Editing Appendix/Ramblings/51BiPolytropeStability/RethinkEnvelope (section)

==Interface Conditions==

Drawing from [[SSC/Structure/BiPolytropes#Table2|Table 2 in our accompanying discussion]], we see that the interface conditions give,

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

<tr>
  <td align="right"><math>\biggl(\frac{\rho_0}{\mu_c}\biggr) \theta_i^5</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\rho_e}{\mu_e}\biggr) \phi_i</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\rho_e}{\rho_0}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^5 \phi_i^{-1}\, ,</math></td>
</tr>
</table>
and,

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

<tr>
  <td align="right"><math>K_c\rho_0^{6/5} \theta_i^6</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>K_e\rho_e^{2} \phi_i^2</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \biggl( \frac{K_e}{K_c}\biggr)^{1 / 2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\rho_0^{3/5} }{\rho_e} \cdot \frac{\theta_i^3}{\phi_i}\, .</math></td>
</tr>
</table>

As a result, throughout the envelope,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{K_e \rho_e^{2}}{K_c\rho_0^{6/5}}\biggr] \phi^{2}
=
\biggl(\frac{\theta_i^6}{\phi_i^2}\biggr) \phi^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho_0^{2/5}\biggl[ \frac{K_e}{2\pi K_c} \biggr]^{1/2} \eta
=
(2\pi)^{-1 / 2} \rho_0^{2/5}\biggl[ \frac{\rho_0^{3/5} }{\rho_e} \cdot \frac{\theta_i^3}{\phi_i} \biggr] \eta
=
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^5 \phi_i^{-1} \biggr]^{-1}(2\pi)^{-1 / 2} \biggl[ \frac{\theta_i^3}{\phi_i} \biggr] \eta
=
(2\pi)^{-1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-2}  \eta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M^*_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \biggl[\rho_e \rho_0^{1 / 5} \biggr]\biggl[ \frac{K_e}{2\pi K_c} \biggr]^{3/2} 
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
=
2(2\pi)^{-1 / 2}\biggl[\rho_e \rho_0^{1 / 5} \biggr]\biggl[ \frac{\rho_0^{3/5} }{\rho_e} \cdot \frac{\theta_i^3}{\phi_i}\biggr]^{3} 
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
=
2(2\pi)^{-1 / 2} \biggl[\frac{\rho_e}{\rho_0}\biggr]^{-2}\biggl[ \frac{\theta_i^3}{\phi_i}\biggr]^{3} 
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(2\pi)^{-1 / 2} \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^5 \phi_i^{-1}\biggr]^{-2}\biggl[ \frac{\theta_i^9}{\phi_i^3}\biggr] 
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
=
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} (\theta_i \phi_i)^{-1} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)\, .
</math>
  </td>
</tr>
</table>

In summary, then,

<table border="1" cellpadding="5" width="80%" align="center">
<tr>
  <td align="center" colspan="1">
<font size="+1" color="darkblue">
'''Core'''
</font>
  </td>
  <td align="center">
<font size="+1" color="darkblue">
'''Envelope'''
</font>
  </td>
</tr>

<tr>
  <td align="center">

<!-- BEGIN LEFT BLOCK details -->
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta^{5} = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-5/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta^{6}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{3}{2\pi} \biggr]^{1/2} \xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl[ \frac{3}{2\pi} \biggr]^{3/2} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{2^4 \cdot 3^3 \pi^2}{2^3\pi^3} \biggr]^{1/2} 
\biggl\{ \frac{\xi^3}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{6}{\pi} \biggr)^{1/2} 
\xi^3\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} </math>
  </td>
</tr>
</table>
<!-- END LEFT BLOCK details -->

  </td>
  <td align="center">

<!-- BEGIN RIGHT BLOCK details -->
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^5 \phi_i^{-1} \phi
=
\frac{A}{\phi_i} \biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^5 \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\theta_i^6}{\phi_i^2}\biggr) \phi^{2}
=
\theta_i^6 \biggl( \frac{A}{\phi_i}\biggr)^{2}\biggl[ \frac{\sin(\eta - B)}{\eta} \biggr]^{2}
\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(2\pi)^{-1 / 2} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\theta_i^{-2}\biggr] \eta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M^*_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}(\theta_i\phi_i)^{-1}
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}\theta_i^{-1}
\biggl\{ \frac{A}{\phi_i} \biggl[ \sin(\eta-B) - \eta\cos(\eta-B)\biggr] \biggr\}
</math>
  </td>
</tr>
</table>
<!-- END RIGHT BLOCK details -->

  </td>
</tr>
</table>
Notice that by setting the pressure to be the same at the interface, we have the relation,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\theta_i^6</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\theta_i^6 \biggl( \frac{A}{\phi_i}\biggr)^{2}\biggl[ \frac{\sin(\eta_i - B)}{\eta_i} \biggr]^{2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{A}{\phi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\eta_i}{\sin(\eta_i - B)} 
\, .
</math>
  </td>
</tr>
</table>

Pulling from [[SSC/Structure/BiPolytropes#Table3|Table 3]] in our accompanying discussion, two other constraints come from making sure that the radius of the configuration and the enclosed mass match at the interface, whether you examine it from the point of view of the core or of the envelope. In principle, these constraints can provide expressions for the two unknown constants, <math>A</math> and <math>B</math>. Let's do the radius first.
<table border="0" align="center" cellpadding="5">

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

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{\eta_i}{\xi_i}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^{1/2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^{2}  \, .</math>
  </td>
</tr>
</table>

Now, from the enclosed mass constraint,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} (\theta_i \phi_i)^{-1} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl[ \frac{3}{2\pi} \biggr]^{3/2} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl(\frac{d\phi}{d\eta} \biggr)_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \biggl[ \frac{3^3}{2^3\pi^3} \biggl(\frac{\pi}{2}\biggr)\biggr]^{1/2} \biggl(\frac{\xi_i}{\eta_i}\biggr)^2 
\times  \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} (\theta_i \phi_i) \biggl(\frac{d\theta}{d\xi} \biggr)_i
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{3/2} \biggl[3^{1/2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^{2}\biggr]^{-2} 
\times  \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} (\theta_i \phi_i) \biggl(\frac{d\theta}{d\xi} \biggr)_i
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1/2} (\theta_i^{-3} \phi_i) \biggl(\frac{d\theta}{d\xi} \biggr)_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\xi_i}{\sqrt{3}} \, .
</math>
  </td>
</tr>
</table>

Alternatively, the ratio of these two expressions gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\frac{\eta_i}{\xi_i} \biggl\{
\biggl(\frac{d\phi}{d\eta} \biggr)_i
\biggr\}^{-1}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1/2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^{2} 
\biggl\{
3^{1/2} (\theta_i^{-3} \phi_i) \biggl(\frac{d\theta}{d\xi} \biggr)_i
\biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\eta_i \phi_i}{(d\phi/d\eta)_i} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr) \frac{\xi_i \theta_i^5}{(d\theta/d\xi)_i} \, ;
</math>
  </td>
</tr>
</table>
and their product gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\frac{\eta_i}{\xi_i}
\biggl(\frac{d\phi}{d\eta} \biggr)_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1/2} (\theta_i^{-3} \phi_i) \biggl(\frac{d\theta}{d\xi} \biggr)_i 
\times 3^{1/2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^{2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\eta_i}{\phi_i}
\biggl(\frac{d\phi}{d\eta} \biggr)_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3\xi_i}{\theta_i} \biggl(\frac{d\theta}{d\xi} \biggr)_i 
\biggl(\frac{\mu_e}{\mu_c}\biggr) \, .
</math>
  </td>
</tr>
</table>