="5" align="right">
<tr>
  <th align="center" colspan="4">Limiting Parameter Values</th>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">min</td>
  <td align="center">max</td>
  <td align="center"><math>~\alpha = \alpha_s</math>
</tr>
<tr>
  <td align="center"><math>~\eta_\mathrm{F}</math></td>
  <td align="center"><math>~\eta_i</math></td>
  <td align="center"><math>~\eta_s</math></td>
  <td align="center"><math>~\frac{8}{\pi} ( \eta_s - \eta_i )^2 + 2\eta_s - \eta_i</math></td>
</tr>
<tr>
  <td align="center"><math>~\alpha</math></td>
  <td align="center"><math>~-\frac{\pi}{2}</math></td>
  <td align="center"><math>~-\frac{5\pi}{8}</math></td>
  <td align="center"><math>~\eta_i - \eta_s - \frac{3\pi}{4}</math></td>
</tr>
<tr>
  <td align="center"><math>~\Lambda</math></td>
  <td align="center"><math>~\eta_i - \frac{\pi}{4}</math></td>
  <td align="center"><math>~\eta_i - \frac{\pi}{8}</math></td>
  <td align="center"><math>~\eta_s</math></td>
</tr>
</table>
we propose,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\mathrm{trial}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b_0}{\Lambda^2} \biggl\{ 1 - \Lambda \biggl[ \frac{\tan(\alpha) + f_\alpha}{1 - f_\alpha \cdot \tan(\alpha)} \biggr] \biggr\} - a_0 \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{f_\alpha} = \tan(\alpha_s)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\tan[ - (\eta_s - \eta_i + \tfrac{3\pi}{4}) ]
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\alpha(\eta_\mathrm{F})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~g_\mathrm{F}  \biggl[ 5\eta_i - 4\eta_s - \eta_\mathrm{F}\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Lambda(\eta_\mathrm{F})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\eta_i + g_\mathrm{F}  \biggl[ \eta_i - 2\eta_s + \eta_\mathrm{F} \biggr] 
\, .</math>
  </td>
</tr>
</table>

<table border="0" align="right">
<tr>
  <th align="center">Figure 8:  Another Analytic Trial</th>
</tr>
<tr><td align="center">[[File:AnalyticAttempt2.png|400px|Second Analytic Trial]]</td></tr>
</table>
This function, <math>~x_\mathrm{trial}</math>, is displayed as the black-dotted curve segment in Figure 8 with the tuning/scaling parameters set to the values, <math>~(a_0, b_0) = (0.31, 0.96)</math>.  We should point out that, when plotting this curve segment in Figure 8, the dimensionless radial coordinate has been defined by the relation, <math>~r^*/R^* = \eta_F/\eta_s</math>.

<font color="red">'''[16 February 2019:  Comment by Tohline]'''</font>  When assessed visually, this ''trial'' function appears to match pretty well the numerically derived eigenfunction for the envelope.  We have not yet critically assessed whether or not the function satisfies the LAWE or whether it satisfies either one (or both) of the required boundary values.  This work is still to be done.

A couple of days after inserting this <font color="red">''Comment''</font>, we recognized for the first time that, quite generally,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_i - \biggl(\Lambda + \frac{3\pi}{4} \biggr) \, .</math>
  </td>
</tr>
</table>
Hence, the parameter, <math>~\alpha</math>, can be straightforwardly removed from the expression for the trial eigenfunction to give,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\mathrm{trial}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b_0}{\Lambda^2} \biggl\{ 1 - \Lambda \biggl[ \frac{\tan(\eta_i - \Lambda - 3\pi/4) + f_\alpha}{1 - f_\alpha \cdot \tan(\eta_i - \Lambda - 3\pi/4)} \biggr] \biggr\} - a_0 \, .
</math>
  </td>
</tr>
</table>


Drawing from our [[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|accompanying discussion of pressure-truncated polytropes]], we need the eigenfunction to satisfy the,
<div align="center">
<font color="maroon"><b>Polytropic LAWE (linear adiabatic wave equation)</b></font><br />
{{ Math/EQ_RadialPulsation02 }}
</div>
(Note that, in order to bring the notation of this ''Key Equation'' in line with the notation used elsewhere in this chapter, we will hereafter adopt the variable mapping <math>~\xi \rightarrow \eta</math> and <math>~\theta \rightarrow \phi</math>.)  Here we are especially focused on finding a solution in the case where <math>~\sigma_c^2 = 0</math> and <math>~n = 1</math>, that is &#8212; see also our [[#Envelope:|above discussion]] &#8212; the relevant envelope 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[ 2 - Q \biggr] \frac{2}{\eta}\cdot \frac{dx}{d\eta} -2\alpha_g Q \cdot \frac{x}{\eta^2} \, ,
</math>
  </td>
</tr>
</table>
where, drawing from our [[SSC/Structure/BiPolytropes/Analytic51#Step_6:__Envelope_Solution|discussion of the n = 1 envelope's equilibrium structure]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ - \frac{d\ln\phi}{d\ln\eta} \biggr]_\mathrm{n=1} = - \frac{\eta}{\phi}\biggl[ \frac{d\phi}{d\eta} \biggr]_\mathrm{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{1}{\sin(\eta - B)} \biggr]\biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 -  \eta\cot(\eta-B) \biggr] \, .</math>
  </td>
</tr>
</table>

In an [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration|accompanying discussion]] &#8212; see also a [[SSC/Stability/InstabilityOnsetOverview#Configurations_Having_an_Index_Less_Than_Three|short summary of the same]] &#8212; we have shown that an analytically specified displacement function that precisely satisfies this LAWE for pressure-truncated configurations (i.e., when B = 0) 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{3}{\eta^2}\biggl[ 1- \eta \cot\eta  \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

In still [[SSC/Stability/n1PolytropeLAWE#What_About_Bipolytropes.3F|another related discussion]], we have attempted to construct an analytic eigenfunction expression that satisfies the LAWE when <math>~B \ne 0</math>.