Editing SSC/Stability/BiPolytropes/RedGiantToPN (section)

===q - &nu; Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show &#8212; via a plot in the <math>(q, \nu)</math> diagram &#8212; how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.


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[[Image:PlotSequencesBest02.png|500px|center]]
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'''Figure 1:'''  Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), &frac12; (red squares), 0.345 (dark purple crosses), &#8531; (pink triangles), 0.309 (light green dashes),  and &frac14; (purple asterisks).  Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values:  0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
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According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
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<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>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} 
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2}  \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>&nbsp; Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist.  We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

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<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}} 
</math>
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<math>=</math>
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<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math> 
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<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>
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<math>\frac{\mu_e}{\mu_c}</math>
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<math>\xi_i</math>
  </td>
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<math>\theta_i</math>
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<math>\eta_i</math>
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<math>\Lambda_i</math>
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<math>A</math>
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<math>\eta_s</math>
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LHS
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RHS
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<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
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<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
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  <td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
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<math>\frac{1}{3}</math>
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<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>
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<math>\frac{2}{\pi}</math> </td>
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0.33
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24.00496  </td>
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0.0719668  </td>
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0.0710624  </td>
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0.2128753  </td>
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0.0726547  </td>
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1.8516032  </td>
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-223.8157  </td>
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-223.8159  </td>
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0.038378833  </td>
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0.52024552  </td>
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0.316943
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10.744571  </td>
  <td align="right">
0.1591479  </td>
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0.1493938  </td>
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0.4903393  </td>
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0.1663869  </td>
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2.1760793  </td>
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-31.55254  </td>
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-31.55254  </td>
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0.068652714  </td>
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0.382383875  </td>
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0.3090
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8.8301772  </td>
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0.1924833  </td>
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0.1750954  </td>
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0.6130669  </td>
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0.2053811  </td>
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2.2958639  </td>
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-18.47809  </td>
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-18.47808  </td>
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0.076265588  </td>
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0.331475715  </td>
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<math>\frac{1}{4}</math>
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4.9379256  </td>
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0.3309933  </td>
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0.2342522  </td>
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1.4179907  </td>
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0.4064595  </td>
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2.761622  </td>
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-2.601255  </td>
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-2.601257  </td>
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0.084824137  </td>
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0.139370157  </td>
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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>
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</table>