Editing SSC/Stability/BiPolytrope00CompareApproaches (section)

==Radial Oscillation Frequencies==

In a [[SSC/Stability/BiPolytrope00#Five_Mode_Summary|separate chapter]], we have summarized some of the quantitative characteristics of five radial oscillation modes that we have determined analytically for bipolytropes that have <math>~(n_c, n_e) = (0,0)</math>.  Table 1 details some of these characteristics; among them is the dimensionless oscillation frequency,
<div align="center">
<math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi \gamma_c G \rho_c} \, .</math>
</div>


<table border="1" cellpadding="6" align="center">
<tr>
  <td align="center" colspan="7"><b>Table 1</b></td>
</tr>
<tr>
  <td align="center" colspan="2">Quantum Numbers</td>
  <td align="center" rowspan="2"><math>~q</math>
  <td align="center" rowspan="2"><math>~\nu</math>
  <td align="center" rowspan="2"><math>~\gamma_c</math>
  <td align="center" rowspan="2"><math>~\gamma_e</math>
  <td align="center" rowspan="2"><math>~\sigma_c^2</math>
</tr>
<tr>
  <td align="center"><math>~\ell</math></td>
  <td align="center"><math>~j</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">1</td>
  <td align="right">0.794385</td>
  <td align="right"> 0.668 </td>
  <td align="right">2.254</td>
  <td align="right">1.194</td>
  <td align="right">16.45</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">2</td>
  <td align="right">0.768375</td>
  <td align="right"> 0.636 </td>
  <td align="right">1.046</td>
  <td align="right">1.209</td>
  <td align="right">34.37</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">1</td>
  <td align="right">0.396061</td>
  <td align="right"> 0.375 </td>
  <td align="right">1.023</td>
  <td align="right">1.344</td>
  <td align="right">12.17</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">2</td>
  <td align="right">0.594040</td>
  <td align="right"> 0.473 </td>
  <td align="right">1.025</td>
  <td align="right">1.056</td>
  <td align="right">34.20</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">3</td>
  <td align="right">0.645515</td>
  <td align="right"> 0.513 </td>
  <td align="right">1.325</td>
  <td align="right">1.840</td>
  <td align="right">65.97</td>
</tr>
</table>

Our [[SSC/Structure/BiPolytropes/Analytic00#Associated_Oscillation_Frequency|free-energy analysis]] of these bipolytropic configurations has shown that each model's characteristic radial oscillation frequency is given by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_\mathfrak{G}^2 \equiv \frac{3\omega_\mathfrak{G}^2}{2\pi \gamma_c G\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3q^2 \nu }{5\gamma_c}\biggl[ 
2( 3\gamma_e - 4)  f 
+ 3(\gamma_e - \gamma_c)(3 - 5 g^2)  \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~f</math> 
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2} - 1 \biggr)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \biggl(\frac{1}{q^5} - 1 \biggr) - \frac{5}{2}\biggl(\frac{1}{q^2} - 1 \biggr) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Here we will restrict our discussion to models that obey the analytic eigenvector constraint, in which case, 

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

<tr>
  <td align="right">
<math>~f</math> 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>1 + \frac{5}{2} \biggl( \frac{2q^3}{1+2q^3} \biggr) \biggl(\frac{1-q^2}{q^2} \biggr)
+ \biggl( \frac{2q^3}{1+2q^3} \biggr)^2 \biggl[ \biggl(\frac{1-q^5}{q^5} \biggr) - \frac{5}{2}\biggl(\frac{1-q^2}{q^2} \biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>1 + 5 \biggl[ \frac{q(1-q^2)}{1+2q^3} \biggr] 
+ \biggl[ \frac{2q}{(1+2q^3)^2} \biggr] \biggl[ 2 (1-q^5 ) - 5q^3 (1-q^2 ) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>1 + 5 \biggl[ \frac{q(1-q^2)}{1+2q^3} \biggr] 
+ \biggl[ \frac{2q}{(1+2q^3)^2} \biggr] \biggl[ 2 - 5q^3 +3q^5  \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\frac{1}{(1+2q^3)^2} \biggl\{ (1 + 4q^3 + 4q^6) + 5 [ q(1-q^2)(1+2q^3) ] 
+ 2q  ( 2 - 5q^3 +3q^5 ) 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\frac{1}{(1+2q^3)^2} \biggl\{ (1 + 4q^3 + 4q^6) 
+ q (5 - 5q^2 + 10q^3 - 10q^5  +  4 - 10q^3 + 6q^5 ) 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\frac{1}{(1+2q^3)^2} \biggl\{ (1 + 4q^3 + 4q^6) 
+ (9q - 5q^3  - 4q^6  ) 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\frac{1}{(1+2q^3)^2} \biggl[ 1 + 9q - q^3   \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(3-5g^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 - 5\biggl[1 + 2\biggl( \frac{2q^3}{1+2q^3} \biggr) - 3\biggl( \frac{2q^3}{1+2q^3} \biggr)^2 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 - \frac{5}{(1+2q^3)^2 }\biggl[(1+2q^3)^2 + 4q^3(1+2q^3) - 12q^6 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(1+2q^3)^2 }\biggl[-2(1+2q^3)^2 - 20q^3(1+2q^3) + 60q^6 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(1+2q^3)^2 }\biggl[-2-8q^3  - 8q^6  -20q^3 - 40q^6 + 60q^6 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2(1 + 14q^3  - 6q^6 ) }{(1+2q^3)^2 } \, .</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\sigma_\mathfrak{G}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3q^2 \nu }{5(1+2q^3)^2}\biggl[ 
2\biggl( \frac{3\gamma_e}{\gamma_c} - \frac{4}{\gamma_c} \biggr)  \biggl( 1 + 9q - q^3  \biggr)
-6 \biggl( \frac{\gamma_e}{\gamma_c} - 1 \biggr)\biggl( 1 + 14q^3  - 6q^6 \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3q^2 \nu }{5(1+2q^3)^2}\biggl[ 
\biggl( - \frac{8}{\gamma_c} \biggr)  \biggl( 1 + 9q - q^3  \biggr)
+\biggl( \frac{6\gamma_e}{\gamma_c} \biggr)  \biggl( 1 + 9q - q^3  
-1 - 14q^3  + 6q^6 \biggr)  
+ 6 \biggl( 1 + 14q^3  - 6q^6 \biggr)  
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3q^2 \nu }{5(1+2q^3)^2}\biggl[ 
6 ( 1 + 14q^3  - 6q^6 ) - \frac{8}{\gamma_c}  \biggl( 1 + 9q - q^3  \biggr)
+6q\biggl( \frac{\gamma_e}{\gamma_c} \biggr)  \biggl( 9  - 15q^2  + 6q^5 \biggr)  
\biggr] 
</math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{q^2 }{5(1+2q^3)}\biggl[ 
6 ( 1 + 14q^3  - 6q^6 ) - \frac{8}{\gamma_c}  \biggl( 1 + 9q - q^3  \biggr)
+6q\biggl( \frac{\gamma_e}{\gamma_c} \biggr)  \biggl( 9  - 15q^2  + 6q^5 \biggr)  
\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, making this last step, we have replaced the leading factor of <math>~\nu</math> with its (above) expression in terms of <math>~q</math>.