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

=Rethink Handling of n = 1 Envelope=

==Solution Steps==
Drawing from an [[SSC/Structure/BiPolytropes#Solution_Steps|accompanying discussion]] &hellip;
* Step 1:  Choose <math>n_c</math> and <math>n_e</math>.
* Step 2:  Adopt boundary conditions at the center of the core (<math>\theta = 1</math> and <math>d\theta/d\xi = 0</math> at <math>\xi=0</math>), then solve the Lane-Emden equation to obtain the solution, <math>\theta(\xi)</math>, and its first derivative, <math>d\theta/d\xi</math> throughout the core; the radial location, <math>\xi = \xi_s</math>, at which <math>\theta(\xi)</math> first goes to zero identifies the natural surface of an [[SSC/Structure/Polytropes#Lane-Emden_Equation|isolated polytrope]] that has a polytropic index <math>n_c</math>.
* Step 3  Choose the desired location, <math>0 < \xi_i < \xi_s</math>, of the outer edge of the core. 
* Step 4:  Specify <math>K_c</math> and <math>\rho_0</math>; the structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the core &#8212; over the radial range, <math>0 \le \xi \le \xi_i</math> and <math>0 \le r \le r_i</math> &#8212; via the relations shown in the <math>2^\mathrm{nd}</math> column of Table 1.
* Step 5:  Specify the ratio <math>\mu_e/\mu_c</math> and adopt the boundary condition, <math>\phi_i = 1</math>; then use the interface conditions as rearranged and presented in Table 3 to determine, respectively:
** The gas density at the base of the envelope, <math>\rho_e</math>;
** The polytropic constant of the envelope, <math>K_e</math>, relative to the polytropic constant of the core, <math>K_c</math>;
** The ratio of the two dimensionless radial parameters at the interface, <math>\eta_i/\xi_i</math>;
** The radial derivative of the envelope solution at the interface, <math>(d\phi/d\eta)_i</math>.
* Step 6:  The last sub-step of solution step 5 provides the boundary condition that is needed &#8212; in addition to our earlier specification that <math>\phi_i = 1</math> &#8212; to derive the desired ''particular'' solution, <math>\phi(\eta)</math>, of the Lane-Emden equation that is relevant throughout the envelope; knowing <math>\phi(\eta)</math> also provides the relevant structural first derivative, <math>d\phi/d\eta</math>, throughout the envelope.
* Step 7:  The surface of the bipolytrope will be located at the radial location, <math>\eta = \eta_s</math> and <math>r=R</math>, at which <math>\phi(\eta)</math> first drops to zero.
* Step 8:  The structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the envelope &#8212; over the radial range, <math>\eta_i \le \eta \le \eta_s</math> and <math>r_i \le r \le R</math> &#8212; via the relations provided in the <math>3^\mathrm{rd}</math> column of Table 1.

==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>

==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>

==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>

==Useful?==
This matches our [[SSC/Structure/BiPolytropes/Analytic51#Step_8:_Throughout_the_envelope_(ηi_≤_η_≤_ηs)|earlier derivation]].  Remember, as well, that <math>\phi_i = 1</math>, that is to say, 
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>A</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl[ \frac{\eta_i}{\sin(\eta_i - B)} \biggr] \, .</math></td>
</tr>
</table>

Suppose we use <math>\eta</math> as the primary abscissa.  Throughout the envelope, for various values of <math>\eta</math>, we set 
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="center"><math>\rho^* = Q_\rho \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr] \, ,
~~~~M^* = Q_m \biggl[ \sin(\eta-B) - \eta\cos(\eta-B)\biggr]  \, ,
~~ \xi = Q_r \eta</math></td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>Q_\rho</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>A \biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^5  </math></td>
</tr>

<tr>
  <td align="right"><math>Q_m</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>A\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}\theta_i^{-1}</math></td>
</tr>

<tr>
  <td align="right"><math>Q_r</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>3^{-1 / 2} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\theta_i^{-2}\biggr] </math></td>
</tr>
</table>
Note as well that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>q</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\frac{\xi_i}{\xi_s} = \frac{\xi_i}{Q_r \eta_s} \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\nu</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\frac{M^*_r|_\mathrm{core}}{M^*_r|_\mathrm{tot}} = \frac{\sin(\eta_i - B) - \eta_i\cos(\eta_i-B)}{\eta_s}\, .</math></td>
</tr>
</table>

==Earlier Example==

In our [[SSC/Stability/BiPolytropes/HeadScratching#Through_the_Envelope|earlier analysis]], we determined that the following relations hold in an equilibrium bipolytrope.

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Keep in mind that, once <math>\mu_e/\mu_c</math> and <math>\xi_i</math> have been specified, other [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values at the interface]] are:
<table border="0" cellpadding="5" align="center">

<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^2_i \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{1}{\eta_i} + \biggl( \frac{d\phi}{d\eta}\biggr)_i 
= 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{1}{\sqrt{3} \xi_i \theta_i^2} - \frac{\xi_i}{\sqrt{3}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta_i(1+\Lambda_i^2)^{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>
B + \pi \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>
As a test case, 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>

The following pair of plots show how the normalized density, <math>\rho^*</math>, and normalized integrated mass, <math>M_r^*</math>, varies over the radial-coordinate range, <math>0 \le \eta \le 3</math>, for both the core description and the envelope description for Model B2.  Both plots present the same four curves except, in the "first plot", the density has been magnified by a factor of 35 to aid in visualizing the shapes of the curves.  In the "first plot" the maximum ordinate value is 40, which comfortably accommodates the maximum value of both mass curves. In the "second plot" the maximum ordinate value is 0.09, which permits us to zoom in on the behavior of the (unmagnified) density curves in the vicinity of the core-envelope interface.


More specifically, here are the expressions that were used to generate each of the four curves (in both plots).

<b>Grey dotted curve:</b>  After setting <math>\xi = Q_r\eta</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>\rho^*\biggr|_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl[1 + \frac{\xi^2}{3}  \biggr]^{-5/2} \, .</math></td>
</tr>
</table>

<b>Orange curve:</b>  After setting <math>\xi = Q_r\eta</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>M_r^*\biggr|_\mathrm{core}</math></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>

<b>Dark-blue dotted curve:</b>  Acknowledging that <math>B = -0.265127</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>\rho^*\biggr|_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>Q_\rho \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr] \, .</math></td>
</tr>
</table>

<b>Red curve:</b>  Acknowledging that <math>B = -0.265127</math> for each value of <math>\eta</math> over the specified range &hellip;
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right"><math>M_r^*\biggr|_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>Q_m \biggl[ \sin(\eta-B) - \eta\cos(\eta-B)\biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">
  <tr>
<td align="center"><b>Model B2</b> &#8212; first plot<br />[[File:ModelB2firstAnnotated.png|400px|First Plot]]</td>
<td align="center"><b>Model B2</b> &#8212; second plot<br />[[File:ModelB2secondAnnotated.png|400px|Second Plot]]</td>
  </tr>
  <tr>
<td align="left" colspan="2">
<b>Things to notice:</b>
<ol>
  <li>Because <math>B \ne 0</math> and <math>\rho^*|_\mathrm{env}</math> is proportional to <math>\eta^{-1}</math>, the envelope-density (dark-blue dotted) curve shoots up to infinity as <math>\eta \rightarrow 0</math>.  Nevertheless, as the red curve in the "first plot" shows, the integrated envelope mass, <math>M_r^*|_\mathrm{env}</math>, is well behaved; it goes to <math>Q_m\sin(-B)=3.5527</math> as <math>\eta \rightarrow 0</math>.</li>
  <li>As seen in the "second plot," the envelope-density (dark-blue dotted curve) first goes to zero when <math>\eta \rightarrow \eta_s = \pi + B = 2.876465</math>.  As the red curve in the "first plot" shows, this is also where <math>M_r^*|_\mathrm{env}</math> reaches its maximum value, <math>Q_m \eta_s = 39.00000</math>.</li>
  <li>The gray-dotted curve in the "first plot" shows how the "core density" varies over the entire examined range.  At the center &#8212; where <math>\eta \rightarrow 0</math> and, hence, <math>\xi \rightarrow 0</math> &#8212; the core density is unity; as <math>\eta</math> climbs, the core density drops smoothly toward zero, but always remains positive.</li>
  <li>As the orange curve in the "first plot" shows, the integrated core mass is zero at <math>\eta =0</math>; as <math>\eta</math> increases, the integrated core mass smoothly increases, heading toward a limiting value of <math>M_r^*|_\mathrm{core} \rightarrow 3^{3/2}(6/\pi)^{1 / 2}= 7.18096</math> as <math>\eta \rightarrow \infty</math> and, hence, <math>\xi \rightarrow \infty</math>.</li>
  <li>As the "first plot" shows, the (red) curve representing the envelope mass intersects the (orange) curve representing the envelope mass ''twice''.  Moving from the center, outward, the first intersection occurs at the <b>Model B2</b> core-envelope interface, where <math>\eta = \eta_i = 0.352159</math> and <math>\xi = \xi_i = 2.47825</math>.  As can be seen in the "second plot," the two "density" curves do not intersect at the interface.  However, by design and construction, at the core-envelope interface the value of <math>\rho^*|_\mathrm{env}</math> is precisely a factor of <math>\mu_e/\mu_c = 0.25</math> smaller than <math>\rho^*|_\mathrm{core}</math>; in the "second plot," the vertical red line-segment highlights this discontinuous drop in the density at the interface.</li>
</ol>
</td>
  </tr>
</table>


<table border="1" align="center" cellpadding="8">
  <tr>
<td align="center"><b>Model B2</b> &#8212; third plot<br />[[File:ModelB2thirdAnnotated.png|400px|Third Plot]]</td>
<td align="center"><b>Model B2</b> &#8212; fourth plot<br />[[File:ModelB2fourthAnnotated.png|400px|Fourth Plot]]</td>
  </tr>
  <tr>
<td align="left" colspan="2">
<b>Things to notice:</b>
</td>
  </tr>
</table>

==Obtain &xi; from &eta;==
Again, let's set <math>\mu_e/\mu_c = 0.25</math> but this time specify the value of <math>\eta_i</math> and work backwards &#8212; through the definition of <math>\theta_i</math> &#8212; to determine <math>\xi_i</math>.  Specifically, we find that,

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

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3 - 3c_0 \xi_i + \xi_i^2
</math>
  </td>
</tr>
</table>
where, <math>c_0 \equiv (\mu_e/\mu_c) 3^{1 / 2} \eta_i^{-1}</math>.  That is,

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

<tr>
  <td align="right">
<math>2\xi_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3c_0 \pm \biggl[3^2c_0^2 - 12  \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
<font color="red"><b>NOTE:</b></font> Real root implies, <math>\eta_i \le \frac{3}{2}\biggl(\frac{\mu_e}{\mu_c}\biggr)</math>; and, at this limit, <math>(\xi_i)_\pm = \sqrt{3}\, .</math>
  </td>
</tr>
</table>

<font color="darkgreen">TEST:</font> &nbsp; As in <b>Model B2</b>, set <math>\mu_e/\mu_c = 0.25</math> and set <math>\eta_i = 0.352159</math>.  Then, <math>c_0 = 1.22959405</math> and, <math>(\xi_i)_+ = 2.47825101</math> while <math>(\xi_i)_- = 1.210531136</math>.  The value for <math>(\xi_i)_+</math> matches the value for <b>Model B2</b>.

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Keep in mind that, once <math>\mu_e/\mu_c</math> and <math>\xi_i</math> have been specified, other [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values at the interface]] are:
<table border="0" cellpadding="5" align="center">

<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^2_i \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{1}{\eta_i} + \biggl( \frac{d\phi}{d\eta}\biggr)_i 
= 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{1}{\sqrt{3} \xi_i \theta_i^2} - \frac{\xi_i}{\sqrt{3}} \, ,
</math>
  </td>
</tr>

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


<table border="1" align="center" cellpadding="8">

<tr><td align="center" colspan="16"><math>\frac{\mu_e}{\mu_c} = 0.25</math></td></tr>
<tr>
  <td align="center" rowspan="1"><math>\eta_i</math></td>
  <td align="center" rowspan="1"><math>c_0</math></td>
  <td align="center" colspan="2"><math>\xi_i</math></td>
  <td align="center"><math>\theta_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>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center"><math>q \equiv \frac{\xi_i}{Q_r\eta_s}</math></td>
  <td align="center"><math>\nu \equiv \frac{M_r^*|_\mathrm{core}}{M_r^*|_\mathrm{tot}}</math></td>
</tr>

<tr>
  <td align="center" rowspan="2">0.352159</td>
  <td align="center" rowspan="2">1.229594</td>
  <td align="center"><math>(+)</math> <b>B2</b></td>
  <td align="center">2.478253</td>
  <td align="center">0.572857</td>
  <td align="center">1.408807</td>
  <td align="center">0.608404</td>
  <td align="center">- 0.265128</td>
  <td align="center">2.875465</td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center">0.00938347</td>
  <td align="center">13.55831</td>
  <td align="center">7.037311</td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center">0.122470</td>
  <td align="center">0.101429</td>
</tr>

<tr>
  <td align="center"><math>(-)</math></td>
  <td align="center">1.210530</td>
  <td align="center">0.819655</td>
  <td align="center">2.140726</td>
  <td align="center">0.832073</td>
  <td align="center">-0.084850</td>
  <td align="center">3.056743</td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center">0.0769585</td>
  <td align="center">12.95956</td>
  <td align="center">3.437456</td>
  <td align="center" bgcolor="grey">&nbsp;</td>
  <td align="center">0.115207</td>
  <td align="center">0.034078</td>
</tr>
</table>
Note as well that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>q</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\frac{\xi_i}{\xi_s} = \frac{\xi_i}{Q_r \eta_s} \, ;</math></td>
  <td align="center" rowspan="3">[[File:QnuPlotB2annotated.png|350px|q-nu plot including Model B2]]</td>
</tr>

<tr>
  <td align="right"><math>\nu</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\frac{M^*_r|_\mathrm{core}}{M^*_r|_\mathrm{tot}} = \frac{\sin(\eta_i - B) - \eta_i\cos(\eta_i-B)}{\eta_s}\, .</math></td>
</tr>
<tr>
  <td align="center" colspan="3">&nbsp;</td>
</tr>
</table>
The figure here, on the right, is intended to illustrate that we can reproduce the results displayed in [[SSC/Stability/BiPolytropes/HeadScratching#Planned_Approach|Figure 2 of our accompanying discussion]] &#8212; see also [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Sequences|here]].  The displayed sequences are, as labeled, for <math>\mu_e/\mu_c = 0.25</math> and for  <math>\mu_e/\mu_c = 0.309</math>.