Editing SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2 (section)

=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Renormalized=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize|Part I: &nbsp; (5, 1) Analytic Renormalize]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2|Part II:&nbsp; Envelope]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt3|III:&nbsp; Interface Pressure Gradient]]<br />&nbsp;</td>
</tr>
</table>

==Numerical Integration Through Envelope==

In an effort to numerically determine the eigenfunction of the envelope, we will follow the procedure described in an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center,_Outward|accompanying stability analysis of pressure-truncated polytropes]] to integrate the <math>n = 1</math> envelope from the core/envelope interface to the surface.  In a [[SSC/Stability/n1PolytropeLAWE#Radial_Oscillations_of_n_=_1_Polytropic_Spheres|closely related chapter titled, ''Radial Oscillations of n = 1 Polytropic Spheres'']], we have tried to find analytic expressions for the eigenvector of marginally unstable configurations.

===Setup===

====Continuous Form  of LAWE====
We begin by writing our generic version of the polytropic LAWE,
<div align="center">
{{ Math/EQ_RadialPulsation02 }}
</div>

then focus on the <math>n=1</math> case &#8212; setting <math>\gamma_g = 1 + 1/n = 2</math> and <math>\alpha = +1</math> &#8212; the relevant LAWE becomes,
<table border=0 cellpadding=2 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\{ 4 - 2 Q \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^2}{\phi} 
- Q\biggr\}  \frac{x}{\eta^2} \, ,</math>
</td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <td align="center" bgcolor="pink">
{{ Chatterji51 }} &#8212; STEP 1
  </td>
</tr>
<tr><td align="left">
If we focus on the <math>n=1</math> case but leave <math>\gamma</math> (and, hence, <math>\alpha</math>) unspecified, the relevant LAWE becomes,
<table border=0 cellpadding=2 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\{ 4 - 2 Q \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{6\gamma} \biggr)  \frac{\eta^2}{\phi} 
- \alpha Q\biggr\}  \frac{x}{\eta^2} \, .</math>
</td>
</tr>
</table>
If, in addition, we make the notation substitutions,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>Q</math></td>
  <td align="center"><math>~~~\rightarrow~~~</math></td>
  <td align="left">
<math>\frac{\mu}{n+1} = \frac{\mu}{2}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\frac{\sigma_c^2}{3\gamma}</math></td>
  <td align="center"><math>~~~\rightarrow~~~</math></td>
  <td align="left">
<math>\omega^2_\mathrm{Chatterji} \, ,</math>
  </td>
</tr>
</table>
the relevant LAWE becomes,
<table border=0 cellpadding=2 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\{ 4 - \mu \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{3\gamma} \biggr)  \frac{\eta^2}{\phi} 
- \alpha \mu\biggr\}  \frac{x}{\eta^2} </math>
</td>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{d^2x}{d\eta^2} + \biggl[ \frac{4 - \mu}{\eta} \biggr] \frac{dx}{d\eta}  
+ \biggl[ \frac{\omega^2_\mathrm{Chatterji}}{\phi} 
- \frac{\alpha \mu}{\eta^2}\biggr]  x \, ,</math>
</td>
</tr>
</tr>
</table>
which, apart from notation, is identical to equation (1) of {{ Chatterji51 }}.
</td></tr>
</table>
Now, in the broadest context (see our [[SSC/Stability/n1PolytropeLAWE/Pt4#Beech88|related discussion]]),

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

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

<tr>
  <td align="center" colspan="3">
[[Appendix/References#Beech88|Beech88]], &sect;3, p. 221, Eq. (6)
  </td>
</tr>
</table>
Therefore, also, we have,
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \eta\cot(\eta - B) \biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <td align="center" bgcolor="pink">
{{ Chatterji51 }} &#8212; STEP 2
  </td>
</tr>
<tr><td align="left">
In the context of an isolated, <math>n=1</math> polytrope, the appropriate parameter values are, <math>A=1</math> and <math>B=0</math>, in which case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\phi(\eta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sin\eta}{\eta} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\mu = 2Q</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(1 - \eta\cot\eta ) \, ,
</math>
  </td>
</tr>
</table>
and the relevant LAWE becomes,
<table border=0 cellpadding=2 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[ \frac{4 + 2(\eta\cot\eta-1)}{\eta} \biggr] \frac{dx}{d\eta}  
+ \biggl[ \frac{\eta ~\omega^2_\mathrm{Chatterji}}{\sin\eta} 
+ \frac{2 \alpha (\eta\cot\eta-1)}{\eta^2}\biggr]  x </math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{d^2x}{d\eta^2} + \biggl[ \frac{2(1 + \eta\cot\eta)}{\eta} \biggr] \frac{dx}{d\eta}  
+ \biggl[ \frac{\eta ~\omega^2_\mathrm{Chatterji}}{\sin\eta} 
+ \frac{2 \alpha (\eta\cot\eta-1)}{\eta^2}\biggr]  x \, ,</math>
</td>
</tr>

</table>
which, again apart from notation, is identical to equation (2) of {{ Chatterji51 }}; see also the (unnumbered) equation in the middle of the left-hand-column of p. 223 in {{ MF85b }}.
</td></tr>
</table>

If we set <math>\gamma = (n+1)/n = 2</math> (and correspondingly set <math>\alpha = [3-4/\gamma] = +1)</math>, the <math>n=1</math> LAWE we becomes,
<table border=0 cellpadding=2 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\{ 4 - 2 \biggl[1 - \eta\cot(\eta - B) \biggr] \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{A\sin(\eta-B)} 
- \biggl[1 - \eta\cot(\eta - B) \biggr]\biggr\}  \frac{x}{\eta^2}</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{d^2x}{d\eta^2} + 2\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{A\sin(\eta-B)} 
- 1 + \eta\cot(\eta - B) \biggr\}  \frac{x}{\eta^2} \, .</math>
</td>
</tr>
</table>
Multiplying through by <math>\phi</math>, we can write,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl[ \frac{A\sin(\eta-B)}{\eta}\biggr]\frac{d^2x}{d\eta^2} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{2A}{\eta}\biggl\{ \sin(\eta - B) + \eta\cos(\eta - B) \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ \frac{2A}{\eta} \biggl\{ \sin(\eta - B) - \eta\cos(\eta - B) \biggr\}  \frac{x}{\eta^2}
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) x \, .
</math>
</td>
</tr>
</table>

====Discrete Form  of LAWE====
In order to integrate this 2<sup>nd</sup>-order ODE numerically, we will build from the [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center,_Outward|more general expression for polytropes used in our separate development of a finite-difference scheme]], namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta_i {x_i''}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \biggl[4\theta_i - (n+1)\xi_i (- \theta^')_i\biggr] \frac{x_i'}{\xi_i}  
- (n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g}  - 
\frac{\alpha}{\xi_i } (- \theta^')_i\biggr]  x_i \, .</math>
  </td>
</tr>
</table>
Making the notation substitutions, <math>(\xi, \theta) \rightarrow (\eta, \phi)</math>, we have instead,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\phi_i {x_i''}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \biggl[4\phi_i - (n+1)\eta_i (- \phi^')_i\biggr] \frac{x_i'}{\eta_i}  
- (n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g}  - 
\frac{\alpha}{\eta_i } (- \phi^')_i\biggr]  x_i \, .</math>
  </td>
</tr>
</table>
Now, adopting the finite-difference expressions,

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

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

<tr>
  <td align="right">
<math>
x_i'' 
</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>
</div>
<span id="DiscreteLAWE">the discrete form of the LAWE becomes,</span>

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

<tr>
  <td align="right">
<math>x_+ \overbrace{\biggl[2\phi_i +\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i\biggr]}^{\mathrm{TERM1}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_- \overbrace{\biggl[\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i - 2\phi_i\biggr]}^{\mathrm{TERM2}} 
+  x_i \overbrace{\biggl\{4\phi_i - \frac{\Delta_\eta^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 
2\alpha \biggl(- \frac{3\phi^'}{\eta}\biggr)_i\biggr]  \biggr\}}^{\mathrm{TERM3}} \, .</math>
  </td>
</tr>
</table>

When applied specifically to an <math>n=1</math>, polytropic configuration, we should insert the following specific expressions: 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\gamma_g</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \frac{1}{n} = 2
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\alpha</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3 - \frac{4}{\gamma_g} = +1
\, ,</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>(-\phi')_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{A}{\eta_i^2} \biggl[\sin(\eta_i-B) - \eta_i \cos(\eta_i - B) \biggr] \, .
</math>
  </td>
</tr>
</table>

===Pressure-Truncated n = 1 Polytrope===

In the case of an ''isolated'' <math>n=1</math> polytrope, we must set <math>B = 0</math>; in addition, it is customary to set <math>A = 1</math>.  The relevant LAWE is, then,
<table border=0 cellpadding=2 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} + 2\biggl\{ 1 + \eta\cot(\eta) \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{\sin(\eta)} 
- 1 + \eta\cot(\eta) \biggr\}  \frac{x}{\eta^2} \, .</math>
</td>
</tr>
</table>

====Review of Trial Analytic Eigenfunction====
This is the same 2<sup>nd</sup>-order ODE that we derived in a [[SSC/Stability/n1PolytropeLAWE#WorkInProgress|separate discussion]]; there it was accompanied by the [[SSC/Stability/Polytropes#Boundary_Conditions|surface boundary condition]],
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>- \frac{d\ln x}{d\ln \xi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\biggl( \frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)}
\biggl[ \frac{\xi}{\theta'}\biggr]_\mathrm{surf}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ - \frac{d\ln x}{d\ln \eta} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
1 + \frac{\sigma_c^2}{12}
\biggl[ \frac{\eta^3}{(\eta \cos\eta - \sin\eta)}\biggr]_{\eta=\pi}
=
1 - \frac{\pi^2 \sigma_c^2}{12} \, .
</math>
  </td>
</tr>
</table>

From, for example, a separate [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration|succinct demonstration]], we appreciate that if the displacement function is assumed to be,

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

<tr>
  <td align="right">
<math>x_P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{\eta^2} \biggl[ 1 - \eta \cot\eta \biggr] </math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
&hellip; that is,

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

<tr>
  <td align="right">
<math>x_P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{\eta^2}  - \frac{3 \cos\eta}{\eta \sin\eta} \, , </math>
  </td>
</tr>
</table>

in which case,

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

<tr>
  <td align="right">
<math>\frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{6}{\eta^3} + 3\biggl[\frac{\cos\eta}{\eta^2\sin\eta} + \frac{1}{\eta} + \frac{\cos^2\eta}{\eta\sin^2\eta}\biggr] 
\, ,
</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right">
<math>\frac{d^2x_P}{d\eta^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
+ 
3\frac{d}{d\eta}\biggl[ \frac{\cos\eta}{\eta^2\sin\eta} \biggr] 
+ 3\frac{d}{d\eta}\biggl[ \frac{1}{\eta} \biggr] 
+ 3\frac{d}{d\eta}\biggl[ \frac{\cos^2\eta}{\eta\sin^2\eta} \biggr]

</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
- 
3\biggl[ \frac{ 2\cos\eta}{\eta^3\sin\eta} + \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{\sin\eta}{\eta^2\sin\eta} \biggr] 
- \biggl[ \frac{3}{\eta^2} \biggr] 
- 3\biggl[ \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{2\cos^3\eta}{\eta\sin^3\eta} + \frac{2\cos\eta}{\eta\sin\eta} \biggr]
\, .
</math>
  </td>
</tr>
</table>
Hence, 
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">LAWE</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \frac{2}{\eta} + \frac{2\cos\eta}{\sin\eta} \biggr\}  \biggl\{
-\frac{6}{\eta^3} + \biggl[\frac{3\cos\eta}{\eta^2\sin\eta} + \frac{3}{\eta} + \frac{3\cos^2\eta}{\eta\sin^2\eta}\biggr]
\biggr\}  
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{\sin(\eta)} 
- 1 + \frac{\eta\cos\eta}{\sin\eta} \biggr\} \biggl[ \frac{3}{\eta^4}  - \frac{3 \cos\eta}{\eta^3 \sin\eta} \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
- 
3\biggl[ \frac{ 2\cos\eta}{\eta^3\sin\eta} + \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{\sin\eta}{\eta^2\sin\eta} \biggr] 
- \biggl[ \frac{3}{\eta^2} \biggr] 
- 3\biggl[ \frac{\cos^2\eta}{\eta^2\sin^2\eta} + \frac{2\cos^3\eta}{\eta\sin^3\eta} + \frac{2\cos\eta}{\eta\sin\eta} \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{2}{\eta} + 2\cot\eta \biggr]  \biggl[
-\frac{6}{\eta^3} + \frac{3\cot\eta}{\eta^2} + \frac{3}{\eta} + \frac{3\cot^2\eta}{\eta}
\biggr]  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{6} \biggr)  \frac{\eta^3}{\sin\eta} 
\biggr\} \biggl[ \frac{3}{\eta^4}  - \frac{3 \cot\eta}{\eta^3 } \biggr]
+ \biggl[- 2 + 2\eta\cot\eta \biggr] \biggl[ \frac{3}{\eta^4}  - \frac{3 \cot\eta}{\eta^3 } \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+\frac{18}{\eta^4}
- 
\biggl[ \frac{ 6\cot\eta}{\eta^3} + \frac{3\cot^2\eta}{\eta^2} + \frac{3}{\eta^2} \biggr] 
- \biggl[ \frac{3}{\eta^2} \biggr] 
- \biggl[ \frac{3\cot^2\eta}{\eta^2} + \frac{6\cot^3\eta}{\eta} + \frac{6\cot\eta}{\eta} \biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{\sigma_c^2}{2} \biggr)  \frac{1}{\eta \sin\eta} \biggr\} 
\biggl[ 1  - \eta \cot\eta \biggr]
+
\biggl[ -\frac{12}{\eta^4} + \frac{6}{\eta^2} \biggr]  
+
\biggl[ 2\cot\eta \biggr]\biggl[ \frac{3}{\eta^3}  + \frac{3\cot\eta}{\eta^2} \biggr]  
+
\biggl[ 2\cot\eta \biggr]  \biggl[
-\frac{6}{\eta^3} + \frac{3\cot\eta}{\eta^2} + \frac{3}{\eta} + \frac{3\cot^2\eta}{\eta}
\biggr]  
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl[ -\frac{6}{\eta^4}  + \frac{6 \cot\eta}{\eta^3 } \biggr]
+ \biggl[ 2\cot\eta \biggr] \biggl[ \frac{3}{\eta^3}  - \frac{3 \cot\eta}{\eta^2 } \biggr]
+\frac{18}{\eta^4}
- \frac{6}{\eta^2} 
+ \biggl[ 2\cot\eta \biggr]\biggl[
- \frac{3\cot^2\eta}{\eta} 
- \frac{3\cot\eta}{\eta^2} 
- \frac{3}{\eta} 
-\frac{ 3}{\eta^3} 
\biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{\sigma_c^2}{2} \biggr)  \frac{1}{\eta \sin\eta} \biggr\} 
\biggl[ 1  - \eta \cot\eta \biggr]
+
\biggl[ 2\cot\eta \biggr]  \biggl[
-\frac{3}{\eta^3}  + \frac{6\cot\eta}{\eta^2} + \frac{3}{\eta} + \frac{3\cot^2\eta}{\eta}
\biggr]  
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl[ 2\cot\eta \biggr]\biggl[\frac{3}{\eta^3} 
- \frac{3\cot^2\eta}{\eta} 
- \frac{6\cot\eta}{\eta^2} 
- \frac{3}{\eta} 
\biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{\sigma_c^2}{2} \biggr)  \frac{1}{\eta \sin\eta} \biggr\} 
\biggl[ 1  - \eta \cot\eta \biggr] \, .
</math>
</td>
</tr>
</table>

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

the <math>n=1</math> LAWE reduces to &hellip;

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">LAWE</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{\sigma_c^2}{2} \biggr)  \frac{1}{\eta \sin\eta} \biggr\} 
\biggl[ 1  - \eta \cot\eta \biggr] \, .
</math>
</td>
</tr>
</table>

ASSESSMENT:&nbsp; 
<ol>
  <li>
If we set <math>\sigma_c^2 = 0</math>, the right-hand-side of this expression goes to zero &#8212; and, hence, the <math>n=1</math> LAWE is satisfied &#8212; for any chosen truncation radius in the range, <math>0 < \eta_i < \pi</math>.  (We have not included the ''isolated'' <math>n=1</math> polytrope because <math>x_P</math> blows up at its surface, <math>\eta_i = \pi</math>.)
  </li>
  <li>
At the surface, <math>\eta_i</math>, the slope of this trial eigenfunction is,

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

<tr>
  <td align="right">
<math>\frac{1}{3} \cdot \frac{dx_P}{d\eta} \biggr|_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{\cot\eta_i}{\eta_i^2} + \frac{1}{\eta_i} + \frac{\cot^2\eta_i}{\eta_i}\biggr] -\frac{2}{\eta_i^3} 
\, .
</math>
  </td>
</tr>
</table>
By contrast, as stated above, the eigenvalue problem will be properly solved only if the surface slope is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>- \frac{d\ln x}{d\ln \eta} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
1 + \frac{\cancelto{0}{\sigma_c^2}}{12}
\biggl[ \frac{\eta^3}{(\eta \cos\eta - \sin\eta)}\biggr]_{\eta=\pi}
= 1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{1}{3} \cdot \frac{dx_P}{d\eta} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
- \frac{x_P}{3\eta}\biggr|_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
- \frac{1}{\eta_i^3} \biggl[ 1 - \eta_i \cot\eta_i \biggr] 
=
\frac{\cot\eta_i}{\eta_i^2} - \frac{1}{\eta_i^3} \, .
</math>
  </td>
</tr>
</table>
These two slopes do not appear to be the same, for any allowed choice of <math>\eta_i</math>.  We conclude, therefore, that no model along the sequence of pressure-truncated <math>n=1</math> polytropes is marginally unstable.
  </li>
</ol>

====Determining Discrete Representation of Eigenfunction====

Let's numerically integrate the [[#DiscreteLAWE|discrete form of the <math>n=1</math> LAWE]] over the radial coordinate range, <math>0 \le \eta_i \le \eta_s</math>.  Following our [[SSC/Stability/Polytropes#KickStart|discussion of the more general polytropic case]], we will kickstart integration from the center, outward, via the expression,

<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_\eta^2}{60} \biggr] \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp; </td>
  <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\biggr] \, .</math>
  </td>
</tr>
</table>
Here, we will restrict our investigation to the case where <math>\gamma_g = (n+1)/n = 2</math>, in which case, <math>\alpha = (3-4/\gamma_g) = +1</math>, <math>\mathfrak{F} = (\sigma_c^2 -4)/2</math>, and 

<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{(\sigma_c^2 -4) \Delta_\eta^2}{60} \biggr] \, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">EXAMPLE:</font> &nbsp; <math>A=1</math>, <math>B=0</math>, <math>\Delta_\eta = \pi/99 = 0.031733259</math>; evaluated over range, <math>0 \le \eta_i \le \pi</math>.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center"><math>\sigma_c^2</math></td>
  <td align="center"><math>\mathfrak{F}</math></td>
  <td align="center"><math>x_2</math></td>
  <td align="center"><math>\phi_2</math></td>
  <td align="center"><math>(-\phi')_2</math></td>
  <td align="center">TERM1</td>
  <td align="center">TERM2</td>
  <td align="center">TERM3</td>
  <td align="center"><math>x_3</math></td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">+0.5</td>
  <td align="center">0.999983217</td>
  <td align="center">0.999832175</td>
  <td align="center">0.010576686</td>
  <td align="center">5.998321784</td>
  <td align="center">1.998993085</td>
  <td align="center">3.998992898</td>
  <td align="center">0.999932829</td>
</tr>
</table>
</td></tr></table>

=====Isolated n = 1 Polytrope=====

If we integrate all the way out to the natural, zero-pressure surface of our <math>n = 1</math> polytrope, then <math>\eta_s = \pi</math> and &#8212; as derived in our [[SSC/Structure/Polytropes#Summary|discussion of the equilibrium structure of n = 1 polytropes]] &#8212; <math>(\rho_c/\bar\rho) = \pi^2/3</math>.  In line with our discussion of [[SSC/Stability/n3PolytropeLAWE#Schwarzschild_(1941)|Schwarzschild's model of oscillations in <math>n=3</math> polytropes]], we therefore expect the boundary condition at the surface of our <math>n=1</math> configurations to be given by the expression,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>-\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\frac{1}{2}\biggl\{ \biggl[ \mathfrak{F} + 2\alpha\biggr]\biggl(\frac{\rho_c}{\bar\rho}\biggr) - 2\alpha \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
1 - \biggl(\frac{\sigma_c^2 \pi^2}{12}\biggr)  \, ,
</math>
  </td>
</tr>
</table>
as [[#Review_of_Trial_Analytic_Eigenfunction|reviewed immediately above]].  


<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <td align="center" bgcolor="pink">
{{ Chatterji51 }} &#8212; STEP 3
  </td>
</tr>
<tr><td align="left">
Here we examine what the boundary condition should be at the surface of an isolated <math>n=1</math> polytrope.  Given that, quite generally in the context of isolated polytropes,

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

<tr>
  <td align="right">
<math>\alpha</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3 - \frac{4}{\gamma} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\mathfrak{F}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sigma_c^2}{\gamma} - 2\alpha \, ,
</math>
  </td>
</tr>
</table>
the surface boundary condition is,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>-\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\frac{1}{2}\biggl\{ \biggl[ \mathfrak{F} + 2\alpha\biggr]\biggl(\frac{\rho_c}{\bar\rho}\biggr) - 2\alpha \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\frac{\sigma_c^2}{2\gamma}\biggl(\frac{\rho_c}{\bar\rho}\biggr)  + 3 - \frac{4}{\gamma}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - \frac{1}{\gamma}\biggl[ 4 + \frac{\sigma_c^2}{2}\biggl(\frac{\rho_c}{\bar\rho}\biggr) \biggr] \, .
</math>
  </td>
</tr>

</table>
<font color="maroon"><b>Considerations:</b></font>
<ol type="A">
<li>For an isolated <math>n=1</math> polytrope, the central-to-mean density is, <math>\rho_c/\bar\rho = \pi^2/3</math>.  Hence,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - \frac{1}{\gamma}\biggl[ 4 + \frac{\pi^2 \sigma_c^2}{6}\biggr] \, .
</math>
  </td>
</tr>

</table>

</li>
<li>If, in addition, we set <math>\gamma = (n+1)/n = 2 ~\Rightarrow ~\alpha=+1</math>, then,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1, ~\alpha=+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - \frac{1}{2}\biggl[ 4 + \frac{\pi^2 \sigma_c^2}{6}\biggr] = 1 - \frac{\pi^2 \sigma_c^2}{12} \, .
</math>
  </td>
</tr>

</table>

</li>
<li>If we set <math>\gamma = [4/(3-\alpha)]</math> for all other values of <math>\alpha</math>, we can write,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1, ~\alpha}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 - (3-\alpha)\biggl[ 1 + \frac{\pi^2 \sigma_c^2}{24}\biggr]
=
\alpha - (3-\alpha)\biggl[ \frac{\pi^2 \sigma_c^2}{24}\biggr]\, .
</math>
  </td>
</tr>

</table>

</li>
</ol>


----


At the bottom of p. 469 of his article, {{ Chatterji51 }} states that, <font color="darkgreen">"&hellip; the condition for the Node to fall at the surface of the star</font> is,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>3f + z~\frac{df}{dz}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
</tr>
</table>
which we interpret to mean,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln f}{d\ln z}\biggr|_\mathrm{surf} \biggr\}_{n=1,~\alpha}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 \, .
</math>
  </td>
</tr>
</table>
He goes on to say, <font color="darkgreen">"As the adiabatic approximation breaks down near the boundary we have not strictly followed this condition."</font>

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


This should be compared with the finite-difference representation of the logarithmic derivative, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>+\frac{\Delta\ln x}{\Delta\ln \xi} \biggr|_\mathrm{surface}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{\xi_\mathrm{max}}{x_N} \biggl[ \frac{x_{N+1}-x_{N-1}}{2\Delta_\xi} \biggr] \, .</math>
  </td>
</tr>
</table>
<font color="red">CAUTION!</font> &nbsp; &nbsp; Because, for each ''guess'' of <math>\sigma_c^2</math>, the eigenfunction climbs (or plummets) rapidly as we approach the surface, in practice we evaluated the finite-difference representation of the logarithmic derivative at a zone location that is a bit inside of the actual surface; for example, when we divided the equilibrium configuration into <math>N = 100</math> grid zones, we evaluated the "surface" derivative at zone number 97.

Here we have adopted an analysis that closely resembles our [[SSC/Stability/n3PolytropeLAWE#Numerical_Integration|discussion of the analysis of <math>n=3</math> polytropes]] that was published by {{ Schwarzschild41 }}. Here we have divided our model into <math>N = 100</math> radial zones and, using this algorithm, integrated the LAWE from the center of the configuration to the surface, for <math>\alpha = +1</math>, and approximately 40 different chosen values of the frequency parameter across the range, <math>-2 \le \mathfrak{F} \le + 18</math>.  The radial displacement functions resulting from these integrations are presented in the following figure as an animation sequence.  The specified value of <math>\mathfrak{F}</math> is displayed at the top of each animation frame, and the resulting displacement function, <math>x(\eta)</math>, is traced by the small, red circular markers in each frame.
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="4">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = n1Oscillations]]Four Modes of Oscillation<br />of an Isolated, <math>n=1</math> Polytrope</td>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\sigma_c^2</math></td>
  <td align="center">Neg. Slope<br><math>1 - (\sigma_c^2\pi^2/12)</math></td>
  <td align="center"><math>\mathfrak{F} = \frac{\sigma_c^2}{\gamma_g} - 2\alpha</math></td>
</tr>
<tr>
  <td align="center">Fundamental</td>
  <td align="center">2.2405295</td>
  <td align="center">3.1287618</td>
  <td align="center">-0.879735</td>
</tr>
<tr>
  <td align="center">1<sup>st</sup> Overtone</td>
  <td align="center">6.340767</td>
  <td align="center">-32.06757</td>
  <td align="center">1.1703835</td>
</tr>
<tr>
  <td align="center">2<sup>nd</sup> Overtone</td>
  <td align="center">13.694927</td>
  <td align="center">-153.2545</td>
  <td align="center">4.8474635</td>
</tr>
<tr>
  <td align="center">3<sup>rd</sup> Overtone</td>
  <td align="center">28.462829</td>
  <td align="center">-665.3074</td>
  <td align="center">12.231415</td>
</tr>

<tr><td align="center" colspan="4">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n1Eigenvectors/MovieFrames/KeyAnimation/n1osc06.gif]]
[[File:N1osc06.gif|600px|Animated gif showing oscillation modes for n = 1 polytrope]]
</td></tr></table>



<table border="1" align="center" cellpadding="8" width="90%">
<tr>
  <td align="center" colspan="3" bgcolor="pink">{{ Chatterji51figure }}</td>
</tr>
<tr>
  <td align="center" colspan="3">''Amplitudes and frequencies of the displacement functions for three modes, assuming'' <math>\alpha = 0.6</math> and <math>\gamma = 5/3</math></td>
</tr>
<tr>
  <td align="center" colspan="1">Amplitudes extracted from Chatterji's Table II (p. 468)</td>
  <td align="center" colspan="1">Plot for comparison with Chatterji's Fig. 1 (p. 469)</td>
  <td align="center" colspan="1">Eigenfrequencies extracted<br />from Chatterji's Table I (p. 468)</td>
</tr>
<tr><td align="left" width="50%">
<div id="Chatterji51Table2Data" style="width: 100%; height: 30em; overflow: auto;">
<pre>
                                  1st            2nd
Abscissa	Fundamental	Overtone	Overtone
0.0	         1.000000	1.000000	1.000000
0.1	         1.000169	0.998882	0.996820
0.2	         1.000677	0.995518	0.987281
0.3	         1.001525	0.989874	0.971386
0.4	         1.002716	0.981890	0.949136
0.5	         1.004251	0.971495	0.920555
0.6	         1.006143	0.958596	0.885671
0.7	         1.008389	0.943052	0.844515
0.8	         1.011009	0.924713	0.797168
0.9	         1.014006	0.903389	0.743728
1.0	         1.017390	0.878859	0.684350
1.1	         1.021175	0.850865	0.619257
1.2	         1.025374	0.819105	0.548757
1.3	         1.030004	0.783221	0.473277
1.4	         1.035102	0.742868	0.393396
1.5	         1.040853	0.697654	0.309941
1.6	         1.047084	0.646920	0.223844
1.7	         1.053810	0.590042	0.136419
1.8	         1.061066	0.526318	0.049379
1.9	         1.068886	0.454939       -0.035065
2.0	         1.077309	0.374974       -0.114079
2.1	         1.086374	0.285357       -0.184050
2.2	         1.096138	0.184876       -0.240444
2.3	         1.106766	0.072271       -0.277903
2.4	         1.118204      -0.054152       -0.288854
2.5	         1.130496      -0.196154       -0.264401
2.6	         1.143708      -0.355598       -0.193931
2.7	         1.157893      -0.534282       -0.065351
2.8	         1.173099      -0.733397	0.133422
2.9	         1.189288      -0.951614	0.408698
3.0	         1.206063      -1.175893	0.734701
3.1	         1.218532      -1.251806	0.780444
</pre>
</div>
  </td>
  <td align="center">
[[File:Chatterji51Summary2.png|400px|Chatterji's Figure 1]]
  </td>
  <td align="center">
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\omega^2_\mathrm{Chatterji}</math></td>
  <td align="center"><math>\sigma_c^2</math></td>
</tr>
<tr>
  <td align="left">Fundamental</td>
  <td align="center" bgcolor="pink">0.231</td>
  <td align="center">1.155</td>
</tr>
<tr>
  <td align="left">1<sup>st</sup> Overtone</td>
  <td align="center" bgcolor="pink">1.517</td>
  <td align="center">7.585</td>
</tr>
<tr>
  <td align="left">2<sup>nd</sup> Overtone</td>
  <td align="center" bgcolor="pink">3.580</td>
  <td align="center">17.900</td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
The solid circular markers in the plot (center panel) show how the amplitude of the displacement function varies with radius <math>0 \le \eta < \pi</math> for three separate radial modes, according to the data provided in Table II of {{ Chatterji51 }}, which has been reproduced here in the ''scrollable'' left-hand panel.  In the plot, ''blue'' is the fundamental mode, ''red'' is the 1<sup>st</sup> overtone, ''green'' is the 2<sup>nd</sup> overtone.  The (square of the) eigenfrequency corresponding to each mode, according to Table I of {{ Chatterji51 }}, is provided in the column of the right-hand panel that is (highlighted in pink and) labeled <math>\omega^2_\mathrm{Chatterji}</math>; also listed are the corresponding values of <math>\sigma_c^2 = 3\gamma \omega^2_\mathrm{Chatterji}</math>.
<br />&nbsp;<br />
The smooth, solid curves in the middle-panel plot are ''not'' fits to Chatterji's data.  Rather, they result from our own, independent numerical integration of the relevant LAWE, assuming that Chatterji's published values of the (square of the) eigenfrequency are correct for all three modes.  In all three cases for the specified eigenfrequency, there is excellent agreement between our determination of the radial eigenfunction and the determination obtained by {{ Chatterji51 }}.
  </td>
</tr>
</table>

We are exceptionally pleased to find that, for each of the three modes of oscillation, the displacement function obtained via our integration of the LAWE (solid curves in the figure) runs through the discrete points recorded by {{ Chatterji51 }} (solid circular markers in the figure).  But in doing so, we find from our higher resolution model that there is an inflection point just inside the surface of the model; this is not the smooth behavior that is expected as the surface is approached.  In an effort to correct this behavior, we have changed the constraint that is applied while integrating the LAWE from the center, outward:  Instead of forcing <math>\omega^2_\mathrm{Chatterji}</math> to match the value published by {{ Chatterji51 }}, we have let the value of this oscillation frequency vary while enforcing the surface boundary condition describe above as <font color="maroon">CONSIDERATION "C"</font>.  The resulting "improved" solution is shown in the figure that follows.


<table border="1" align="center" cellpadding="8" width="90%">
<tr>
  <td align="center" colspan="3" bgcolor="yellow">Our Imposed Surface Boundary Condition:  <br />
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ -\frac{d\ln x}{d\ln \eta}\biggr|_\mathrm{surf} \biggr\}_{n=1, ~\alpha}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\alpha - (3-\alpha)\biggl[ \frac{\pi^2 \sigma_c^2}{24}\biggr]\, .
</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">''Amplitudes and frequencies of the displacement functions for three modes, assuming'' <math>\alpha = 0.6</math> and <math>\gamma = 5/3</math></td>
</tr>
<tr>
  <td align="center" colspan="1">Amplitudes determined from our numerical integration of the LAWE</td>
  <td align="center" colspan="1">Our version of Chatterji's Fig. 1</td>
  <td align="center" colspan="1">Eigenfrequencies determined from our integration</td>
</tr>
<tr><td align="left" width="50%">
<div id="Chatterji51Table2Data" style="width: 100%; height: 30em; overflow: auto;">
<pre>
Abscissa	Fundamental	1st Overtone	2nd Overtone
0.000000000	1.000000000	 1.000000000	 1.000000000
0.031733259	1.000047844	 0.999943498	 0.999786141
0.063466518	1.000089016	 0.999601959	 0.998867632
0.095199777	1.000174709	 0.999061021	 0.997382877
0.126933037	1.000294726	 0.998302963	 0.995304066
0.158666296	1.000449107	 0.997327028	 0.992631032
0.190399555	1.000637906	 0.996132244	 0.989363571
0.222132814	1.000861191	 0.994717421	 0.985501447
0.253866073	1.001119039	 0.993081149	 0.981044389
0.285599332	1.001411544	 0.991221792	 0.975992105
0.317332591	1.001738809	 0.989137492	 0.970344285
0.34906585	1.002100951	 0.986826157	 0.964100608
0.38079911	1.002498098	 0.984285466	 0.957260757
0.412532369	1.002930394	 0.981512860     0.949824425
0.444265628	1.003397993	 0.978505538	 0.94179133
0.475998887	1.003901064	 0.975260454	 0.933161228
0.507732146	1.004439786	 0.971774312	 0.923933924
0.539465405	1.005014356	 0.968043557	 0.914109296
0.571198664	1.005624981	 0.964064374	 0.903687306
0.602931923	1.006271884	 0.959832677	 0.892668024
0.634665183	1.006955299	 0.955344103	 0.881051648
0.666398442	1.007675478	 0.950594005	 0.868838527
0.698131701	1.008432684	 0.945577446	 0.856029188
0.72986496	1.009227198	 0.940289184	 0.842624363
0.761598219	1.010059312	 0.934723671	 0.828625017
0.793331478	1.010929336	 0.928875035	 0.814032388
0.825064737	1.011837596	 0.922737076	 0.798848013
0.856797996	1.01278443	 0.916303251	 0.783073774
0.888531256	1.013770197	 0.909566661	 0.766711937
0.920264515	1.01479527	 0.902520043	 0.749765195
0.951997774	1.015860038	 0.895155752	 0.732236722
0.983731033	1.016964908	 0.887465751	 0.714130222
1.015464292	1.018110307	 0.879441592	 0.695449985
1.047197551	1.019296676	 0.8710744	 0.676200956
1.07893081	1.020524478	 0.862354861	 0.656388792
1.110664069	1.021794193	 0.853273198	 0.636019943
1.142397329	1.023106322	 0.843819155	 0.615101726
1.174130588	1.024461383	 0.833981977	 0.593642407
1.205863847	1.025859919	 0.82375039	 0.5716513
1.237597106	1.027302491	 0.813112573	 0.549138858
1.269330365	1.028789682	 0.802056142	 0.526116785
1.301063624	1.030322099	 0.790568118	 0.502598146
1.332796883	1.03190037	 0.778634906	 0.478597499
1.364530142	1.033525148	 0.766242261	 0.454131023
1.396263402	1.035197111	 0.753375263	 0.429216666
1.427996661	1.03691696	 0.740018284	 0.403874305
1.45972992	1.038685425	 0.726154954	 0.378125913
1.491463179	1.04050326	 0.711768126	 0.351995745
1.523196438	1.042371249	 0.696839839	 0.325510537
1.554929697	1.044290203	 0.681351278	 0.298699722
1.586662956	1.046260964	 0.665282732	 0.271595661
1.618396215	1.048284403	 0.64861355	 0.244233898
1.650129475	1.050361424	 0.631322093	 0.216653434
1.681862734	1.052492964	 0.613385687	 0.188897019
1.713595993	1.054679993	 0.594780567	 0.161011472
1.745329252	1.056923518	 0.575481822	 0.133048033
1.777062511	1.059224581	 0.555463336	 0.105062731
1.80879577	1.061584262	 0.534697725	 0.077116794
1.840529029	1.064003681	 0.513156272	 0.049277088
1.872262289	1.066484	 0.490808851	 0.021616595
1.903995548	1.069026422	 0.467623859	-0.005785076
1.935728807	1.071632194	 0.443568129	-0.032841123
1.967462066	1.074302612	 0.418606853	-0.059456951
1.999195325	1.077039017	 0.392703483	-0.085529513
2.030928584	1.079842802	 0.365819646	-0.110946599
2.062661843	1.082715411	 0.337915031	-0.135586067
2.094395102	1.085658345	 0.308947292	-0.159315006
2.126128362	1.08867316	 0.278871926	-0.181988834
2.157861621	1.091761474	 0.247642155	-0.203450318
2.18959488	1.094924965	 0.215208794	-0.223528513
2.221328139	1.098165381	 0.181520118	-0.242037616
2.253061398	1.101484536	 0.146521711	-0.258775724
2.284794657	1.104884319	 0.110156314	-0.273523491
2.316527916	1.108366697	 0.072363659	-0.286042681
2.348261175	1.111933717	 0.033080298	-0.296074601
2.379994435	1.115587515	-0.007760587	-0.303338425
2.411727694	1.11933032	-0.050229374	-0.307529379
2.443460953	1.12316446	-0.094400211	-0.308316806
2.475194212	1.127092372	-0.140351229	-0.305342096
2.506927471	1.131116609	-0.188164774	-0.298216493
2.53866073	1.135239852	-0.237927637	-0.286518776
2.570393989	1.139464924	-0.289731302	-0.269792846
2.602127248	1.143794808	-0.343672193	-0.247545241
2.633860508	1.148232666	-0.39985192	-0.219242641
2.665593767	1.152781872	-0.458377516	-0.184309461
2.697327026	1.157446048	-0.519361653	-0.142125667
2.729060285	1.162229122	-0.582922804	-0.092025093
2.760793544	1.167135405	-0.649185328	-0.033294642
2.792526803	1.172169712	-0.718279394	 0.034824895
2.824260062	1.17733755	-0.790340637	 0.113135186
2.855993321	1.18264542	-0.865509304	 0.202471616
2.887726581	1.188101346	-0.943928474	 0.303685139
2.91945984	1.193715839	-1.025740391	 0.417605887
2.951193099	1.199503802	-1.111078809	 0.54496891
2.982926358	1.205488688	-1.200051936	 0.686253524
3.014659617	1.2117128	-1.292700149	 0.841298052
3.046392876	1.21826791	-1.388871861	 1.008209416
3.078126135	1.225416475	-1.487741753	 1.179292479
3.109859394	1.234427405	-1.584566325	 1.314705175
</pre>
</div>
  </td>
  <td align="center">
[[File:OurImprovedSummary2.png|400px|Chatterji's Figure 1]]
  </td>
  <td align="center">
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\omega^2_\mathrm{Chatterji}</math></td>
  <td align="center"><math>\sigma_c^2</math></td>
</tr>
<tr>
  <td align="left">Fundamental</td>
  <td align="center" bgcolor="yellow">0.2298579</td>
  <td align="center">1.1492896</td>
</tr>
<tr>
  <td align="left">1<sup>st</sup> Overtone</td>
  <td align="center" bgcolor="yellow">1.4733124</td>
  <td align="center">7.366562</td>
</tr>
<tr>
  <td align="left">2<sup>nd</sup> Overtone</td>
  <td align="center" bgcolor="yellow">3.3484654</td>
  <td align="center">16.742327</td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
The solid circular markers in the plot (center panel) show how the amplitude of the displacement function varies with radius <math>0 \le \eta < \pi</math> for three separate radial modes, according to the data provided in Table II of {{ Chatterji51 }}, which has been reproduced here in the ''scrollable'' left-hand panel.  In the plot, ''blue'' is the fundamental mode, ''red'' is the 1<sup>st</sup> overtone, ''green'' is the 2<sup>nd</sup> overtone.  The (square of the) eigenfrequency corresponding to each mode, according to Table I of {{ Chatterji51 }}, is provided in the column of the right-hand panel that is (highlighted in pink and) labeled <math>\omega^2_\mathrm{Chatterji}</math>; also listed are the corresponding values of <math>\sigma_c^2 = 3\gamma \omega^2_\mathrm{Chatterji}</math>.
<br />&nbsp;<br />
The smooth, solid curves in the middle-panel plot are ''not'' fits to Chatterji's data.  Rather, they result from our own, independent numerical integration of the relevant LAWE, assuming that Chatterji's published values of the (square of the) eigenfrequency are correct for all three modes.  In all three cases for the specified eigenfrequency, there is excellent agreement between our determination of the radial eigenfunction and the determination obtained by {{ Chatterji51 }}.
  </td>
</tr>
</table>

=====Pressure-Truncated n = 1 Polytrope=====
Drawing from an [[SSC/Stability/NeutralMode#Part_2|accompanying discussion]], if the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,

<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>3</math> &nbsp; &nbsp; at &nbsp; &nbsp; <math>\eta = \tilde\eta</math>.
  </td>
</tr>
</table>

===Bipolytropic Envelope (Trial Simplification)===

For the <math>n=1</math> envelope of a <math>(n_c, n_e) = (5, 1)</math> bipolytrope, the relevant LAWE is,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">LAWE</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{d^2x}{d\eta^2} 
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 2\biggl\{ 1 + \frac{\eta\cos(\eta - B)}{\sin(\eta-B)} \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta^3}{A\sin(\eta-B)} - 1 + \frac{\eta\cos(\eta - B)}{\sin(\eta-B)} \biggr\}  \frac{x}{\eta^2}
\, .</math>
</td>
</tr>
</table>
Three terms in this expression blow up at the surface, where <math>(\eta - B) \rightarrow \pi</math> and, hence, <math>\sin(\eta - B) \rightarrow 0</math>.  We can improve the behavior of this LAWE expression by assuming that the eigenfunction is of the form,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
f(\eta) \biggl[ \sin(\eta - B) \biggr]^m \, ,
</math>
</td>
</tr>
</table>
in which case,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\frac{dx}{d\eta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \sin(\eta - B) \biggr]^m \frac{df}{d\eta} + m f(\eta) \biggl[ \sin(\eta - B) \biggr]^{m-1}\cos(\eta-B) \, ;</math> &nbsp; &nbsp; and,
</td>
</tr>

<tr>
  <td align="right"><math>\frac{d^2x}{d\eta^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \sin(\eta - B) \biggr]^m \frac{d^2f}{d\eta^2} + 2m\biggl[ \sin(\eta - B) \biggr]^{m-1} \cos(\eta - B) \cdot \frac{df}{d\eta} 
+ 
m (m-1)f(\eta) \biggl[ \sin(\eta - B) \biggr]^{m-2}\cos^2(\eta-B)  - m f(\eta) \biggl[ \sin(\eta - B) \biggr]^{m-1}\sin(\eta-B) \, .
</math>
</td>
</tr>
</table>

This gives,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">LAWE</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \sin(\eta - B) \biggr]^m \frac{d^2f}{d\eta^2} + 2m\biggl[ \sin(\eta - B) \biggr]^{m-1} \cos(\eta - B) \cdot \frac{df}{d\eta} 
+ 
m (m-1)f(\eta) \biggl[ \sin(\eta - B) \biggr]^{m-2}\cos^2(\eta-B)  - m f(\eta) \biggl[ \sin(\eta - B) \biggr]^{m-1}\sin(\eta-B) 
</math>
</td>
</tr>

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta}{A\sin(\eta-B)} 
- \frac{1}{\eta^2} + \frac{\cos(\eta - B)}{\eta \sin(\eta-B)} \biggr\} f(\eta) \biggl[ \sin(\eta - B) \biggr]^m 
</math>
</td>
</tr>

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 2m\biggl\{ \frac{1}{\eta} + \frac{\cos(\eta - B)}{\sin(\eta-B)} \biggr\}  
\biggl[ \sin(\eta - B) \biggr]^{m-1}\cos(\eta-B)   
\cdot f(\eta) 
+ 2 \biggl\{ \biggl( \frac{\sigma_c^2}{12} \biggr)  \frac{\eta}{A\sin(\eta-B)} 
- \frac{1}{\eta^2} + \frac{\cos(\eta - B)}{\eta \sin(\eta-B)} \biggr\} \biggl[ \sin(\eta - B) \biggr]^m f(\eta) 
</math>
</td>
</tr>

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
\frac{2m}{\eta} \biggl[ \sin(\eta - B) \biggr]\cos(\eta-B)   
+
2m \cos^2(\eta-B)   
+ \biggl[ \biggl( \frac{\sigma_c^2}{6} \biggr)  \frac{\eta}{A} \biggr] \biggl[ \sin(\eta - B) \biggr] 

- \frac{2}{\eta^2} \biggl[ \sin(\eta - B) \biggr]^2 

+ \frac{2\cos(\eta - B)}{\eta} \biggl[ \sin(\eta - B) \biggr] 
\biggr\} \biggl[ \sin(\eta - B) \biggr]^{m-2}\cdot f(\eta) 
\, .</math>
</td>
</tr>
</table>

====Try m = 1 and m = 2====
If we set <math>m=1</math>, there are still terms in the LAWE expression that blow up at the surface, where <math>(\eta-B)\rightarrow \pi</math> and, hence, <math>\sin(\eta-B) \rightarrow 0</math>.  Instead, let's try <math>m=2</math>:
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\mathrm{LAWE}\biggr|_{m=2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \sin(\eta - B) \biggr]^2 \frac{d^2f}{d\eta^2} 
+ 2\biggl\{
\frac{1}{\eta} \biggl[ \sin(\eta - B) \biggr]     
+ 3 \cos(\eta - B) 
\biggr\}\biggl[ \sin(\eta - B) \biggr]\frac{df}{d\eta}
</math>
</td>
</tr>

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
\frac{4}{\eta} \biggl[ \sin(\eta - B) \biggr]\cos(\eta-B)   
+
4 \cos^2(\eta-B)   
+ \biggl[ \biggl( \frac{\sigma_c^2}{6} \biggr)  \frac{\eta}{A} \biggr] \biggl[ \sin(\eta - B) \biggr] 

- \frac{2}{\eta^2} \biggl[ \sin(\eta - B) \biggr]^2 

+ \frac{2\cos(\eta - B)}{\eta} \biggl[ \sin(\eta - B) \biggr] 
\biggr\} \cdot f(\eta) \, ,
</math>
</td>
</tr>
</table>

which, at the surface <math>\eta \rightarrow \eta_s = (\pi + B)</math>, reduces to &hellip;

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl\{ \mathrm{LAWE}\biggr|_{m=2} \biggr\}_{\eta_s}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
6f(\eta_s) \, .
</math>
</td>
</tr>
</table>
Hence, this LAWE will be satisfied for any function, <math>f(\eta)</math>, that goes to zero at the surface.


====Try m = 3====
Setting <math>m=3</math>, we obtain,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\mathrm{LAWE}\biggr|_{m=3}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \sin(\eta - B) \biggr]^3 \frac{d^2f}{d\eta^2} 
+ 2\biggl\{
\frac{1}{\eta} \biggl[ \sin(\eta - B) \biggr]     
+ 4 \cos(\eta - B) 
\biggr\}\biggl[ \sin(\eta - B) \biggr]^{2}\frac{df}{d\eta}
</math>
</td>
</tr>

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
\frac{6}{\eta} \biggl[ \sin(\eta - B) \biggr]\cos(\eta-B)   
+
6 \cos^2(\eta-B)   
+ \biggl[ \biggl( \frac{\sigma_c^2}{6} \biggr)  \frac{\eta}{A} \biggr] \biggl[ \sin(\eta - B) \biggr] 

- \frac{2}{\eta^2} \biggl[ \sin(\eta - B) \biggr]^2 

+ \frac{2\cos(\eta - B)}{\eta} \biggl[ \sin(\eta - B) \biggr] 
\biggr\} \biggl[ \sin(\eta - B) \biggr]\cdot f(\eta) 
\, .</math>
</td>
</tr>
</table>

which trivially reduces to zero at the surface because, <math>\eta \rightarrow \eta_s = (\pi + B) ~~~\Rightarrow ~~~ \sin(\eta - B) \rightarrow 0</math>.  For all other relevant radial positions in the envelope, <math>\eta_i \le \eta < \eta_s</math>, we can divide through by <math>\sin^3(\eta-B)</math> to obtain,

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>[\sin(\eta-B)]^{-3} \times \mathrm{LAWE}\biggr|_{m=3}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{d^2f}{d\eta^2} 
+ 2\biggl\{
\frac{1}{\eta}      
+ 4 \cot(\eta - B) 
\biggr\}\cdot \frac{df}{d\eta}
+ \biggl\{
12\cot^2(\eta-B)   
- 3   
+ 
\frac{8}{\eta} \biggl[ \cot(\eta-B)\biggr]   
+ \biggl[ \biggl( \frac{\sigma_c^2}{6} \biggr)  \frac{\eta}{A\sin(\eta - B)} \biggr]  
- \frac{2}{\eta^2}  
\biggr\} \cdot f(\eta) \, ,
</math>
</td>
</tr>
</table>

====Boundary Condition====

In addition, there is a (boundary condition) constraint on the slope of the eigenfunction at the surface.  So, let's examine &hellip;
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\frac{d\ln x}{d\ln\eta}\biggr|_{m=2} \equiv \biggl[ \frac{\eta}{x} \cdot \frac{dx}{d\eta}\biggr]_{m=2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\eta}{f(\eta) \sin^2(\eta-B)}
\biggl\{
\sin^2(\eta - B) \frac{df}{d\eta} + 2 f(\eta) \sin(\eta - B) \cos(\eta-B)
\biggr\}</math>
</td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{d\ln f}{d\ln\eta} + 2\eta \cot(\eta-B)
</math>
</td>
</tr>
</table>
Now, [[#Setup|from above]], we appreciate that when <math>\phi = A\sin(\eta-B)/\eta</math>,

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\eta \cot(\eta-B)</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
1 + \frac{d\ln\phi}{d\ln \xi}
</math>
</td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{d\ln x}{d\ln\eta}\biggr|_{m=2} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{d\ln f}{d\ln\eta} + 2\biggl[ 1 + \frac{d\ln\phi}{d\ln \xi} \biggr] \, .
</math>
</td>
</tr>
</table>
It therefore appears as though we should adopt the function relation,

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\frac{d\ln f}{d\ln\eta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-2\frac{d\ln \phi}{d\ln\eta} 
</math>
</td>
</tr>
</table>


<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
f(\eta) \biggl[ \sin(\eta - B) \biggr]^m \, ,
</math>
</td>
</tr>
</table>

----


Let's now examine "model <b>A</b>" [[#Model_Pairings|from above]], for which, <math>A = 0.200812422</math> and <math>B = -0.859270052</math>.  If we set <math>\sigma_c^2 = 0</math>, this LAWE becomes,
<table border=0 cellpadding=2 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[ 4 - 2 Q \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
- 2 \biggl[ \alpha Q\biggr]  \frac{x}{\eta^2} \, .</math>
</td>
</tr>
</table>

===Discrete Determination of Bipolytropic Envelope===
Here we focus on the ''specific'' <math>(n_c, n_e) = (5, 1)</math> equilibrium model sequence that has <math>(\mu_e/\mu_c) = 0.31</math>; and along this sequence, we attempt to analyze the dynamical stability of "model <b>A</b>" [[#Model_Pairings|from above]], which sits along the sequence at the maximum-core-mass turning point for which &hellip;
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="8"><math>(n_c, n_e) = (5, 1)</math> and <math>(\mu_e/\mu_c) = 0.31</math></td>
</tr>
<tr>
  <td align="center">Model</td>
  <td align="center"><math>A</math></td>
  <td align="center"><math>B</math></td>
  <td align="center"><math>\xi_i</math></td>
  <td align="center"><math>\theta_i = (1 + \xi_i^2/3)^{-1 / 2}</math></td>
  <td align="center"><math>\eta_i = 3^{1 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \theta_i^2</math></td>
  <td align="center"><math>\eta_s = B + \pi</math></td>
  <td align="center"><math>\mathcal{m}_\mathrm{surf}=\biggl( \frac{2}{\pi}\biggr)^{1 / 2}\frac{A\eta_s}{\theta_i}</math></td>
</tr>
<tr>
  <td align="center"><b>A</b></td>
  <td align="center">0.200812422</td>
  <td align="center">- 0.859270052</td>
  <td align="center">9.0149598</td>
  <td align="center">0.188679805</td>
  <td align="center">0.17232050</td>
  <td align="center">2.28232260</td>
  <td align="center">1.9381270</td>
</tr>
</table>


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3">Key Parameter-Parameter Ratios</td>
</tr>
<tr>
  <td align="center"><math>\frac{\xi}{\tilde{r}} = \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \mathcal{m}_\mathrm{surf}^2 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4}</math></td>
  <td align="center"><math>\frac{\eta}{\tilde{r}} = \mathcal{m}_\mathrm{surf}^{2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{2}_i (2\pi)^{1/2}</math></td>
  <td align="center"><math>\frac{\eta}{\xi}</math></td>
</tr>
<tr>
  <td align="center">588.6362811</td>
  <td align="center">11.25175286</td>
  <td align="center">0.019114950</td>
</tr>
</table>

As [[#Core|presented above]], when <math>\sigma_c^2 = 0</math>, the eigenfunction for the core that we have deduced via the B-KB74 conjecture appears to be well represented by the expressions,

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

<tr>
  <td align="right">
<math>x_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>C_0\biggl[ 1 - \frac{\xi^2}{15} \biggr] \, ,</math> &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; <math>\frac{dx}{d\xi}\biggr|_\mathrm{core} = - \frac{2 C_0 \xi}{15}\, ,</math> &nbsp; &nbsp; &nbsp; with, &nbsp; &nbsp; &nbsp; <math>C_0 = - 0.0011 \, ,</math>
  </td>
</tr>
</table>
over the radial-parameter range, <math>0 \le \xi \le \xi_i\, .</math>


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="5">At the Core/Envelope Interface<br /><font size="-1">(as viewed from the core)</font></td>
</tr>
<tr>
  <td align="center"><math>x_i</math></td>
  <td align="center"><math>\frac{dx}{d\xi}\biggr|_i</math></td>
  <td align="center"><math>\tilde{r}_i</math></td>
  <td align="center"><math>\frac{dx}{d\tilde{r}}\biggr|_i</math></td>
  <td align="center"><math>\biggl[\frac{d\ln x}{d\ln \xi}\biggr]_i = \biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_i</math></td>
</tr>
<tr>
  <td align="center">+ 0.004859763</td>
  <td align="center">+ 0.001322194</td>
  <td align="center">0.015314992</td>
  <td align="center">+ 0.778291359</td>
  <td align="center">+ 2.4526969</td>
</tr>
</table>


Copying from our [[#Envelope|earlier discussion of the envelope]] for "model <b>A</b>", the range of the radial parameter is,
<div align="center">
<math>(\eta_i = 0.1723205) \le \eta \le (\eta_s = 2.282322601) \, .</math>
</div>

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">

<font color="red">SLOPE:</font>&nbsp;  As we have [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Eigenfunction_Details|detailed elsewhere]], <font color="red">we expect</font> that the slope of the function, <math>x_\mathrm{env}(\tilde{r})</math>, is related to the slope of <math>x_\mathrm{core}(\tilde{r})</math> at the interface via the expression,

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

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i \biggr\}_\mathrm{env}
=
\biggl\{ \frac{d\ln x}{d\ln \eta}\biggr|_i \biggr\}_\mathrm{env}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl\{ \frac{d\ln x}{d\ln \xi}\biggr|_i \biggr\}_\mathrm{core} 
\, .</math>
  </td>
</tr>
</table>
In our case, <math>\gamma_c = 6/5</math> and <math>\gamma_e = 2 ~~\Rightarrow \gamma_c/\gamma_e = 3/5</math>.  Hence, from the point of view of the envelope displacement function, at the interface,

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

<tr>
  <td align="right">
<math>
\biggl[ \frac{\tilde{r}}{x_\mathrm{env}} \cdot \frac{d x_\mathrm{env}}{d \tilde{r} } \biggr]_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{5} \biggl\{ \biggl[ \frac{d\ln (x_\mathrm{core})}{d\ln \xi}\biggr]_i  - 2\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{5} \biggl\{ \biggl[ 2.452697  - 2\biggr\} = 0.271618 \, .
</math>
  </td>
</tr>
</table>
Now, at the interface of any bipolytrope, the ratio <math>\tilde{r}/x</math> should have the same numerical value whether it is viewed from the point of view of the core or the envelope.  Given that, for our particular "model <b>A</b>",
<div align="center">
<math>\biggl[ \frac{\tilde{r}}{x} \biggr]_i = \frac{0.015315}{0.00485976} = 3.15139 \, ,</math>
</div>
we should expect the slope of the envelope's displacement function at the interface to be,

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

<tr>
  <td align="right">
<math>
\frac{d x_\mathrm{env}}{d \tilde{r} } \biggr|_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0.08619 \, .
</math>
  </td>
</tr>
</table>

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

[[#Discrete_Form_of_LAWE|As above]], we will integrate the discrete LAWE ''outward'' using the finite-difference expression,

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

<tr>
  <td align="right">
<math>x_+ \overbrace{\biggl[2\phi_i +\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i\biggr]}^{\mathrm{TERM1}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_- \overbrace{\biggl[\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i - 2\phi_i\biggr]}^{\mathrm{TERM2}} 
+  x_i \overbrace{\biggl\{4\phi_i - \frac{\Delta_\eta^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 
2\alpha \biggl(- \frac{3\phi^'}{\eta}\biggr)_i\biggr]  \biggr\}}^{\mathrm{TERM3}} \, .</math>
  </td>
</tr>
</table>
When we started the integration at the center of the configuration, we kickstarted the process by, first, setting <math>x_1 = 1</math>; then, [[#Determining_Discrete_Representation_of_Eigenfunction|second, setting]],

<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_\eta^2}{60} \biggr] \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp; </td>
  <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\biggr] \, .</math>
  </td>
</tr>
</table>
Having obtained <math>x_1 \rightarrow x_-</math> and <math>x_2 \rightarrow x_i</math>, we then used the finite-difference expression to calculate <math>x_+ \rightarrow x_3</math>, as well as all subsequent "<math>x_+</math>" values, all the way to the surface.  

Here, instead, we want to start the ''envelope'' integration at the core/envelope interface as follows:
<ol>
  <li>The displacement function for the core gives us the value of the displacement function, <math>x_i = + 0.004859763</math>, at <math>\xi_i = 9.0149598</math>, that is, at <math>\eta_i = 0.17232050</math>; we recognize that this value of <math>x_i</math> (at the interface) also furnishes the value of <math>x_i</math> in the first integration step of the finite-difference expressions.</li>
  <li>We will then "guess" the slope of the envelope's displacement function, <math>q \equiv [dx/d\eta]_i</math>, at the interface.</li>
  <li>Our discrete representation of this first derivative permits us to write,

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

<tr>
  <td align="right">
<math>
q
</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>
\Rightarrow ~~~ x_-
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
x_+ - 2q\Delta_\eta \, .
</math>
  </td>
</tr>
</table>
Inserting this expression into the finite-difference approximation to the LAWE gives <b>for the first integration step only!</b>

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

<tr>
  <td align="right">
<math>x_+ \cdot \mathrm{TERM1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(x_+ - 2q\Delta_\eta) \cdot \mathrm{TERM2} 
+  x_i \cdot \mathrm{TERM3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \cdot [\mathrm{TERM1} - \mathrm{TERM2}]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i \cdot \mathrm{TERM3}
- 2q\Delta_\eta \cdot \mathrm{TERM2} \, .
</math>
  </td>
</tr>
</table>

  </li>
</ol>
<font color="red">NOTE:</font> &nbsp; Judging by the behavior of the B-KB74 generated displacement function, at the interface we expect the slope, <math>[d\ln x/d\ln \tilde{r}]_\mathrm{env}</math>, from the envelope's perspective to be shallower than the slope, <math>[d\ln x/d\ln \tilde{r}]_\mathrm{core}</math>, from the core's perspective.  That is to say, we expect to "guess" values of <math>q</math> such that at the interface,

<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[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
- \biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{env}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{d\ln x}{d\ln \eta}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{\eta}{x} \cdot \frac{dx}{d \eta}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{dx}{d \eta}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\frac{x_i}{\eta_i} \cdot \biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~q</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{+ 0.004859763}{+ 0.1723205}\biggr] \cdot 2.4526969 = +0.06917068 \, .
</math>
  </td>
</tr>
</table>