Editing Appendix/Ramblings/BiPolytropeStability (section)

==Model 10==
As we have [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29|reviewed in an accompanying discussion]], equilibrium Model 10 from [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy &amp; Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)] is defined by setting <math>(\xi_i, m) = (2.5646, 1)</math>.  Drawing directly from [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29|our reproduction of their Table 1]], we see that a few relevant structural parameters of Model 10 are, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~6.5252876</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{r_i}{R} = \frac{\xi_i}{\xi_s}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.39302482</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\rho_c}{\bar\rho} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~34.346</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{env}}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5.89 \times 10^{-4}</math>
  </td>
</tr>
</table>
Here we list a few other model parameter values that will aid in our attempt to correctly integrate the LAWE to find various radial oscillation eigenvectors.
<table border="1" cellpadding="5" align="center">

<tr>
  <td align="center" colspan="12">
'''A Sampling of Model 10's Equilibrium Parameter Values'''<sup>&dagger;</sup>
</td>
</tr>
<tr>
  <td align="center">Grid<br />Line</td>
  <td align="center"><math>~\frac{r}{R}</math></td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\eta</math></td>
  <td align="center"><math>~\Delta</math></td>
  <td align="center"><math>~\phi</math></td>
  <td align="center"><math>~- \frac{d\phi}{d\eta}</math></td>
  <td align="center"><math>~r^*</math></td>
  <td align="center"><math>~\rho^*</math></td>
  <td align="center"><math>~P^*</math></td>
  <td align="center"><math>~M_r^*</math></td>
  <td align="center"><math>~g_0^*\equiv \frac{M_r^*}{(r^*)^2}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">25</td>
  <td align="right">0.12093071</td>
  <td align="right">0.789108</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">0.31480842</td>
  <td align="right">0.89940188</td>
  <td align="right">0.80892374</td>
  <td align="right">0.122726799</td>
  <td align="right">1.23835945</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">40</td>
  <td align="right"> 0.19651241</td>
  <td align="right">1.2823</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right"> 0.51156369</td>
  <td align="right"> 0.74761972</td>
  <td align="right"> 0.55893525</td>
  <td align="right"> 0.473819194</td>
  <td align="right"> 1.81056130</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">79</td>
  <td align="right"> 0.393025</td>
  <td align="right">2.5646</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right"> 1.02312737</td>
  <td align="right"> 0.21270605</td>
  <td align="right"> 0.04524386</td>
  <td align="right"> 2.150231108</td>
  <td align="right"> 2.05411964</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">79</td>
  <td align="right"> 0.393025</td>
  <td align="right">&nbsp;</td>
  <td align="right">1.4806725</td>
  <td align="right">2.6746514</td>
  <td align="right">1.000000</td>
  <td align="right">1.112155</td>
  <td align="right"> 1.02312737</td>
  <td align="right"> 0.21270605</td>
  <td align="right"> 0.04524386</td>
  <td align="right"> 2.15023111</td>
  <td align="right"> 2.0541196</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">100</td>
  <td align="right"> 0.49883919</td>
  <td align="right">&nbsp;</td>
  <td align="right">1.8793151</td>
  <td align="right">2.7938569</td>
  <td align="right">0.6505914</td>
  <td align="right">0.69070815</td>
  <td align="right"> 1.2985847</td>
  <td align="right"> 0.0247926</td>
  <td align="right"> 0.0034309</td>
  <td align="right"> 2.15127319</td>
  <td align="right"> 1.2757189</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">150</td>
  <td align="right"> 0.7507782</td>
  <td align="right">&nbsp;</td>
  <td align="right">2.8284641</td>
  <td align="right">2.9982701</td>
  <td align="right">0.2149684</td>
  <td align="right">0.30495637</td>
  <td align="right"> 1.95443562</td>
  <td align="right"> 9.7646E-05</td>
  <td align="right"> 4.4649E-06</td>
  <td align="right"> 2.15149752</td>
  <td align="right">0.563246</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">199</td>
  <td align="right"> 0.9976784</td>
  <td align="right">&nbsp;</td>
  <td align="right">3.7586302</td>
  <td align="right">3.1404305</td>
  <td align="right">0.00150695</td>
  <td align="right">0.17269514</td>
  <td align="right">2.59716948</td>
  <td align="right"> 1.653E-15</td>
  <td align="right"> 5.2984E-19</td>
  <td align="right"> 2.15149876</td>
  <td align="right">0.31896316</td>
</tr>
<tr>
  <td align="left" colspan="12">
<sup>&dagger;</sup>Our chosen (uniform) grid spacing is,
<div align="center">
<math>~\frac{\delta r}{R} = \frac{1}{78}\biggl( \frac{r_i}{R} \biggr) \approx 0.00503878 \, ;</math>
</div>
as a result, the center is at zone 1, the interface is at grid line 79, and the surface is just beyond grid line 199.
  </td>
</tr>
</table>