Editing SSC/Structure/BiPolytropes/Analytic51/Pt4 (section)

===Stability Condition===

According to the accompanying [[SSC/BipolytropeGeneralization#Stability|free-energy based, generalized formulation of stability in bipolytropes]], our above derived <math>~(n_c, n_e) = (5,1)</math> bipolytropes &#8212; or, equivalently, <math>~(\gamma_c, \gamma_e) = (6/5, 2)</math> bipolytropes &#8212; will be dynamically stable only if,

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<math>~ - (W_\mathrm{core} + W_\mathrm{env}) \biggl( \gamma_e - \frac{4}{3}\biggr) 
</math>
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<math>~>~</math>
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<math> ~ 2(\gamma_e-\gamma_c) S_\mathrm{core}  \, .</math>
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Otherwise, they will be dynamically unstable toward radial perturbations.  For various values of the <math>~\mu_e/\mu_c</math> ratio, Table 3 identifies the value of <math>~\xi_i</math> &#8212; and the corresponding values of <math>~q</math> and <math>~\nu</math> &#8212; at which the left-hand side of this stability relation equals the right-hand side.  The locus of points provided by Table 3 defines the curve that separates stable from unstable regions of the <math>~q-\nu</math> parameter space. The red-dashed curve drawn in Figure 3 graphically depicts this demarcation:  the region below the curve identifies bipolytrope models that are dynamically stable while the region above the curve identifies unstable models.

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<table border="0" cellpadding="5" width="85%">

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Table 3:  Points Defining Stability Curve
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<tr>
  <td align="center" width="25%">
<math>~\mu_e/\mu_c</math>
  </td>
  <td align="center" width="25%">
<math>~\xi_i</math>
  </td>
  <td align="center" width="25%">
<math>~q </math>
  </td>
  <td align="center">
<math>~\nu </math>
  </td>

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&nbsp;
  </td>

  <td align="center" width="25%">
<math>~\mu_e/\mu_c</math>
  </td>
  <td align="center" width="25%">
<math>~\xi_i</math>
  </td>
  <td align="center" width="25%">
<math>~q </math>
  </td>
  <td align="center">
<math>~\nu </math>
  </td>
</tr>

<tr>
  <td align="center">
1
  </td>
  <td align="center">
2.416
  </td>
  <td align="center">
0.5952
  </td>
  <td align="center">
0.6830
  </td>

  <td align="center">
0.375
  </td>
  <td align="center">
6.259
  </td>
  <td align="center">
0.1695
  </td>
  <td align="center">
0.6054
  </td>
</tr>

<tr>
  <td align="center">
0.95
  </td>
  <td align="center">
2.500
  </td>
  <td align="center">
0.5805
  </td>
  <td align="center">
0.6884
  </td>

  <td align="center">
0.350
  </td>
  <td align="center">
7.341
  </td>
  <td align="center">
0.1284
  </td>
  <td align="center">
0.5439
  </td>
</tr>

<tr>
  <td align="center">
0.90
  </td>
  <td align="center">
2.594
  </td>
  <td align="center">
0.5642
  </td>
  <td align="center">
0.6937
  </td>

  <td align="center">
0.340
  </td>
  <td align="center">
7.991
  </td>
  <td align="center">
0.1109
  </td>
  <td align="center">
0.5081
  </td>
</tr>

<tr>
  <td align="center">
0.80
  </td>
  <td align="center">
2.816
  </td>
  <td align="center">
0.5255
  </td>
  <td align="center">
0.7031
  </td>

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&#8531;
  </td>
  <td align="center">
8.548
  </td>
  <td align="center">
0.0990
  </td>
  <td align="center">
0.4790
  </td>
</tr>

<tr>
  <td align="center">
0.70
  </td>
  <td align="center">
3.109
  </td>
  <td align="center">
0.4775
  </td>
  <td align="center">
0.7104
  </td>

  <td align="center">
0.32
  </td>
  <td align="center">
10.2
  </td>
  <td align="center">
0.0744
  </td>
  <td align="center">
0.4038
  </td>
</tr>

<tr>
  <td align="center">
0.65
  </td>
  <td align="center">
3.296
  </td>
  <td align="center">
0.4481
  </td>
  <td align="center">
0.7124
  </td>

  <td align="center">
0.31
  </td>
  <td align="center">
12.4
  </td>
  <td align="center">
0.05536
  </td>
  <td align="center">
0.3264
  </td>
</tr>

<tr>
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0.60
  </td>
  <td align="center">
3.523
  </td>
  <td align="center">
0.4142
  </td>
  <td align="center">
0.7125
  </td>

  <td align="center">
0.305
  </td>
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14.4
  </td>
  <td align="center">
0.04494
  </td>
  <td align="center">
0.2772
  </td>
</tr>

<tr>
  <td align="center">
0.55
  </td>
  <td align="center">
3.809
  </td>
  <td align="center">
0.3748
  </td>
  <td align="center">
0.7096
  </td>

  <td align="center">
0.3
  </td>
  <td align="center">
17.733
  </td>
  <td align="center">
0.03412
  </td>
  <td align="center">
0.2186
  </td>
</tr>

<tr>
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&frac12;
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4.186
  </td>
  <td align="center">
0.3284
  </td>
  <td align="center">
0.7014
  </td>

  <td align="center">
0.295
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  <td align="center">
25.737
  </td>
  <td align="center">
0.02165
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  <td align="center">
0.14347
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</tr>

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0.45
  </td>
  <td align="center">
4.719
  </td>
  <td align="center">
0.2733
  </td>
  <td align="center">
0.6830
  </td>

  <td align="center">
0.291
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  <td align="center">
75.510
  </td>
  <td align="center">
0.00666
  </td>
  <td align="center">
0.0450
  </td>
</tr>

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0.40
  </td>
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5.574
  </td>
  <td align="center">
0.2073
  </td>
  <td align="center">
0.6429
  </td>

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&nbsp;
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&nbsp;
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&nbsp;
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&nbsp;
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</tr>
</table>

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[[Image:PlotStabilityBest02.png|500px|center]]
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'''Figure 3:'''  Largely the same as Figure 1, above, but a red-dashed curve has been added that separates the <math>~q - \nu</math> domain into regions that contain stable models (lying below the curve) from dynamically unstable models (lying above the curve), as determined by the virial stability analysis presented here.
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</table>
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