Editing SSC/Structure/BiPolytropes/51RenormaizePart2 (section)

===Basic Equilibrium Structure===
Most of the details underpinning the following summary relations can be [[SSC/Structure/BiPolytropes/Analytic51Renormalize#BiPolytrope_with_(nc,_ne)_=_(5,_1)|found here]].

<div align="center"><b>New Normalization</b></div>

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

<tr>
  <td align="right"><math>\tilde\rho</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\rho \biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{-5} \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{P}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>P \biggl[K_c^{-10} G^{9} M_\mathrm{tot}^{6}  \biggr] \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{r}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>r \biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2}  \biggr]\, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{M}_r</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\frac{M_r}{M_\mathrm{tot}} \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{H}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>H \biggl[K_c^{-5 / 2} G^{3 / 2} M_\mathrm{tot} \biggr] \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{t}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>t \biggl[K_c^{15} G^{-13} M_\mathrm{tot}^{-10} \biggr]^{1 / 4} \, .</math></td>
</tr>
</table>

<span id="VariableProfiles">Note:&nbsp;</span>  For an n = 5 polytrope (like our bipolytrope's core), the units of the polytropic constant, <math>K_c</math>, are <math>\biggl[ \frac{\mathrm{length}^{13}}{\mathrm{mass} \cdot \mathrm{time}^{10}} \biggr]^{1 / 5}</math>.

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center">
Quantity
  </td>
  <td align="center">
[[SSC/Structure/BiPolytropes/Analytic51#Steps_2_&_3|Core]]<br />
<math>0 \le \xi \le \xi_i</math><br />
----
<math>\theta = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1/2}</math><br />
<math>\frac{d\theta}{d\xi} = - \frac{\xi}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2}</math>
  </td>
  <td align="center">
[[SSC/Structure/BiPolytropes/Analytic51#Step_6:_Envelope_Solution|Envelope]]<br />
<math>\eta_i \le \eta \le \eta_s</math><br />
----
<math>\phi = A \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr]</math><br />
<math>\frac{d\phi}{d\eta} = -\frac{A}{\eta^2} \biggl[ \sin(\eta-B) - \eta\cos(\eta-B)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\tilde{r}</math>
  </td>
  <td align="center">
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi</math>
  </td>
  <td align="center">
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta </math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\tilde{\rho}</math>
  </td>
  <td align="center">
<math>\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</math>
  </td>
  <td align="center">
<math>\mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi </math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\tilde{P}</math>
  </td>
  <td align="center">
<math>\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} </math>
  </td>
  <td align="center">
<math>\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \theta^{6}_i \phi^{2}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\tilde{M}_r</math>
  </td>
  <td align="center">
<math>
\mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
</math>
  </td>
  <td align="center">
<math>\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="left" colspan="3">
Note that, for a given specification of the molecular-weight ratio, <math>\mu_e/\mu_c</math>, and the interface location, <math>\xi_i</math>, 
<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>B</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\eta_i - \frac{\pi}{2} + \tan^{-1}(\Lambda_i) \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_s</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\pi + B = 
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \, ,</math></td>
</tr>
</table>
in which case,
<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>

<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} \, ,
</math>
  </td>
</tr>

<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>

  </td>
</tr>
</table>