Editing SSC/BipolytropeGeneralization (section)

==Solution Strategy==
For a given set of free-energy coefficients, <math>~\mathcal{A}, \mathcal{B},</math> and <math>~\mathcal{C}</math>, along with a choice of the two adiabatic exponents <math>~(\gamma_c, \gamma_e)</math>, here's how to determine all of the physical parameters that are detailed in the above example table.

* '''<font color="darkgreen">Step 1:</font>'''  Guess a value of <math>~0 < q < 1</math>.

* '''<font color="darkgreen">Step 2:</font>'''  Given the pair of parameter values, <math>~(\mathcal{A}, q)</math>, determine the interface-density ratio, <math>~\rho_e/\rho_c</math>, by finding the appropriate root of the expression that defines the function, <math>~\mathcal{A}(q, \rho_e/\rho_c)</math>.  This can be straightforwardly accomplished because, [[#Explain_Logic|as demonstrated below]], the relevant expression can be written as a quadratic function of <math>~(\rho_e/\rho_c)</math>.

* '''<font color="darkgreen">Step 3:</font>'''  Given the pair of parameter values, <math>~(q, \rho_e/\rho_c)</math>, determine the value of the core-to-total mass ratio, <math>~\nu</math>, from the expression that was obtained from an integration over the mass, namely,
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<table border="0" cellpadding="5" align="center">
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<math>~\frac{1}{\nu}</math>
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<math>~=</math>
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<math>~1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{1}{q^3} - 1 \biggr) \, .</math>
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</tr>
</table>
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* '''<font color="darkgreen">Step 4:</font>'''  Given the value of <math>~\mathcal{B}</math> along with the pair of parameter values, <math>~(q, \nu)</math>, the above expression that defines <math>~\mathcal{B}</math> can be solved to give the relevant value of the dimensionless parameter, <math>~\Lambda_\mathrm{eq}.</math>

* '''<font color="darkgreen">Step 5:</font>'''  The value of <math>~\mathcal{C}^'</math> &#8212; the coefficient that appears on the right-hand-side of the above expression that defines <math>~\mathcal{C}</math> &#8212; can be determined, given the values of parameter triplet, <math>~(q, \nu, \Lambda_\mathrm{eq})</math>.

* '''<font color="darkgreen">Step 6:</font>'''  Given the value of <math>~\mathcal{C}</math> and the just-determined value of the coefficient <math>~\mathcal{C}^'</math>, the normalized equilibrium radius, <math>~\chi_\mathrm{eq},</math> that corresponds to the value of <math>~q</math> that was ''guessed'' in Step #1 can be determined from the above definition of <math>~\mathcal{C}</math>, specifically,
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<math>~\chi_\mathrm{eq}\biggr|_\mathrm{guess} </math>
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<math>~=</math>
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<math>~\biggl( \frac{\mathcal{C}}{\mathcal{C}^'} \biggr)^{1/(3\gamma_e - 3\gamma_c)} \, .</math>
  </td>
</tr>
</table>
</div>

* '''<font color="darkgreen">Step 7:</font>'''  But, independent of this ''guessed'' value of <math>~\chi_\mathrm{eq},</math> the condition for virial equilibrium &#8212; which identifies extrema in the free-energy function &#8212; gives the following expression for the normalized equilibrium radius:
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<table border="0" cellpadding="5" align="center">
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<math>~\chi_\mathrm{eq}\biggr|_\mathrm{virial} </math>
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<math>~=</math>
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<math>~\biggl[ \frac{\mathcal{A}}{\mathcal{B} + \mathcal{C}^'}  \biggr]^{1/(4 - 3\gamma_c)} \, .</math>
  </td>
</tr>
</table>
</div>

* '''<font color="darkgreen">Step 8:</font>'''  If <math>~\chi_\mathrm{eq}|_\mathrm{guess} \ne \chi_\mathrm{eq}|_\mathrm{virial}</math>, return to Step #1 and guess a different value of <math>~q</math>.  Repeat Steps #1 through #7 until the two independently derived values of the normalized radius match, to a desired level of precision.

* '''<font color="darkgreen">Keep in mind:</font>'''  (A) A graphical representation of the free-energy function, <math>~\mathfrak{G}(\chi)</math>, can also be used to identify the "correct" value of <math>~\chi_\mathrm{eq}</math> and, ultimately, the above-described iteration loop should converge on this value.  (B) The free-energy function may exhibit more than one (or, actually, no) extrema, in which case more than one (or no) value of <math>~q</math> should lead to convergence of the above-described iteration loop.