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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 ≤ χ ≤ χ<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 specify 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_i</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>

===Envelope Solution (χ ≥ χ<sub>i</sub>)===
After specifying the envelope-to-core density ratio, <math>\rho_e/\rho_0</math>, the envelope's equilibrium radial profile of the density, pressure, and integrated mass are, respectively,

<table border="0" cellpadding="3" align="center">
<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
  <math>~=</math> 
</td>
<td align="left">
<math>~\rho_e</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{P}{P_0}</math>
</td>
<td align="center">
  <math>~=</math> 
</td>
<td align="left">
<math>1 - \frac{2\pi}{3}\chi_i^2 +
\frac{2\pi}{3} \biggl(\frac{\rho_e}{\rho_0}\biggr) \chi_i^2
\biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( \frac{\chi_i}{\chi} -
1\biggr) - \frac{\rho_e}{\rho_0} \biggl(\frac{\chi^2}{\chi_i^2} - 1 \biggr) \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_i^3\biggl[1 +\frac{\rho_e}{\rho_0}
\biggl( \frac{\chi^3}{\chi_i^3} - 1\biggr) \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, n_e) = (5,1)</math>.

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Appendix/Ramblings/Interrelating51and00Bipolytropes/Organization - Revision history