Editing Appendix/Ramblings/PPToriPt1A (section)

==Goldreich, Goodman and Narayan (1986)==

===Unperturbed Slim Torus Structure===
[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 only the slimmest tori, so overlap between the GGN86 and Blaes85 work is easiest to recognize if we begin with the enthalpy distribution prescribed for a "slim torus" by Blaes (1985), as [[#Blaes85|discussed above]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H = H_0\Theta_H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~H_0 - \frac{H_0}{\beta^2}\biggl[r^2 + r^3(3\cos\theta - \cos^3\theta) + \mathcal{O}(r^4)  \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
[<font color="red"><b>Note:</b></font> &nbsp; Here we have replaced the variable name, <math>~x</math>, as used in Blaes85, with the variable name, <math>~r</math>, in order (1) to emphasize that the variable represents a dimensionless ''radial'' coordinate, and (2) to avoid conflict with the GGN86 variable, <math>~x</math>, which is a Cartesian coordinate with the standard dimension of length.] 

Now, from our [[#Equilibrium_Configuration|above discussion of equilibrium PP tori]] and recognizing that the Keplerian angular frequency at the location of the enthalpy maximum is, 

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<math>\Omega_K \equiv \frac{GM_\mathrm{pt}}{\varpi_0^3} \, ,</math>
</div>

we can set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_\mathrm{pt}\beta^2}{2\varpi_0} = \tfrac{1}{2} \Omega_K^2 \varpi_0^2 \beta^2 \, .</math>
  </td>
</tr>
</table>
</div>
Hence, for the slimmest tori &#8212; that is, keeping only the lowest order term in <math>~r</math> &#8212; the enthalpy distribution becomes,

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

<tr>
  <td align="right">
<math>~H </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2} \Omega_K^2 \varpi_0^2 \beta^2 
- \tfrac{1}{2} \Omega_K^2 \varpi_0^2\biggl[r^2 + \cancelto{0}{r^3}(3\cos\theta - \cos^3\theta) + \cancelto{0}{\mathcal{O}(r^4)} ~~\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{\Omega_K^2}{2} [\varpi_0^2 \beta^2  -   \varpi_0^2 r^2] \, .</math>
  </td>
</tr>
</table>
</div>

Following GGN86, the surface of the torus &#8212; where the enthalpy drops to zero &#8212; occurs at <math>~r = a/\varpi_0</math>.  Hence, we recognize that,

<div align="center">
<math>\beta = \frac{a}{\varpi_0} \, ,</math>
</div>

and we can rewrite the expression for the unperturbed enthalpy distribution as,

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

<tr>
  <td align="right">
<math>~H </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Omega_K^2}{2} [a^2  - \varpi_0^2 r^2 ] \, .</math>
  </td>
</tr>
</table>
</div>
This expression exactly matches equation (2.13) of GGN86 &#8212; which, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_0(x,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Omega^2}{2} \biggl[ (2q-3)(a^2 - x^2) - z^2 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>

once it is appreciated that, in moving from the Blaes85 discussion to the GGN86 discussion, <math>~\varpi_0^2 r^2 \rightarrow (x^2 + z^2)</math>, and it is recognized that Blaes85 restricted his investigation to tori that have uniform specific angular momentum <math>~(q = 2)</math>.

===Additional Notation===

<div align="center">
<math>~(ky)_\mathrm{GGN} = \biggl( \frac{my}{\varpi_0} \biggr)_\mathrm{GGN} ~~\leftrightarrow ~~ (m\phi)_\mathrm{Blaes}</math>
</div>

<div align="center">
<math>~\beta_\mathrm{GGN} \equiv \biggl( \frac{ma}{\varpi_0} \biggr)_\mathrm{GGN} ~~\leftrightarrow ~~ m\beta_\mathrm{Blaes}</math>
</div>

From equation (5.16) of GGN86 we obtain "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>
</table>
</div>

Working on the imaginary part of this expression to put it in the terminology of Blaes85, we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Im}(\psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp 4\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>~\mp 4\biggl(\frac{3}{2}\biggr)^{1/2} \biggl(\frac{m}{\varpi_0}\biggr) [\varpi_0 (\eta\beta_\mathrm{Blaes})\cos\theta ](m\beta_\mathrm{Blaes}) </math>
  </td>
</tr>

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

which exactly matches <math>~\mathrm{Im}(f_m)</math> as derived by Blaes85 and [[#Incompressible_Slim_Tori|summarized above]].  Similarly,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\frac{1}{4} \biggl(\frac{m}{\varpi_0}\biggr)^2[\varpi_0^2 r^2(5\cos^2\theta - 3\sin^2\theta)] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\frac{1}{4} \eta^2 m^2 \beta^2_\mathrm{Blaes}[8\cos^2\theta - 3] </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] \, .</math>
  </td>
</tr>
</table>
</div>

This exactly matches <math>~\mathrm{Re}(f_m)</math> as derived by Blaes85 and [[#Incompressible_Slim_Tori|summarized above]].  This is in line with the following statement that appears in the acknowledgement section of GGN86:  "We note that Omar Blaes &hellip; [has] independently derived many of the results reported in this paper."