Editing Apps/PapaloizouPringle84 (section)

====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>, 
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<math>~\eta \approx \frac{x}{\beta} \, ,</math>
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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,
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<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>
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and our latest PDE expression becomes,

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

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

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<math>~0</math>
  </td>
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<math>~\approx</math>
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<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>
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</table>
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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''.

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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
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[[File:Blaes85Eq1.6.png|600px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
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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>
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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.