Editing Appendix/Ramblings/PatrickMotl (section)

===Summary===

For three separate equilibrium sequences &#8212; specifically, for the cases of <math>~\mu_e/\mu_c = 1, \tfrac{1}{2}, \tfrac{1}{3}</math> &#8212; the table on the left, below, identifies the location <math>~(\xi_i)</math> along the sequence where the model is marginally unstable toward a global, dynamical instability as ''estimated'' by the our free-energy analysis (column 2) and as determined by our numerical integration of the coupled pair of governing LAWEs (column 6).  For each identified model, the corresponding values of <math>~q, \nu</math> and <math>~\rho_c/\bar\rho</math> are also provided.  Additional details can be found in our accompanying discussion of, respectively, our [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|free-energy analysis]] and our numerical solution of the relevant [[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models|eigenvalue problem]].

<table border="0" align="center" width="90%" cellpadding="8"><tr><td align="left">
<table border="1" align="left" cellpadding="9">
<tr>
  <td align="center" colspan="10">Marginally Unstable (dynamically)<br />assuming <math>~\gamma_c = \tfrac{6}{5}</math> and <math>~\gamma_e = 2</math></td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" colspan="4">Determined via<br />[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Free-Energy Analysis]]</td>
  <td align="center" rowspan="5" bgcolor="lightgrey">&nbsp;</td>
  <td align="center" colspan="4">Determined via<br />[[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models|Eigenvector Analysis]]</td>
</tr>
<tr>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~q</math></td>
  <td align="center"><math>~\nu</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~q</math></td>
  <td align="center"><math>~\nu</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">2.416</td>
  <td align="center">0.595</td>
  <td align="center">0.683</td>
  <td align="center">16.4</td>
  <td align="center">1.66865</td>
  <td align="center">0.539</td>
  <td align="center">0.498</td>
  <td align="center">8.42</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="center">4.186</td>
  <td align="center">0.328</td>
  <td align="center">0.701</td>
  <td align="center">354.</td>
  <td align="center">2.27926</td>
  <td align="center">0.185</td>
  <td align="center">0.234</td>
  <td align="center">63.3</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="center">8.548</td>
  <td align="center">0.099</td>
  <td align="center">0.479</td>
  <td align="center">63,000.</td>
  <td align="center">2.58201</td>
  <td align="center">0.176</td>
  <td align="center">0.218</td>
  <td align="center">230.</td>
</tr>
</table>

<table border="1" align="right" cellpadding="5">
<tr><td align="center">'''Figure 9'''</td></tr>
<tr><td align="center"><math>~\gamma_c = 6/5</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>~\gamma_e = 2</math><br />
[[File:SmoothAndUnstable01.png|375px|Convectively stable while dynamically unstable]]
</td></tr></table>
</td></tr></table>

In the figure shown here on the right, the orange-dashed curve shows how <math>~\xi_i</math> (table column 6) varies with <math>~\mu_e/\mu_c</math> (table column 1) for marginally unstable models as determined by our numerical eigenvector analysis.  The square of the fundamental-mode eigenfrequency is negative for all models above this orange-dashed curve, indicating that they should be ''globally'' dynamically unstable.  The solid blue curve shows how <math>~[\xi_i]_\mathrm{smooth}</math> varies with <math>~\mu_e/\mu_c</math>, as [[#Pick_a_Different_Molecular-Weight_Ratio|derived above]].  Models above and to the left of this blue curve should be stable against convection at the core-envelope interface.