Editing Apps/PapaloizouPringle84 (section)

===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>"