User:Tohline/Apps/Blaes85SlimLimit
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,
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<math>~0</math> |
<math>~=</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. Making the substitution,
<math>~\delta W^{(0)} ~\rightarrow~ V(\eta) \exp (ik\theta) \, ,</math>
this governing equation — now, a one-dimensional, 2nd-order ODE — becomes,
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<math>~0</math> |
<math>~=</math> |
<math>~ \eta^2 (1-\eta^2)\cdot \frac{d^2V}{d \eta^2} - k^2(1-\eta^2) V + \biggl[ \eta (1-\eta^2) -2 n \eta^3 \biggr] \cdot \frac{d V}{d \eta} + 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 V \, . </math> |
Making the additional substitution,
<math>~V ~\rightarrow~ \eta^{|k|} \Upsilon(\eta) \, ,</math>
and appreciating that,
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<math>~\frac{dV}{d\eta}</math> |
<math>~=</math> |
<math>~|k|\eta^{|k|-1} \Upsilon + \eta^{|k|} \frac{d\Upsilon}{d\eta} \, ,</math> |
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<math>~\frac{d^2V}{d\eta^2}</math> |
<math>~=</math> |
<math>~ |k|[|k|-1] \eta^{|k|-2}\Upsilon + 2|k|\eta^{|k|-1} \frac{d\Upsilon}{d\eta} + \eta^{|k|} \frac{d^2\Upsilon}{d\eta^2}\, ,</math> |
the governing ODE becomes,
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<math>~ \biggl\{k^2(1-\eta^2) - 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2\biggr\} \eta^{|k|}\Upsilon </math> |
<math>~=</math> |
<math>~ \eta^2 (1-\eta^2)\cdot \biggl[ |k|[|k|-1] \eta^{|k|-2}\Upsilon + 2|k|\eta^{|k|-1} \frac{d\Upsilon}{d\eta} + \eta^{|k|} \frac{d^2\Upsilon}{d\eta^2} \biggr] + \biggl[ \eta (1-\eta^2) -2 n \eta^3 \biggr] \cdot \biggl[ |k|\eta^{|k|-1} \Upsilon + \eta^{|k|} \frac{d\Upsilon}{d\eta} \biggr] </math> |
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<math>~=</math> |
<math>~(1-\eta^2) \biggl[ |k|[|k|-1] \eta^{|k|}\Upsilon + 2|k|\eta^{|k|+1} \frac{d\Upsilon}{d\eta} + \eta^{|k|+2} \frac{d^2\Upsilon}{d\eta^2}\biggr] + \biggl[ (1-\eta^2) -2 n \eta^2 \biggr] \cdot \biggl[ |k|\eta^{|k|} \Upsilon + \eta^{|k|+1} \frac{d\Upsilon}{d\eta} \biggr] </math> |
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<math>~=</math> |
<math>~\eta^{|k|}(1-\eta^2) \biggl[ k^2 \Upsilon + (2|k|+1)\eta \frac{d\Upsilon}{d\eta} + \eta^{2} \frac{d^2\Upsilon}{d\eta^2} \biggr] - \eta^{|k|}\biggl[ 2 n \eta^2 \biggr] \cdot \biggl[ |k| \Upsilon + \eta \frac{d\Upsilon}{d\eta} \biggr] </math> |
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<math>~\Rightarrow~~~ - 2n\eta^2 \biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math> |
<math>~=</math> |
<math>~(1-\eta^2) \biggl[ \eta^{2} \frac{d^2\Upsilon}{d\eta^2} + (2|k|+1)\eta \frac{d\Upsilon}{d\eta} \biggr] - \biggl[ 2 n \eta^3 \frac{d\Upsilon}{d\eta} \biggr] \, . </math> |
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Finally, then, making the independent variable substitution,
<math>~\eta^2 ~\rightarrow ~ y</math> <math>~\Rightarrow</math> <math>~dy = 2\eta d\eta</math> <math>~\Rightarrow</math> <math> ~~ </math> and <math>\frac{d^2}{d\eta^2} ~\rightarrow~ 2y^{1/2}\frac{d}{dy}</math>
in which case,
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<math>~\frac{d}{d\eta}</math> |
<math>~\rightarrow</math> |
<math>~2y^{1/2}\frac{d}{dy}</math> |
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<math>~\frac{d^2}{d\eta^2}</math> |
<math>~\rightarrow</math> |
<math>~4y^{1/2}\frac{d}{dy} + 4y^{3/2}\frac{d^2}{dy^2}</math> |
and,
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<math>~ - 2ny \biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math> |
<math>~=</math> |
<math>~ (1-y)y \biggl[ 4y^{1/2}\frac{d\Upsilon}{dy} + 4y^{3/2}\frac{d^2\Upsilon}{dy^2}\biggr] + 2(2|k|+1)(1-y)y \frac{d\Upsilon}{dy} - \biggl[ 4 n y^{2} \frac{d\Upsilon}{dy} \biggr] </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |