Editing SSC/Stability/BiPolytrope00 (section)

===The Envelope's LAWE===

Subsequently, we also [[SSC/Stability/BiPolytrope00Details#More_General_Solution|have deduced]] that, for the envelope, the governing LAWE becomes,

<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 + \frac{(g^2-\mathcal{B}) \xi}{\mathcal{A}} - \mathcal{D} \xi^3\biggr] \frac{d^2x}{d\xi^2} 
+ \biggl\{ 3  + \frac{4(g^2-\mathcal{B}) \xi}{\mathcal{A}} - 6\mathcal{D} \xi^3 \biggr\} 
\frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+ \biggl[ 
\mathcal{D}   \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e  
\biggr]\frac{x}{\xi^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{A}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2\biggl(\frac{\rho_e}{\rho_c}\biggr)  \biggl(1 - \frac{\rho_e}{\rho_c} \biggr)  \, ;
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>\mathcal{D}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{\mathcal{A}}\biggl( \frac{\rho_e}{\rho_c}\biggr)^2  = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr]
\, ,
</math>
  </td>
</tr>

<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>

<span id="KeyConstraint">We have been unable</span> to demonstrate that this governing equation can be solved analytically for ''arbitrary'' pairs of the key model parameters, <math>q</math> and <math>\rho_e/\rho_c</math>.  But, if we choose parameter value pairs that satisfy the constraint,
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<math>g^2 = \mathcal{B} </math>
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp; 
<math>g^2 = \frac{1+8q^3}{(1+2q^3)^2} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; &nbsp; 
<math>q^3 = \mathcal{D} =  \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr] </math>
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\frac{\rho_e}{\rho_c} = \frac{2q^3}{1+2q^3} \, ,</math>
</div>

then the LAWE that is relevant to the envelope simplifies.  Specifically, it takes the form,

<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>
( 1  - q^3 \xi^3 ) \frac{d^2x}{d\xi^2} + ( 3  - 6q^3 \xi^3 ) \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>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{x}{\xi^2}\biggl\{
( 1  - q^3 \xi^3 ) 
\biggl[ \frac{d}{d\ln\xi} \biggl( \frac{d\ln x}{d\ln \xi} \biggr) - \biggl(  1 -  \frac{d\ln x}{d\ln \xi} \biggr)\cdot \frac{d\ln x}{d\ln \xi}\biggr]
+ ( 3  - 6q^3 \xi^3 )  \frac{d\ln x}{d\ln \xi} 
+ \biggl[ q^3  \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e  
\biggr] \biggr\}
\, .
</math>
  </td>
</tr>
</table>
</div>
Shortly after deriving this last expression, we realized that one possible solution is a simple power-law eigenfunction of the form,
<div align="center">
<math>x=a_0 \xi^{c_0} \, ,</math>
</div>
where the (constant) exponent is one of the roots of the quadratic equation,
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<math>c_0^2 + 2c_0 - \alpha_e = 0 \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp; 
<math>c_0 = -1 \pm \sqrt{1+\alpha_e} \, .</math>
</div>
This power-law eigenfunction must be paired with the associated, dimensionless eigenfrequency parameter,
<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(c_0+5) = 3c_0 + \alpha_e</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e}   </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3(c_0 + \alpha_e) = 3[\alpha_e -1 \pm \sqrt{1+\alpha_e}] \, .</math>
  </td>
</tr>
</table>
</div>

Next, [[SSC/Stability/BiPolytrope00Details#Eureka_Regarding_Prasad.27s_1948_Paper|we noticed]] the strong similarities between the mathematical properties of this eigenvalue problem and the one that was studied by {{ Prasad48full }} in connection with, what we now recognize to be, a closely related problem.  Drawing heavily from {{ Prasad48hereafter }}'s analysis, we discovered an infinite number of eigenfunctions (each, a truncated polynomial expression) and associated eigenfrequencies that satisfy this governing envelope LAWE.  The eigenvectors associated with the lowest few modes are tabulated, below.