Editing Apps/PapaloizouPringle84 (section)

==Formulation of Eigenvalue Problem==
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! style="height: 125px; width: 125px; background-color:lightgreen;" |[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like_2|<b>Defining the<br />Eigenvalue Problem</b>]]
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In an effort to better understand the full three-dimensional structure of the eigenvector that characterizes nonaxisymmetric instabilities in Papaloizou-Pringle tori, we have studied in considerable depth five published analyses:  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)].  In order to quantitatively compare and contrast the results of these separate analyses, it has been necessary to understand the terminology, technical approaches, and, in particular, the variable names adopted by each group.  In order to accomplish this, here we show how the two-dimensional, 2<sup>nd</sup>-order PDE that fundamentally defines the PP-torus eigenvalue problem is derived in each case; in each of the four subsequently published papers, the derivation has its roots in the original PP84 derivation, as it should.


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="5">
<math>~q(\varpi,\varphi,z,t) = q_0(\varpi,z) + q^'(\varpi,z)f_\sigma</math>
  </td>
</tr>
<tr>
  <th align="center">
<font size="+1">Source</font>
  </th>
  <td align="center">
<math>~q^'</math>
  </td>
  <th align="center">
<font size="+1">Eq. #</font>
  </th>
  <th align="center">
<font size="+1">Units</font>
  </th>
  <td align="center">
<math>~f_\sigma</math>
  </td>
</tr>
<tr>
  <td algin="center" colspan="5" bgcolor="lightgreen">&nbsp;</td>
</tr>
<tr>
  <td align="center">
Here
  </td>
  <td align="center">
<math>~W^' \equiv \frac{P^'}{\rho_0(\sigma + m{\dot\varphi}_0)} </math>
  </td>
  <td align="center">
n/a
  </td>
  <td align="center">
specific<p></p>
angular<p></p>
momentum
  </td>
  <td align="center">
<math>~\exp[i(m\varphi + \sigma t)]</math>
  </td>
</tr>
<tr>
  <td align="center">
[http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84]
  </td>
  <td align="center">
<math>~W \equiv p^'/[\rho(\sigma + m\Omega)]</math>
  </td>
  <td align="center">
(3.13)
  </td>
  <td align="center">
specific<p></p>
angular<p></p>
momentum
  </td>
  <td align="center">
[[File:expPP84.png|100px|center|Papaloizou and Pringle (1984, MNRAS, 208, 721)]]
  </td>
</tr>
<tr>
  <td align="center">
[http://adsabs.harvard.edu/abs/1985MNRAS.213..799P PP85]
  </td>
  <td align="center">
<math>~W \equiv p^'/(\rho\bar\sigma)</math><p></p>
where, <math>~\bar\sigma\equiv \sigma + m\Omega</math>
  </td>
  <td align="center">
Near (2.12)
  </td>
  <td align="center">
specific<p></p>
angular<p></p>
momentum
  </td>
  <td align="center">
[[File:expPP84.png|100px|center|Papaloizou and Pringle (1985, MNRAS, 213, 799)]]
  </td>
</tr>
<tr>
  <td align="center">
[http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]
  </td>
  <td align="center">
<math>~W \equiv \delta p/[\rho(\sigma + m\Omega)]</math>
  </td>
  <td align="center">
(1.5)
  </td>
  <td align="center">
specific<p></p>
angular<p></p>
momentum
  </td>
  <td align="center">
[[File:expBlaes85.png|100px|center|Blaes (1985, MNRAS, 216, 553)]]
  </td>
</tr>
<tr>
  <td align="center">
[http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima86]
  </td>
  <td align="center">
<math>~\mathcal{W} \equiv \delta p/[-\rho(\omega - m\Omega)]</math>
  </td>
  <td align="center">
(13)
  </td>
  <td align="center">
specific<p></p>
angular<p></p>
momentum
  </td>
  <td align="center">
[[File:expKojima86.png|100px|center|Kojima (1986, Progress of Theoretical Physics, 75, 251)]]
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">
[http://adsabs.harvard.edu/abs/1986MNRAS.221..339G GGN86]
  </td>
  <td align="center">
<math>~Q = \rho^'(\Gamma p/\rho^2)</math>
  </td>
  <td align="center">
Near (2.25)
  </td>
  <td align="center" rowspan="2">
enthalpy
  </td>
  <td align="center" rowspan="2">
[[File:expGGN86.png|100px|center|Goldreich, Goodman &amp; Narayan (1986, MNRAS, 221, 339)]]<p></p>
where, <math>~ky = \tfrac{m}{\varpi_0}(\varpi_0 \phi) = m\phi</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~Q \equiv \sigma_{GGN} \cdot W_{PP85}</math><p></p>
where, <math>~\sigma_{GGN} \equiv -{\bar\sigma}_{PP85}</math>&nbsp; &nbsp; (?) [[#Direct_Comparison_of_Derived_Equations|see above]] (?)
  </td>
  <td align="center">
Near (2.27)
  </td>
</tr>
</table>

The second column of the above table identifies, for example, the definition of the principal perturbation parameter, <math>~q^'</math>, that appears in our derivation as well as in the governing PDE of each of the five above-identified publications; we have purposely closely aligned our notation with the notation found in [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84].  The third column of the table identifies the equation number within each paper where &#8212; or at least near where &#8212; the definition of this principal perturbation parameter can be found.  The fourth column lists the physical units of the chosen principal perturbation parameter.  It is fair to say that, in each case, the parameter was ''intended'' to be the ratio,<math>~P^'/\rho</math>, that is, the perturbed fluid enthalpy with units of velocity-squared, but PP84 found it mathematically "convenient" also to divide this quantity by a frequency that characterizes each unstable mode.  Hence, the units of their principal perturbation parameter is "velocity-squared &times; time", which is also the units of specific angular momentum. Following the lead of PP84, three of the other four published analyses also adopted a principal perturbation parameter with units of specific angular momentum.  Finally, the last column of the table identifies the specific expression that was adopted in each case for the factor, <math>~f_\sigma</math>, which facilitates a Fourier-analysis-based isolation of separate azimuthal eigenmodes.  Note that the different adopted sign conventions for the mode eigenfrequency <math>~(\sigma</math> or <math>~\omega)</math> can introduce confusion when attempting to quantitatively compare different published derivations and eigenvector solutions.

While it is the case, as indicated in the above table, that [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] begins his derivation (see his equation 3.1) using the same principal perturbation parameter, <math>~W</math>, as was introduced in [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84], by equation (3.2) he has shifted to a new ''dimensionless'' principal parameter &#8212; which he still calls <math>~W</math>, but which we will refer to as <math>~\delta W</math> (see [[#Equivalent_Dimensionless_Expression|below]]).  We have deduced that this dimensionless perturbation parameter is defined as follows:

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

<tr>
  <td align="right">
<math>~\delta W</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{\Omega_0 \rho_\mathrm{max}}{P_\mathrm{max}} \biggr]W^' </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{(\sigma + m{\dot\varphi}_0)\rho_0}{\Omega_0 \rho_\mathrm{max}}\biggr]^{-1}  \frac{P^'}{P_\mathrm{max}} \, .</math>
  </td>
</tr>
</table>
</div>
Consistent with this, immediately after his equation (3.2), Blaes defines the dimensionless eigenfrequency, 

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

<tr>
  <td align="right">
<math>~\nu</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\sigma}{\Omega_0} \, .</math>
  </td>
</tr>
</table>
</div>




===Seminal Work by Papaloizou &amp; Pringle===
Let's plug the three expressions for the components of the perturbed velocity into the linearized continuity equation and, as well, replace <math>~\rho^'</math> in favor of <math>~W^'</math> via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W^' </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P^'}{\rho_0(\sigma + m{\dot\varphi}_0)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{ (\sigma + m{\dot\varphi}_0)} \biggl(\frac{\gamma P_0 \rho^' }{\rho_0^2}\biggr) \, .</math>
  </td>
</tr>

</table>
</div>

Also multiplying through by <math>~-i</math>, we obtain,

<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>~\rho^'[(\sigma + m{\dot\varphi}_0)]  + \frac{m\rho_0}{\varpi} (\varpi {\dot\varphi}^' )
- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi (i {\dot\varpi}^' ) \biggr] 
- \frac{\partial}{\partial z} \biggl[ \rho_0 (i{\dot{z}}^') \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\bar\sigma}^2 \biggl( \frac{\rho_0^2 W^'}{\gamma P_0  } \biggr)  
+ \frac{m\rho_0}{\varpi ({\bar\sigma}^2 - \kappa^2 )} \biggl\{ -  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{({\bar\sigma}^2 - \kappa^2 )} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr] \biggr\} 
+ \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} \, .
</math>
  </td>
</tr>
</table>

</div>

Multiplying through by <math>~D^2 \equiv ({\bar\sigma}^2 - \kappa^2)^2</math> and reorganizing terms gives,

<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>~
\frac{D^2}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{D} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0}  \biggr)   \biggr] \biggr\} 
+ \frac{D m\rho_0}{\varpi} \biggl\{ -  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ D^2 \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
+ \biggl( \frac{D^2 {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } \biggr)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{D^2}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{D} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0}  \biggr)   \biggr] \biggr\} + \rho_0 D \biggl\{
- \frac{m^2W^' }{\varpi^2 }  \biggl[{\bar\sigma}^2 + \frac{\kappa^2 \varpi  }{ 2 {\dot\varphi}_0 } \biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr)   \biggr]  
- \frac{m\kappa^2 {\bar\sigma} }{ 2\varpi{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ D^2 \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
+ \biggl( \frac{D^2 {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } \biggr) \, .  
</math>
  </td>
</tr>
</table>

</div>

As can be confirmed by comparing it to equation (3.18) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image &#8212; this expression matches the 2<sup>nd</sup>-order, two-dimensional PDE that defines the eigenvalue problem discussed by Papaloizou and Pringle in their seminal 1984 paper.

<div align="center" id="EigenvaluePP84">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equation (3.18) extracted without modification from p. 726 of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle (1984)]<p></p>
"''The dynamical stability of differentially rotating discs with constant specific angular momentum''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 208, pp. 721-750 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:PP84Eq3.18.png|500px|center|Papaloizou and Pringle (1984, MNRAS, 208, 721)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

After dividing through by <math>~D^2</math> and, again, rearranging terms, we also have,


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

<tr>
  <td align="right">
<math>~
\frac{ {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\}
-\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{\frac{\rho_0 \varpi}{D} \biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0}  \biggr)   \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\rho_0}{D} \biggl\{
\frac{m^2 {\bar\sigma}^2 W^' }{\varpi^2 } \biggr\}
+ \frac{\rho_0}{D} \biggl\{
\frac{m^2W^' }{\varpi^2 }  \biggl[\frac{\kappa^2 \varpi  }{ 2 {\dot\varphi}_0 } \biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr)   \biggr]  \biggr\}
+ \frac{\rho_0}{D} \biggl\{\frac{m\kappa^2 {\bar\sigma} }{ 2\varpi{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\}
+ \biggl\{ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} \biggr\}
- \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0 \kappa^2m }{2 {\dot\varphi}_0 D} \biggl(  W^' \bar\sigma \biggr)   \biggr] 
+ \frac{\rho_0 \kappa^2 m}{2 {\dot\varphi}_0 D} \biggl[ \frac{W^' }{\varpi} \cdot \frac{\partial (m{\dot\varphi}_0)}{\partial\varpi}   \biggr]
+ \frac{\kappa^2 m \rho_0}{2{\dot\varphi}_0 D} \biggl[\frac{ {\bar\sigma} }{\varpi} \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\}
+ \biggl\{ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} \biggr\}
- \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0 \kappa^2m }{2 {\dot\varphi}_0 D} (  W^' \bar\sigma ) \biggr] 
+ \frac{\rho_0 \kappa^2 m}{2 {\dot\varphi}_0 D} \biggl[ \frac{1}{\varpi} \cdot \frac{\partial (W^' \bar\sigma)}{\partial\varpi}   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl( 
\frac{\rho_0 \varpi  {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr)
+ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} 
- \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W^'}{\partial z} \biggr) 
- \frac{m W^' \bar\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{\varpi \kappa^2 }{2 {\dot\varphi}_0 } \biggr)\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Here, it is advantageous to note that, in place of the [[#epicyclic|definition of the (square of the) epicyclic frequency provided above]], we could have equally well written,
<div align="center" id="epicyclic2">
<math>~\kappa^2 = \frac{1}{\varpi^3} \frac{d j_0^2}{d \varpi} \, ,</math>
</div>
where, <math>~j_0(\varpi)\equiv \varpi^2{\dot\varphi}_0(\varpi)</math> is a function that specifies how the fluid's specific angular momentum varies with radius in the initial, unperturbed, equilibrium configuration.  (See our related discussion of [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple Rotation Profiles'']].)  From this relation, we recognize as well that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dj_0}{d\varpi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\kappa^2 \varpi^3}{2j_0} = \frac{\kappa^2 \varpi}{2{\dot\varphi}_0} \, .</math>
  </td>
</tr>
</table>
</div>
So the last term inside the square brackets of our expression could meaningfully be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{\varpi \kappa^2 }{2 {\dot\varphi}_0 } \biggr)\biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~\rightarrow~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{dj_0}{d\varpi} \biggr)\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

If we let <math>~h^'</math> represent the radial derivative of the specific angular momentum in the initial, unperturbed, equilibrium configuration (Papaloizou &amp; Pringle use "h" instead of "j" to denote the specific angular momentum, and they use a prime to denote differentiation with respect to the radial coordinate), we see that our derived expression matches equation (2.19) of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image.  Clearly this mathematical definition of the eigenvalue problem discussed by PP85 is fundamentally the same as the one introduced and discussed in PP84.  

<div align="center" id="EigenvaluePP85">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equation (2.19) extracted without modification from p. 803 of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)]<p></p>
"''The dynamical stability of differentially rotating discs. II''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 213, pp. 799-820 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:PP85Eq2.19.png|500px|center|Papaloizou and Pringle (1985, MNRAS, 213, 799)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

===Analyses of Configurations with Uniform Specific Angular Momentum===
====Kojima's Setup====
Presumably PP85 rewrote the equation in the latest form presented above in order to help make it clear how the equation simplifies (specifically, the last term on the right-hand side vanishes) for configurations that initially have uniform specific angular momentum &#8212; that is, for configurations in which <math>~h^' = 0</math>.  But other simplifications arise as well because the epicyclic frequency, <math>~\kappa</math>, also goes to zero in configurations with uniform specific angular momentum.  This means that the frequency ratio, <math>~{\bar\sigma}^2/D</math>, that appears in two terms of our derived expression goes to unity, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{{\bar\sigma}^2}{D}\biggr|_{j_0-\mathrm{constant}} = \biggl[ \frac{{\bar\sigma}^2}{{\bar\sigma}^2 - \kappa^2}\biggr]_{j_0-\mathrm{constant}}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~\rightarrow~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~1 \, .</math>
  </td>
</tr>
</table>
</div>

Implementing both of these simplifications, the latest form of our "eigenvalue problem" equation becomes,

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

<tr>
  <td align="right">
<math>~
\frac{ {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \cancelto{1}{\biggl(\frac{{\bar\sigma}^2}{D} \biggr)} \rho_0 \varpi \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
+ \cancelto{1}{\biggl(\frac{{\bar\sigma}^2}{D} \biggr)} \frac{\rho_0 m^2  W^' }{\varpi^2} 
- \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W^'}{\partial z} \biggr) 
- \frac{m W^' \bar\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{\varpi D} \cancelto{0}{\biggl( \frac{dj_0 }{d\varpi } \biggr)} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[  \rho_0 \varpi \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W^'}{\partial z} \biggr) 
-  \frac{\rho_0 m^2  W^' }{\varpi^2} 
+ \frac{ {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } \, .
</math>
  </td>
</tr>
</table>
</div>

As can be confirmed by comparing it to equation (12) of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image &#8212; this expression matches the 2<sup>nd</sup>-order, two-dimensional PDE that defines the eigenvalue problem discussed by Kojima.

<div align="center" id="EigenvalueKojima86">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equations (12) &amp; (13) extracted without modification from p. 254 of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)]<p></p>
"''The Dynamical Stability of a Fat Disk with Constant Specific Angular Momentum''"<p></p>
Progress of Theoretical Physics, vol. 75, pp. 251-261 &copy; The Physical Society of Japan
</td></tr>
<tr>
  <td align="center">
[[File:Kojima86Eq12.png|500px|center|Kojima (1986, Progress of Theoretical Physics, 75, 251)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

This expression also serves as the starting point for the stability analysis presented by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; see his equation (3.1), but note that he has replaced the adiabatic exponent with the polytropic index via the relation, <math>~\gamma = (n+1)/n</math>.

====Equivalent Dimensionless Expression====

Now, as should be clear from [[SR#Barotropic_Structure|our introductory description of barotropic structures]], because the initial, unperturbed [[Apps/PapaloizouPringleTori#Solution|Papaloizou-Pringle torus]] is a polytropic configuration, the functions <math>~P_0(\varpi,z)</math> and <math>~\rho_0(\varpi,z)</math> can both be expressed in terms of the [[Appendix/Ramblings/PPToriPt1A#Equilibrium_Configuration|dimensionless enthalpy distribution]],

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

<tr>
  <td align="right">
<math>~\Theta_H(\varpi,z) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{\beta^2}\biggl[ \chi^{-2} 
- 2 ( \chi^2 + \zeta^2 )^{-1/2} + 1 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~\chi \equiv \frac{\varpi}{\varpi_0} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\zeta \equiv \frac{z}{\varpi_0} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\beta^2 \equiv \frac{2n}{\mathfrak{M}_0^2} \, ,</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~</math>
  </td>
</tr>

</table>
</div>
and <math>~\mathfrak{M}_0</math> is the Mach number of the circular, azimuthal flow at the pressure and density maximum.  Specifically (see also equation 1.1 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]),

<div align="center">
<math>~P_0 = P_\mathrm{max} \Theta_H^{n+1} </math> 
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\rho_0 = \rho_\mathrm{max} \Theta_H^{n} \, .</math> 
</div>

Making these state-variable substitutions in the PDE that we have just presented for comparison with Kojima's work, we have,

<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>~
\frac{\rho_\mathrm{max}}{\varpi} \frac{\partial}{\partial\varpi} \biggl[  \Theta_H^n \varpi \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
+ \rho_\mathrm{max}\frac{\partial}{\partial z} \biggl(\Theta_H^n \frac{\partial W^'}{\partial z} \biggr) 
+ \rho_\mathrm{max}\biggl\{ \frac{ n{\bar\sigma}^2  \Theta_H^{(n-1)} \rho_\mathrm{max} }{(n+1) P_\mathrm{max}  } 
-  \frac{\Theta_H^n m^2 }{\varpi^2} \biggr\} W^' \, .
</math>
  </td>
</tr>
</table>
</div>

In an effort to make this entire expression dimensionless, let's define a dimensionless enthalpy perturbation via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W</math>
  </td>
  <td align="center">
<math>~\equiv \biggl[ \frac{\Omega_0 \rho_\mathrm{max}}{P_\mathrm{max}} \biggr]W^' \, ,</math>
  </td>
  <td align="left">
<math>~</math>
  </td>
</tr>
</table>
</div>
and multiply our expression through by <math>~(\varpi_0^2 \Omega_0/P_\mathrm{max})</math>.  This gives,

<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>~
\frac{1}{\chi} \frac{\partial}{\partial\chi} \biggl[  \Theta_H^n \chi \cdot \frac{\partial (\delta W) }{\partial \chi} \biggr]
+ \frac{\partial}{\partial \zeta} \biggl[ \Theta_H^n \frac{\partial (\delta W) }{\partial \zeta} \biggr] 
+ \biggl[ 
\Theta_H^{n-1} \biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2
\frac{ n\Omega_0^2 \rho_\mathrm{max} \varpi_0^2}{(n+1)P_\mathrm{max}  }-  \frac{\Theta_H^n m^2 }{\chi^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H^n \cdot \frac{\partial^2 (\delta W) }{\partial \chi^2} 
+\Theta_H^n \cdot  \frac{\partial^2 (\delta W) }{\partial \zeta^2} 
+ \biggl[ \Theta_H^{n-1} \biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2 \frac{2n}{\beta^2} -  \frac{\Theta_H^n m^2 }{\chi^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\frac{1}{\chi} \frac{\partial (\delta W) }{\partial \chi} \biggl[\Theta_H^n + n\chi \Theta_H^{n-1} \frac{\partial \Theta_H}{\partial\chi} \biggr]
+ n \Theta_H^{n-1} ~\frac{\partial (\delta W) }{\partial \zeta} \frac{\partial \Theta_H}{\partial \zeta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Theta_H^{n-1} \biggl\{
\Theta_H \cdot \frac{\partial^2 (\delta W) }{\partial \chi^2} 
+\Theta_H \cdot  \frac{\partial^2 (\delta W) }{\partial \zeta^2} 
+ \biggl[\frac{\Theta_H}{ \chi } + n \frac{\partial \Theta_H}{\partial\chi} \biggr]\frac{\partial (\delta W) }{\partial \chi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ n \frac{\partial \Theta_H}{\partial \zeta} \biggr] \frac{\partial (\delta W) }{\partial \zeta} 
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{\chi^2} \biggr]\delta W 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

[It appears as though I'm on the right track because this expression is very similar to equation (3.2) of Blaes85!]

====Change to Off-Axis Polar-Coordinate System====
In his effort to derive an ''analytic'' solution to this eigenvalue problem, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] adopted a different meridional-plane coordinate system.  As is illustrated in his Figure 1, Blaes shifted from the (dimensionless) rectilinear <math>~(\chi,\zeta)</math> system to a (dimensionless) polar-coordinate <math>~(x,\theta)</math> system whose origin sits at the pressure-maximum of the initial, unperturbed Papaloizou-Pringle torus.  Mapping between these two coordinate systems is accomplished via the relations (see equation 2.1 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]),

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x^2 = (1-\chi)^2 + \zeta^2</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\theta = \tan^{-1}\biggl[\frac{\zeta}{1-\chi}\biggr] \, ;</math>
  </td>
</tr>
<tr><td align="center" colspan="3">&nbsp; &nbsp; or &nbsp; &nbsp; &nbsp; &nbsp;</td></tr>
<tr>
  <td align="right">
<math>~\chi = 1 - x\cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\zeta = x\sin\theta \, .</math>
  </td>
</tr>
</table>
</div>

Mapping of partial derivatives is accomplished via the relations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial\chi}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
- \cos\theta \cdot \frac{\partial}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial}{\partial\theta} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial\zeta}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
\sin\theta \cdot \frac{\partial}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial}{\partial\theta} \, .
</math>
  </td>
</tr>
</table>
</div>

This means, for example, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial^2(\delta W)}{\partial \chi^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial\chi}\biggl[  
- \cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\cos\theta \cdot\frac{\partial}{\partial x}\biggl[  
- \cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr] +
\frac{\sin\theta}{x} \cdot \frac{\partial}{\partial\theta}\biggl[  
- \cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\cos\theta \biggl[  
- \cos\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
-\frac{\sin\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\sin\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta}
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\frac{\sin\theta}{x} \biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
- \cos\theta \cdot \frac{\partial^2(\delta W)}{\partial\theta~\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\sin\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[  
\cos^2\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{\sin\theta\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
-\frac{\sin\theta\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta}
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[  
\frac{\sin^2\theta}{x} \cdot \frac{\partial(\delta W)}{\partial x}
- \frac{\sin\theta\cos\theta}{x} \cdot \frac{\partial^2(\delta W)}{\partial\theta~\partial x}
+\frac{\sin\theta\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\sin^2\theta}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos^2\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{\sin^2\theta}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \frac{\sin^2\theta}{x} \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{2\sin\theta\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
-\frac{2\sin\theta\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta} \, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{\partial^2(\delta W)}{\partial \zeta^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial\zeta}\biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sin\theta \cdot \frac{\partial}{\partial x}\biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr] +
\frac{\cos\theta}{x} \cdot \frac{\partial}{\partial\theta}\biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sin\theta \biggl[  
\sin\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
-\frac{\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta}
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\frac{\cos\theta}{x} \biggl[  
\cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\sin\theta \cdot \frac{\partial^2(\delta W)}{\partial\theta~\partial x}
-\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\sin^2\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{\cos^2\theta}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{\cos^2\theta}{x}  \cdot \frac{\partial(\delta W)}{\partial x}
-\frac{2\sin\theta \cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{2\sin\theta \cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta} \, .
</math>
  </td>
</tr>
</table>
</div>

A significant amount of simplification occurs when these two expressions are added.  Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial^2(\delta W)}{\partial \chi^2}
+\frac{\partial^2(\delta W)}{\partial \zeta^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{1}{x}  \cdot \frac{\partial(\delta W)}{\partial x} \, .
</math>
  </td>
</tr>
</table>
</div>

With this as a start, a coordinate mapping of our above-derived dimensionless "eigenvalue problem" expression gives,
<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>~
\Theta_H \biggl[ \frac{\partial^2 (\delta W) }{\partial \chi^2} + \frac{\partial^2 (\delta W) }{\partial \zeta^2} \biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{\chi^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{\Theta_H}{ \chi } + n \frac{\partial \Theta_H}{\partial\chi} \biggr]\frac{\partial (\delta W) }{\partial \chi} 
+ \biggl[ n \frac{\partial \Theta_H}{\partial \zeta} \biggr] \frac{\partial (\delta W) }{\partial \zeta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H \biggl[ \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{1}{x}  \cdot \frac{\partial(\delta W)}{\partial x}  \biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{\frac{\Theta_H}{ \chi } + n \biggl[ - \cos\theta \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr]\biggr\} 
\biggl[- \cos\theta \cdot \frac{\partial (\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial (\delta W)}{\partial\theta} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ n\biggl[ \sin\theta \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial\Theta_H}{\partial\theta} \biggr] \biggl[ \sin\theta \cdot \frac{\partial (\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial (\delta W)}{\partial\theta} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H \biggl[ \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{1}{x}  \cdot \frac{\partial(\delta W)}{\partial x}  \biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{- \frac{\cos\theta\cdot\Theta_H}{ (1-x\cos\theta)} + n \biggl[ \cos^2\theta \cdot \frac{\partial\Theta_H}{\partial x}
-\frac{\sin\theta \cos\theta}{x}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr]\biggr\} 
\cdot \frac{\partial (\delta W)}{\partial x}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ n\biggl[ \sin^2\theta \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\sin\theta\cos\theta}{x}\cdot \frac{\partial\Theta_H}{\partial\theta} \biggr] \cdot \frac{\partial (\delta W)}{\partial x}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{\frac{\sin\theta\cdot\Theta_H}{ x(1-x\cos\theta) } + n \biggl[ - \frac{\sin\theta\cos\theta}{x} \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\sin^2\theta}{x^2}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr]\biggr\} 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ n\biggl[ \frac{\sin\theta\cos\theta}{x} \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\cos^2\theta}{x^2}\cdot \frac{\partial\Theta_H}{\partial\theta} \biggr] \cdot \frac{\partial (\delta W)}{\partial\theta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H \biggl[ \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}\biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{\frac{\Theta_H}{x}- \frac{\cos\theta\cdot\Theta_H}{ (1-x\cos\theta)} + n \biggl[ \frac{\partial\Theta_H}{\partial x}
\biggr]\biggr\} 
\cdot \frac{\partial (\delta W)}{\partial x}
+ \biggl[\frac{\sin\theta\cdot\Theta_H}{ x(1-x\cos\theta) } + 
\frac{n}{x^2}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} \, .
</math>
  </td>
</tr>

</table>
</div>

Finally, after multiplying through by <math>~x^2</math> and rearranging terms, we obtain,

<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>~
\Theta_H x^2\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\Theta_H \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{\Theta_H x \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] + nx^2 \cdot \frac{\partial\Theta_H}{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{\Theta_Hx\sin\theta}{ (1-x\cos\theta) } + 
n\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2}{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 x^2\Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W \, .
</math>
  </td>
</tr>

</table>
</div>

Note that, because [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] is only considering tori with a uniform specific angular momentum distribution, <math>~q=2</math>, the frequency ratio, <math>~(\bar\sigma/\Omega_0)</math>, may be rewritten as,

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

<tr>
  <td align="right">
<math>~\frac{\bar\sigma}{\Omega_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sigma + m{\dot\varphi}_0(\varpi)}{\Omega_0} = \frac{\sigma}{\Omega_0} + m\chi^{-q}
= \frac{\sigma}{\Omega_0} + \frac{m}{(1-x\cos\theta)^2} \, .</math>
  </td>
</tr>
</table>
</div>

As can be confirmed by comparing it to equation (3.2) of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image &#8212; our just-derived PDE matches the 2<sup>nd</sup>-order, two-dimensional PDE that defines the ''dimensionless'' eigenvalue problem discussed by Blaes (1985).

<div align="center" id="EigenvalueBlaes85">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 216, Issue 3, October 1985, pp. 553-563<br />&copy; Royal Astronomical Society<br />
https://doi.org/10.1093/mnras/216.3.553
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq3.2.png|600px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
<tr>
  <td align="left">
This digital image has been posted by permission of author, O.M. Blaes (via email dated 19 July 2020), and by permission of Oxford University Press on behalf of the Royal Astronomical Society (via email dated 31 July 2020). 
<div align="center">Original publication website:  https://academic.oup.com/mnras/article/216/3/553/1005987</div>
  </td>
</tr>
</table>
</div>

<span id="isolatingBlaes85">Isolating on the right-hand side terms that explicitly involve the dimensionless eigenfrequency, <math>~\nu \equiv (\sigma/\Omega_0)</math>, this governing PDE may also be written in the form (see equations 4.1 and 4.2 of Blaes85),</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{L} (\delta W)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2n(1-\Theta_H)(M\nu^2 + N\nu)(\delta W) \, ,</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~\hat{L} (\delta W)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\Theta_H x^2\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\Theta_H \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{\Theta_H x \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] + nx^2 \cdot \frac{\partial\Theta_H}{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{\Theta_Hx\sin\theta}{ (1-x\cos\theta) } + 
n\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2 m^2}{\beta^2(1-x\cos\theta)^4}  -  \frac{m^2 x^2\Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{x^2}{(1-\Theta_H)\beta^2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~N</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2mx^2}{(1-\Theta_H)\beta^2(1-x\cos\theta)^2} \, .</math>
  </td>
</tr>

</table>
</div>

====Introduce Coordinate-Parameter &eta;====

Following the lead of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], and without loss of generality, we can everywhere replace the dimensionless function representing the unperturbed equilibrium enthalpy distribution,  <math>~\Theta_H</math> &#8212; that varies from unity at the cross-sectional center of the torus to zero at the torus surface &#8212; with the parameter,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 - \Theta_H \, ,</math>
  </td>
</tr>
</table>
</div>
that varies from zero at the (cross-sectional) center to unity at the surface.  Making this substitution, our governing PDE 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>~
(1-\eta^2) x^2\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+(1-\eta^2) \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{(1-\eta^2) x \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] + nx^2 \cdot \frac{\partial(1-\eta^2) }{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{(1-\eta^2) x\sin\theta}{ (1-x\cos\theta) } + 
n\cdot \frac{\partial (1-\eta^2) }{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2}{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 x^2(1-\eta^2) }{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^2 (1-\eta^2)\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+ (1-\eta^2) \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{x (1-\eta^2) \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] -2 nx^2 \eta \cdot \frac{\partial \eta }{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{(1-\eta^2) x\sin\theta}{ (1-x\cos\theta) } -2n\eta \cdot \frac{\partial \eta }{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2}{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 x^2(1-\eta^2) }{(1-x\cos\theta)^2} \biggr]\delta W \, .
</math>
  </td>
</tr>
</table>
</div>

===Slender Torus Approximation===

====Blaes85====

Generally, in Papaloizou-Pringle tori, the equilibrium enthalpy distribution is a function of both <math>~x</math> and <math>~\theta</math>, hence also, <math>~\eta = \eta(x,\theta)</math>.  However, as [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] explains &#8212; see the discussion associated with his equation (2.6) &#8212; to lowest order in <math>~x</math>, 
<div align="center">
<math>~\eta \approx \frac{x}{\beta} \, ,</math>
</div>
and the function <math>~\eta</math> has no dependence on <math>~\theta</math>.  Hence, near the (cross-sectional) center of each torus, we can make the substitutions,
<div align="center">
<math>\frac{\partial \eta}{\partial \theta} = 0 \, ,</math>
&nbsp;  &nbsp; &nbsp; &nbsp; and &nbsp;  &nbsp; &nbsp; &nbsp;
<math>~x = \beta\eta </math> 
&nbsp;  &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp;  &nbsp; &nbsp; &nbsp;
<math>\frac{\partial}{\partial x} = \frac{1}{\beta}\cdot\frac{\partial}{\partial \eta} \, ,</math>
</div>

and our latest PDE expression becomes,

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\beta\eta\frac{(1-\eta^2)\sin\theta}{ (1-\beta\eta\cos\theta) } \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl\{ 2n\eta^2 \biggl[   \frac{\sigma}{\Omega_0} + \frac{m}{(1-\beta\eta\cos\theta)^2}  \biggr]^2  -  \frac{m^2 \beta^2 \eta^2(1-\eta^2) }{(1-\beta\eta\cos\theta)^2} \biggr\}\delta W \, .
</math>
  </td>
</tr>
</table>
</div>

Finally, the slim-torus approximation results from setting <math>~\beta = 0</math>, in which case the eigenvalue problem is defined by the expression,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~\approx</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, quoting Blaes (1985), "<font color="darkgreen">the superscript (0) denotes the infinitely slender limit.</font>"   As can be confirmed by comparing it to equation (1.6) of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image &#8212; our just-derived PDE matches the 2<sup>nd</sup>-order, two-dimensional PDE that defines the ''dimensionless'' eigenvalue problem in the ''slender torus approximation''.

<div align="center" id="EigenvalueBlaes85">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equation (1.6) &#8212; identical to Eq. (3.5) &#8212; extracted without modification from p. 555 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 216, Issue 3, October 1985, pp. 553-563<br />&copy; Royal Astronomical Society<br />
https://doi.org/10.1093/mnras/216.3.553
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq1.6.png|600px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
<tr>
  <td align="left">
This digital image has been posted by permission of author, O.M. Blaes (via email dated 19 July 2020), and by permission of Oxford University Press on behalf of the Royal Astronomical Society (via email dated 31 July 2020). 
<div align="center">Original publication website:  https://academic.oup.com/mnras/article/216/3/553/1005987</div>
  </td>
</tr>
</table>
</div>

In a [[Apps/Blaes85SlimLimit|separate chapter]], we dissect the Blaes85 assertion that Jacobi Polynomials provide an analytic eigenvector solution to this specific "slim torus" eigenvalue problem.

====GGN86====

The discussion in this subsection builds on our [[#Seminal_Work_by_Papaloizou_.26_Pringle|above derivation]] of equation (3.18) in PP84 and equation (2.19) in PP85.  Following along the lines of the variable substitution that [[#Preamble|was made earlier in our "preamble"]], let's replace <math>~W^'</math> with <math>~Q_{JT} \equiv \bar\sigma W^'</math> throughout this expression.  As was also [[#Preamble|pointed out above]], because <math>~\bar\sigma</math> is a function of the radial coordinate, we note that,

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

<tr>
  <td align="right">
<math>~\frac{\partial \bar\sigma}{\partial\varpi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
m \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \, ,
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{\partial W^'}{\partial\varpi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\bar\sigma}^{-2} \biggl[ {\bar\sigma} \frac{\partial  Q_{JT} }{\partial \varpi}
- m Q_{JT} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Making these substitutions in equation (2.19) of PP85, we have,

<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>~
\frac{ {\bar\sigma}^2  \rho_0^2 Q_{JT} }{\gamma P_0 \bar\sigma } 
+ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi }{D} \cdot \biggl[ {\bar\sigma} \frac{\partial  Q_{JT} }{\partial \varpi}
- m Q_{JT} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr]\biggr\}
- \frac{\rho_0 m^2 {\bar\sigma} Q_{JT}  }{\varpi^2 D} 
+ \frac{1}{\bar\sigma}\frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial Q_{JT} }{\partial z} \biggr) 
+ \frac{m Q_{JT} }{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{\varpi \kappa^2 }{2 {\dot\varphi}_0 } \biggr)\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Then, multiplying through by <math>~\bar\sigma</math> and rearranging terms, gives,

<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>~\biggl\{
\frac{ {\bar\sigma}^2  \rho_0^2 }{\gamma P_0} 
- \frac{{\bar\sigma}^2\rho_0  }{D} \biggl( \frac{m}{\varpi} \biggr)^2
+ \bar\sigma\biggl(\frac{m  }{\varpi} \biggr) \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{ D} \biggl( \frac{ \kappa^2 }{2 {\dot\varphi}_0 } \biggr)\biggr] 
\biggr\}Q_{JT} ~
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial Q_{JT} }{\partial z} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \frac{m\bar\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi Q_{JT} }{D} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr\}
+~ \frac{\bar\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\bar\sigma \rho_0 \varpi }{D} \cdot \biggl[ \frac{\partial  Q_{JT} }{\partial \varpi}\biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
\frac{ {\bar\sigma}^2  \rho_0^2 }{\gamma P_0} 
- \frac{{\bar\sigma}^2\rho_0  }{D} \biggl( \frac{m}{\varpi} \biggr)^2
+ \bar\sigma\biggl(\frac{m  }{\varpi} \biggr) \frac{\rho_0}{ D} \frac{\partial}{\partial\varpi} \biggl[ 2{\dot\varphi}_0 +\varpi \frac{\partial \dot\varphi_0}{\partial\varpi} \biggr] 
+ \bar\sigma\biggl(\frac{m  }{\varpi} \biggr) \biggl[ 2{\dot\varphi}_0 +\varpi \frac{\partial \dot\varphi_0}{\partial\varpi} \biggr] \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{ D} \biggr] 
\biggr\}Q_{JT} ~
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial Q_{JT} }{\partial z} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \frac{m\bar\sigma \rho_0 }{D} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \frac{\partial  Q_{JT}}{\partial\varpi} 
-~ \frac{m\bar\sigma Q_{JT} }{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi }{D} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr\}
+~ {\bar\sigma}^2 \biggl(\frac{\partial  Q_{JT} }{\partial \varpi}\biggr) \frac{\partial}{\partial\varpi} \biggl(\frac{\rho_0 }{D}\biggr)
+~ \frac{ {\bar\sigma}^2\rho_0 }{D} \biggl(\frac{\partial^2  Q_{JT} }{\partial \varpi^2}\biggr)
+~ \frac{\bar\sigma}{\varpi}\frac{\rho_0 }{D} \cdot \frac{\partial  Q_{JT} }{\partial \varpi} \biggl[\bar\sigma + m\varpi \biggl( \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
\frac{ {\bar\sigma}^2  \rho_0^2 }{\gamma P_0} 
- \frac{{\bar\sigma}^2\rho_0  }{D} \biggl( \frac{m}{\varpi} \biggr)^2
+ \bar\sigma\biggl(\frac{m  }{\varpi} \biggr) \frac{\rho_0}{ D} \frac{\partial}{\partial\varpi} \biggl[ 2{\dot\varphi}_0 +\varpi \frac{\partial \dot\varphi_0}{\partial\varpi} \biggr] 
+ \bar\sigma\biggl(\frac{m  }{\varpi} \biggr) \biggl[ 2{\dot\varphi}_0 +\varpi \frac{\partial \dot\varphi_0}{\partial\varpi} \biggr] \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{ D} \biggr] 
-~ \frac{\bar\sigma }{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \biggl(\frac{\rho_0 }{D}\biggr)m\varpi \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr]
\biggr\}Q_{JT} ~
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ \biggl\{
-~ \frac{m\bar\sigma \rho_0 }{D} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) 
+~ {\bar\sigma}^2  \frac{\partial}{\partial\varpi} \biggl(\frac{\rho_0 }{D}\biggr)
+~ \frac{\bar\sigma}{\varpi}\frac{\rho_0 }{D} \biggl[\bar\sigma + m\varpi \biggl( \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \biggr]
\biggr\}\frac{\partial  Q_{JT}}{\partial\varpi} 
+~ \frac{ {\bar\sigma}^2\rho_0 }{D} \biggl(\frac{\partial^2  Q_{JT} }{\partial \varpi^2}\biggr)
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial Q_{JT} }{\partial z} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
\frac{ {\bar\sigma}^2  \rho_0^2 }{\gamma P_0} 
- \frac{{\bar\sigma}^2\rho_0  }{D} \biggl( \frac{m}{\varpi} \biggr)^2
+ \bar\sigma\biggl(\frac{m  }{\varpi} \biggr) \biggl[ 2{\dot\varphi}_0 \biggr] \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{ D} \biggr] 
\biggr\}Q_{JT} ~
+ {\bar\sigma}^2  \frac{\partial}{\partial\varpi} \biggl(\frac{\rho_0 }{D}\biggr) \frac{\partial  Q_{JT}}{\partial\varpi} +~ \frac{ {\bar\sigma}^2\rho_0 }{D} \biggl(\frac{\partial^2  Q_{JT} }{\partial \varpi^2}\biggr)
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial Q_{JT} }{\partial z} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\bar\sigma\biggl(\frac{m  }{\varpi} \biggr) \biggl\{
\frac{\rho_0}{ D} \frac{\partial}{\partial\varpi} \biggl[ 2{\dot\varphi}_0 \biggr] 
+\frac{\rho_0}{ D} \frac{\partial}{\partial\varpi} \biggl[ \varpi \frac{\partial \dot\varphi_0}{\partial\varpi} \biggr] 
+ \biggl[ \varpi \frac{\partial \dot\varphi_0}{\partial\varpi} \biggr] \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{ D} \biggr] 
-~ \frac{\partial}{\partial\varpi} \biggl[ \biggl(\frac{\rho_0 }{D}\biggr)\varpi \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr]
\biggr\}Q_{JT} ~
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ \biggl\{
\frac{\bar\sigma^2\rho_0}{\varpi D}
+ \frac{m\bar\sigma \rho_0}{D} \biggl( \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) 
-~ \frac{m\bar\sigma \rho_0 }{D} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) 
\biggr\}\frac{\partial  Q_{JT}}{\partial\varpi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\frac{ {\bar\sigma}^2  \rho_0^2 }{\gamma P_0} 
- \frac{{\bar\sigma}^2\rho_0  }{D} \biggl( \frac{m}{\varpi} \biggr)^2
+ 2 \bar\sigma {\dot\varphi}_0\biggl(\frac{m  }{\varpi} \biggr) \frac{\partial}{\partial\varpi} \biggl( \frac{\rho_0}{ D} \biggr)
\biggr] Q_{JT} ~
+ {\bar\sigma}^2  \frac{\partial}{\partial\varpi} \biggl(\frac{\rho_0 }{D}\biggr) \frac{\partial  Q_{JT}}{\partial\varpi} +~ \frac{ {\bar\sigma}^2\rho_0 }{D} \biggl(\frac{\partial^2  Q_{JT} }{\partial \varpi^2}\biggr)
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial Q_{JT} }{\partial z} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\biggl(\frac{\bar\sigma}{\varpi} \cdot \frac{\rho_0}{D}\biggr) \biggl\{ Q_{JT}  \cdot
\frac{\partial (2m {\dot\varphi}_0 )}{\partial\varpi} ~
+ \bar\sigma \cdot \frac{\partial  Q_{JT}}{\partial\varpi} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>


Each term in the first line of this two-line expression can be found in equation (2.27) of [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman &amp; Narayan (1986)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image.  But notice that the sign on a number of the terms is flipped.  Notice, as well, that neither one of the terms in the second line of our expression appears in equation (2.27) of GGN86.

<div align="center" id="EigenvalueGGN86">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equations (2.27) and (2.23) extracted without modification from p. 343 of [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman &amp; Narayan (1986)]<p></p>
"''The stability of accretion tori. I - Long-wavelength modes of slender tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 221, pp. 339-364 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:GGN86Eq2.23.png|500px|center|Goldreich, Goodman &amp; Narayan (1986, MNRAS, 221, 339)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

It appears as though the sign on various terms is flipped because GGN86 adopted different sign conventions from Papaloizou &amp; Pringle.  Specifically, as is made clear by equation (2.23) of GGN86, 
<div align="center">
<math>~D~~~\rightarrow ~~~ -D_\mathrm{GGN} \, .</math>
</div>
Furthermore, as was [[#Direct_Comparison_of_Derived_Equations|deduced above when we compared expressions for the components of the perturbed velocity]],
<div align="center">
<math>~\bar\sigma~~~\rightarrow ~~~ -\sigma_\mathrm{GGN} \, .</math>
</div>

Making this pair of substitutions, our derived two-line expression 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>~\biggl[
\frac{ \sigma_\mathrm{GGN}^2  \rho_0^2 }{\gamma P_0} 
+ \frac{\sigma_\mathrm{GGN}^2\rho_0  }{D_\mathrm{GGN}} \biggl( \frac{m}{\varpi} \biggr)^2
+ 2 \sigma_\mathrm{GGN}{\dot\varphi}_0\biggl(\frac{m  }{\varpi} \biggr) \frac{\partial}{\partial\varpi} \biggl( \frac{\rho_0}{ D_\mathrm{GGN}} \biggr)
\biggr] Q_{JT} ~
- \sigma_\mathrm{GGN}^2  \frac{\partial}{\partial\varpi} \biggl(\frac{\rho_0 }{D_\mathrm{GGN}}\biggr) \frac{\partial  Q_{JT}}{\partial\varpi} 
-~ \frac{ \sigma_\mathrm{GGN}^2\rho_0 }{D_\mathrm{GGN}} \biggl(\frac{\partial^2  Q_{JT} }{\partial \varpi^2}\biggr)
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial Q_{JT} }{\partial z} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\biggl(\frac{\sigma_\mathrm{GGN}}{\varpi} \cdot \frac{\rho_0}{D_\mathrm{GGN}}\biggr) \biggl\{ Q_{JT}  \cdot
\frac{\partial (2m {\dot\varphi}_0 )}{\partial\varpi} ~
- \sigma_\mathrm{GGN} \cdot \frac{\partial  Q_{JT}}{\partial\varpi} \biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>

and the first line appears to exactly match equation (2.27) from GGN86.  In an effort to explain why the pair of terms in the second line of our expression do not appear in the GGN86 equation, we quote directly from the text that immediately follows equation (2.27) in [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman &amp; Narayan (1986)]:  &nbsp; "<font color="darkgreen">This is very similar to equation (2.19) for <math>~W = Q/\sigma</math> derived in PPII.  The few small differences can be traced to our neglect of the azimuthal curvature of the torus through our Cartesian approximation.  These terms are not important and so we expect equation (2.27) to capture all the essential physics of the narrow torus.</font>"