Editing SSC/Stability/BiPolytropes/51Models (section)

===SearchMuRatio===

Adding models to the [[#Introduction_&_Summary|above table]], here we choose <math>\xi_i</math> and iterate until we have found the value of <math>\mu_e/\mu_c</math> that corresponds to the fundamental-mode.  At the interface, we expect,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\gamma_e \biggl[3 + \biggl(\frac{d\ln x}{d \ln \xi}\biggr)_\mathrm{env}  \biggr]_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\gamma_c \biggl[3 + \biggl(\frac{d\ln x}{d \ln \xi}\biggr)_\mathrm{core}  \biggr]_i \, .
</math>
  </td>
</tr>
</table>
Throughout the core, for the ''neutral'' (i.e., <math>\sigma_c^2 = 0</math>) fundamental mode of oscillation, we expect that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>x_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
1 - \frac{\xi^2}{15}</math> &nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; <math> \frac{dx_\mathrm{core}}{d\xi} = -\frac{2\xi}{15}\, .
</math>
  </td>
</tr>
</table>
Given that <math>(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math> at the interface, we expect,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[\biggl(\frac{d\ln x}{d \ln \xi}\biggr)_\mathrm{env}  \biggr]_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\gamma_c}{\gamma_e} \biggl[3 + \frac{\xi}{x_\mathrm{core}}\biggl(\frac{d x_\mathrm{core}}{d \xi}\biggr)  \biggr]_i -3
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3}{5} \biggl[3 - \frac{15\xi}{(15-\xi^2)}\biggl(\frac{2\xi}{15}\biggr)  \biggr]_i -3
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-\frac{3}{5} \biggl[2+ \frac{2\xi^2}{(15-\xi^2)} \biggr]_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[\frac{18}{\xi_i^2-15} \biggr] \, . 
</math>
  </td>
</tr>
</table>
Similarly at the surface of the envelope for the ''neutral'' (i.e., <math>\sigma_c^2 = 0</math>) fundamental mode of oscillation, we expect that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[\biggl(\frac{d\ln x}{d \ln \xi}\biggr)_\mathrm{env}  \biggr]_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\cancelto{0}{\frac{\sigma_c^2}{4}} \biggl(\frac{\rho_c}{\bar\rho}\biggr) - 1 = -1 \, .
</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" colspan="13">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = ConvectiveBoundary]]<b>Properties of ''Neutral'' Fundamental Mode for Various Sequences</b></td>
</tr>

<tr>
  <td align="center" rowspan="14">[[File:FundModeLocations05Labels.png|500px|Fundamental Model Locations]]</td>
  <td align="center" rowspan="3"><math>\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" rowspan="3"><math>\xi_i</math></td>
  <td align="center" rowspan="3"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center" rowspan="3"><math>\nu \equiv \frac{M_c}{M_\mathrm{tot}}</math></td>
  <td align="center" rowspan="3"><math>q \equiv \frac{r_c}{R}</math></td>
  <td align="center" rowspan="3"><math>\sigma_c^2</math></td>
  <td align="center" colspan="4"><math>[d\ln x/d\ln\xi]_\mathrm{env}</math></td>
</tr>

<tr>
  <td align="center" colspan="2">Interface</td>
  <td align="center" colspan="2">Surface</td>
</tr>

<tr>
  <td align="center" colspan="1">expected<br /><math>18/(\xi_i^2-15)</math></td>
  <td align="center" colspan="1">measured</td>
  <td align="center" colspan="1">expected<br /><math>-1</math></td>
  <td align="center" colspan="1">measured</td>
</tr>

<tr>
  <td align="right">1.000</td>
  <td align="right">1.6639103365</td>
  <td align="right">8.4811731</td>
  <td align="right">0.49622717</td>
  <td align="right">0.53833097</td>
  <td align="right">0.000000</td>
  <td align="right">-1.471622</td>
  <td align="right">-1.471622</td>
  <td align="right">-1</td>
  <td align="right">-1.0062</td>
</tr>

<tr>
  <td align="right">0.681590377</td>
  <td align="right">2.0</td>
  <td align="right">23.176456</td>
  <td align="right">0.476716895</td>
  <td align="right">0.418529653</td>
  <td align="right">0.000000</td>
  <td align="right">-1.636364</td>
  <td align="right">-1.636364</td>
  <td align="right">-1</td>
  <td align="right">-1.0078</td>
</tr>

<tr>
  <td align="right">0.500</td>
  <td align="right">2.2703111897</td>
  <td align="right">62.666493</td>
  <td align="right">0.399760079</td>
  <td align="right">0.305764976</td>
  <td align="right">0.000000</td>
  <td align="right">-1.828212</td>
  <td align="right">-1.828212</td>
  <td align="right">-1</td>
  <td align="right">-1.0093</td>
</tr>

<tr>
  <td align="right">0.425426009</td>
  <td align="right">2.4</td>
  <td align="right">108.10495</td>
  <td align="right">0.332967203</td>
  <td align="right">0.248624189</td>
  <td align="right">0.000000</td>
  <td align="right">-1.948052</td>
  <td align="right">-1.948052 </td>
  <td align="right">-1</td>
  <td align="right">-1.0100</td>
</tr>

<tr>
  <td align="right">0.345</td>
  <td align="right">2.546385206</td>
  <td align="right">205.77394</td>
  <td align="right">0.232779379</td>
  <td align="right">0.185262833</td>
  <td align="right">0.000000</td>
  <td align="right">-2.113688</td>
  <td align="right">-2.113688</td>
  <td align="right">-1</td>
  <td align="right">-1.0108</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{1}{3}</math></td>
  <td align="right">2.5675774773</td>
  <td align="right">225.75664</td>
  <td align="right">0.216806201</td>
  <td align="right">0.176420918</td>
  <td align="right">0.000000</td>
  <td align="right">-2.140934</td>
  <td align="right">-2.140934</td>
  <td align="right">-1</td>
  <td align="right">-1.0110</td>
</tr>

<tr>
  <td align="center"><math>0.310</math></td>
  <td align="right">2.6095097538</td>
  <td align="right">270.59221</td>
  <td align="right">0.184909369</td>
  <td align="right">0.159274</td>
  <td align="right">0.000000</td>
  <td align="right">-2.197679</td>
  <td align="right">-2.197679</td>
  <td align="right">-1</td>
  <td align="right">-1.0112</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{1}{4}</math></td>
  <td align="right">2.712384289</td>
  <td align="right">415.67338</td>
  <td align="right">0.109935743</td>
  <td align="right">0.1192667</td>
  <td align="right">0.000000</td>
  <td align="right">-2.355105</td>
  <td align="right">-2.355105</td>
  <td align="right">-1</td>
  <td align="right">-1.0117</td>
</tr>

<tr>
  <td align="center"><math>0.156419569</math></td>
  <td align="right">2.85</td>
  <td align="right">757.45344</td>
  <td align="right">0.034014631</td>
  <td align="right">0.068440082</td>
  <td align="right">0.000000</td>
  <td align="right">-2.61723</td>
  <td align="right">-2.61723 </td>
  <td align="right">-1</td>
  <td align="right">-1.0123</td>
</tr>

<tr>
  <td align="center"><math>0.067984979</math></td>
  <td align="right">2.95</td>
  <td align="right">1688.1377</td>
  <td align="right">0.005065202</td>
  <td align="right">0.028486668</td>
  <td align="right">0.000000</td>
  <td align="right">-2.858277</td>
  <td align="right">-2.858277</td>
  <td align="right">-1</td>
  <td align="right">-1.0148</td>
</tr>

<tr>
  <td align="center"><math>0.012591194</math></td>
  <td align="right">2.995</td>
  <td align="right">8547.1981</td>
  <td align="right">0.000151797</td>
  <td align="right">0.005211544</td>
  <td align="right">0.000000</td>
  <td align="right">-2.985087</td>
  <td align="right">-2.985087 </td>
  <td align="right">-1</td>
  <td align="right">-1.0132</td>
</tr>
</table>