Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

==Free-Energy Stability Evaluation==

Here we pull together excerpts from several different [[H_BookTiledMenu#Tiled_Menu|H_Book Chapters]] in which we have presented, from several different perspectives, an analysis of the free-energy of bipolytropes.  
* In [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|one chapter]], using purely analytic techniques, we have derived expressions that detail the structural properties of bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>.  Among these are analytic expressions for various terms that make up the free-energy expression:  <math>~\mathfrak{s}_\mathrm{core}</math>, <math>~\mathfrak{s}_\mathrm{env}</math>, <math>~\mathfrak{w}_\mathrm{core}</math>, <math>~\mathfrak{w}_\mathrm{core}</math>, and <math>~P_iV_\mathrm{core}</math>. Equilibrium model ''sequences'' are defined by fixing the ratio, <math>~\mu_e/\mu_c</math>, then varying the radial location, <math>~r_i</math>, of the core-envelope interface; note that the volume of the core is, then, <math>~V_\mathrm{core} \equiv 4\pi r_i^3/3</math>.  
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* <font color="red">[Virial Equilibrium]</font> In a subsection of this same chapter titled, ''[[SSC/Structure/BiPolytropes/Analytic51#Global|Equilibrium Condition: Global]]'', we have shown that a statement of virial equilibrium &#8212; obtained by setting the first derivative of the free-energy expression to zero &#8212; is,<table border="0" cellpadding="4" align="center">
<tr>
  <td align="right">
<math>~( 2S + W )_\mathrm{tot} ~=~ 2(S_\mathrm{core} + S_\mathrm{env}) + (W_\mathrm{core} + W_\mathrm{env})</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 0 \, .</math>
  </td>
</tr>
</table>In another subsection of this same chapter titled, ''[[SSC/Structure/BiPolytropes/Analytic51#Global|Equilibrium Condition: In Parts]]'', we showed that for each bipolytropic equilibrium structure, the statements
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~2S_\mathrm{core} + W_\mathrm{core} = 3P_i V_\mathrm{core}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~2S_\mathrm{env} + W_\mathrm{env} = - 3P_i V_\mathrm{core}  \, ,</math>
  </td>
</tr>
</table>also hold separately. Therefore, for every equilibrium configuration we should expect the <b>CASE1</b> expression (see Table XXX) to precisely sum to unity.
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* <font color="red">[Marginally Unstable Model]</font> Near the bottom of this same chapter, in a subsection titled, ''[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Stability Condition]]'', we point out that the model along each sequence that is marginally (dynamically) unstable &#8212; obtained setting the second derivative of the free-energy expression to zero &#8212; is identified by the configuration for which,<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
2(\gamma_e - \gamma_c) \mathfrak{s}_\mathrm{core} + ( \mathfrak{w}_\mathrm{core} +  \mathfrak{w}_\mathrm{env} )(\gamma_e - \tfrac{4}{3})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  \, .</math>
  </td>
</tr>
</table>Therefore, along each equilibrium sequence, the marginally unstable model can be identified by the configuration for which the <b>CASE2</b> expression (see Table XXX) precisely sums to zero.  Immediately above, in a subsection titled, ''[[SSC/Stability/BiPolytropes#What_to_Expect_for_Equilibrium_Configurations|What to Expect for Equilibrium Configurations]]'', we have shown that this same marginally unstable model can be identified by the configuration for which the <b>CASE3</b> energy ratio,  <math>~\mathfrak{s}_\mathrm{core}/\mathfrak{s}_\mathrm{env}</math>, has a value that is precisely 5. And, thirdly, as highlighted in our accompanying [[SSC/SynopsisStyleSheet#Bipolytropes|Tabular Overview]], this same marginally unstable model can be identified by the configuration for which the <b>CASE4</b> expression precisely sums to zero.
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<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="16">
'''Table XXX:'''  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 Free-Energy Arguments
  </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>~P_i</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>
  <td align="center"><math>~3P_i V_\mathrm{core}</math></td>
  <td align="center"><b>CASE1</b></td>
  <td align="center"><b>CASE2</b></td>
  <td align="center"><b>CASE3</b></td>
  <td align="center"><b>CASE4</b></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">2.41610822</td>
  <td align="right">2.8049</td>
  <td align="center">0.59520261</td>
  <td align="center">0.68306067</td>
  <td align="center">16.3788</td>
  <td align="center">0.039116848</td>
  <td align="center">4.446748782</td>
  <td align="center">- 6.606135366</td>
  <td align="center">0.889349762</td>
  <td align="center">- 4.066061722</td>
  <td align="center">2.287362198</td>
  <td align="center">1</td>
  <td align="center">0</td>
  <td align="center">5.0000000</td>
  <td align="center">2.1 &times; 10<sup>-8</sup></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">4.1853093</td>
  <td align="right">8.8058</td>
  <td align="center">0.328419479</td>
  <td align="center">0.70131896</td>
  <td align="center">354.089</td>
  <td align="center">0.003126324</td>
  <td align="center">5.76978580</td>
  <td align="center">- 10.58931853</td>
  <td align="center">1.153956968</td>
  <td align="center">- 3.258165567</td>
  <td align="center">0.95025163</td>
  <td align="center">1</td>
  <td align="center">6.3 &times; 10<sup>-8</sup></td>
  <td align="center">5.0000002</td>
  <td align="center">0</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">7.64325</td>
  <td align="right">44.116</td>
  <td align="center">0.119714454</td>
  <td align="center">0.52700045</td>
  <td align="center">2.85 &times; 10<sup>4</sup></td>
  <td align="center">0.000116533</td>
  <td align="center">6.230343527</td>
  <td align="center">- 12.24495934</td>
  <td align="center">1.1246068658</td>
  <td align="center">- 2.707865028</td>
  <td align="center">0.215727713</td>
  <td align="center">1</td>
  <td align="center">6.4 &times; 10<sup>-8</sup></td>
  <td align="center">5.0000002</td>
  <td align="center">0</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">8.548103</td>
  <td align="right">59.643</td>
  <td align="center">0.099032423</td>
  <td align="center">0.47901529</td>
  <td align="center">6.30 &times; 10<sup>4</sup></td>
  <td align="center">6.1337 &times; 10<sup>-5</sup></td>
  <td align="center">6.261548334</td>
  <td align="center">- 12.36425897</td>
  <td align="center">1.252309682</td>
  <td align="center">- 2.663457063</td>
  <td align="center">0.158837699</td>
  <td align="center">1</td>
  <td align="center">0</td>
  <td align="center">4.9999999</td>
  <td align="center">6.0 &times; 10<sup>-8</sup></td>
</tr>
<tr>
  <td align="center">0.316943</td>
  <td align="left">10.7441565</td>
  <td align="right">108.14</td>
  <td align="center">0.068655205</td>
  <td align="center">0.38238387</td>
  <td align="center">2.93 &times; 10<sup>5</sup></td>
  <td align="center">1.6252 &times; 10<sup>-5</sup></td>
  <td align="center">6.301810768</td>
  <td align="center">- 12.52005323</td>
  <td align="center">1.260362204</td>
  <td align="center">- 2.604292714</td>
  <td align="center">0.083568307</td>
  <td align="center">1</td>
  <td align="center">0</td>
  <td align="center">4.9999998</td>
  <td align="center">2.0 &times; 10<sup>-7</sup></td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">12.77156</td>
  <td align="right">166.06</td>
  <td align="center">0.053145011</td>
  <td align="center">0.31696879</td>
  <td align="center">8.70 &times; 10<sup>5</sup></td>
  <td align="center">5.8905 &times; 10<sup>-6</sup></td>
  <td align="center">6.318902171</td>
  <td align="center">- 12.58692884</td>
  <td align="center">1.26378042</td>
  <td align="center">- 2.57843634</td>
  <td align="center">0.050875500</td>
  <td align="center">1</td>
  <td align="center">1.9 &times; 10<sup>-8</sup></td>
  <td align="center">5.0000001</td>
  <td align="center">0</td>
</tr>
<tr>
  <td align="center" colspan="16">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<b>CASE1</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{6P_i V_\mathrm{core}}\biggl[
(2 \mathfrak{s}_\mathrm{core} + \mathfrak{w}_\mathrm{core})  
-(2 \mathfrak{s}_\mathrm{core} + \mathfrak{w}_\mathrm{core})  
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<b>CASE2</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
2(\gamma_e - \gamma_c) \mathfrak{s}_\mathrm{core} + ( \mathfrak{w}_\mathrm{core} +  \mathfrak{w}_\mathrm{env} )(\gamma_e - \tfrac{4}{3})
</math>
  </td>
</tr>

<tr>
  <td align="right">
<b>CASE3</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\mathfrak{s}_\mathrm{core}}{\mathfrak{s}_\mathrm{env}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<b>CASE4</b>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
(4 - 3\gamma_c)\mathfrak{w}_\mathrm{core} + (4 - 3\gamma_e)\mathfrak{w}_\mathrm{env} + 3^2(\gamma_c - \gamma_e)P_i V_\mathrm{core}
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>
</div>


The left-hand panel of Figure 3 is identical to Figure 2, above.  It displays in the <math>~q-\nu</math> parameter space, the behavior of <math>~(n_c, n_e) = (5, 1)</math> bipolytropic equilibrium sequences that have, as labeled, seven different values of the ratio of mean-molecular-weights, <math>~\mu_e/\mu_c</math>.  Using a numerical root-finding technique, we have determined where the virial stability condition, <math>~\mathfrak{s}_\mathrm{core}/\mathfrak{s}_\mathrm{env} = 5</math>, is satisfied along each of these sequences &#8212; as well as along a number of additional equilibrium sequences.  Key properties of each of these identified models have been recorded in [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Table 1 of an accompanying discussion]]; see also an [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|associated discussion of the free-energy of these configurations]].  Pulling from this tabulated data, the solid-red circular markers that appear in the right-hand panel of Figure 3 identify where this virial stability condition is satisfied along the separate equilibrium sequences while the accompanying dashed red curve identifies more broadly how the <math>~q-\nu</math> parameter space is divided into stable (below and to the right) versus unstable (above and to the left) regions.  

<table border="0" align="center" cellpadding="8">
<tr>
  <th align="center">Figure 3</th>
</tr>
<tr>
  <td align="center">
[[File:CompositeAlabeled.png|800px|dynamical stability in qNu plane]]
  </td>
</tr>
</table>

In what follows we use a complementary &#8212; and more quantitatively rigorous &#8212; approach to evaluating the stability of equilibrium models, and contrast the results of that analysis with the virial-analysis results presented graphically here in Figure 3.