Editing SSC/Stability/BiPolytropes/HeadScratching (section)

=Rethink Evolution and Stability=
<table border="0" align="left" cellpadding="8">
<tr>
<th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = muratio0.25]]Figure 4: &nbsp; Mass vs. Radius</th>
<th align="center">Figure 5: &nbsp; &nu; vs. q</th>
</tr>
<tr>
<td align="center">
[[File:MassVradius0.25B.png|center|320px|Mass versus Radius m_e = 0.93]]
</td>
<td align="center">
[[File:NuVq0.25B.png|center|320px|Mass versus Radius m_e = 0.93]]
</td>
</tr>
</table>
Consider a system that has <math>m_3 = (3\mu_e/\mu_c) = 0.75</math> and that slowly evolves along the appropriate (light blue), <math>m_3 =</math> constant equilibrium sequence shown in Figure 3.  It begins its evolution with a very small core &#8212; that is, with <math>\xi_i</math> nearly zero, and with <math>q</math> and <math>\nu</math> both very small.  The location of the core-envelope interface, <math>\xi_i</math>, moves slowly outward (in Lagrangian mass space) as the ashes left from hydrogen burning build up the mass of the core at the expense of the envelope.  This means that (slow) evolution proceeds along the <math>q - \nu</math> equilibrium sequence in a counter-clockwise direction.  

[[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Schwarzschild and his collaborators]] noticed that, as an evolution proceeds along an equilibrium sequence for which (in our example case) <math>m_3 \le 1</math>, the fractional core mass increases only up to a limiting value, <math>\nu_\mathrm{max}</math>; in our case, <math>\nu_\mathrm{max} \approx 0.139</math>.  As the core-envelope interface location, <math>\xi_i</math>, attempts to increase to a value larger than the value associated with the model at <math>\nu_\mathrm{max}</math>, something rather drastic must happen &#8212; at least on a secular time scale associated with nuclear burning.  We have wondered whether a ''dynamical'' instability is also encountered at this "turning point" along the equilibrium sequence.  Up to now, all of our (inadequate) detailed analyses have been focused on securing an answer to this question.

As Figure 4 illustrates, along this same sequence, the normalized radius <math>(R^*)</math> starts off small and it steadily grows as the evolution proceeds, but the normalized total mass <math>(M^*_\mathrm{tot})</math> starts off large and initially decreases.  In reality, we expect the system to conserve its total mass throughout the evolution.  Given that the mass has been normalized via the expression,
<div align="center">
<math>M^*_\mathrm{tot} = M_\mathrm{tot} \biggl[ \frac{G^{3/2} \rho_c^{1 / 5}}{K_c^{3/2}} \biggr] \, ,</math>
</div>
we appreciate that a decrease in the dimensionless mass (as depicted in Fig. 4) can quite naturally be attributed to a steady increase in the specific entropy of the core material, <math>K_c</math>.  This evolution along the equilibrium sequence will happen on a secular, rather than dynamical, time scale that is set by the rate at which <math>K_c</math> increases &#8212; that is, at a rate set by nuclear burning.  But at each point along the sequence, we can check to see whether the equilibrium configuration is dynamically stable.  We expect that the turning point along the <math>M^*(R^*)</math> sequence is an indication of transition from a (dynamically) stable to (dynamically) unstable state.  We should be able to apply the B-KB74 conjecture to get a good idea of what the unstable eigenfunction looks like at this turning point.

==Differentiate M<sup>*</sup> With Respect to &ell;<sub>i</sub>==

In an accompanying discussion titled, [[SSC/StabilityConjecture/Bipolytrope51#New_Derivation|''New Derivation'']], we examined how the core mass-fraction <math>(\nu)</math> varies with <math>\ell_i \equiv \xi_i/\sqrt{3}</math>.  Here, we want to examine how the total mass <math>(M^*_\mathrm{tot})</math> varies with <math>\ell_i</math>.  In what follows, we borrow heavily from various analytic expressions that have been obtained via this separate ''New Derivation''; and, as in this earlier analysis, numerical evaluations (in parentheses) come from '''Example #1''' for which, <math>\mu_e/\mu_c = 0.25</math> and <math>\xi_i = 0.5</math>, which implies that <math>m_3 = 0.75</math> and <math>\ell_i = (12)^{-1 / 2}</math>.

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

<tr>
  <td align="right">
<math>M^*_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl[- \frac{\eta_s^2}{\theta_i} \cdot \biggl(\frac{d\phi}{d\eta}\biggr)_s \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \biggl[\frac{A\eta_s}{\theta_i} \biggr]
= 40.09338625</math>,
  </td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(1 + \ell_i^2)^{-1 / 2} = 0.960768923</math>,
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{m_3 \ell_i}{(1+\ell_i^2)} = 0.199852016</math>,
  </td>
</tr>

<tr>
  <td align="right">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{m_3\ell_i}\biggl[1 + (1 - m_3)\ell_i^2 \biggr] 
= \frac{49}{6\sqrt{3}}
= 4.715027199\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\pi}{2} +  \tan^{-1}\Lambda_i\biggr) +  \frac{m_3\ell_i }{(1 + \ell_i^2)} 
= 3.132453649\, , 
</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}
=
\frac{m_3\ell_i}{(1+\ell_i^2)}\biggl\{
1 + \frac{1}{m_3^2 \ell_i^2}\biggl[1 + (1 - m_3)\ell_i^2 \biggr]^2
\biggr\}^{1 / 2}
= 0.963267676 \, .
</math>
  </td>
</tr>
</table>
Now, the differentiation:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\frac{d\theta_i}{d\ell_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\ell_i (1 + \ell_i^2)^{-3 / 2} 
= - 12 (13)^{-3/2}
= - 0.256015475 (7)
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\Lambda_i}{d\ell_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2(1-m_3)\ell_i}{m_3\ell_i} - \frac{1}{m_3\ell_i^2}\biggl[1 + (1 - m_3)\ell_i^2 \biggr] 
=
\frac{1}{m_3\ell_i^2}\biggl\{2(1-m_3)\ell_i^2 - [1 + (1 - m_3)\ell_i^2 ] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{48}{3}\biggl\{\frac{1}{24} - \frac{49}{48} \biggr\} = -\frac{47}{3}
= -15.66666666
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\eta_s}{d\ell_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{d\ell_i} \biggl[
\frac{m_3\ell_i }{(1 + \ell_i^2)}
\biggr]
+
\frac{d}{d\ell_i} 
\biggl(\tan^{-1}\Lambda_i\biggr) 
=
\biggl[
\frac{m_3}{(1 + \ell_i^2)}
\biggr]
-
\biggl[
\frac{2m_3\ell_i^2 }{(1 + \ell_i^2)^2}
\biggr]
+
\biggl[1 + \Lambda_i^2\biggr]^{-1} \frac{d\Lambda_i}{d\ell_i}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{9}{13}
-
\frac{2\cdot 3^2}{13^2}
+
\biggl[\frac{2509}{2^2\cdot 3^3}\biggr]^{-1} \biggl( - \frac{47}{3}\biggr)
=
\frac{1}{13^2}\biggl\{9\cdot 13 - 2\cdot 3^2 - \biggl[\frac{2^2\cdot 3^2 \cdot 13 \cdot 47}{193}\biggr] 
\biggr\}
=
\frac{3^2}{13^2 \cdot 193}\biggl\{11 \cdot 193 - 2^2 \cdot 13 \cdot 47 
\biggr\}
=
-0.088573443
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[
\frac{m_3}{(1 + \ell_i^2)}
\biggr]
-
\biggl[
\frac{2m_3\ell_i^2 }{(1 + \ell_i^2)^2}
\biggr]
+
\biggl\{ [1 + (1 - m_3)\ell_i^2 ]^2 + m_3^2 \ell_i^2 \biggr\}^{-1} \biggl\{
2m_3(1-m_3)\ell_i^2
- m_3 [1 + (1 - m_3)\ell_i^2 ]
\biggr\} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{dA}{d\ell_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1+\Lambda_i^2)^{1 / 2} \frac{d}{d\ell_i}\biggl[ \frac{m_3 \ell_i}{(1+\ell_i^2)} \biggr]
+
\biggl[ \frac{m_3 \ell_i}{(1+\ell_i^2)} \biggr] \frac{d}{d\ell_i}(1+\Lambda_i^2)^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1+\Lambda_i^2)^{1 / 2} \biggl[ \frac{m_3 }{(1+\ell_i^2)} - \frac{2m_3 \ell_i^2}{(1+\ell_i^2)^2} \biggr]
+
\biggl[ \frac{m_3 \ell_i}{(1+\ell_i^2)} \biggr]\frac{\Lambda_i }{(1+\Lambda_i^2)^{1 / 2}} \frac{d\Lambda_i}{d\ell_i}
= 4.819904715(4)\biggl[ 0.585798817(5) \biggr] + 0.195503386(6)\cdot \biggl(-\frac{47}{3}\biggr)
= -0.239391902
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{13\cdot 193}{2^2 \cdot 3^3} \biggr)^{1 / 2} \biggl[ 3^2 \cdot 13  - 2 \cdot 3^2  \biggr]\frac{1}{13^2}
+
\frac{49}{2\cdot 3^{3/2}}\biggl[ \frac{3^2 }{13} \biggr]\cdot \frac{1}{2\cdot 3^{1 / 2}}
\biggl(\frac{2^2 \cdot 3^3}{13\cdot 193} \biggr)^{1 / 2} \biggl( - \frac{47}{3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3\cdot 11^2\cdot 193}{2^2 \cdot 13^3} \biggr)^{1 / 2} 
-
\biggl(\frac{3 \cdot 7^4 \cdot 47^2}{2^2 \cdot 13^{3}\cdot 193} \biggr)^{1 / 2}  
=
\frac{3^{1 / 2}}{2\cdot 13^{3/2} \cdot 193^{1 / 2}}\biggl[11\cdot 193  - 7^2 \cdot 47  \biggr]
=
- \biggl[ \frac{2^2 \cdot 3^{5} \cdot 5^2}{13^{3} \cdot 193} \biggr]^{1 / 2}
= - 0.239391901
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(1+\Lambda_i^2)^{1 / 2}} \biggl\{
(1+\Lambda_i^2) \biggl[ \frac{m_3 }{(1+\ell_i^2)} - \frac{2m_3 \ell_i^2}{(1+\ell_i^2)^2} \biggr]
+
\frac{\Lambda_i}{\ell_i} \biggl[ \frac{1}{(1+\ell_i^2)} \biggr]
\biggl[  2(1-m_3)\ell_i^2 - [1 + (1 - m_3)\ell_i^2 ] \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \biggl(\frac{\pi}{2}\biggr)^{1 / 2}  \frac{dM^*_\mathrm{tot}}{d\ell_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{d\ell_i}\biggl[\frac{A\eta_s}{\theta_i}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\eta_s}{\theta_i}\frac{dA}{d\ell_i}
+
\frac{A}{\theta_i}\frac{d\eta_s}{d\ell_i}
-
\frac{A\eta_s}{\theta_i^2}\frac{d\theta_i}{d\ell_i}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
-\biggl[\frac{(3.132453649)}{(0.960768923)} \biggr]0.239391901
-
\biggl[ \frac{( 0.963267676 )}{(0.960768923)} \biggr]0.088573443
+
\biggl[ \frac{(0.963267676 )(3.132453649)}{(0.960768923)^2} \biggr]0.256015475
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
- 0.78050(8)
- 0.088803
+ 0.836874
=
-0.032426 \, .
</math>
  </td>
</tr>
</table>

==Selected Models==

Via a crude iterative technique, we have determined that the derivative, <math>dM^*_\mathrm{tot}/d\ell_i</math>, goes to zero when <math>\xi_i = 2.27276626</math> (to eight significant digits); this is therefore the minimum-mass model &#8212; identified by the light-blue diamond-shaped marker &#8212; along the <math>M^*_\mathrm{tot}(R^*)</math> sequence shown above in Figure 4. A few other properties of this model "<b>A</b>" are recorded in Table 2.  For example, <math>(M^*_\mathrm{tot}, R^*) = (38.97032951, 12.598233)</math>; and its position (also marked by a light-blue diamond) along the Figure 5 <math>(q, \nu) = (0.1246568, 0.0927131)</math>. The lower-left figure in Table 2 shows how <math>(r^*)</math> varies with enclosed mass-fraction for this minimum-mass model "<b>A</b>"; the core-envelope interface &#8212; where the blue and red segments of the plotted curve meet &#8212; is located at <math>(M_r/M_\mathrm{tot}, r^*) = (0.092713145, 1.5704549)</math>.

<table border="1" align="center" cellpadding="5">

<tr>
  <th align="center" colspan="8">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/Bipolytrope/Stability/qAndNuMax.xlsx --- worksheet = K-BK74 thru MinuPreparation]]Table 2<br />Bipolytrope with <math>(n_c, n_e) = (5, 1)</math><br />Selected Pairings along the <math>\mu_e/\mu_c = 0.25</math> Sequence</th>
</tr>

<tr>
  <td align="center">Pairing</td>
  <td align="center"><math>(\xi_i)_+</math></td>
  <td align="center"><math>\Lambda_i</math></td>
  <td align="center"><math>M^*_\mathrm{tot}</math></td>
  <td align="center"><math>\frac{dM^*_\mathrm{tot}}{d\ell_i}</math></td>
  <td align="center"><math>R^*</math></td>
  <td align="center"><math>q</math></td>
  <td align="center"><math>\nu</math></td>
</tr>

<tr>
  <td align="center">'''Example #1'''</td>
  <td align="center"><math>0.5</math></td>
  <td align="center"><math>4.715027199</math></td>
  <td align="center"><math>40.09338625</math></td>
  <td align="center"><math>-0.413955</math></td>
  <td align="center">--</td>
  <td align="center">--</td>
  <td align="center">--</td>
</tr>

<tr>
  <td align="center">'''A''' <font size="-1">(degenerate)</font></td>
  <td align="center"><math>2.27276626</math></td>
  <td align="center"><math>1.4535131</math></td>
  <td align="center"><math>38.97032951</math></td>
  <td align="center"><math>6.20\times 10^{-9}</math></td>
  <td align="center"><math>12.598233</math></td>
  <td align="center"><math>0.1246568</math></td>
  <td align="center"><math>0.0927131</math></td>
</tr>

<tr>
  <td align="center">'''B1'''</td>
  <td align="center"><math>2.0653386</math></td>
  <td align="center"><math>1.5156453</math></td>
  <td align="center"><math>39.00000000</math></td>
  <td align="center"><math>-0.491175</math></td>
  <td align="center"><math>11.31459</math></td>
  <td align="center"><math>0.1261314</math></td>
  <td align="center"><math>0.082829</math></td>
</tr>

<tr>
  <td align="center">'''B2'''</td>
  <td align="center"><math>2.4782510</math></td>
  <td align="center"><math>1.4088069</math></td>
  <td align="center"><math>39.00000000</math></td>
  <td align="center"><math>+0.500086</math></td>
  <td align="center"><math>13.987375</math></td>
  <td align="center"><math>0.1224277</math></td>
  <td align="center"><math>0.1013938</math></td>
</tr>

<tr>
  <td align="center">'''C1'''</td>
  <td align="center"><math>1.83343536</math></td>
  <td align="center"><math>1.612448</math></td>
  <td align="center"><math>39.10000000</math></td>
  <td align="center"><math>-0.990185582</math></td>
  <td align="center"><math>10.019034</math></td>
  <td align="center"><math>0.1264476</math></td>
  <td align="center"><math>0.0705448</math></td>
</tr>

<tr>
  <td align="center">'''C2'''</td>
  <td align="center"><math>2.70235958</math></td>
  <td align="center"><math>1.3746562</math></td>
  <td align="center"><math>39.10000000</math></td>
  <td align="center"><math>+1.042782519</math></td>
  <td align="center"><math>15.637446</math></td>
  <td align="center"><math>0.119412</math></td>
  <td align="center"><math>0.1095988</math></td>
</tr>

<tr>
  <td align="center" colspan="4">
[[File:MinMassProfile0.25.png|400px|Radius vs. Mass for Minimum-Mass Bipoltrope having &mu;-ratio = 0.250]]
  </td>
  <td align="center" colspan="4">
'''Eigenfunction Obtained Via B-KB74 Conjecture'''<br />
[[File:EigenfunctionCorrected.png|450px|Eigenfunction for Minimum-Mass Bipoltrope having &mu;-ratio = 0.250]]
  </td>
</tr>
</table>


In the context of our analysis of the stability of pressure-truncated n = 5 polytropes, we showed how the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|B-KB74 conjecture]] can be used to illustrate the approximate shape of the radial eigenfunction of the marginally unstable mode.  Proceeding along the lines of this independent discussion, here we have identified two equilibrium models &#8212; labeled "<b>B1</b>" and "<b>B2</b>" in Table 2 &#8212; that lie near to, but on either side of, the minimum-mass model along the equilibrium sequence and that have identical total masses: in this case, <math>M^*_\mathrm{tot} = 39.00000000</math> (identical, to nine significant digits).  Using the mass-fraction, <math>m_r \equiv M_r/M_\mathrm{tot}</math>, as the Lagrangian coordinate for both models, we subtracted the profile of model "<b>B1</b>" from the profile of model "<b>B2</b>" and divided this difference by the average profile, we obtained the approximate neutral-mode eigenfunction, <math>x(m_r)</math>, displayed in the lower-right figure of Table 2.

Things to note about this iteratively derived, approximate neutral-mode eigenfunction:
<ol>
  <li>The radial-displacement function, <math>x(m_r)</math>, has been normalized to unity at the surface.</li>
  <li>The location of the model "<b>A</b>" core-envelope interface <math>(m_r = \nu_{A} = 0.0927131)</math> has been marked by the vertical, red-dashed line segment.</li>
  <li>Throughout the core, <math>x</math> is very small; consistent with being zero throughout.</li>
  <li>Moving inward through the envelope, <math>x</math> appears to drop smoothly from "plus" one (at the surface) to approximately "minus" one (at the interface).</li>
  <li>Because <math>x</math> passes through zero one time inside the envelope, this cannot be the eigenfunction of the fundamental mode of radial oscillation; instead, it is likely associated with the 1st overtone, as discussed for example in connection with [[SSC/Stability/n3PolytropeLAWE#Fig1|Schwarzschild's modeling of radial eigenfunctions of n = 3 polytropes]].</li>
</ol>

With regard to the second itemized note, we should point out that, although models "<b>B1</b>" and "<b>B2</b>" have identical total masses, their core mass-fraction &#8212; that is, the location of the core-envelope interface as defined by the Lagrangian mass marker &#8212; is different: <math>\nu_{B1} = 0.082829</math> and <math>\nu_{B2} = 0.101394</math>.  As a result, the B-KB74 conjecture should not be expected to apply in the immediate vicinity of the core-envelope interface.

==LAWE==

Let's perform the LAWE integration in two parts: (1) Integrate from the center (where the derivative of the displacement function must be zero), through the core, up to the core-envelope interface; and (2) integrate from the surface (where the logarithmic derivative of the displacement function is negative one), through the envelope, down to the core-envelope interface.  Examine the discontinuity that results and see whether it makes sense in terms of the required "matching conditions" at the interface.

===Throughout the Configuration===

From the last couple of lines of an [[SSC/Stability/BiPolytropes#Foundation|accompanying ''Foundation'' presentation]], the relevant LAWE may be written as,

<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\{ \frac{4}{r^*} -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)^2}\biggr\} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{\omega^2}{\gamma_\mathrm{g} G\rho_c} + \biggl(\frac{4}{\gamma_\mathrm{g}} - 3\biggr)\frac{ M_r^*}{(r^*)^3}\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 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\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} + \frac{\mathcal{H}}{r^*} \frac{dx}{dr*} 
+ 
\biggl[\biggl(\frac{\sigma_c^2}{\gamma_g}\biggr) \mathcal{K}_1 - \alpha_g \mathcal{K}_2\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\sigma_c^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3\omega^2}{2\pi G\rho_c}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl(3 - \frac{4}{\gamma_g}\biggr) \, .
</math>
  </td>
<td align="center" colspan="4">&nbsp; </td>
</tr>
</table>
From a related discussion of [[SSC/Stability/BiPolytropes#Profile|interior structural profiles]], we appreciate that throughout the core we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{1}{3}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}  \, ;</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{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]\biggl( \frac{2\pi}{3}\biggr)^{1 / 2} \frac{1}{\xi} 
= 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{(r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{2\pi}{3}\biggr) \xi^{-2}
\, ;</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~1
\, ;
</math>
  </td>
</tr>

<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^{-1}_i \phi(\eta)^{-1}
\, ;
</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} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \biggr]^{-1}
=
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{\theta_i}{\eta} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{(r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \eta^{-2}
\, .
</math>
  </td>
</tr>
</table>

===Surface Boundary Condition===
In an effort to [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations|ensure finite-amplitude fluctuations]] at the surface, we will enforce the condition,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r_0 \frac{d\ln x}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, ,</math>
  </td>
</tr>
</table>
that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^* \frac{d\ln x}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl(\frac{2\pi}{3}\biggr)\frac{\sigma_c^2 (R^*)^3}{\gamma_g M^*_\mathrm{tot}} -\alpha_g \biggr]</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r^* = R^* \, ,</math>
  </td>
</tr>
</table>
where the asterisks <math>(*)</math> signal that we have employed the same variable normalizations as have been adopted in our [[SSC/Stability/BiPolytropes#Foundation|accompanying ''Foundations'' discussion]].  Since our analysis, here, is focused on the marginally unstable (minimum-mass) configuration in which we expect <math>\sigma_c^2 = 0</math>, the surface (envelope) constraint becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\alpha_g = -~1</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~\eta = \eta_s \, .</math>
  </td>
</tr>
</table>

===Interface===

Drawing from an [[SSC/Stability/BiPolytropes#Interface_Conditions|accompanying discussion]], the matching condition at the interface is given by the expression,
<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>\biggl[ \gamma_c x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln r^*} \biggr) - \gamma_e x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d \ln r^*} \biggr)\biggr]_i \, .</math>
  </td>
</tr>
</table>
Given that <math>\gamma_c = 6/5</math> and <math>\gamma_e = 2</math>, this becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln r^*} \biggr) \biggr]_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{5}{3}\biggl[ x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d \ln r^*} \biggr)\biggr]_i \, .</math>
  </td>
</tr>
</table>

===Central Boundary Condition===

The central boundary condition is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dx_\mathrm{core}}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>

In order to kick-start the integration outward from the center of the configuration, we will following the procedure that has been detailed in an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|accompanying discussion]].  At the center of the configuration <math>(\xi_1 = 0)</math>, we label the fractional displacement function as <math>x_1</math> &#8212; value to be set later, perhaps in an effort to help secure the proper matching conditions at the interface &#8212; then we will draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \Delta_\xi</math>, away from the center.  Specifically, given that <math>n = 5, \gamma_g = 6/5</math>, and <math>\alpha_g = -1/3</math> in the core, we will set,

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

<tr>
  <td align="right">
<math>
x_2 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F}  \Delta_\xi^2}{60} \biggr] 
=
x_1 \biggl[ 1 - \frac{\mathfrak{F}  \Delta_\xi^2}{10} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\mathfrak{F}
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 2\alpha_g \biggr]
=
\frac{1}{6}\biggl[ 5 \sigma_c^2  + 4 \biggr]
\, .</math>
  </td>
</tr>
</table>

===Numerical Integration===
====Through the Core====
Throughout the core, the governing LAWE 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}{d\xi^2} + \frac{\mathcal{H}}{\xi} \frac{dx}{d\xi} 
+ 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi }{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 - 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{4\pi}{3}\biggr) \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \, .
</math>
  </td>
</tr>
</table>

Now, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the pair of substitutions,

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

<tr>
  <td align="right">
<math>x_i' = \frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2 \Delta_\xi}   \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
x_i'' = \frac{d^2x}{d\xi^2}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} \, ,</math>
  </td>
</tr>
</table>
which will provide an approximate expression for <math>~x_+ \equiv x_{i+1}</math>, given the values of <math>~x_- \equiv x_{i-1}</math> and <math>~x_i</math>.  Specifically, if the center of the configuration is denoted by the grid index, <math>~i=1</math>, then for zones, <math>~i = 2 \rightarrow N</math>,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{x_+ - 2x_i + x_-}{\Delta_\xi^2}  \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ - x_-}{2 \Delta_\xi} \biggr]
- 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{x_+ }{\Delta_\xi^2}  \biggr] + \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ }{2 \Delta_\xi} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{2x_i - x_-}{\Delta_\xi^2}  \biggr]
+ \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_-}{2 \Delta_\xi} \biggr]
- 
\biggl(\frac{1}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr]  x_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\mathcal{H}}{\xi} \biggl[ \frac{\Delta_\xi }{2 } \biggr] \biggr\}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2
- 
\biggl(\frac{\Delta_\xi^2}{4\pi}\biggr)\biggl[5\sigma_c^2 \mathcal{K}_1 + 2 \mathcal{K}_2\biggr] \biggr\} 
-  x_-\biggl\{1 - \frac{\mathcal{H}}{\xi} \biggl[ \frac{\Delta_\xi}{2 } \biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
<div align="center">'''Check Against Independent Derivation'''</div>
We have dealt with this identical LAWE in connection with our [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|analysis of the stability of pressure-truncated n = 5 Polytropic configurations]].  Let's see whether that derivation matches our current one.  In that case, we found,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_+ \biggl\{ 2\theta +\frac{4\Delta_\xi \theta}{\xi} - \Delta_\xi (n+1)(- \theta^')\biggr\} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{4\theta - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 
2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr) \biggr]  \biggr\}
- x_- \biggl[2\theta - \frac{4\Delta_\xi \theta}{\xi} + \Delta_\xi (n+1)(- \theta^') \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\Delta_\xi }{2\xi} 
\biggl[ 4 - \frac{6\xi (- \theta^')}{\theta} \biggr]\biggr\}2\theta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2 - \frac{\Delta_\xi^2}{\theta}\biggl[ \frac{5\sigma_c^2}{6}  + 
\frac{2}{3} \biggl(- \frac{3\theta^'}{\xi}\biggr) \biggr]  \biggr\}2\theta
- x_- \biggl\{ 1 - \frac{\Delta_\xi}{2\xi}\biggl[ 4 - \frac{6\xi (- \theta^')}{\theta}\biggr] \biggr\} 2\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + \frac{\Delta_\xi }{2\xi} 
\biggl[ \mathcal{H} \biggr]\biggr\}2\theta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i\biggl\{2 - \frac{\Delta_\xi^2}{4\pi}\biggl[ 5\sigma_c^2 \cdot \frac{4\pi}{6\theta}  + 
\frac{8\pi}{3} \biggl(- \frac{3\theta^'}{\xi \theta}\biggr) \biggr]  \biggr\}2\theta
- x_- \biggl\{ 1 - \frac{\Delta_\xi}{2\xi}\biggl[ \mathcal{H}\biggr] \biggr\} 2\theta \, .
</math>
  </td>
</tr>
</table>
Given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi}{6\theta} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2\pi}{3}\biggl(1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}
~~\rightarrow ~~ \mathcal{K}_1
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp;&nbsp; &nbsp; </td>
  <td align="right">
<math>\frac{4\pi}{3}\biggl(-\frac{3\theta'}{\xi\theta}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3}\biggl(1 + \frac{1}{3}\xi^2 \biggr)^{-1}
~~\rightarrow ~~ \mathcal{K}_2 \, ,
</math>
  </td>
</tr>
</table>
we can confirm that the two expressions are identical.
</td>
</tr>
</table>

====Through the Envelope====
Throughout the envelope &#8212; that is, for <math>\eta_i \le \eta \le \eta_s</math> &#8212; the governing LAWE 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}{d\eta^2} 
+ 
\frac{\mathcal{H}}{\eta} \frac{dx}{d\eta} 
+ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}  \biggl[\biggl(\frac{\sigma_c^2}{2}\biggr) \mathcal{K}_1 -  \mathcal{K}_2\biggr]  x\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, .
</math>
  </td>
</tr>
</table>

<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>

From a related discussion of [[SSC/Stability/BiPolytropes#Profile|interior structural profiles]], we appreciate that throughout the envelope we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\alpha_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~1
\, ;
</math>
  </td>
</tr>

<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^{-1}_i \phi(\eta)^{-1}
\, ;
</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} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta \biggr]^{-1}
=
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \frac{\theta_i}{\eta} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{(r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \eta^{-2}
\, .
</math>
  </td>
</tr>
</table>

Hence,

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

<tr>
  <td align="right">
<math>\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1} 
=
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{\rho^*}{P^*}\biggr)\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 2\cdot \frac{d\ln \phi}{d\ln \eta}
=
2\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{H}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}
=
4 -2\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]
=
2\biggl[1 + \frac{\eta}{\tan(\eta-B)}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2} 
\, .
</math>
  </td>
</tr>
</table>

Finally, restructuring the radially dependent coefficient of the linear term in the LAWE, we have,

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

<tr>
  <td align="right">
<math>\frac{d^2x}{d\eta^2} 
+ 
\frac{\mathcal{H}}{\eta} \frac{dx}{d\eta} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}  \mathcal{K}_2
-
\biggl(\frac{\sigma_c^2}{2}\biggr)\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}   \mathcal{K}_1 
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}\cdot
4\pi \biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \theta_i^4 \biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{2}\biggr)\frac{1}{2\pi \theta_i^4}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \cdot
\frac{2\pi }{3}\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{2}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x \, .
</math>
  </td>
</tr>
</table>

Again, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the pair of substitutions,

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

<tr>
  <td align="right">
<math>x_i' = \frac{dx}{d\eta}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2 \Delta_\eta}   \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
x_i'' = \frac{d^2x}{d\eta^2}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} \, .</math>
  </td>
</tr>
</table>
For the envelope, we will integrate from the surface, into the core-envelope interface.  So, this time these "finite-difference" expressions will provide an approximate expression for <math>x_- \equiv x_{i-1}</math>, given the values of <math>x_+ \equiv x_{i+1}</math> and <math>x_i</math>.  If the surface of the configuration is denoted by the grid index, <math>i=N</math>, then for zones, <math>i = (N-1) \rightarrow ??</math>,

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

<tr>
  <td align="right">
<math>
\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2 \Delta_\eta}  \biggr]
+ 
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
\biggl[ \frac{x_-}{\Delta_\eta^2} \biggr]
-
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_- }{2 \Delta_\eta}  \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-
\biggl[ \frac{x_+ - 2x_i }{\Delta_\eta^2} \biggr] 
-
\frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+}{2 \Delta_\eta}  \biggr]
+ 
\biggl\{ 
\biggl[4 - \mathcal{H}\biggr]\frac{1}{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{1}{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~x_- \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\} x_i  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] x_+  \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">'''[[#Surface_Boundary_Condition|Surface Boundary Condition]]'''</div>

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

<tr>
  <td align="right">
<math>\frac{d\ln x}{d\ln \eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-1
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; at, &nbsp; &nbsp; &nbsp; <math>\eta = \eta_s</math>.</td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>
\biggl[\frac{x_+ - x_-}{2 \Delta_\eta}\biggr]_s
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\biggl[\frac{dx}{d\eta} \biggr]_s = -\frac{x_s}{\eta_s}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl[\frac{x_{N+1} - x_{N-1}}{2 \Delta_\eta}\biggr]
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>-\frac{x_N}{\eta_s}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
x_{N+1}
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>x_{N-1} - \frac{2\Delta_\eta x_N}{\eta_s} \, .</math>
  </td>
</tr>
</table>
Inserting this expression for "<math>x_+</math>" in the finite-difference representation of the envelope's LAWE allows us to determine the value for <math>x_- = x_{N-1}</math>.  Specifically,

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

<tr>
  <td align="right">
<math>x_{N-1} \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_s x_N  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] \biggl[ x_{N-1} - \frac{2\Delta_\eta x_N}{\eta_s} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_{N-1} \biggl\{
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr\}
+
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] x_{N-1} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_s x_N  
+
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr]_s \biggl[ \frac{2\Delta_\eta }{\eta_s} \biggr] x_N
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_{N-1} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 1 
+
\biggl[ \frac{\Delta_\eta }{\eta_s} \biggr]
+
2\biggl[ \frac{\Delta_\eta^2 }{\eta^2_s}\biggr]  
+
\biggl(\frac{\sigma_c^2}{2^2\cdot 3}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta_s}{A}\biggr]  
\biggr\} x_N  \, .
</math>
  </td>
</tr>
</table>
Note that, in the last term of this last expression, we have acknowledged that, <math>(\eta_s - B) = \pi ~~\Rightarrow ~~ \sin(\eta_s - B) = -1</math>.

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


<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<div align="center">'''Slope at the Interface'''</div>

We will need to determine the slope that is associated with the envelope's eigenfunction, <math>[dx/d\eta]_\mathrm{env}</math>, precisely at the interface. While the envelope's eigenfunction does not actually exist on the "core" side of the interface, we can ''project'' what its value at <math>x_-</math> ''would'' be if the envelope's eigenfunction were to continue smoothly just one small step beyond the interface, then use this ''projected'' value to determine the function's slope ''at'' the interface location.  Labeling the interface at <math>i = J</math>, first we have, 

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

<tr>
  <td align="right">
<math>\biggl[
1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
\biggr]_J\biggl[x_-\biggr]_\mathrm{project} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 2 +
\biggl[4 - \mathcal{H}\biggr]\frac{\Delta_\eta^2 }{\eta^2}  
-
\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} 
\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]  
\biggr\}_J x_J  
-
\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr]_J x_{J+1}  \, .
</math>
  </td>
</tr>
</table>
Then we conclude that,

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

<tr>
  <td align="right">
<math>[x_\mathrm{env}^']_J \equiv \biggl[ \frac{dx_\mathrm{env}}{d\eta} \biggr]_J</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{x_{J+1} }{2 \Delta_\eta}
- 
\frac{1}{2 \Delta_\eta} \biggl[ x_- \biggr]_\mathrm{project} \, .
</math>
  </td>
</tr>
</table>

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

===Feeble Analytic Attempt===

Noice that if we assume <math>\sigma_c^2 = 0</math>, the governing envelope LAWE 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}{d\eta^2} 
+ 
\biggl[1 + \frac{\eta}{\tan(\eta-B)}\biggr]\frac{2}{\eta} \frac{dx}{d\eta}
-
\biggl\{ 
\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]\frac{2}{\eta^2}  
\biggr\} x \, .
</math>
  </td>
</tr>
</table>
Let's try &hellip;

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

<tr>
  <td align="right">
<math>x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\eta^{m}[\tan(\eta-B)]^{k}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{dx}{d\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m\eta^{m-1}[\tan(\eta-B)]^{k}
+
k\eta^{m}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d^2x}{d\eta^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
km\eta^{m-1}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-3}\sin(\eta-B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>
</table>

Hence,

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

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ 
\biggl[\frac{2}{\eta} + \frac{2}{\tan(\eta-B)}\biggr] 
\biggl\{ 
m\eta^{m-1}[\tan(\eta-B)]^{k}
+
k\eta^{m}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-2
\biggl[1 - \frac{\eta}{\tan(\eta-B)}\biggr]  
\eta^{m-2}[\tan(\eta-B)]^{k}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ 
2m\eta^{m-2}[\tan(\eta-B)]^{k}
+
2k\eta^{m-1}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ 
2m\eta^{m-1}[\tan(\eta-B)]^{k-1}
+
2k\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-
2  \eta^{m-2}[\tan(\eta-B)]^{k}
+
2 \eta^{m-1}[\tan(\eta-B)]^{k-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m(m-1)\eta^{m-2}[\tan(\eta-B)]^{k}
+
2k\eta^{m}[\tan(\eta-B)]^{k} [\cos(\eta-B)]^{-2}
+ 
2m\eta^{m-2}[\tan(\eta-B)]^{k}
-
2  \eta^{m-2}[\tan(\eta-B)]^{k}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
2m\eta^{m-1}k[\tan(\eta-B)]^{k-1}[\cos(\eta-B)]^{-2}
+
2k\eta^{m-1}[\tan(\eta-B)]^{k-1} [\cos(\eta-B)]^{-2}
+ 
2m\eta^{m-1}[\tan(\eta-B)]^{k-1}
+
2 \eta^{m-1}[\tan(\eta-B)]^{k-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
k(k-1)\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-4}
+
2k\eta^{m}[\tan(\eta-B)]^{k-2} [\cos(\eta-B)]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}\eta^{m-2}[\cos(\eta-B)]^{-2} \biggl\{
[m(m-1) +2m - 2][\cos(\eta-B)]^{2}
+
2k\eta^{2} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ [\tan(\eta-B)]^{k-1}2(m+1)\eta^{m-1}[\cos(\eta-B)]^{-2}\biggl\{
k + [\cos(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ k[\tan(\eta-B)]^{k-2}[\cos(\eta-B)]^{-4}\eta^{m}
\biggl\{
(k-1) 
+
2[\cos(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>
</table>
If <math>m = -1</math>, the second group of terms disappears and we have,

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

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}[\cos(\eta-B)]^{-2} \eta^{-3}\biggl\{\biggl[
2k\eta^{2} 
- 2[\cos(\eta-B)]^{2}
\biggr]
+ k[\sin(\eta-B)]^{-2}\eta^{2}
\biggl[(k-1) +2[\cos(\eta-B)]^{2}\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}[\cos(\eta-B)]^{-2} \eta^{-3}[\sin(\eta-B)]^{-2}\biggl\{
\biggl[
2k\eta^{2} 
\biggr][\sin(\eta-B)]^{2}
-
\biggl[
2[\cos(\eta-B)]^{2}
\biggr][\sin(\eta-B)]^{2}
+ k\eta^{2}(k-1) 
+ 2k\eta^{2}
[\cos(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\tan(\eta-B)]^{k}[\cos(\eta-B)]^{-2} \eta^{-3}[\sin(\eta-B)]^{-2}\biggl\{
k(k+1)\eta^{2} 
-
2[\cos(\eta-B)]^{2}[\sin(\eta-B)]^{2}
\biggr\}
</math>
  </td>
</tr>
</table>