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

==Structure==

<ol>
  <li>Individual model profiles, taken from: 
      <ul><li>[[SSC/Structure/BiPolytropes/Analytic51#Examples|SSC/Structure/BiPolytropes/Analytic51#Examples]]</li></ul>
  </li>

  <li><math>(q, \nu)</math> sequences of fixed <math>\mu_e/\mu_c</math>, taken from: 
      <ul><li>[[SSC/Structure/BiPolytropes/Analytic51#Model_Sequences|SSC/Structure/BiPolytropes/Analytic51#Model_Sequences]]</li></ul>
  </li>

  <li><math>\nu_\mathrm{max}</math> model, taken from:
      <ul><li>[[SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass|SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass]]
<br />&nbsp;<br />

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="13">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
  </td>
</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>\theta_i</math>
  </td>
  <td align="center">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>A</math>
  </td>
  <td align="center">
<math>\eta_s</math>
  </td>
  <td align="center">
LHS
  </td>
  <td align="center">
RHS
  </td>
  <td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
  </td>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="right">
<math>[\xi_i]_\mathrm{smooth}</math>  </td>
  <td align="center" rowspan="7">[[File:TurningPoints51BipolytropesLabels.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{3}</math>
  </td>
  <td align="center">
<math>\infty</math> </td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">0.0 </td>
  <td align="center">
<math>\frac{2}{\pi}</math> </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.33
  </td>
  <td align="right">
24.00496  </td>
  <td align="right">
0.0719668  </td>
  <td align="right">
0.0710624  </td>
  <td align="right">
0.2128753  </td>
  <td align="right">
0.0726547  </td>
  <td align="right">
1.8516032  </td>
  <td align="right">
-223.8157  </td>
  <td align="right">
-223.8159  </td>
  <td align="right">
0.038378833  </td>
  <td align="right">
0.52024552  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.316943
  </td>
  <td align="right">
10.744571  </td>
  <td align="right">
0.1591479  </td>
  <td align="right">
0.1493938  </td>
  <td align="right">
0.4903393  </td>
  <td align="right">
0.1663869  </td>
  <td align="right">
2.1760793  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
0.068652714  </td>
  <td align="right">
0.382383875  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.31
  </td>
  <td align="right">
9.014959766  </td>
  <td align="center">
0.1886798  </td>
  <td align="center">
0.172320503  </td>
  <td align="right">
0.59835053  </td>
  <td align="center">
0.20081242  </td>
  <td align="center">
2.2823226  </td>
  <td align="center">
---  </td>
  <td align="center">
---  </td>
  <td align="right">
0.0755022550  </td>
  <td align="right">
0.3372170064  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.3090
  </td>
  <td align="right">
8.8301772  </td>
  <td align="right">
0.1924833  </td>
  <td align="right">
0.1750954  </td>
  <td align="right">
0.6130669  </td>
  <td align="right">
0.2053811  </td>
  <td align="right">
2.2958639  </td>
  <td align="right">
-18.47809  </td>
  <td align="right">
-18.47808  </td>
  <td align="right">
0.076265588  </td>
  <td align="right">
0.331475715  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{4}</math>
  </td>
  <td align="right">
4.9379256  </td>
  <td align="right">
0.3309933  </td>
  <td align="right">
0.2342522  </td>
  <td align="right">
1.4179907  </td>
  <td align="right">
0.4064595  </td>
  <td align="right">
2.761622  </td>
  <td align="right">
-2.601255  </td>
  <td align="right">
-2.601257  </td>
  <td align="right">
0.084824137  </td>
  <td align="right">
0.139370157  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="left" colspan="13">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
Also, go [[Appendix/Ramblings/PatrickMotl#Pick_a_Different_Molecular-Weight_Ratio|here]] for definition of <math>[\xi_i]_\mathrm{smooth}</math>, which identifies the location of the specific-entropy step function; stability against convection is ensured whenever <math>\xi_i > [\xi_i]_\mathrm{smooth}</math>.
  </td>
</tr>
</table>
</li>

<br />&nbsp;</br />
      </li>
      <li>
[[SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings|SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings]]
<br />&nbsp;<br />

<table border="1" align="center" cellpadding="5">

<tr>
  <th align="center" colspan="5">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/Bipolytrope/Stability/qAndNuMax.xlsx --- worksheet = B-KB74 thru MinuPreparation]]Bipolytrope with <math>(n_c, n_e) = (5, 1)</math><br />Selected Pairings along the <math>\mu_e/\mu_c = 0.31</math> Sequence</th>
</tr>

<tr>
  <td align="center">Pairing</td>
  <td align="center"><math>\xi_i</math></td>
  <td align="center"><math>\Lambda_i</math></td>
  <td align="center"><math>\nu</math></td>
  <td align="center"><math>q</math></td>
</tr>

<tr>
  <td align="center">'''A'''</td>
  <td align="center"><math>9.014959766</math></td>
  <td align="center"><math>0.59835053</math></td>
  <td align="center"><math>0.3372170064</math></td>
  <td align="center"><math>0.0755022550</math></td>
</tr>

<tr>
  <td align="center">'''B1'''</td>
  <td align="center"><math>9.12744</math></td>
  <td align="center"><math>0.60069262</math></td>
  <td align="center"><math>0.3372001445</math></td>
  <td align="center"><math>0.0746451491</math></td>
</tr>

<tr>
  <td align="center">'''B2'''</td>
  <td align="center"><math>8.90394</math></td>
  <td align="center"><math>0.59610192</math></td>
  <td align="center"><math>0.33720014467</math></td>
  <td align="center"><math>0.0763642133</math></td>
</tr>
</table>


<table border="1" align="center" cellpadding="10">
<tr>
  <td align="center">[[File:TurningPoints51BipolytropesLabels.png|right|350px|Bipolytropic (5, 1) Equilibrium Sequences]]</td>
  <td align="center">[[File:TurningPoints51Bpairing.png|right|350px|Bipolytropic (5, 1) Equilibrium Sequences]]</td>
</tr>
</table>

<table border="1" align="center" cellpadding="10">
<tr>
  <td align="center">[[File:FundModeLocations02Labels.png|right|350px|Bipolytropic (5, 1) Neutral Fundamental Mode Locations]]</td>
  <td align="center">[[File:ConvectiveBoundary2Labeled.png|right|350px|Bipolytropic (5, 1) Equilibrium Sequences]]</td>
</tr>
</table>

      </li>
      </ul>
</li>
</ol>