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

==Shift from &eta; to &Delta;==

Again, let's shift from the envelope's standard radial coordinate, <math>\eta</math>, to
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\Delta</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\eta - B</math></td>
</tr>
<tr>
  <td align="right"><math>\Rightarrow~~~\phi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>A\biggl[ \frac{\sin\Delta}{\Delta + B}\biggr] \, ;</math></td>
</tr>
</table>
and,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{d\phi}{d\Delta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{A}{(\Delta + B)^2}\biggl[
(\Delta + B)\cos\Delta - \sin\Delta
\biggr] 
</math>
  </td>
</tr>
<tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{1}{\phi} \cdot \frac{d\phi}{d\Delta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[  \frac{\Delta + B}{\sin\Delta}\biggr]
\frac{1}{(\Delta + B)^2}\biggl[
(\Delta + B)\cos\Delta - \sin\Delta
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{(\Delta + B)}\biggl[
(\Delta + B)\cot\Delta - 1
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\cot\Delta - \frac{1}{(\Delta + B)} \, .
</math>
  </td>
</tr>
</table>
The pair of constraints obtained from matching the radius and the enclosed mass, respectively, are,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\eta_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^{1/2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \theta_i^{2} </math>
  </td>
</tr>

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

and,

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

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

<tr>
  <td align="right">
<math>\Rightarrow~~~
- \frac{\xi_i}{\sqrt{3}} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\cot\Delta_i - \frac{1}{(\Delta_i + B)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
\cot\Delta_i  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}  \, .
</math>
  </td>
</tr>
</table>

Cross-checking against our [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|earlier tabulation of parameter values]] &#8212; specifically the parameter, <math>\Lambda_i</math> &#8212; we recognize that,

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

<tr>
  <td align="right">
<math>
\Lambda_i  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~   \cot\Delta_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Lambda_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~   \Delta_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\tan^{-1}\biggl(\frac{1}{\Lambda_i } \biggr) + m\pi \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
For the record:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\Lambda_i = \frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sqrt{3} - \eta_i\xi_i}{\sqrt{3}\eta_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sqrt{3} - \sqrt{3}(\mu_e/\mu_c)\theta_i^2\xi_i^2}{3(\mu_e/\mu_c)\theta_i^2\xi_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1 - (\mu_e/\mu_c)\theta_i^2\xi_i^2}{\sqrt{3}(\mu_e/\mu_c)\theta_i^2\xi_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(1 + \xi_i^2/3) - (\mu_e/\mu_c)\xi_i^2}{\sqrt{3}(\mu_e/\mu_c)\xi_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3 + \xi_i^2 [1 - 3(\mu_e/\mu_c)]}{3^{3 / 2}(\mu_e/\mu_c)\xi_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{1}{\Lambda_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3^{3 / 2}(\mu_e/\mu_c)\xi_i}{3 + \xi_i^2 [1 - 3(\mu_e/\mu_c)]} \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

NOTE:  By adding the additional term, <math>m\pi</math>, we are able to take advantage of the oscillatory nature of the density function, <math>\phi</math>.  As a result, we see that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
B 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i - \tan^{-1}\biggl(\frac{1}{\Lambda_i } \biggr) - m\pi \, ;
</math>
  </td>
</tr>
</table>
and, given that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\sin\Delta_i 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\sin\biggl\{\tan^{-1}\biggl(\frac{1}{\Lambda_i } \biggr) + m\pi\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\sin \biggl[ \tan^{-1}\biggl(\frac{1}{\Lambda_i }\biggr) \biggr] \cdot \cos(m\pi)
+
\cos \biggl[ \tan^{-1}\biggl(\frac{1}{\Lambda_i }\biggr) \biggr] \cdot \cancelto{0}{\sin(m\pi)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{1/\Lambda_i}{ (1 + 1/\Lambda_i^2)^{1 / 2}} \biggr] \cdot \cos(m\pi)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\cos(m\pi) \cdot \biggl[1 + \Lambda_i^2  \biggr]^{-1 / 2} \, ,
</math>
  </td>
</tr>
</table>

the other constant is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\frac{A}{\phi_i} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\Delta_i + B}{\sin\Delta_i} = \frac{\eta_i}{\sin\Delta_i} = \frac{\eta_i (1 + \Lambda_i^2)^{1 / 2}}{\cos(m\pi)} \, .
</math>
  </td>
</tr>
</table>

As in the [[#Earlier_Example|earlier case depicted below]], let's draw from the [[SSC/Stability/BiPolytropes/HeadScratching#Selected_Models|accompanying <b>B2</b> model]] for which, <math>\mu_e/\mu_c = 0.25</math> and <math>\xi_i = 2.4782510</math> and &hellip;
<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center"><math>\theta_i</math></td>
  <td align="center"><math>\eta_i</math></td>
  <td align="center"><math>\Lambda_i</math></td>
  <td align="center"><math>A</math></td>
  <td align="center"><math>B</math></td>
  <td align="center"><math>\eta_s</math></td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center"><math>Q_\rho</math></td>
  <td align="center"><math>Q_m</math></td>
  <td align="center"><math>Q_r</math></td>
</tr>

<tr>
  <td align="center">0.572857</td>
  <td align="center">0.352159</td>
  <td align="center">1.408807</td>
  <td align="center">0.608404</td>
  <td align="center">-0.265127</td>
  <td align="center"><math>2.876465</math></td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center">0.00938349</td>
  <td align="center">13.558308</td>
  <td align="center">7.0373055</td>
</tr>
</table>
<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center"><math>\theta_i</math></td>
  <td align="center"><math>\biggl(\frac{d\theta}{d\xi} \biggr)_i</math></td>
  <td align="center"><math>\eta_i</math></td>
  <td align="center"><math>b_i</math></td>
  <td align="center"><math>(y_i)_+</math></td>
  <td align="center"><math>(y_i)_-</math></td>
  <td align="center"><math>A</math></td>
  <td align="center"><math>B</math></td>
  <td align="center"><math>\eta_s</math></td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center"><math>Q_\rho</math></td>
  <td align="center"><math>Q_m</math></td>
  <td align="center"><math>Q_r</math></td>
</tr>

<tr>
  <td align="center">0.572857</td>
  <td align="center">-0.1552971</td>
  <td align="center">0.352159</td>
  <td align="center">-1.430819</td>
  <td align="center">0.672019</td>
  <td align="center">-0.987752</td>
  <td align="center">0.608404</td>
  <td align="center">-0.265127</td>
  <td align="center"><math>2.876465</math></td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center">0.00938349</td>
  <td align="center">13.558308</td>
  <td align="center">7.0373055</td>
</tr>
</table>