Editing SSC/Structure/BiPolytropes/FreeEnergy00 (section)

==Mass Profile==

In this case, <math>~\rho_\mathrm{core}(x) = \rho_c = </math> constant &#8212; hence, also, <math>~[\rho(x)/\bar\rho]_\mathrm{core} = 1</math> &#8212; and <math>~\rho_\mathrm{env}(x) = \rho_e = </math> constant &#8212; hence, also, <math>~[\rho(x)/\bar\rho]_\mathrm{env} = 1</math> &#8212; but in general <math>~\rho_e \ne \rho_c</math>.  Performing the separate integrals to obtain expressions for <math>~M_r(r)</math> inside the core and the envelope, [[SSC/BipolytropeGeneralizationVersion2#Partitioning_the_Mass|as established in our accompanying overview]], we obtain:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(\mathrm{For}~0 \leq x \leq q)</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~M_r </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} \biggl( \frac{\nu}{q^3} \biggr) \int_0^{x}  3x^2 dx = \nu M_\mathrm{tot} \biggl( \frac{x}{q} \biggr)^3 \, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(\mathrm{For}~q \leq x \leq 1)</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~M_r </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} \biggl\{\nu + \biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{q}^{x}  3
x^2 dx \biggr\} 
= M_\mathrm{core}  + (1-\nu) M_\mathrm{tot}\biggl( \frac{x^3 - q^3}{1-q^3} \biggr)\, .
</math>
  </td>
</tr>
</table>
</div>
When <math>~x = q</math>, both expressions give,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_\mathrm{core} = \nu M_\mathrm{tot} \, ,</math>
  </td>
</tr>
</table>
</div>
as they should.  We deduce, as well, that the mass contained in the envelope is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~M_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_\mathrm{tot} - M_\mathrm{core} = (1-\nu) M_\mathrm{tot} \, ,</math>
  </td>
</tr>
</table>
</div>
and that the volumes occupied by the core and envelope are, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~V_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^3 R_\mathrm{edge}^3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~V_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1- q^3) R_\mathrm{edge}^3 \, .</math>
  </td>
</tr>

</table>
</div>

Hence, the ratio of envelope density to core density is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\rho_e}{\rho_c} = \frac{\bar\rho_\mathrm{env}}{\bar\rho_\mathrm{core}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{M_\mathrm{env}/V_\mathrm{env}}{M_\mathrm{core}/V_\mathrm{core}} = \frac{q^3(1-\nu)}{\nu(1-q^3)} \, .
</math>
  </td>
</tr>
</table>
</div>

These relations should be compared to &#8212; and ultimately must match &#8212; the prescriptions for <math>~M_r</math> that have been presented elsewhere in connection with [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|detailed force-balance models of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes]] and in our introductory discussion of [[SSC/VirialStability#Expressions_for_Mass|the virial stability of bipolytropes]].