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

==Setup==
Drawing from the [[SSC/Structure/BiPolytropes#Setup|accompanying Table 1]], we have &hellip;

<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">
<math>n_c = 5</math>
  </td>
  <td align="center">
<math>n_e = 1</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\theta}{d\xi} \biggr) = - \theta^{5}
</math>

sol'n:
<math>
\theta(\xi)
</math>
  </td>
  <td align="center">
<math>
\frac{1}{\eta^2} \frac{d}{d\eta} \biggl( \eta^2 \frac{d\phi}{d\eta} \biggr) = - \phi
</math>

sol'n:
<math>
\phi(\eta)
</math>
  </td>
</tr>

<tr>
  <td align="center">

<!-- BEGIN LEFT BLOCK details -->
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Specify:  <math>K_c</math> and <math>\rho_0 ~\Rightarrow</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta^{5}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \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{3K_c}{2\pi G} \biggr]^{1/2} \rho_0^{-2/5} \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{3K_c}{2\pi G} \biggr]^{3/2} \rho_0^{-1/5} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
  </td>
</tr>
</table>
<!-- END LEFT BLOCK details -->

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

<!-- BEGIN RIGHT BLOCK details -->
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Knowing:  <math>K_e</math> and <math>\rho_e ~\Rightarrow</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_e \phi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_e \rho_e^{2} \phi^{2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_e}{2\pi G} \biggr]^{1/2} \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[ \frac{K_e}{2\pi G} \biggr]^{3/2} \rho_e \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
</tr>
</table>
<!-- END RIGHT BLOCK details -->

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


From an [[SSC/Structure/BiPolytropes/Analytic51#Steps_2_&_3|accompanying discussion]] of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, we know that the solution to the pair of Lane-Emden equations is &hellip;
<div align="center">
<math>
\theta(\xi) = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1/2} ~~~~\Rightarrow ~~~~  \theta_i = \biggl[ 1 + \frac{1}{3}\xi_i^2 \biggr]^{-1/2} \,,
</math>

<math>
\frac{d\theta}{d\xi} = - \frac{\xi}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} ~~~~\Rightarrow ~~~~  \biggl(\frac{d\theta}{d\xi}\biggr)_i = - \frac{\xi_i}{3}\biggl[ 1 + \frac{1}{3}\xi_i^2 \biggr]^{-3/2} \, ;
</math>
</div>
and,
<div align="center">
<math>
\phi = A \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr] \, ,
</math>

<math>
\frac{d\phi}{d\eta} = \frac{A}{\eta^2} \biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr] \, .
</math>
</div>


<table border="1" cellpadding="8" align="center" width="75%">
<tr><td align="left">
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>
The solid blue curve in the following plot shows how <math>\phi</math> varies with <math>\Delta</math> when <math>(A, B) = (0.608404, - 0.265127)</math>. Notice that <math>\phi</math> is zero when <math>\Delta = \pi, 2\pi, 3\pi, 4\pi, 5\pi</math>; more generally, it crosses zero when <math>\Delta = \pm m\pi</math>, for all positive values of the integer, <math>m</math>. Notice as well that when <math>\Delta = -B</math> &hellip; (that is, when <math>\eta = 0</math>) &hellip; <math>\phi \rightarrow \pm \infty</math>.  <br />
[[File:PhsVSdelta.png|800px|center|phi vs Delta]]

The solid blue curve exhibits an extremum when,
<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] ~~~\rightarrow ~~~ 0 \, .
</math>
  </td>
</tr>
<tr>
</table>

This occurs when the quantity, <math>[\phi - A\cos\Delta]</math> (the dotted grey curve) goes to zero and/or when the quantity, <math>[\tan\Delta - (\Delta+B)]</math> (the dotted orange curve) goes to zero.


----

<font color="red">NOTE:</font>&nbsp; A very similar expression arises in our [[SSC/Structure/BiPolytropes/Analytic15#Caution|accompanying discussion of bipolytropes with <math>(n_c, n_e) = (1, 5)</math>]].  Specifically,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\frac{\xi_\mathrm{trans}}{\tan(\xi_\mathrm{trans})}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
1 - \frac{3}{2}\biggl( \frac{\mu_e}{\mu_c}\biggr)^{-1} \, .
</math>
  </td>
</tr>
<tr>
</table>
I'm not sure whether this is relevant information or not!

----


The solid black, vertical line segments in this plot bracket the regime <math>2\pi < \Delta < 3\pi</math>; the function, <math>\phi</math> is positive everywhere across this interval.  In this interval, one extremum arises at <math>\Delta = \Delta_\mathrm{ext} \approx 7.72832</math>; it is identified in the plot by the dashed red, vertical line segment.  The function, <math>\phi(\Delta)</math>, can be used to construct a physically viable envelope over the interval, <math>\Delta_\mathrm{ext} \le \Delta \le 3\pi</math>, because, across this interval, <math>\phi</math> is everywhere positive (if not zero) and <math>d\phi/d\Delta</math> is everywhere negative (if not zero).

The blue curve in the following plot is identical to the one depicted in the previous plot, except the function, <math>\phi</math>, is plotted versus <math>\eta</math> rather than versus <math>\Delta</math>. This is analogous to the blue curve shown in Figure 3 of our [[SSC/Structure/Polytropes#Fig3|accompanying discussion of Shrivastava's <math>\theta_{5F}</math> Function]].

[[File:PhiVSmu.png|800px|center|phi vs eta]]<br />

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

Adopting [[SSC/Structure/BiPolytropes/Analytic51#Normalization|the same normalizations as before]], namely,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{\rho}{\rho_0}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>M_r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
  </td>
</tr>

</table>
</div>

we have,

<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^* \equiv \frac{\rho}{\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta^{5}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^* \equiv \frac{P}{K_c\rho_0^{6/5}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\theta^{6}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r^* \equiv r \biggl[\frac{G^{1/2}\rho_0^{2 / 5}}{K_c^{1 / 2}} \biggr]</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^* \equiv M_r\biggl[\frac{G^{3/2}\rho_0^{1 / 5}}{K_c^{3 / 2}} \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>
</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{\rho_e}{\rho_0}\biggr) \phi</math>
  </td>
</tr>

<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}</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</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)</math>
  </td>
</tr>
</table>
<!-- END RIGHT BLOCK details -->

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