Editing Appendix/Ramblings/PPToriPt2A (section)

===Summary===
[[#KeyExpression|As stated above]], the eigenvalue problem that must be solved in order to identify the eigenfunction, <math>~\Lambda(x,\theta)</math>, and eigenfrequency, <math>~(\nu/m)</math>, of unstable (as well as stable) nonaxisymmetric modes in slim <math>~(\beta \ll 1)</math>, polytropic <math>~(n)</math> PP tori with uniform specific angular momentum is defined by the following two-dimensional <math>~(x,\theta)</math>, 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>~f (1-x\cos\theta)^2 \biggl\{
~\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3}
\biggr\}
+ ~\frac{n}{\beta^2} \biggl\{ \mathrm{TERM4}
~+~ \mathrm{TERM5}\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~f(x,\theta)</math> is the enthalpy distribution in the unperturbed, axisymmetric torus, and
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{TERM1}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{TERM2}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x} 
+ \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{TERM3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- [ 2^2(n+1)^2 + m^2\Lambda ] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{TERM4}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(1-x\cos\theta)^4\biggl[ \frac{\partial \Lambda}{\partial x} \cdot \frac{\partial (\beta^2 f)}{\partial x}
~+~ \frac{\partial \Lambda}{\partial \theta} \cdot \frac{\partial (\beta^2 f/x^2)}{\partial \theta} \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{TERM5}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ (1-x\cos\theta)^4\biggl(\frac{\nu}{m}\biggr)^2 + 2(1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)+ 1 \biggr] 
[ 2^3(n+1)^2 + 2m^2\Lambda ] \, .</math>
  </td>
</tr>
</table>
</div>

We also should appreciate that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f\ell^2 \equiv f(1-x\cos\theta)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\eta^2)(1-2x\cos\theta + x^2\cos^2\theta)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 - \beta\biggl(\frac{x}{\beta}\biggr)^3 b\biggr]
\biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
\biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr]
-~\beta\biggl(\frac{x}{\beta}\biggr)^3 b
\biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
\biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr] + \mathcal{O}(\beta^3) \, .
</math>
  </td>
</tr>
</table>
</div>

If an exact solution, <math>~(\Lambda,\nu/m)</math>, to this eigenvalue problem were plugged into this governing PDE, we would expect that ''both'' of the following summations would be exactly zero at all meridional-plane <math>~(x,\theta)</math> locations throughout the torus:
<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>~\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathrm{TERM4} + \mathrm{TERM5}  \, .</math>
  </td>
</tr>
</table>
</div>

While an exact analytic solution to this eigenvalue problem is not (yet) known, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] has determined that a good approximate solution is an eigenvector defined by the complex eigenfrequency,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\nu}{m}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~-1 ~\pm ~ i~\biggl[ \frac{3}{2(n+1)} \biggr]^{1/2} \beta \, ,
</math>
  </td>
</tr>
</table>
</div>

and, simultaneously, the complex eigenfunction,

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

<tr>
  <td align="right">
<math>~\Lambda</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- (4n+1)\beta^2 + (\beta\eta)^2 (n+1)^2[
2^3 \cos^2\theta - 3]  ~\pm~i~\beta [  2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\beta\eta)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^2[1+x(3\cos\theta - \cos^3\theta )] \, .</math>
  </td>
</tr>
</table>
</div>


<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="6"><font size="+1">''Real'' Components of Various Terms</font></th>
</tr>
<tr>
  <td align="center">Order</td>
  <td align="center"><math>~f\ell^2\cdot \mathrm{TERM1}</math></td>
  <td align="center"><math>~f\ell^2\cdot \mathrm{TERM2}</math></td>
  <td align="center"><math>~f\ell^2\cdot \mathrm{TERM3}</math></td>
  <td align="center"><math>~\frac{n}{\beta^2} \cdot\mathrm{TERM4}</math></td>
  <td align="center"><math>~\frac{n}{\beta^2} \cdot\mathrm{TERM5}</math></td>
</tr>
<tr>
  <td align="center"><math>~\mathcal{O}(\beta^{-2})</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~\frac{n}{\beta^2}(1-2+1)</math></td>
</tr>
<tr>
  <td align="center"><math>~\mathcal{O}(\beta^{-1})</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~\frac{n}{\beta^2}(4-4)</math></td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\mathcal{O}(\beta^0)</math></td>
  <td align="center"><math>~(n+1) [ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta ]f\ell^2 </math></td>
  <td align="center"><math>~(n+1) [-6 + 2^4(n+1)\cos^2\theta ]f\ell^2 </math></td>
  <td align="center"><math>~-  2^2(n+1)^2f\ell^2</math> </td>
  <td align="center"><math>~-~n \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta -3]</math></td>
  <td align="center"><math>~2^3 n (n+1)^2\biggl[ 4\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta-\frac{3}{2(n+1)} \biggr]</math></td>
</tr>
<tr>
  <td align="left" colspan="5">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(n+1)\biggl\{ \biggl[-6+2^4(n+1) - 2^4(n+1)\cos^2\theta 
~-6 + 2^4(n+1)\cos^2\theta 
-  2^2(n+1) \biggr]\cdot \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
-~n \biggl( \frac{x}{\beta}\biggr)^2 [2^5(n+1)\cos^2\theta]
+~12n \biggl( \frac{x}{\beta}\biggr)^2
+~2^5 n (n+1)\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta  
-~12 n \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(n+1)\biggl\{ \biggl[
-12 + 12(n+1)\biggr]\cdot \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
+~12n \biggl( \frac{x}{\beta}\biggr)^2 - 12n\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math> &nbsp; &nbsp; Amazing!
  </td>
</tr>
</table>
  </td>
</tr>
</table>


[[File:BetaErrorPlot02.png|right|350px|Beta Error Plot]]We have plugged this "Blaes85" approximate eigenvector into the five separate "TERM" expressions &#8212; analytically evaluating partial (1<sup>st</sup> and 2<sup>nd</sup>) derivatives along the way, as appropriate &#8212; then, with the aid of an Excel spreadsheet, have numerically evaluated each of the expressions over a range of coordinate locations <math>~(0 < x/\beta < 1; 0 \le \theta \le 2\pi)</math>.  The appropriate numerical sums of these TERMs are, indeed, nearly zero for slim <math>~(\beta \ll 1)</math> configurations.  


The log-log plot shown here, on the right, illustrates the behavior of the "TERM4 + TERM5" sum for the example parameter set, <math>~(n, \theta, x/\beta) = (1, \tfrac{\pi}{3}, \tfrac{1}{4})</math>.  As the blue diamonds illustrate, the real part of this sum drops by approximately two orders of magnitude for every factor of ten drop in <math>~\beta</math>.  The total drop is roughly eight orders of magnitude over the displayed range, <math>~\beta = 1 ~\rightarrow~ 10^{-4}</math>.  As the salmon-colored squares in the same plot indicate, the imaginary part of the sum, "TERM4 + TERM5," is even closer to zero, dropping roughly 12 orders of magnitude over the same range of <math>~\beta</math>.  This indicates that, with the Blaes85 eigenvector, the real part of the sum of this pair of terms differs from zero by a residual whose leading-order term varies as <math>~\beta^{2}</math> while the corresponding imaginary part of the sum differs from zero by a residual whose leading-order term varies as <math>~\beta^{3}</math>.


As our [[#Imaginary_Parts|above analytic analysis]] shows, when each of the expressions for TERM4 and TERM5 is rewritten as a power series in <math>~\beta</math>, a sum of the two analytically specified TERMs results in precise cancellation of leading-order terms.  For the imaginary component of this sum, our derived expression for the residual is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Im}(\mathcal{R}_{45})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\mathrm{Im}[\mathrm{TERM4}+\mathrm{TERM5}]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \beta^3 \biggl(\frac{x}{\beta}\biggr)^2 [2^7\cdot 3 (n+1)^3]^{1/2}[ 
3  + 6\cos^2\theta - 2\cos^4\theta ] + \mathcal{O}(\beta^4) \, .
</math>
  </td>
</tr>
</table>
</div>

The dotted, salmon-colored line of slope 3 that has been drawn in our accompanying log-log plot was generated using this analytic expression for the <math>~\beta^3</math>-residual term.  It appears to precisely thread through the points (the salmon-colored squares) whose plot locations have been determined via our numerical spreadsheet evaluation of the imaginary component of the "TERM4 + TERM5" sum.  Additional confirmation that we have derived the correct analytic  expression for <math>~\mathrm{Im}(\mathcal{R}_{45})</math> comes from subtracting this analytically defined <math>~\beta^3</math> residual from the numerically determined sum:  The result is the green-dashed curve in the accompanying log-log plot, which appears to be a line of slope 4.


Analogously, for the real component of this sum, the precise expression for the residual is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}(\mathcal{R}_{45})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\mathrm{Re}[\mathrm{TERM4}+\mathrm{TERM5}]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\beta^22^2\cdot 3(n+1)\biggl[1-\biggl( \frac{x}{\beta}\biggr)^2 \biggr] + \mathcal{O}(\beta^3) \, .
</math>
  </td>
</tr>
</table>
</div>
The dotted, light blue line of slope 2 that has been drawn in our accompanying log-log plot was generated using this analytic expression for the <math>~\beta^2</math>-residual term.  It appears to precisely thread through the points (the light blue diamonds) whose plot locations have been determined via our numerical spreadsheet evaluation of the real part of the "TERM4 + TERM5" sum.  Notice that at the surface of the torus &#8212; that is, when <math>~x/\beta = 1</math> &#8212; this <math>~\beta^2</math>-residual goes to zero, in which case the leading order term in the "real" component residual will be drop to <math>~\mathcal{O}(\beta^3)</math>.