Editing SSC/BipolytropeGeneralization/Pt4 (section)

===Explain Logic===
[[Image:FreeEnergyExample.jpg|right|400px]]  The figure presented here, on the right, shows a plot of the free energy, as a function of the dimensionless radius, <math>~\mathfrak{G}^*(\chi)</math>, where,
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<math>~\mathfrak{G}^*</math>
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<math>~=</math>
  </td>
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<math>~ -~ 3\mathcal{A} \chi^{-1} - \frac{\mathcal{B}}{(1-\gamma_c)} ~\chi^{3-3\gamma_c} - \frac{\mathcal{C}}{(1-\gamma_e)} ~\chi^{3-3\gamma_e} \, ,</math>
  </td>
</tr>
</table>
and, where we have used the parameter values,
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<math>~\gamma_c = 6/5; ~~~ \gamma_e = 2</math>
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</tr>

<tr>
  <th align="center">
<math>~\mathcal{A}</math>
  </th>
  <th align="center">
<math>~\mathcal{B}</math>
  </th>
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<math>~\mathcal{C}</math>
  </th>
</tr>

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  <td align="center">
0.201707
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  <td align="center">
0.0896
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  <td align="center">
0.002484
  </td>
</tr>
</table>
Directly from this plot we deduce that this free-energy function exhibits a minimum at <math>~\chi_\mathrm{eq} = 0.1235</math> and that, at this equilibrium radius, the configuration has a free-energy value, <math>~\mathfrak{G}^*(\chi_\mathrm{eq} ) = -2.0097</math>.  Via the steps described below, we demonstrate that this identified equilibrium radius is appropriate for an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope (with the just-specified core and envelope adiabatic indexes) that has the following physical properties:
* Fractional core mass, <math>~\nu = 0.1</math>;
* Core-envelope interface located at <math>~r_i/R = q = 0.435</math>;
* Density jump at the core-envelope interface, <math>~\rho_e/\rho_c = 0.8</math>.


'''<font color="red">Step 1:</font>'''  Because the ratio, <math>~q^3/\nu</math>, is a linear function of the density ratio, <math>~\rho_e/\rho_c</math>, the full definition of the free-energy coefficient, <math>~\mathcal{A}</math>, can be restructured into a quadratic equation that gives the density ratio for any choice of the parameter pair, <math>~(q, \mathcal{A})</math>.  Specifically,
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<math>~5 \biggl( \frac{q^3}{\nu} \biggr)^2 \mathcal{A}</math>
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<math>~=</math>
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  <td align="left">
<math> 
q^5 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (1 - q^2 )q^3
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl( 1 - \frac{5}{2}q^3 + \frac{3}{2}q^5\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ 5\mathcal{A} \biggl[ \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q^3) + q^3 \biggr]^2 </math>
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  <td align="center">
<math>~=</math>
  </td>
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<math> 
q^5 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (1 - q^2 )q^3
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl( 1 - \frac{5}{2}q^3 + \frac{3}{2}q^5\biggr) \, ,
</math>
  </td>
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</table>
</div>
and this can be written in the form,
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<math> \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 a + \biggl( \frac{\rho_e}{\rho_c} \biggr) b + c</math>
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<math>~=</math>
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<math> 
0 \, ,
</math>
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</tr>
</table>
</div>
where,
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  <td align="right">
<math>~a</math>
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<math>~\equiv</math>
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<math>~5\mathcal{A} (1-q^3)^2 - 1 + \frac{5}{2}q^3 - \frac{3}{2}q^5 \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~b</math>
  </td>
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<math>~\equiv</math>
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<math>~10\mathcal{A} q^3(1-q^3) - \frac{5}{2}q^3 (1-q^2) \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~c</math>
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<math>~\equiv</math>
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<math>~5\mathcal{A} q^6 - q^5 \, .</math>
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</tr>
</table>
</div>
Hence,
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<math>~\frac{\rho_e}{\rho_c}</math>
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<math>~=</math>
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<math>~\frac{1}{2a} \biggl[\pm ( b^2 - 4ac)^{1/2} - b \biggr] \, .</math>
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</table>
</div>
(For our physical problem it appears as though only the positive root is relevant.)  For the purposes of this example, we set <math>~\mathcal{A} = 0.2017</math> and examined a range of values of <math>~q</math> to find a physically interesting value for the density ratio.  We picked:
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  <td align="center">
<math>~\mathcal{A}</math>
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<math>~q</math>
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&nbsp;
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<math>~a</math>
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<math>~b</math>
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<math>~c</math>
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&nbsp;
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<math>~\frac{\rho_e}{\rho_c}</math>
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&nbsp;
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  <td align="center">
<math>~\nu</math>
  </td>
</tr>

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<math>~0.2017</math>
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<math>~0.435</math>
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<math>\Rightarrow</math>
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<math>~0.03173</math>
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<math>~-0.01448</math>
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<math>~-0.008743</math>
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<math>\Rightarrow</math>
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<math>~0.80068</math>
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<math>\Rightarrow</math>
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<math>~0.10074</math>
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</table>


'''<font color="red">Step 2:</font>'''  Next, we chose the parameter pair, 
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<math>
~\biggl(q, \frac{\rho_e}{\rho_c} \biggr) = (0.43500, 0.80000)
</math>
</div>
and determined the following parameter values from the known analytic solution:
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<math>~\nu</math>
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<math>~f\biggl(q, \frac{\rho_e}{\rho_c} \biggr)</math>
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<math>~g^2\biggl(q, \frac{\rho_e}{\rho_c} \biggr)</math>
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  <td align="center">
<math>~\Lambda_\mathrm{eq}</math>
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  <td align="center">
<math>~\chi_\mathrm{eq}</math>
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  <td align="center">
<math>~\mathcal{A}</math>
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<math>~\mathcal{B}</math>
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<math>~\mathcal{C}</math>
  </td>
</tr>

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<math>~0.100816</math>
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<math>~43.16365</math>
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<math>~3.923017</math>
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<math>~0.13684</math>
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<math>~0.12349</math>
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<math>~0.201707</math>
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<math>~0.089625</math>
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<math>~0.002484</math>
  </td>
</tr>
</table>