Editing SSC/Stability/BiPolytrope00 (section)

===Example Solutions===

<div align="center" id="Table3">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="6"><font size="+1">Table 3:</font>&nbsp; Example Analytic Model Parameters for <math>~(\ell,j) = (2,1)</math><br />NOTE:  &nbsp;<math>\mathfrak{F}_\mathrm{core} = 14</math></th></tr>
<tr>
  <td align="center">Eigenfunction</td>
  <td align="center"><math>~\gamma_c ~(n_c)</math></td>
  <td align="center"><math>~\gamma_e</math></td>
  <td align="center"><math>~q</math></td>
  <td align="center"><math>~\frac{\rho_e}{\rho_c}</math></td>
  <td align="center"><math>~\sigma_c^2</math></td>
</tr>
<tr>
  <td>[[File:A21.png|center|thumb|100px|Model A21]]
  <td align="center">&nbsp;<math>~\frac{5}{3} ~~\biggl(\frac{3}{2} \biggr)</math></td>
  <td align="center">1.1340607</td>
  <td align="center">0.6684554</td>
  <td align="center">0.3739731</td>
  <td align="center">25.333333</td>
</tr>
<tr>
  <td>[[File:B21.png|center|thumb|100px|Model B21]]
  <td align="center">&nbsp;<math>~\frac{4}{3} ~~\biggl( 3 \biggr)</math></td>
  <td align="center">1.0263212</td>
  <td align="center">0.6385711</td>
  <td align="center">0.3424445</td>
  <td align="center">18.666666</td>
</tr>
<tr>
  <td>[[File:C21.png|center|thumb|100px|Model C21]]
  <td align="center">&nbsp;<math>~\frac{6}{5} ~~\biggl( 5 \biggr)</math></td>
  <td align="center">1.0028319</td>
  <td align="center">0.6187646</td>
  <td align="center">0.3214875</td>
  <td align="center">16.000000</td>
</tr>
</table>
</div>


It appears as though the eigenvectors (eigenfunction and eigenfrequency) of other radial oscillation modes can be identified by holding all other parameters fixed but changing the value of the quantum number, <math>~\ell</math>, in the [[#OtherSigmas|expression provided below]].  Picking the configuration identified as model '''C21''' in [[#Table3|Table 3]], for example, in addition to the parameter values provided in the table we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha_e = 3 - \frac{4}{\gamma_e} = -0.9887044</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~c_0 = \sqrt{1+\alpha_e} - 1 = -0.8937192 \, ,</math>
  </td>
</tr>
</table>
</div>

so we expect the variation in (the square of) the eigenfrequency, <math>~\sigma_c</math>, with <math>~\ell</math> to be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_c^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~12\biggl[\frac{c_0^2 + c_0(2\ell+3) + \ell(3\ell+5) }{(3-\alpha_e) } \biggr]\frac{\rho_e}{\rho_c} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.9671938[c_0^2 + c_0(2\ell+3) + \ell(3\ell+5) ] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.9671938[-1.8824236 +(3.2125616)\ell + 3\ell^2 ] \, .</math>
  </td>
</tr>
</table>

<div align="center" id="Table4">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="6"><font size="+1">Table 4:</font>&nbsp; Additional Hypothesized Oscillation Modes for Model C21</th>
</tr>
<tr>
  <td align="center"><math>~\ell = 0</math><p></p><math>~\sigma_c^2 = -1.821</math></td>
  <td align="center"><math>~\ell = 1</math><p></p><math>~\sigma_c^2 = +4.188</math></td>
  <td align="center"><math>~\ell = 2</math><p></p><math>~\sigma_c^2 = +16</math></td>
  <td align="center"><math>~\ell = 3</math><p></p><math>~\sigma_c^2 = +33.615</math></td>
  <td align="center"><math>~\ell = 4</math><p></p><math>~\sigma_c^2 = +57.033</math></td>
  <td align="center"><math>~\ell = 5</math><p></p><math>~\sigma_c^2 = +86.255</math></td>
</tr>
<tr>
  <td align="center">[[File:C01hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C11hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C21hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C31hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C41hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C51hypothetical.png|150 px|center]]</td>
</tr>
</table>
</div>

Each of the six plots displayed in [[#Table4|Table 4]] (click on a panel in order to view a larger image) was [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface.  At the interface: &nbsp; the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4|according to the above "Step 4" discussion]].  Each of the graphically illustrated Table 4 eigenfunctions has been scaled in such a way that the central value is unity; note that the panel labeled <math>(\ell=2; \sigma_c^2 = +16)</math> displays an eigenfunction that is identical to the ''analytically defined'' eigenfunction displayed as Model C21 in [[#Table3|Table 3]], but it has been rescaled &#8212; and by necessity inverted &#8212; to provide a central value of unity.


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