Editing SSC/BipolytropeGeneralization (section)

==Old Stuff==
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~W_\mathrm{grav} + \mathfrak{S}_A\biggr|_\mathrm{core} + \mathfrak{S}_A\biggr|_\mathrm{env} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~W_\mathrm{grav} + \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] S_\mathrm{core} + \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] S_\mathrm{env}  \, .
</math>
  </td>
</tr>
</table>
</div>
In addition to the gravitational potential energy, which is naturally written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \cdot \mathfrak{f}_{WM} \, ,</math>
  </td>
</tr>
</table>
</div>
it seems reasonable to write the separate thermal energy contributions as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~S_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{3}{2}\biggl[ M_\mathrm{core} \biggl( \frac{P_{ic}}{\rho_{ic}} \biggr) \biggr] s_\mathrm{core} 
= \frac{3}{2} \biggl[ \nu M_\mathrm{tot}  P_{ic} \biggl( \frac{\rho_{ic}}{\bar\rho} \biggr)^{-1} \biggl( \frac{4\pi R^3}{3 M_\mathrm{tot}} \biggr) \biggr] s_\mathrm{core}
= 2\pi R^3 P_{ic} \biggl[ q^3 s_\mathrm{core} \biggr]
\, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~S_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{3}{2}\biggl[ M_\mathrm{env} \biggl( \frac{P_{ie}}{\rho_{ie}} \biggr) \biggr] s_\mathrm{env} 
= \frac{3}{2} \biggl[ (1-\nu) M_\mathrm{tot}  P_{ie} \biggl( \frac{\rho_{ie}}{\bar\rho} \biggr)^{-1} \biggl( \frac{4\pi R^3}{3 M_\mathrm{tot}} \biggr) \biggr] s_\mathrm{env}
= 2\pi R^3 P_{ie} \biggl[ (1-q^3) s_\mathrm{env} \biggr]
\, ,</math>
  </td>
</tr>
</table>
</div>
where the subscript "<math>i</math>" means "at the interface," and <math>~\mathfrak{f}_{WM},</math> <math>~s_\mathrm{core},</math> and <math>~s_\mathrm{env}</math> are dimensionless functions of order unity (all three functions to be determined) akin to the [[SSCpt1/Virial#Structural_Form_Factors|structural form factors]] used in our examination of isolated polytropes.  

While exploring how the free-energy function varies across parameter space, we choose to hold <math>~M_\mathrm{tot}</math> and <math>~K_c</math> fixed.  By dimensional analysis, it is therefore reasonable to normalize all energies, length scales, densities and pressures by, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{G^{3(\gamma_c-1)} M_\mathrm{tot}^{5\gamma_c-6}}{K_c}  \biggr]^{1/(3\gamma_c -4)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} \biggr]^{1/(3\gamma_c -4)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3}{4\pi} \biggl[ \frac{G^3 M_\mathrm{tot}^2}{K_c^3} \biggr]^{1/(3\gamma_c -4)}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{G^{3\gamma_c} M_\mathrm{tot}^{2\gamma_c}}{K_c^4} \biggr]^{1/(3\gamma_c -4)}  \, .</math>
  </td>
</tr>
</table>
</div>

As is detailed below &#8212; [[#Detailed_Derivations|first, here]], and via [[#Another_Derivation_of_Free_Energy|an independent derivation, here]] &#8212; quite generally the expression for the normalized free energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{E_\mathrm{norm}} \biggr) \biggl( \frac{1}{R} \biggr) \cdot \mathfrak{f}_{WM}
+ \biggl[ \frac{4\pi q^3 s_\mathrm{core} }{3(\gamma_c - 1)} \biggr] \biggl[ \frac{ R^3 P_{ic} }{E_\mathrm{norm}} \biggr]  
+ \biggl[ \frac{4\pi (1-q^3) s_\mathrm{env} }{3(\gamma_e - 1)} \biggr] \biggl[ \frac{ R^3 P_{ie} }{E_\mathrm{norm}} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3\cdot \mathfrak{f}_{WM}}{5} \chi^{-1} 
+ \biggl[ \frac{4\pi q^3 s_\mathrm{core} }{3(\gamma_c - 1)} \biggr] \biggl[ \frac{ R_\mathrm{norm}^4 P_\mathrm{norm} }{E_\mathrm{norm} R_\mathrm{norm}} \biggr]  
\biggl[ \biggl( \frac{P_{ic}}{P_\mathrm{norm}} \biggr) \chi^3 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ \biggl[ \frac{4\pi (1-q^3) s_\mathrm{env} }{3(\gamma_e - 1)} \biggr] 
\biggl[ \frac{ R_\mathrm{norm}^4 P_\mathrm{norm} }{E_\mathrm{norm} R_\mathrm{norm}} \biggr]  
\biggl[ \biggl( \frac{P_{ie}}{P_\mathrm{norm}} \biggr) \chi^3 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \frac{3\cdot \mathfrak{f}_{WM}}{5} \chi^{-1} 
+ \biggl[ \frac{4\pi q^3 s_\mathrm{core} }{3(\gamma_c - 1)} \biggr]   
\biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3 -3\gamma_c}
+ \biggl[ \frac{4\pi (1-q^3) s_\mathrm{env} }{3(\gamma_e - 1)} \biggr] 
\biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3 -3\gamma_e}
</math>
  </td>
</tr>

</table>
</div>
where we have introduced the parameter, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>.  After defining the normalized (and dimensionless) configurarion radius, <math>~\chi \equiv R/R_\mathrm{norm}</math>, we can write the normalized free energy of a bipolytrope in the following compact form:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \cdot \mathfrak{f}_{WM} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{C}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\biggl( \frac{4\pi }{3} \biggr) (1-q^3) s_\mathrm{env} 
\biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \, .</math>
  </td>
</tr>
</table>
</div>

As is [[#pressureMatch|further detailed below]], the second expression for the coefficient, <math>~\mathcal{C}</math>, ensures that the pressure at the "surface" of the core matches the pressure at the "base" of the envelope; but it should only be employed ''after an equilibrium radius'', <math>~\chi_\mathrm{eq}</math>, ''has been identified by locating an extremum in the free energy.''