Editing Appendix/Ramblings/BiPolytrope51AnalyticStability (section)

===Equilibrium Properties of Marginally Unstable Models===
Table 2 summarizes some of the equilibrium properties of these six models.  For example, the second column of the table gives the value of the core-envelope interface radius, <math>~\xi_i</math>, associated with each marginally unstable model.  The table also lists:  the value of the model's dimensionless radius, <math>~R^*_\mathrm{surf}</math>, the key structural parameters, <math>~q</math> &amp; <math>~\nu</math>, and the central-to-mean density associated with each model; and in each case the dimensionless thermal energy <math>~(\mathfrak{s})</math> and dimensionless gravitational potential energy <math>~(\mathfrak{w})</math> associated, separately, with the core and the envelope.  Note that, once the pair of parameters, <math>~(\mu_e/\mu_c, \xi_i)</math>, has been specified, we can legitimately assign high-precision values to all of the other model parameters because they are [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|analytically prescribed]].

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="10">
'''Table 2:'''  Properties of Marginally Unstable Bipolytropes Having<br /><br /><math>~(n_c, n_e) = (5, 1)</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math><br /><br />Determined from Integration of the LAWE
  </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"><math>~R^*_\mathrm{surf}</math></td>
  <td align="center"><math>~q \equiv \frac{r_\mathrm{core}}{R_\mathrm{surf}}</math></td>
  <td align="center"><math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{env}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">2.139737</td>
  <td align="center">0.53885819</td>
  <td align="center">0.497747626</td>
  <td align="center">8.51704656</td>
  <td align="center">3.021916335</td>
  <td align="center">-3.356583022</td>
  <td align="center">1.47780476</td>
  <td align="center">-5.642859167</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">5.146499</td>
  <td align="center">0.306021732</td>
  <td align="center">0.401776274</td>
  <td align="center">63.29514949</td>
  <td align="center">4.241287117</td>
  <td align="center">-6.074241035</td>
  <td align="center">4.284931508</td>
  <td align="center">-10.97819621</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">9.554041</td>
  <td align="center">0.185160563</td>
  <td align="center">0.234302525</td>
  <td align="center">209.7739052</td>
  <td align="center">4.639705843</td>
  <td align="center">-7.125754184</td>
  <td align="center">11.72861751</td>
  <td align="center">-25.61089252</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">10.120558</td>
  <td align="center">0.176288391</td>
  <td align="center">0.218241608</td>
  <td align="center">230.4125398</td>
  <td align="center">4.667042505</td>
  <td align="center">-7.200966267</td>
  <td align="center">13.15887139</td>
  <td align="center">-28.45086152</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="center">11.464303</td>
  <td align="center">0.158362807</td>
  <td align="center">0.184796947</td>
  <td align="center">279.0788798</td>
  <td align="center">4.722277318</td>
  <td align="center">-7.354156963</td>
  <td align="center">17.1374434</td>
  <td align="center">-36.36528446</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="center">15.895632</td>
  <td align="center">0.118924863</td>
  <td align="center">0.11071211</td>
  <td align="center">430.0444648</td>
  <td align="center">4.84592201</td>
  <td align="center">-7.70305421</td>
  <td align="center">37.84289623</td>
  <td align="center">-77.67458196</td>
</tr>
</table>
</div>

<!--
OLD VERSION; IGNORE!

<div align="center">
<font color="red">'''OLD Table 2'''</font>

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="9">
Properties of Marginally Unstable Bipolytropes Having<br /><br /><math>~(n_c, n_e) = (5, 1)</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math><br /><br />Determined from Integration of the LAWE
  </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"><math>~q \equiv \frac{r_\mathrm{core}}{R_\mathrm{surf}}</math></td>
  <td align="center"><math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{core}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">2.467359668</td>
  <td align="center">0.597684036</td>
  <td align="center">0.692367564</td>
  <td align="center">17.09749847</td>
  <td align="center">4.518031091</td>
  <td align="center">-6.79580606</td>
  <td align="center">0.857904827</td>
  <td align="center">-3.956065776</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.65925887</td>
  <td align="center">0.314759926</td>
  <td align="center">0.481811565</td>
  <td align="center">95.04044773</td>
  <td align="center">4.759771212</td>
  <td align="center">-7.459080087</td>
  <td align="center">3.20977926</td>
  <td align="center">-8.480020858</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.77457272408</td>
  <td align="center">0.183201946</td>
  <td align="center">0.25725514</td>
  <td align="center">281.7482802</td>
  <td align="center">4.887554727</td>
  <td align="center">-7.822410223</td>
  <td align="center">10.22665208</td>
  <td align="center">-22.40600339</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.7843424754</td>
  <td align="center">0.17412463</td>
  <td align="center">0.237677946</td>
  <td align="center">305.4881577</td>
  <td align="center">4.897826446</td>
  <td align="center">-7.852004624</td>
  <td align="center">11.6114224</td>
  <td align="center">-25.16649306</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.8050375512</td>
  <td align="center">0.156021514</td>
  <td align="center">0.197918988</td>
  <td align="center">359.3276918</td>
  <td align="center">4.91930991</td>
  <td align="center">-7.914090174</td>
  <td align="center">15.49005222</td>
  <td align="center">-32.90463409</td>
</tr>
</table>
</div>

OLD VERSION; IGNORE!
-->

As was expected from our [[#What_to_Expect_for_Equilibrium_Configurations|above discussion of virial equilibrium conditions]], we found that to high precision for each of these equilibrium models,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
(\mathfrak{w}_\mathrm{core}  
~+~\mathfrak{w}_\mathrm{env})
~+~2(\mathfrak{s}_\mathrm{core}  
~+~\mathfrak{s}_\mathrm{env})  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
<span id="Figure4">However, contrary to expectations,</span> in no case did we find that <math>~\mathfrak{s}_\mathrm{core}/\mathfrak{s}_\mathrm{env} = 5</math>.  That is to say, we found that ''none'' of the models lies on the (red-dashed) curve in the <math>~q-\nu</math> parameter space that separates stable from unstable models as defined by our [[#What_to_Expect_for_Equilibrium_Configurations|above free-energy-based stability analysis]]. The left-hand panel of Figure 4 shows this (red-dashed) demarcation curve; for all intents and purposes, it is a reproduction of the right-hand panel of [[#Virial_Stability_Evaluation|Figure 3, above]] &#8212; turning-point markers have been removed to minimize clutter, the equilibrium sequences have been labeled, and the horizontal axis has been extended to unity in order to include a longer portion of the <math>~\mu_e/\mu_c = 1</math> sequence.  The orange triangular markers that appear in the right-hand panel of Figure 4 pinpoint where each of the Table 2 "marginally unstable" models resides in this <math>~q-\nu</math> plane.  Clearly, all six of the orange triangles lie well off of &#8212; and to the ''stable'' side of &#8212; the red-dashed demarcation curve.  This discrepancy, which has resulted from our use of two separate approaches to stability analysis, will be discussed further and gratifyingly resolved, below.

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

<tr>
  <th align="center">Figure 4</th>
</tr>
<tr>
  <td align="center" colspan="10">[[File:NEWCompositeDlabeled.png|800px|Marginally unstable models]]</td>
</tr>
</table>