Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

===Implementation===

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="15">
Table 3  
  </th>
</tr>
<tr>
  <td rowspan="3" align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td rowspan="3" align="center"><math>~\xi_i</math></td>
  <td colspan="6" align="center" bgcolor="lightblue">Core</td>
  <td colspan="6" align="center" bgcolor="lightgreen">Envelope</td>
  <td align="center">Virial</td>
</tr>
<tr>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dS^*_\mathrm{therm}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dW^*_\mathrm{grav}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dS^*_\mathrm{therm}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dW^*_\mathrm{grav}</math></td>
  <td rowspan="2" align="center">Numerical<p></p><math>~\biggl[\frac{2\mathfrak{s}_\mathrm{tot}}{|\mathfrak{w}_\mathrm{tot}|}-1\biggr] </math></td>
</tr>
<tr>
  <td align="center">Analytic<br /><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center">'''TERM1'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center">'''TERM2'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center">'''TERM3'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center">'''TERM4'''</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">3.021916335</td>
  <td align="center">3.021921</td>
  <td align="center">0.116389175</td>
  <td align="center">-3.356583022</td>
  <td align="center">-3.35666</td>
  <td align="center">-2.649752079</td>
  <td align="center">1.47780476</td>
  <td align="center">1.47791</td>
  <td align="center">1.0720821</td>
  <td align="center">-5.642859167</td>
  <td align="center">-5.642820</td>
  <td align="center">-1.91142893</td>
  <td align="center">0.000020</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">4.241287117</td>
  <td align="center">4.241410819</td>
  <td align="center">0.440878529</td>
  <td align="center">-6.074241035</td>
  <td align="center">-6.074317546</td>
  <td align="center">-4.150731169</td>
  <td align="center">4.284931508</td>
  <td align="center">4.28547195</td>
  <td align="center">1.44651932</td>
  <td align="center">-10.97819621</td>
  <td align="center">-10.97847622</td>
  <td align="center">-0.92598634</td>
  <td align="center">0.000057</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">4.639705843</td>
  <td align="center">4.6399114</td>
  <td align="center">0.6794857</td>
  <td align="center">-7.125754184</td>
  <td align="center">-7.125854025</td>
  <td align="center">-4.5487829</td>
  <td align="center">11.72861751</td>
  <td align="center">11.730381</td>
  <td align="center">1.51410084</td>
  <td align="center">-25.61089252</td>
  <td align="center">-25.6115597</td>
  <td align="center">-0.4496513</td>
  <td align="center">0.000097</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">4.667042505</td>
  <td align="center">4.667254935</td>
  <td align="center">0.700414598</td>
  <td align="center">-7.200966267</td>
  <td align="center">-7.201068684</td>
  <td align="center">-4.57274936</td>
  <td align="center">13.15887139</td>
  <td align="center">13.1608467</td>
  <td align="center">1.51408246</td>
  <td align="center">-28.45086152</td>
  <td align="center">-28.45153761</td>
  <td align="center">-0.4170461</td>
  <td align="center">0.000101</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="right">4.722277318</td>
  <td align="center">4.722504339</td>
  <td align="center">0.744964507</td>
  <td align="center">-7.354156963</td>
  <td align="center">-7.3542507</td>
  <td align="center">-4.61961058</td>
  <td align="center">17.1374434</td>
  <td align="center">17.1399773</td>
  <td align="center">1.51055838</td>
  <td align="center">-36.36528446</td>
  <td align="center">-36.36591543</td>
  <td align="center">-0.3524855</td>
  <td align="center">0.000110</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="right">4.84592201</td>
  <td align="center">4.846185027</td>
  <td align="center">0.857001395</td>
  <td align="center">-7.70305421</td>
  <td align="center">-7.703178009</td>
  <td align="center">-4.7163542</td>
  <td align="center">37.84289623</td>
  <td align="center">37.8479208</td>
  <td align="center">1.47966673</td>
  <td align="center">-77.67458196</td>
  <td align="center">-77.67408155</td>
  <td align="center">-0.2194152</td>
  <td align="center">0.000128</td>
</tr>
<tr>
  <td colspan="15" align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{r^*_\mathrm{core}}  dS^*_\mathrm{therm}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM1'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{w}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^{r^*_\mathrm{core}}  dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM2'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_{r^*_\mathrm{core}}^{R^*} dS^*_\mathrm{therm}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM3'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{w}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_{r^*_\mathrm{core}}^{R^*} dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM4'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}</math>
  </td>
</tr>
<tr>
  <td colspan="7" align="center">NOTE:  In all integrals, the fractional radial-displacement function, <math>~x</math>, has been normalized<br />to unity at the center of the spherical model.</td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{2\pi \cdot \mathbf{TERM5} } \biggl\{
\frac{2\gamma_c}{3} [\mathbf{TERM1}]
- (3\gamma_c - 4) [\mathbf{TERM2}]
+ \frac{2 \gamma_e}{3} [\mathbf{TERM3}]
- (3\gamma_e - 4)   [\mathbf{TERM4}]
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^* [3(\gamma_e - \gamma_c)] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.036312577\times \biggl[0.093111340 - 1.059900832 + 1.4294428 - (-3.82285786) - 1.782200484\times (2.4) \biggr] = 0.036312577\times [0.00819] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0002973 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.020704186\times \biggl[0.352702823 - 1.660292468 + 1.928692426 - (-1.85197268) - 1.029029184\times (2.4) \biggr] = 0.020704186\times [0.003405] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0000705 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 0.345}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019377703\times \biggl[0.54358856 - 1.81951316 + 2.01880112 - (-0.8993026) - 0.68747441\times (2.4) \biggr] = 0.019377703\times [0.003585] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0000695 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019467503\times \biggl[0.560331678 - 1.81099744 + 2.018776613 - (-0.8340922) - 0.658354814\times (2.4) \biggr] = 0.019467503\times [1.602203052 - 1.580051555] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.00043123 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 0.309}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019781992\times \biggl[0.595971606 - 1.847844232 + 2.01407784 - (-0.704971) - 0.60905486\times (2.4) \biggr] = 0.019781992\times [1.467176214 - 1.461731665] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0001077 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.021344909\times \biggl[0.685601116 - 1.88654168 + 1.972888973 - (-0.4388304) - 0.499262049\times (2.4) \biggr] = 0.021344909\times [1.210778809 - 1.198228918] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.000267876 \, .</math>
  </td>
</tr>
</table>


<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="11">
Table 4
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center">Analytic<br /><math>~\mathfrak{m}_\mathrm{tot}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{m}_\mathrm{tot}</math></td>
  <td align="center">'''TERM5'''</td>
  <td align="center"><math>~x_i</math></td>
  <td align="center"><math>~r^*_\mathrm{core}</math></td>
  <td align="center"><math>~P^*_i</math></td>
  <td align="center"><math>~\biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}</math></td>
  <td align="center"><math>~\biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env}</math></td>
  <td align="center"><math>~\biggl[\sigma_c^2\biggr]^\mathrm{VP}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">4.818155928</td>
  <td align="center">4.818145</td>
  <td align="center">13.14874521</td>
  <td align="center">0.814374698</td>
  <td align="center">1.153014872</td>
  <td align="center">0.139506172</td>
  <td align="center">+0.455871977</td>
  <td align="center">+1.47352</td>
  <td align="center">0.0002973</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">9.020985415</td>
  <td align="center">9.021084268</td>
  <td align="center">23.0612702</td>
  <td align="center">0.653665497</td>
  <td align="center">1.574940686</td>
  <td align="center">0.049058481</td>
  <td align="center">+1.059668912</td>
  <td align="center">+1.835801347</td>
  <td align="center">0.0000705</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">17.41399388</td>
  <td align="center">17.4141672</td>
  <td align="center">24.63990825</td>
  <td align="center">0.563043202</td>
  <td align="center">1.769031527</td>
  <td align="center">0.030957085</td>
  <td align="center">+1.552125296</td>
  <td align="center">+2.131275177</td>
  <td align="center">0.0000695</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">18.8449906</td>
  <td align="center">18.8451614</td>
  <td align="center">24.52624951</td>
  <td align="center">0.555549156</td>
  <td align="center">1.78413696</td>
  <td align="center">0.029889634</td>
  <td align="center">+1.600041467</td>
  <td align="center">+2.16002488</td>
  <td align="center">0.0004312</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="right">22.61541791</td>
  <td align="center">22.6155686</td>
  <td align="center">24.13633675</td>
  <td align="center">0.539776219</td>
  <td align="center">1.815519219</td>
  <td align="center">0.027798189</td>
  <td align="center">+1.705239188</td>
  <td align="center">+2.223143513</td>
  <td align="center">0.0001077</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="right">39.12088278</td>
  <td align="center">39.1208075</td>
  <td align="center">22.36902623</td>
  <td align="center">0.501037082</td>
  <td align="center">1.890385851</td>
  <td align="center">0.023427588</td>
  <td align="center">+1.991720516</td>
  <td align="center">+2.39503231</td>
  <td align="center">0.0002679</td>
</tr>
<tr>
  <td colspan="11" align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{m}_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}dM_r^*</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~I_\mathrm{sphere}^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}(r^*)^2   dM_r^*</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM5'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}(r^*)^2   x^2 dM_r^*</math>
  </td>
</tr>
<tr>
  <td colspan="11" align="center">NOTE:  In the '''TERM5''' integral (as elsewhere), the fractional radial-displacement function, <math>~x</math>, <br />has been normalized to unity at the center of the spherical model.</td>
</tr>
</table>
  </td>

</tr>
</table>