Editing SSC/Structure/BiPolytropes/Analytic51Renormalize (section)

==Behavior of Central Density Along Equilibrium Sequence==
Each equilibrium sequence will be defined as a sequence of models having the same jump in the mean-molecular weight, <math>\mu_e/\mu_c</math>.  Along a given sequence, we vary the location of the core/envelope interface, <math>\xi_i</math>.  Our desire is to analyze the behavior of the central density, while holding the total mass fixed, as the location of the interface is varied.

The central density is given by the expression,

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

<tr>
  <td align="right"><math>{\tilde\rho}_c</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl[ \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr]_{\xi = 0}
=
\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathcal{m}_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i} \, .</math></td>
</tr>
</table>
In order to evaluate <math>\mathcal{m}_\mathrm{surf}</math> for a given specification of the interface location, <math>\xi_i</math>, we need to know that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\theta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(1+\frac{1}{3}\xi_i^2\biggr)^{-1 / 2} \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\Lambda_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\xi_i}{\sqrt{3}} 
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr] \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>A</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\eta_i\biggl(1 + \Lambda_i^2\biggr)^{1 / 2} \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_s</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \, .</math></td>
</tr>
</table>
Keep in mind, as well, that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} ~ \biggl[ \frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr] \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>q \equiv \frac{r_\mathrm{core}}{R}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3} ~ \biggl[ \frac{\xi_i \theta_i^2}{\eta_s}\biggr] \, .</math></td>
</tr>
</table>

<span id="Sequences">&nbsp;</span>
<table border="1" align="center" cellpadding="10">
<tr>
  <td align="center">[[File:TurningPoints51BipolytropesLabels.png|right|350px|Bipolytropic (5, 1) Equilibrium Sequences]]</td>
  <td align="center">[[File:RhoC.png|350px|Central Density versus xi_i (mu_ratio = 0.3100)]]</td>
</tr>
</table>