View source for Appendix/Ramblings/Interrelating51and00Bipolytropes/Organization

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=Interrelating (5, 1) and (0, 0) Bipolytropes=

==Structure of (n<sub>c</sub>, n<sub>e</sub>) = (0, 0) Bipolytropes==
Here we draw heavily from an [[SSC/Structure/BiPolytropes/Analytic00|accompanying discussion]] to construct a [[SSC/Structure/BiPolytropes#BiPolytropes|bipolytrope]] in which both the core and the envelope have uniform densities, that is, the structure of both the core and the envelope will be modeled using an <math>n = 0</math> polytropic index.

Assuming that the central density, <math>\rho_0</math>, and central pressure, <math>P_0</math>, are specified, the natural dimensionless radius is given by the expression,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\chi</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
r \biggl[ \frac{G\rho_0^2}{P_0} \biggr]^{1 / 2} 
 \, .</math>
  </td>
</tr>
</table>

===Throughout the core (0 &le; &chi; &le; &chi;<sub>i</sub>)===
In equilibrium, the radial profile of the density, pressure, and integrated mass are, respectively,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math> 
  </td>
  <td align="left">
<math>\rho_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math> 
  </td>
  <td align="left">
<math>P_0 \biggl( 1 - \frac{2\pi}{3}\chi^2 \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math> 
 </td>
  <td align="left">
<math>\frac{4\pi}{3} \biggl[ \frac{P_0^3}{G^3 \rho_0^4} \biggr]^{1/2} \chi^3 \, .</math>
  </td>
</tr>
</table>

===Interface Conditions===

In terms of the (as yet unspecified) total radius, <math>R</math>, we use <math>q</math> to define the fractional radial location of the core/envelope interface, that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>q</math>
  </td>
  <td align="center">
<math>\equiv</math> 
  </td>
  <td align="left">
<math>
\frac{r_i}{R}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \chi_i</math>
  </td>
  <td align="center">
<math>\equiv</math> 
  </td>
  <td align="left">
<math>
q ~ \biggl[ \frac{G\rho_0^2 R^2}{P_0} \biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
And whether viewed from the perspective of the core or the envelope, the pressure at the interface is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math> 
  </td>
  <td align="left">
<math>P_0 \biggl( 1 - \frac{2\pi}{3}\chi_i^2 \biggr)</math>
  </td>
</tr>
</table>

=Related Discussions=
* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic solution]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.


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