Editing Appendix/Ramblings/BiPolytropeStability (section)

==Profile==

Now, referencing the [[SSC/Structure/BiPolytropes/Analytic15#Profile|derived bipolytropic model profile]], we should incorporate the following relations:

<div align="center">
<table border="1" cellpadding="6">
<tr>
  <td align="center" rowspan="2">
Variable
  </td>
  <td align="center" rowspan="2">
Throughout the Core<br>
<math>0 \le r^* \le \frac{\xi_i}{\sqrt{2\pi}}</math>
  </td>
  <td align="center" rowspan="2">
Throughout the Envelope<sup>&dagger;</sup><br>
<math>\frac{\xi_i}{\sqrt{2\pi}} \le r^* \le \frac{\xi_i e^{2(\pi - \Delta_i)}}{\sqrt{2\pi}}</math>
  </td>

  <td align="center" colspan="3">
Plotted Profiles
  </td>
</tr>

<tr>
  <td align="center">
<math>\xi_i = 0.5</math>
  </td>
  <td align="center">
<math>\xi_i = 1.0</math>
  </td>
  <td align="center">
<math>\xi_i = 3.0</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>\xi = \sqrt{2\pi}~r^*</math>
  </td>
  <td align="center">
<math>\eta = \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2}~r^*</math>
  </td>

  <td align="center" colspan="3">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>\frac{\sin\xi}{\xi}</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i [\phi(\eta)]^5</math>
  </td>

  <td align="center">
<!--  [[File:PlotDensity_xi_0.5.jpg|thumb|75px]] -->
[[Image:DenXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>~P^*</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^2</math>
  </td>
  <td align="center">
<math>\theta^{2}_i [\phi(\eta)]^{6}</math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:PresXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>\biggl( \frac{2}{\pi}\biggr)^{1/2} (\sin\xi - \xi\cos\xi)</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{2\cdot 3^3 }{\pi} \biggr)^{1/2} \theta_i 
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) </math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:MassXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="left" colspan="6">
<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution|presented in Step 6 of our accompanying discussion]].
  </td>
</tr>
</table>
</div>


Therefore, throughout the core we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\xi}{\sin\xi} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{2\pi}}{\xi}\biggl( \frac{2}{\pi}\biggr)^{1/2} (\sin\xi - \xi\cos\xi) = \frac{2\sin\xi}{\xi} (1 - \xi\cot\xi) \, .</math>
  </td>
</tr>
</table>
In which case the governing LAWE throughout the core is,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ 4 -2(1-\xi\cot\xi )\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \frac{\xi}{\sin\xi} \biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g}~\frac{4\pi \sin\xi}{\xi^3} (1 - \xi\cot\xi) \biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ 4 -2(1-\xi\cot\xi )\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \frac{2\pi \xi}{\sin\xi} \biggl\{ \frac{\sigma_c^2}{3\gamma_\mathrm{g}}  ~+~\frac{2\alpha_\mathrm{g}}{\xi^3} \biggl(\xi\cos\xi - \sin\xi \biggr) \biggr\}  x \, .
</math>
  </td>
</tr>
</table>

Next, throughout the envelope we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i [\phi(\eta)]^5
\biggl\{\theta^{2}_i [\phi(\eta)]^{6}\biggr\}^{-1}
=
\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) }
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{2\cdot 3^3 }{\pi} \biggr)^{1/2} \theta_i \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl\{ \frac{1}{\eta} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \biggr\}
=
6\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr)
</math>
  </td>
</tr>
</table>
So, the governing LAWE throughout the envelope is,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ 4 - 6\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) } \biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) }
\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~6\alpha_\mathrm{g}~\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr) 
\biggl[ \frac{1}{\eta} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \biggr]^2 \biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl[ 4 - 6  \biggl(-\frac{d\ln\phi}{d\ln\eta} \biggr)  \biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) }
\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~6\alpha_\mathrm{g}~\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{ \theta_i}{\eta} \biggl(- \frac{d\phi}{d\eta} \biggr) 
\biggl(\frac{2\pi}{3}\biggr) \biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl[ 4 - 6  \biggl(-\frac{d\ln\phi}{d\ln\eta} \biggr)  \biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \frac{2\pi}{3}\biggl( \frac{\mu_e}{\mu_c} \biggr)^2\frac{ 1}{\eta^2} 
\biggl\{ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \biggl( \frac{\sigma_c^2}{\gamma_\mathrm{g}}\biggr) \frac{\eta^2}{\theta_i \phi(\eta) }  ~-~6\alpha_\mathrm{g}~\biggl(- \frac{d\ln \phi}{d\ln\eta} \biggr)  
\biggr\}  x \, .
</math>
  </td>
</tr>
</table>