Editing SSC/BipolytropeGeneralization/Pt4 (section)

===Construction Multiple Curves to Define a Free-Energy Surface===
Okay.  Now that we have the hang of this, let's construct a sequence of curves that represent physical evolution at a fixed interface-density ratio, <math>~\rho_e/\rho_c</math>, but for steadily increasing core-to-total mass ratio, <math>~\nu</math>.  Specifically, we choose,
<div align="center">
<math>
~\frac{\rho_e}{\rho_c} = \frac{1}{2} \, .
</math>
</div>
From the known analytic solution, here are parameters defining several different equilibrium models:
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="9">
<font size="+1">Identification of Local ''Minimum'' in Free Energy</font>
  </th>
</tr>
<tr>
  <td align="center">
<math>~\nu</math>
  </td>
  <td align="center">
<math>~q</math>
  </td>
  <td align="center">
<math>~f\biggl(q, \frac{\rho_e}{\rho_c} \biggr)</math>
  </td>
  <td align="center">
<math>~g^2\biggl(q, \frac{\rho_e}{\rho_c} \biggr)</math>
  </td>
  <td align="center">
<math>~\Lambda_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~\chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\mathcal{C}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~0.2</math>
  </td>
  <td align="center">
<math>~9^{-1/3} = 0.48075</math>
  </td>
  <td align="center">
<math>~12.5644</math>
  </td>
  <td align="center">
<math>~2.091312</math>
  </td>
  <td align="center">
<math>~0.366531</math>
  </td>
  <td align="center">
<math>~0.037453</math>
  </td>
  <td align="center">
<math>~0.2090801</math>
  </td>
  <td align="center">
<math>~0.2308269</math>
  </td>
  <td align="center">
<math>~2.06252 \times 10^{-4}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~0.4</math>
  </td>
  <td align="center">
<math>~4^{-1/3} = 0.62996</math>
  </td>
  <td align="center">
<math>~4.21974</math>
  </td>
  <td align="center">
<math>~1.56498</math>
  </td>
  <td align="center">
<math>~0.707989</math>
  </td>
  <td align="center">
<math>~0.0220475</math>
  </td>
  <td align="center">
<math>~0.2143496</math>
  </td>
  <td align="center">
<math>~0.5635746</math>
  </td>
  <td align="center">
<math>~4.4626 \times 10^{-5}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~0.5</math>
  </td>
  <td align="center">
<math>~3^{-1/3} = 0.693361</math>
  </td>
  <td align="center">
<math>~2.985115</math>
  </td>
  <td align="center">
<math>~1.42334</math>
  </td>
  <td align="center">
<math>~0.9448663</math>
  </td>
  <td align="center">
<math>~0.0152116</math>
  </td>
  <td align="center">
<math>~0.2152641</math>
  </td>
  <td align="center">
<math>~0.791882</math>
  </td>
  <td align="center">
<math>~1.5464 \times 10^{-5}</math>
  </td>
</tr>

<tr>
  <td align="left" colspan="5">
Here we are examining the behavior of the free-energy function for bipolytropic models having <math>~(n_c, n_e) = (0, 0)</math>, <math>~(\gamma_c, \gamma_e) = (6/5, 2)</math>, and a density ratio at the core-envelope interface of <math>~\rho_e/\rho_c = 1/2</math>.  The figure shown here, on the right, displays the three separate free-energy curves, <math>~\mathfrak{G}^*(\chi)</math> &#8212; where, <math>~\chi \equiv R/R_\mathrm{norm}</math> is the normalized configuration radius &#8212; that correspond to the three values of <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math> given in the first column of the above table.  Along each curve, the local free-energy minimum corresponds to the the equilibrium radius, <math>~\chi_\mathrm{eq}</math>, recorded in column 6 of the above table.
  </td>
  <td align="center" colspan="4">
[[Image:ThreeFreeEnergyCurves.png|center|300px]]
  </td>
</tr>
</table>

Each of the free-energy curves shown above has been entirely defined by our specification of the three coefficients in the free-energy function, <math>~\mathcal{A}, \mathcal{B}</math>, and <math>~\mathcal{C}</math>.  In each case, the values of these three coefficients was judiciously chosen to ''produce'' a curve with a local minimum at the correct value of <math>~\chi_\mathrm{eq}</math> corresponding to an equilibrium configuration having the desired <math>~(\nu, \rho_e/\rho_c)</math> model parameters.  Upon plotting these three curves, we noticed that two of the curves &#8212; curves for <math>~\nu = 0.4</math> and <math>~\nu = 0.5</math> &#8212; also display a local ''maximum''.  Presumably, these maxima also identify equilibrium configurations, albeit unstable ones.   From a careful inspection of the plotted curves, we have identified the value of <math>~\chi_\mathrm{eq}</math> that corresponds to the two newly discovered (unstable) equilibrium models; these values are recorded in the table that immediately follows this paragraph.  By construction, we also know what values of <math>~\mathcal{A}, \mathcal{B}</math>, and <math>~\mathcal{C}</math> are associated with these two identified equilibria; these values also have been recorded in the table.  But it is not immediately obvious what the values are of the <math>~(\nu, \rho_e/\rho_c)</math> model parameters that correspond to these two equilibrium models.

<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="9">
Subsequently Identified Local Energy ''Maxima''
  </td>
</tr>
<tr>
  <td align="center">
<math>~\chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~\mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>~\chi_\mathrm{eq}^{4-3\gamma_c}</math>
 </td>
  <td align="center">
<math>~\therefore</math>
  </td>
  <td align="center">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\mathcal{C}</math>
  </td>
  <td align="center">
<math>~\mathcal{C}^' = \mathcal{A} \chi_\mathrm{eq}^{3\gamma_c-4} - \mathcal{B}</math>
  </td>
  <td align="center">
<math>~\biggl( \frac{\mathcal{C}}{\mathcal{C}^'} \biggr)^{1/(3\gamma_e - 3\gamma_c)}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~0.08255</math>
  </td>
  <td align="center">
<math>~+ 4.87562</math>
  </td>
  <td align="center">
<math>~0.368715</math>
  </td>
  <td align="center">
<math>~0</math>
  </td>
  <td align="center">
<math>~0.2143496</math>
  </td>
  <td align="center">
<math>~0.5635746</math>
  </td>
  <td align="center">
<math>~4.4626 \times 10^{-5}</math>
  </td>
  <td align="center">
<math>~1.7768 \times 10^{-2}</math>
  </td>
  <td align="center">
<math>~0.08254</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~0.032196</math>
  </td>
  <td align="center">
<math>~+11.5187</math>
  </td>
  <td align="center">
<math>~0.25300</math>
  </td>
  <td align="center">
<math>~0</math>
  </td>
  <td align="center">
<math>~0.2152641</math>
  </td>
  <td align="center">
<math>~0.791882</math>
  </td>
  <td align="center">
<math>~1.5464 \times 10^{-5}</math>
  </td>
  <td align="center">
<math>~5.8964 \times 10^{-2}</math>
  </td>
  <td align="center">
<math>~0.032196</math>
  </td>
</tr>
</table>