Editing Apps/Blaes85SlimLimit (section)

==Statement of the Eigenvalue Problem==

Here, we build on [[Apps/PapaloizouPringle84#Nonaxisymmetric_Instability_in_Papaloizou-Pringle_Tori|our discussion in an accompanying chapter]] in which  five published analyses of nonaxisymmetric instabilities  in Papaloizou-Pringle tori were reviewed:  The discovery paper, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84], and papers by four separate groups that were published within a couple of years of the discovery paper &#8212; [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)], [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)], and [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman &amp; Narayan (1986)].  Following the lead of  [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes] (1985; hereafter Blaes85), in particular, we have shown that the relevant eigenvalue problem is defined by the following 2<sup>nd</sup>-order PDE,

<div align="center">
<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>~
\eta^2 (1-\eta^2)\cdot \frac{\partial^2(\delta W)^{(0)}}{\partial \eta^2}
+ (1-\eta^2) \cdot \frac{\partial^2(\delta W)^{(0)}}{\partial\theta^2}
+ \biggl[ \eta (1-\eta^2)  -2 n \eta^3 
\biggr] \cdot \frac{\partial (\delta W)^{(0)}}{\partial \eta} 
+ 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 (\delta W)^{(0)} \, ,
</math>
  </td>
</tr>
</table>
</div>

where, <math>~\delta W^{(0)}</math> is the dimensionless enthalpy perturbation.  Making the substitution,
<div align="center">
<math>~\delta W^{(0)} ~\rightarrow~ V(\eta) \exp (ik\theta) \, ,</math>
</div>
this governing equation &#8212; now, a one-dimensional, 2<sup>nd</sup>-order ODE &#8212; becomes,

<div align="center">
<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>~
\eta^2 (1-\eta^2)\cdot \frac{d^2V}{d \eta^2}
- k^2(1-\eta^2) V
+ \biggl[ \eta (1-\eta^2)  -2 n \eta^3 
\biggr] \cdot \frac{d V}{d \eta} 
+ 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 V \, .
</math>
  </td>
</tr>
</table>
</div>

Making the additional substitution,
<div align="center">
<math>~V ~\rightarrow~ \eta^{|k|} \Upsilon(\eta) \, ,</math>
</div>
and appreciating that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dV}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|k|\eta^{|k|-1} \Upsilon + \eta^{|k|} \frac{d\Upsilon}{d\eta} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d^2V}{d\eta^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
|k|[|k|-1] \eta^{|k|-2}\Upsilon 
+ 2|k|\eta^{|k|-1} \frac{d\Upsilon}{d\eta} 
+ \eta^{|k|} \frac{d^2\Upsilon}{d\eta^2}\, ,</math>
  </td>
</tr>
</table>
</div>

the governing ODE becomes,

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

<tr>
  <td align="right">
<math>~
\biggl\{k^2(1-\eta^2)  - 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2\biggr\} \eta^{|k|}\Upsilon 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\eta^2 (1-\eta^2)\cdot \biggl[ |k|[|k|-1] \eta^{|k|-2}\Upsilon 
+ 2|k|\eta^{|k|-1} \frac{d\Upsilon}{d\eta} 
+ \eta^{|k|} \frac{d^2\Upsilon}{d\eta^2}
 \biggr]
+ \biggl[ \eta (1-\eta^2)  -2 n \eta^3 
\biggr] \cdot \biggl[ |k|\eta^{|k|-1} \Upsilon + \eta^{|k|} \frac{d\Upsilon}{d\eta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\eta^2)
\biggl[ |k|[|k|-1] \eta^{|k|}\Upsilon 
+ 2|k|\eta^{|k|+1} \frac{d\Upsilon}{d\eta} 
+ \eta^{|k|+2} \frac{d^2\Upsilon}{d\eta^2}\biggr]
+ \biggl[ (1-\eta^2)  -2 n \eta^2 
\biggr] \cdot \biggl[ |k|\eta^{|k|} \Upsilon + \eta^{|k|+1} \frac{d\Upsilon}{d\eta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta^{|k|}(1-\eta^2)
\biggl[ k^2 \Upsilon 
+ (2|k|+1)\eta \frac{d\Upsilon}{d\eta} 
+ \eta^{2} \frac{d^2\Upsilon}{d\eta^2} \biggr]
- \eta^{|k|}\biggl[ 2 n \eta^2 \biggr] \cdot \biggl[ |k| \Upsilon + \eta \frac{d\Upsilon}{d\eta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
- 2n\eta^2 \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\eta^2)
\biggl[ \eta^{2} \frac{d^2\Upsilon}{d\eta^2} + (2|k|+1)\eta \frac{d\Upsilon}{d\eta} \biggr]
- \biggl[ 2 n \eta^3  \frac{d\Upsilon}{d\eta} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Finally, then, making the independent variable substitution,
<div align="center">
<math>~\eta^2 ~\rightarrow ~ y</math> &nbsp; &nbsp; &nbsp;
<math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; 
<math>~dy = 2\eta d\eta</math> &nbsp; &nbsp; &nbsp;
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\eta}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~2y^{1/2}\frac{d}{dy}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d^2}{d\eta^2}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~2\frac{d}{dy} + 4y\frac{d^2}{dy^2} \, .</math>
  </td>
</tr>
</table>
</div>
and,

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

<tr>
  <td align="right">
<math>~
- 2ny \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-y) y \frac{d^2\Upsilon}{d\eta^2} + (2|k|+1)(1-y)y^{1/2} \frac{d\Upsilon}{d\eta} 
- 2 n y^{3/2}  \frac{d\Upsilon}{d\eta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4(1-y)y^2 \frac{d^2\Upsilon}{dy^2} + 2(1-y) y \frac{d\Upsilon}{dy} + 2(2|k|+1)(1-y)y \frac{d\Upsilon}{dy} 
- 4 n y^{2}  \frac{d\Upsilon}{dy} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
- \frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-y)y \frac{d^2\Upsilon}{dy^2} + \frac{1}{2}(1-y) \frac{d\Upsilon}{dy} + \frac{1}{2}(2|k|+1)(1-y)\frac{d\Upsilon}{dy} 
- n y  \frac{d\Upsilon}{dy} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-y)y \frac{d^2\Upsilon}{dy^2} + (|k|+1)(1-y)\frac{d\Upsilon}{dy} 
- n y  \frac{d\Upsilon}{dy} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-y)y \frac{d^2\Upsilon}{dy^2} + (|k|+1)\frac{d\Upsilon}{dy} -y (|k|+1+n)\frac{d\Upsilon}{dy} \, .
</math>
  </td>
</tr>
</table>
</div>
This matches equation (3.9) of Blaes85.  According to [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], this equation "<font color="green">&hellip; is a standard eigenvalue problem whose only solutions are the Jacobi polynomials &hellip;</font>"