Editing SSC/Stability/n1PolytropeLAWE/Pt3 (section)

====Following Up on the Brute-Force Trial Fit====
In an [[SSC/Stability/BiPolytropes#Is_There_an_Analytic_Expression_for_the_Eigenfunction.3F|accompanying discussion]] &#8212; see especially [[SSC/Stability/BiPolytropes#Attempt_2|Attempt #2]] &#8212; we have determined by visual inspection that a decent fit to the envelope's eigenfunction is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\mathrm{trial}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b_0}{\Lambda^2} \biggl\{ 1 - \Lambda \biggl[ \frac{\tan(\eta_i - \Lambda - 3\pi/4) + f_\alpha}{1 - f_\alpha \cdot \tan(\eta_i - \Lambda - 3\pi/4)} \biggr] \biggr\} - a_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b_0}{\Lambda^2} \biggl\{ 1 - \Lambda \cot(\Lambda - E)\biggr\} - a_0 \, ,
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="5" align="right">
<tr>
  <th align="center" colspan="4">Limiting Parameter Values</th>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">min</td>
  <td align="center">max</td>
  <td align="center"><math>~\alpha = \alpha_s</math>
</tr>
<tr>
  <td align="center"><math>~\eta_\mathrm{F}</math></td>
  <td align="center"><math>~\eta_i</math></td>
  <td align="center"><math>~\eta_s</math></td>
  <td align="center"><math>~\frac{8}{\pi} ( \eta_s - \eta_i )^2 + 2\eta_s - \eta_i</math></td>
</tr>
<tr>
  <td align="center"><math>~\alpha</math></td>
  <td align="center"><math>~-\frac{\pi}{2}</math></td>
  <td align="center"><math>~-\frac{5\pi}{8}</math></td>
  <td align="center"><math>~\eta_i - \eta_s - \frac{3\pi}{4}</math></td>
</tr>
<tr>
  <td align="center"><math>~\Lambda</math></td>
  <td align="center"><math>~\eta_i - \frac{\pi}{4}</math></td>
  <td align="center"><math>~\eta_i - \frac{\pi}{8}</math></td>
  <td align="center"><math>~\eta_s</math></td>
</tr>
</table>
where, over the range, <math>~\eta_i \le \eta \le \eta_s \, ,</math>

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

<tr>
  <td align="right">
<math>~E</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\eta_i - \frac{5\pi}{4} + \tan^{-1} f_\alpha \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Lambda(\eta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\eta_i + g_\mathrm{F}  \biggl[ \eta_i - 2\eta_s + \eta \biggr] = \Lambda_0 + g_\mathrm{F}\eta 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{f_\alpha} = \tan(\alpha_s)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\tan[ - (\eta_s - \eta_i + \tfrac{3\pi}{4}) ]
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~g_\mathrm{F}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\pi}{8(\eta_s - \eta_i)}
\, .</math>
  </td>
</tr>
</table>


----

Here, we reference a [[SSC/Stability/BiPolytropes#Attempt_1|separate discussion of the bipolytrope's underlying equilibrium structure]]

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" width="50%"><math>~B = \eta_i - \frac{\pi}{2} + \tan^{-1}f</math></td>
  <td align="center"><math>~E = \eta_i - \frac{5\pi}{4} + \tan^{-1}f_\alpha</math></td>
</tr>
<tr>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Rightarrow~~~\cot(\eta_i - B)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\tfrac{\pi}{2} - (\eta_i - B)]</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\tfrac{\pi}{2} - (\tfrac{\pi}{2} - \tan^{-1}f)]</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~f</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan(B + \tfrac{\pi}{2} - \eta_i )</math>
  </td>
</tr>
</table>
  </td>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Rightarrow~~~\cot(\eta_i - E)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\tfrac{\pi}{2} - (\eta_i - E)]</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\tfrac{\pi}{2} - (\tfrac{5\pi}{4} - \tan^{-1}f_\alpha)]</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan( \tan^{-1}f_\alpha - \tfrac{3\pi}{4} )</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\tan( \tfrac{3\pi}{4} - \tan^{-1}f_\alpha )</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\cot( \tan^{-1}f_\alpha )</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{1}{f_\alpha}</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
Hence &hellip; &nbsp; &nbsp; <math>~\cot(\eta - B)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\tfrac{\pi}{2} - (\eta - B)]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\tfrac{\pi}{2} - \eta + \eta_i - \tfrac{\pi}{2} + \tan^{-1}f]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\eta_i - \eta + \tan^{-1}f]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ \tan(\eta_i-\eta) + f }{ 1 - f \cdot \tan(\eta_i - \eta)}</math>
  </td>
</tr>
</table>
  </td>
  <td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
Hence &hellip; &nbsp; &nbsp; <math>~\cot(\Lambda - E)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\tfrac{\pi}{2} - (\Lambda - E)]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan[\eta_i - \Lambda - \tfrac{3\pi}{4} + \tan^{-1}f_\alpha]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tan(\eta_i - \Lambda - \tfrac{3\pi}{4}) + f_\alpha  }{1 - f_\alpha \cdot \tan(\eta_i - \Lambda - \tfrac{3\pi}{4})  }</math>
  </td>
</tr>
</table>
  </td>
 </tr>
<tr>
  <td align="center">Also &hellip; &nbsp; &nbsp;<math>~B = \eta_s - \pi</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Rightarrow ~~~ f = \cot(\eta_i - B)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cot(\eta_i - \eta_s + \pi)</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{f}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan(\eta_i - \eta_s + \pi)</math>
  </td>
</tr>
</table>
  </td>
  <td align="center"><math>~</math></td>
</tr>
</table> 


----

Let's examine the first and second derivatives of this trial eigenfunction, recognizing that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dx_\mathrm{trial}}{d\eta} = \frac{d\Lambda}{d\eta} \cdot \frac{dx_\mathrm{trial}}{d\Lambda}= g_\mathrm{F} \cdot  \frac{dx_\mathrm{trial}}{d\Lambda}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{d^2x_\mathrm{trial}}{d\eta^2} 
= \frac{d\Lambda}{d\eta} \cdot \frac{d}{d\Lambda} \biggl[ g_\mathrm{F}\cdot \frac{dx_\mathrm{trial}}{d\Lambda} \biggr]
= g_\mathrm{F}^2 \cdot \frac{d^2x_\mathrm{trial}}{d\Lambda^2} \, .
</math>
  </td>
</tr>
</table>
and drawing from the [[#Prior_to_the_Brute-Force_Trial_Fit|derivative expressions already derived, above]].  For the first derivative, we have,

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

<tr>
  <td align="right">
<math>~\frac{dx_\mathrm{trial}}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
g_\mathrm{F} \biggl( \frac{b_0}{\Lambda^3} \biggr) 
\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E)  +  \Lambda^2\cot^2(\Lambda - E) \biggr] \, .
</math>
  </td>
</tr>
</table>
And the second derivative gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2x_\mathrm{trial}}{d\eta^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
g_\mathrm{F}^2 \biggl(\frac{2b_0}{\Lambda^4} \biggr) \biggl[ 3 - \Lambda^2 
- (\Lambda + \Lambda^3)\cot(\Lambda-E)  
-  \Lambda^2\cot^2(\Lambda - E) 
-  \Lambda^3\cot^3(\Lambda-E) 
\biggr]  \, .
</math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x_\mathrm{trial}}{d\eta^2} + \biggl\{ 4 -2Q \biggr\}\frac{1}{\eta} \cdot \frac{dx_\mathrm{trial}}{d\eta} 
~-~  2 Q \cdot \frac{x_\mathrm{trial}}{\eta^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x_\mathrm{trial}}{d\eta^2} + \biggl\{ 4 -2 \biggl[1- \eta\cot(\eta-B) \biggr]\biggr\}\frac{1}{\eta} \cdot \frac{dx_\mathrm{trial}}{d\eta} 
~-~  \biggl[1- \eta\cot(\eta-B) \biggr]   \frac{2x_\mathrm{trial}}{\eta^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{b_0}{\eta^4} \biggl\{
\frac{\eta^4}{b_0} \cdot \frac{d^2x_\mathrm{trial}}{d\eta^2} + \biggl[ 1 + \eta\cot(\eta-B) \biggr] \frac{2\eta^3}{b_0} \cdot \frac{dx_\mathrm{trial}}{d\eta} 
~-~  \biggl[1- \eta\cot(\eta-B) \biggr]  \frac{2\eta^2 x_\mathrm{trial}}{b_0} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{b_0}{\eta^4} \biggl\{
\frac{\eta^4}{b_0} \cdot g_\mathrm{F}^2 \biggl(\frac{2b_0}{\Lambda^4} \biggr) \biggl[ 3 - \Lambda^2 
- (\Lambda + \Lambda^3)\cot(\Lambda-E)  
-  \Lambda^2\cot^2(\Lambda - E) 
-  \Lambda^3\cot^3(\Lambda-E) 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ 1 + \eta\cot(\eta-B) \biggr] \frac{2\eta^3}{b_0} \cdot 
g_\mathrm{F} \biggl( \frac{b_0}{\Lambda^3} \biggr) 
\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E)  +  \Lambda^2\cot^2(\Lambda - E) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~  \biggl[1- \eta\cot(\eta-B) \biggr]  \frac{2\eta^2 }{b_0} \cdot 
\biggl[\frac{b_0}{\Lambda^2} \biggl\{ 1 - \Lambda \cot(\Lambda - E)\biggr\} - a_0\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{b_0}{\eta^4} \biggl\{
g_\mathrm{F}^2 \biggl(\frac{2\eta^4}{\Lambda^4} \biggr) \biggl[ 3 - \Lambda^2 
- (\Lambda + \Lambda^3)\cot(\Lambda-E)  
-  \Lambda^2\cot^2(\Lambda - E) 
-  \Lambda^3\cot^3(\Lambda-E) 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ 1 + \eta\cot(\eta-B) \biggr] \cdot 
g_\mathrm{F} \biggl( \frac{2\eta^3}{\Lambda^3} \biggr) 
\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E)  +  \Lambda^2\cot^2(\Lambda - E) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~  \biggl[1- \eta\cot(\eta-B) \biggr]  
\biggl[\frac{2\eta^2}{\Lambda^2} [ 1 - \Lambda \cot(\Lambda - E) ] - \frac{2\eta^2 a_0}{b_0} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2b_0}{\Lambda^4\eta^2} \biggl\{
g_\mathrm{F}^2 \eta^2 \biggl[ 3 - \Lambda^2 
- (\Lambda + \Lambda^3)\cot(\Lambda-E)  
-  \Lambda^2\cot^2(\Lambda - E) 
-  \Lambda^3\cot^3(\Lambda-E) 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ 1 + \eta\cot(\eta-B) \biggr] \cdot 
g_\mathrm{F} \Lambda \eta 
\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E)  +  \Lambda^2\cot^2(\Lambda - E) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~  \biggl[1- \eta\cot(\eta-B) \biggr]  
\biggl[\Lambda^2 [ 1 - \Lambda \cot(\Lambda - E) ] - \frac{a_0\Lambda^4}{b_0} \biggr]
\biggr\}
</math>
  </td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow~~~\biggl(\frac{\Lambda^4}{2b_0}\biggr) \cdot</math> &nbsp;LAWE
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
g_\mathrm{F}^2 \biggl[ 3 - \Lambda^2 
- (\Lambda + \Lambda^3)\cot(\Lambda-E)  
-  \Lambda^2\cot^2(\Lambda - E) 
-  \Lambda^3\cot^3(\Lambda-E) 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  
\frac{g_\mathrm{F} \Lambda}{ \eta }
\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E)  +  \Lambda^2\cot^2(\Lambda - E) \biggr] 
~-~    \biggl(\frac{\Lambda}{\eta}\biggr)^2\biggl[  1 - \Lambda \cot(\Lambda - E)  - \frac{a_0\Lambda^2}{b_0} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \eta\cot(\eta-B) \biggr] \biggl\{ 
\frac{g_\mathrm{F} \Lambda }{\eta }
\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E)  +  \Lambda^2\cot^2(\Lambda - E) \biggr] 
~+~ \biggl( \frac{\Lambda}{\eta}\biggr)^2 \biggl[ 1 - \Lambda \cot(\Lambda - E)  - \frac{a_0\Lambda^2}{b_0 }\biggr]  \biggr\}
</math>
  </td>
</tr>
</table>