Editing Appendix/Ramblings/ToHadleyAndImamura (section)

====For n = 0 Configurations====

As we have [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29|discussed elsewhere]], [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman &amp; Narayan (1986, MNRAS, 221, 339)] &#8212; hereafter, GGN86 &#8212; also used analytic techniques to analyze the properties of unstable, nonaxisymmetric eigenmodes in Papaloizou-Pringle tori.  They restricted their discussion to slim, ''incompressible'' tori, so in order to assess the overlap between the GGN86 and Blaes85 works, we will set <math>~n=0</math> in the general expressions presented in Blaes85.

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{2^2}\biggl[a_0(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{2^2}\biggl[b_0(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_0(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~[8\cos^2\theta - 3]\eta^2  - 1 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_0(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3 \eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^3 (2\cdot 3)^{1/2} \eta \cos\theta  \, .
</math>
  </td>
</tr>
</table>
</div>

From the Blaes85 analysis, then, we conclude that the unstable eigenfunction for slim, ''incompressible'' PP-tori is,

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{W}{W_0}\biggr]_{n=0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^2}\biggl[(8\cos^2\theta - 3)\eta^2  - 1\biggr]
+ i \frac{ (\beta m)^2}{2^2}\biggl[ 2^3 (2\cdot 3)^{1/2} \eta \cos\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \tfrac{1}{4}(\beta m)^2\biggr] + (\beta m)^2 \biggl[\biggl(2\cos^2\theta - \frac{3}{4}\biggr)\eta^2 \biggr]
+ 4i (\beta m)^2\biggl[ \biggl(\frac{3}{2}\biggr)^{1/2} \eta \cos\theta \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

This should be compared with equation (5.16) of GGN86, which is "the lowest order [complex] expression for the [perturbed] velocity potential," namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\psi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\tfrac{1}{4} k^2(5x^2 - 3z^2) \mp 4i\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+m^2 \beta^2_\mathrm{Blaes}\biggl[2\eta^2\cos^2\theta - \frac{3\eta^2}{4}\biggr]  
\mp 4i \biggl(\frac{3}{2}\biggr)^{1/2} m^2\beta^2_\mathrm{Blaes} \eta\cos\theta \, ,</math>
  </td>
</tr>
</table>
</div>

where the last expression results from [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29|our mapping of the GGN86 terminology to the Blaes85 terminology]].  The two derived expressions match in every detail, except one:  The constant term, <math>~\tfrac{1}{4}(\beta m)^2</math>, that is subtracted from unity on the right-hand side of the Blaes85 function is missing from the function derived by GGN86.  Hopefully, additional study of this problem will rectify this apparent difference between the two published analyses.

<!-- OMIT
In addition, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 + a(\eta,\theta) }{  b(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 - 1 
+ [8\cos^2\theta - 3]\eta^2  }{  2^3 (2\cdot 3)^{1/2} \eta \cos\theta } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

and the square of the amplitude is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{\delta W}{W_0} \biggr|^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^2}\biggl[a(\eta,\theta)\biggr] \biggr\}^2 +  
\biggl\{ \frac{(\beta m)^2}{2^2}\biggl[b(\eta,\theta)\biggr] \biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ 2(\beta m)^2a}{2^2}  
+ \frac{ (\beta m)^4 }{2^4} (a^2 + b^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2}  \biggl\{ [8\cos^2\theta - 3]\eta^2  - 1 \biggr\}
+ \mathcal{O}[(\beta m)^4] \, .
</math>
  </td>
</tr>
</table>
</div>
END OMIT -->