Difference between revisions of "User:Tohline/Apps/Blaes85SlimLimit"
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= | =Stability of PP Tori in the Slim Torus Limit= | ||
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== | ==Statement of the Eigenvalue Problem== | ||
Here, we build on [[User:Tohline/Apps/PapaloizouPringle84#Nonaxisymmetric_Instability_in_Papaloizou-Pringle_Tori|our discussion in an accompanying chapter]] in which five published analyses of nonaxisymmetric instabilities in Papaloizou-Pringle tori were reviewed: 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 — [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou & 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 & Narayan (1986)]. Following the lead of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes] (1985; hereafter Blaes85), in particular, we have shown that the relevant eigenvalue problem is defined by the following 2<sup>nd</sup>-order PDE, | |||
<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, <math>~\delta W^{(0)}</math> is the dimensionless enthalpy perturbation. | |||
Revision as of 21:21, 3 May 2016
Stability of PP Tori in the Slim Torus Limit
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Statement of the Eigenvalue Problem
Here, we build on our discussion in an accompanying chapter in which five published analyses of nonaxisymmetric instabilities in Papaloizou-Pringle tori were reviewed: The discovery paper, PP84, and papers by four separate groups that were published within a couple of years of the discovery paper — Papaloizou & Pringle (1985), Blaes (1985), Kojima (1986), and Goldreich, Goodman & Narayan (1986). Following the lead of Blaes (1985; hereafter Blaes85), in particular, we have shown that the relevant eigenvalue problem is defined by the following 2nd-order PDE,
|
<math>~0</math> |
<math>~\approx</math> |
<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> |
where, <math>~\delta W^{(0)}</math> is the dimensionless enthalpy perturbation.
See Also
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© 2014 - 2021 by Joel E. Tohline |