Editing SSC/Stability/BiPolytrope00 (section)

==Two Separate LAWEs==

In an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{ Math/EQ_RadialPulsation01 }}
</div>

<!--
<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave Equation'''</font><br />

<math>
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x = 0 \, ,
</math>
</div>
-->
For both regions of the bipolytrope, we define the dimensionless (Lagrangian) radial coordinate,
<div align="center">
<math>\xi  \equiv \frac{r_0}{r_i} \, .</math>
</div>
So, the interface is, by definition, located at <math>\xi = 1</math>; and, the surface is necessarily at <math>\xi = q^{-1}</math>.  As the material in the bipolytrope's core (envelope) is compressed/de-compressed during a radial oscillation, we will assume that heating/cooling occurs in a manner prescribed by an adiabat of index <math>\gamma_c ~(\gamma_e)</math>; in general, <math>~\gamma_e \ne \gamma_c</math>.  For convenience, we will also adopt the frequently used shorthand "alpha" notation,
<div align="center">
<math>~\alpha_c \equiv 3 - \frac{4}{\gamma_c} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\alpha_e \equiv 3 - \frac{4}{\gamma_e} \, .</math>
</div>

===The Core's LAWE===
After adopting, for convenience, the function notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>g^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
1  + \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]  \, ,
</math>
  </td>
</tr>
</table>
</div>
we [[SSC/Stability/BiPolytrope00Details#Match_Prasad-like_Envelope_Eigenvector_to_the_Core_Eigenvector|have deduced]] that, for the core, the LAWE may be written in 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 - \eta^2)\frac{d^2x}{d\eta^2} +  
( 4 - 6\eta^2 )  \frac{1}{\eta} \cdot \frac{dx}{d\eta} 
+ \mathfrak{F}_\mathrm{core} x \, .
</math>
  </td>
</tr>
</table>
</div>

where,

<div align="center">
<math>\eta \equiv \frac{\xi}{g} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c\, .</math>
</div>

Not surprisingly, this is identical in form to the eigenvalue problem that was first presented &#8212; and solved analytically &#8212; by {{ Sterne37full }} in connection with his examination of radial oscillations in ''isolated'' uniform-density spheres.  As is demonstrated below, for the core of our zero-zero bipolytrope, we can in principle adopt any one of the [[SSC/Stability/UniformDensity#Sterne.27s_General_Solution|polynomial eigenfunctions and corresponding eigenfrequencies]] derived by {{ Sterne37 }}.  


===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,
<div align="center">
<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,
<div align="center">
<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,
<div align="center">
<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.