Editing SSC/BipolytropeGeneralizationVersion2 (section)

===Review of Isolated Polytrope===
If we were configuring isolated polytropes &#8212; instead of ''bi''polytropes &#8212; the free-energy expression would be, simply,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_A \, ,
</math>
</div> 
and, following the detailed steps presented in our [[SSCpt1/Virial#Implementation|introductory discussion of the free energy of spherically symmetric, configurations]], properly normalized expressions for the two contributing energy terms would be,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \int_0^{\chi = R_\mathrm{edge}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* 
</math>
  </td>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \biggl\{ \biggl( \frac{\rho_c}{\bar\rho} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \biggr\}_\mathrm{eq} \cdot \chi^{-1}  \, ,
</math>
  </td>

<tr>
  <td align="right">
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{({\gamma_g}-1)}  \int_0^{\chi=R_\mathrm{edge}^*}  4\pi (r^*)^2 P^* dr^* </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{({\gamma_g}-1)} \biggl( \frac{4\pi}{3} \biggr)^{1-\gamma}
\biggl\{ \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{\gamma}
\int_0^{1}  3x^2 \biggl[ \frac{P(x)}{P_c} \biggr]  dx \biggr\}_\mathrm{eq} \cdot \chi^{3-3\gamma} \, ,</math>
  </td>
</tr>

</table>
</div>
where, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{M_r(x)}{M_\mathrm{tot} }  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx \, .</math>
  </td>
</tr>
</table>
</div>
We note that, because <math>~M_r(x)/M_\mathrm{tot} = 1</math> in the limit, <math>~x \rightarrow 1</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq}   </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \int_0^{1}  3x^2 \biggl( \frac{\rho(x)}{\rho_c} \biggr)  dx \biggr]^{-1} \, ,</math>
  </td>
</tr>
</table>
</div>
or, if desired, the central-to-mean density ratio in one or both energy terms could be replaced by a term involving the normalized central pressure and the dimensionless equilibrium radius, <math>~\chi_\mathrm{eq}</math>, via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi^{3\gamma} \biggr]_\mathrm{eq} \, .</math>
  </td>
</tr>
</table>
</div>