Editing SSC/Stability/BiPolytrope00Old (section)

==Examining Alignment with Surface Boundary Condition==

===Expectation===
As we have reviewed in [[SSC/Stability/BiPolytrope00Details#Boundary_Condition|an accompanying discussion]], one astrophysically reasonable surface boundary condition provides a mathematical relationship between the logarithmic derivative of the eigenfunction with respect to the radius, in terms of the eigenfrequency as follows:


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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \xi}\biggr|_{\xi = 1/q}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3\omega^2 }{4\pi G\rho_c \gamma_e} \biggl( \frac{\nu}{q^3}\biggr) - \biggl( 3 - \frac{4}{\gamma_e}\biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3\omega^2 }{4\pi G\rho_c \gamma_e} \biggl( \frac{1+2q^3}{3q^3}\biggr) - \alpha_e </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{3} \biggl[\frac{3\omega^2 }{2\pi G\rho_c \gamma_e} \biggr] \biggl( \frac{\rho_c}{\rho_e}\biggr) - \alpha_e </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{3} \biggl[\mathfrak{F}_\mathrm{env} + 2\alpha_e \biggr]  - \alpha_e </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{3} \biggl[\mathfrak{F}_\mathrm{env} - \alpha_e \biggr]  \, . </math>
  </td>
</tr>
</table>
</div>

Now, according to our [[#Envelope_Segment|above-described envelope segment of the eigenfunction]], we established the analytic prescription,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathfrak{F}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(c_0 + 3\ell)(c_0 + 3\ell+5) \, ,</math>
  </td>
</tr>
</table>
</div>
in which case the desired surface boundary condition is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>3 \cdot \frac{d\ln x}{d\ln \xi}\biggr|_{\xi = 1/q}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(c_0 + 3\ell)(c_0 + 3\ell+5) - \alpha_e  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ [c_0^2 + c_0(6\ell + 5 ) + 3\ell(3\ell+5)] - (c_0^2 + 2c_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 3[ c_0(2\ell + 1 ) + \ell(3\ell+5)]</math>
  </td>
</tr>
</table>
</div>

That is, we expect to find the following,
<div align="center" id="DesiredBoundaryCondition">
<table border="1" align="center"><tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="3">
Desired Boundary Condition
  </th>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln \xi}\biggr|_{\xi = 1/q}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ c_0(2\ell + 1 ) + \ell(3\ell+5) \, .</math>
  </td>
</tr>
</table>

</td></tr></table>
</div>

===Analytic2===
Continuing, from above, a discussion specifically of the case, <math>\ell = 2</math>, the analytically specified envelope eigenfunction is,

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

<tr>
  <td align="right">
<math>x_{\ell=2} |_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{\ell=2} \xi^{3} +  q^6 A_{\ell=2}B_{\ell=2}\xi^{6} }{ 1 +  q^3 A_{\ell=2}  +  q^6 A_{\ell=2}B_{\ell=2}}\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the values of the newly introduced coefficients,
<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>A_{\ell=2}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{c_0(c_0+5) - (c_0 + 6)(c_0 + 11)}{(c_0 + 3)(c_0+5) - \alpha_e}\biggr] = \frac{-2(2c_0+11)}{(2c_0+5)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_{\ell=2}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(c_0+3)(c_0+8) - (c_0 + 6)(c_0 + 11)}{(c_0 + 6)(c_0+8) - \alpha_e}\biggr] = \frac{-(c_0+7)}{2(c_0+4)} \, ,</math>
  </td>
</tr>
</table>
</div>
in which case,

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi} \biggl|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\xi}{x} \biggl\{
c_0\xi^{c_0-1}\biggl[ \frac{ 1 +  q^3 A \xi^{3} +  q^6 AB\xi^{6} }{ 1 +  q^3 A  +  q^6 AB}\biggr] 
+ \xi^{c_0}\biggl[ \frac{ 3q^3 A \xi^{2} +  6q^6 AB\xi^{5} }{ 1 +  q^3 A  +  q^6 AB}\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0
+ \biggl[ \frac{ 3q^3 A \xi^{3} +  6q^6 AB\xi^{6} }{ 1 +  q^3 A\xi^3  +  q^6 AB\xi^6}\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, at the surface <math>(\xi = 1/q)</math>, we find,

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \xi} \biggl|_{\xi=1/q}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0
+\biggl[ \frac{ 3 A  +  6 AB }{ 1 +  A  +  AB}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0
+\biggl[ \frac{ -12(2c_0+11)(c_0+4)  + 12(2c_0+11)(c_0+7) }{ 2(2c_0 + 5)(c_0+4) -  4(2c_0+11)(c_0+4)  +  2(2c_0+11) (c_0+7)}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0
+6\biggl[ \frac{ (2c_0^2 + 25c_0 + 77) -(2c_0^2 + 19c_0 +44)  }{ (2c_0^2 + 13c_0 + 20) -  2(2c_0^2 + 19c_0 + 44)  +  (2c_0^2 + 25c_0 + 77)}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0
+6 \biggl[ \frac{ 6c_0 +33 }{ 9}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
5c_0 + 22 \, .
</math>
  </td>
</tr>
</table>
</div>

It is gratifying &#8212; although, somewhat surprising (to me!) &#8212; to find that this precisely matches the [[#DesiredBoundaryCondition|above-defined, desired boundary condition]] for the case of <math>\ell = 2</math>.

===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]].

===Broader Analysis===

Let's, then, examine the behavior of the envelope's LAWE at the surface in the most general case &#8212; that is, when ''not'' constrained to <math>~g^2 = \mathcal{B}</math>.  First, we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g^2 - \mathcal{B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\rho_e}{\rho_c}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( 1-q \biggr) + 
\frac{\rho_e}{\rho_c} \biggl(\frac{1}{q^2} - 1\biggr) \biggr]  -2\biggl(\frac{\rho_e}{\rho_c}\biggr)  + 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2
</math>
  </td>
</tr>

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

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

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

Hence, at the surface quite generally, the coefficient of the second derivative is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\mathcal{A}}\biggl[\mathcal{A} + (g^2 - \mathcal{B})\xi - \mathcal{A}\mathcal{D} \xi^3 \biggr]_{\xi=1/q}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\mathcal{A}}\biggl\{
2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1-\frac{\rho_e}{\rho_c}\biggr)
+ \frac{1}{q^3} \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[ \biggl( \frac{\rho_e}{\rho_c} \biggr) + 2q^3 \biggl(\frac{\rho_e}{\rho_c}  - 1  \biggr) \biggr] 
- \frac{1}{q^3}\biggl(\frac{\rho_e}{\rho_c}\biggr)^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\mathcal{A}} \biggl(\frac{\rho_e}{\rho_c}\biggr)\biggl\{
- 2\biggl(\frac{\rho_e}{\rho_c} -1\biggr)
+ \frac{1}{q^3}  \biggl[ \biggl( \frac{\rho_e}{\rho_c} \biggr) + 2q^3 \biggl(\frac{\rho_e}{\rho_c}  - 1  \biggr) \biggr] 
- \frac{1}{q^3}\biggl(\frac{\rho_e}{\rho_c}\biggr)
\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>
</div>

And, at the surface quite generally, the coefficient of the first derivative is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\mathcal{A}}\biggl[3\mathcal{A} + 4(g^2 - \mathcal{B})\xi - 6\mathcal{A}\mathcal{D} \xi^3 \biggr]_{\xi=1/q}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\mathcal{A}}\biggl\{
6\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1-\frac{\rho_e}{\rho_c}\biggr)
+ \frac{4}{q^3} \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[ \biggl( \frac{\rho_e}{\rho_c} \biggr) + 2q^3 \biggl(\frac{\rho_e}{\rho_c}  - 1  \biggr) \biggr] 
- \frac{6}{q^3}\biggl(\frac{\rho_e}{\rho_c}\biggr)^2
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{\mathcal{A}} \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[
\biggl(\frac{\rho_e}{\rho_c} - 1\biggr)
- \frac{1}{q^3}\biggl(\frac{\rho_e}{\rho_c}\biggr)
\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, at the surface quite generally, the envelope's LAWE becomes,

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

<tr>
  <td align="right">
<math>~
- \biggl[ 3\mathcal{A}  + 4(g^2-\mathcal{B}) \xi - 6\mathcal{A} \mathcal{D} \xi^3 \biggr]_{\xi=1/q} \frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surface}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathcal{A}\biggl[ 
\mathcal{D}   \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e  \biggr]_{\xi=1/q} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
- 2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[
\biggl(\frac{\rho_e}{\rho_c} - 1\biggr)
- \frac{1}{q^3}\biggl(\frac{\rho_e}{\rho_c}\biggr)
\biggr]  \frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surface}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \mathfrak{F}_\mathrm{env} \cdot \frac{1}{q^3} - 2\alpha_e \biggl(\frac{\rho_e}{\rho_c}\biggr)\biggl(1-  \frac{\rho_e}{\rho_c}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl[2\biggl(\frac{\rho_e}{\rho_c}\biggr) +
2q^3\biggl(1 - \frac{\rho_e}{\rho_c} \biggr)
\biggr]  \frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surface}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho_e}{\rho_c}\biggr) \mathfrak{F}_\mathrm{env}  - 2q^3 \alpha_e \biggl(1-  \frac{\rho_e}{\rho_c}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surface}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[2 + 2q^3\biggl(1 - \frac{\rho_e}{\rho_c} \biggr)\biggl(\frac{\rho_e}{\rho_c}\biggr)^{-1} \biggr]  ^{-1}
\biggl[ \mathfrak{F}_\mathrm{env}  - 2q^3 \alpha_e \biggl(1-  \frac{\rho_e}{\rho_c}\biggr)\biggl(\frac{\rho_e}{\rho_c}\biggr)^{-1} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\mathfrak{F}_\mathrm{env} - \Kappa \alpha_e}{2+\Kappa} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\Kappa \equiv 2q^3\biggl(1 - \frac{\rho_e}{\rho_c} \biggr)\biggl(\frac{\rho_e}{\rho_c}\biggr)^{-1} \, .</math>
</div>

Notice that in the special case for which we have been able to identify analytically specifiable eigenvectors, namely, when
<div align="center">
<math>~g^2 = \mathcal{B} </math>&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; <math>~\Kappa = 1 \, ,</math>
</div>

this surface boundary condition simplifies to the ''expected'' expression,

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

<tr>
  <td align="right">
<math>~
\frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surface}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{3}
\biggl[ \mathfrak{F}_\mathrm{env}  - \alpha_e \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Under what condition &#8212; other than when <math>~g^2=\mathcal{B}</math> &#8212; does the general expression generate the ''expected'' expression?  We need,

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

<tr>
  <td align="right">
<math>~\frac{1}{3}
\biggl[ \mathfrak{F}_\mathrm{env}  - \alpha_e \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[2 + \Kappa \biggr]  ^{-1}
\biggl[ \mathfrak{F}_\mathrm{env}  - \alpha_e \Kappa \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (2 + \Kappa )
\biggl[ \mathfrak{F}_\mathrm{env}  - \alpha_e \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3\biggl[ \mathfrak{F}_\mathrm{env}  - \alpha_e \Kappa \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
(2 + \Kappa ) \mathfrak{F}_\mathrm{env}  - (2 + \Kappa )  \alpha_e 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3\mathfrak{F}_\mathrm{env}  - 3\Kappa \alpha_e 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
(\Kappa -1) \mathfrak{F}_\mathrm{env}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2(\Kappa-1 )  \alpha_e 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\mathfrak{F}_\mathrm{env}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2\alpha_e \, .
</math>
  </td>
</tr>
</table>
</div>

But, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{F}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e}   - 2\alpha_e
\, ,
</math>
  </td>
</tr>
</table>
</div>

we see that the ''expected'' boundary condition will result only for <math>~\omega_\mathrm{env}^2 = 0</math>, that is, only for, <math>~\sigma_c^2 = 0</math>.  This is what we have been noticing as we have played with numerically generated eigenvectors:  When integrating from the center of the zero-zero bipolytrope, to its surface, the naturally resulting (first) derivative of the eigenfunction at the surface of the configuration matches the ''expected'' surface boundary condition &hellip;
* for all values of <math>\sigma_c^2</math>, when <math>~g^2= \mathcal{B}</math>, that is, when <math>~\Kappa=1</math>;
* only for <math>~\sigma_c^2 = 0</math> in all other configurations, that is, for all <math>~\Kappa \ne 1</math>.

What do we make of this?