Editing SSC/Stability/BiPolytrope00Old (section)

===Duh!===

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 4 February 2017:  This numerical determination of surface boundary conditions was carried out inside spreadsheet "FDflex22" of Excel file ''analyticeigenvectorcorrected.xlsx''.]]After also checking conformance with the expected boundary condition in <!-- Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes -->[[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Check_Surface_Boundary_Condition|the case of analytic eigenfunctions having <math>~\ell = 3</math>]] and, separately (not shown), for numerically generated eigenfunctions having a wide range of oscillation frequencies, it dawned on us that the [[#DesiredBoundaryCondition|"desired" surface boundary condition]] may actually be a natural outcome of the envelope's LAWE.  

By constraining our discussion to models for which <math>~g^2 = \mathcal{B}</math> and <math>~\mathcal{D} = q^3</math>, the [[#The_Envelope.27s_LAWE|envelope's LAWE]] is,

<div align="center">
<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[ 1  - q^3 \xi^3\biggr] \frac{d^2x}{d\xi^2} 
+ \biggl\{ 3   - 6q^3 \xi^3 \biggr\} 
\frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+ \biggl[ 
q^3  \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e  
\biggr]\frac{x}{\xi^2} \, .
</math>
  </td>
</tr>
</table>
</div>
At the surface <math>~(\xi = 1/q)</math>, the coefficient of the second derivative term goes to zero, in which case the LAWE reduces in form to,

<div align="center">
<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{3}{\xi} \cdot \frac{dx}{d\xi} 
+ \biggl[ 
\mathfrak{F}_\mathrm{env}  -\alpha_e  
\biggr]\frac{x}{\xi^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 3\cdot \frac{d\ln x}{d\ln \xi}\biggr|_\mathrm{surface} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{F}_\mathrm{env}  -\alpha_e  \, .
</math>
  </td>
</tr>
</table>
</div>

And this is ''precisely'' the condition that derives from the astrophysically reasonable boundary condition that we have [[SSC/Stability/BiPolytrope00Details#Boundary_Condition|discussed separately]] and that has been [[#Expectation|reviewed, above]].