Editing Appendix/Ramblings/PatrickMotl (section)

====Our Numerical Analysis of Bipolytropes Having (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)====
In an [[SSC/Stability/BiPolytropes#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|accompanying discussion]] we have shown that we can integrate the linear adiabatic wave equation (LAWE) &#8212; that is, we effectively have been able to solve the eigenvalue problem &#8212; to obtain the eigenvector associated with marginally unstable models along equilibrium sequences for bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>.  The marginally unstable models that have been identified via this more rigorous approach &#8212; see the orange triangles in the following figure &#8212; fall at different points along each equilibrium sequence than the points that were identified via our free-energy analysis &#8212; see the red-dashed demarcation curve.  Assuming that the algorithm that we developed to integrate the LAWE was basically error-free, we trust the model identifications generated via this more rigorous technique.  

Our analysis indicates that, along the <math>~\mu_e/\mu_c = 1</math> equilibrium sequence, the core-envelope interface of the marginally unstable model is located at <math>~\xi_i = 1.6686460157</math>; in the following figure, the red arrow points to this location along that sequence.  In the brief table that accompanies the figure, we have listed values for the dimensionless specific entropy in the core and, separately, in the envelope, along with the interface location, <math>~\xi_i</math>.


<table border="0" align="center" width="80%"><tr><td align="center">
<table border="1" align="right" cellpadding="8">
<tr><td align="center">'''Figure 8'''</td></tr>
<tr><td align="center"><math>~\gamma_c = 6/5</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>~\gamma_e = 2</math><br />&nbsp;<br />[[File:Entropy04Annotated.png|350px|Entropy distribution]]</td></tr></table>
<table border="1" align="left" cellpadding="8" width="40%">
<tr><td align="center" colspan="4">'''Figure 7'''</td></tr>
<tr>
  <td align="center" colspan="4" width="80%"><b>Initial Model Parameters<br />for<br />LAWE-Determined<br />Marginally Unstable Model</b></td>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" width="25%"><math>~\xi_i</math></td>
  <td align="center" width="25%"><math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math></td>
  <td align="center" width="25%"><math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">1.6686460157</td>
  <td align="center">8.04719</td>
  <td align="center">1.31310</td>
</tr>
<tr>
  <td align="center" colspan="4" width="100%">[[File:MotlLAWEdetermination02.png|350px|LAWE determination of marginally unstable model]]</td>
</tr>
<tr>
  <td align="left" colspan="4" width="100%">For more details, see the accompanying discussion titled, ''[[SSC/Stability/BiPolytropes#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|Eigenvectors from Solution of LAWE]]''</td>
</tr>
</table>
</td></tr></table>