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

====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>