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> |
Finally, then, making the independent variable substitution,
<math>~\eta^2 ~\rightarrow ~ y</math> <math>~\Rightarrow</math> <math>~dy = 2\eta d\eta</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>~2\frac{d}{dy} + 4y\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 \frac{d^2\Upsilon}{d\eta^2} + (2|k|+1)(1-y)y^{1/2} \frac{d\Upsilon}{d\eta} - 2 n y^{3/2} \frac{d\Upsilon}{d\eta} </math> |
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<math>~=</math> |
<math>~ 4(1-y)y^2 \frac{d^2\Upsilon}{dy^2} + 2(1-y) y \frac{d\Upsilon}{dy} + 2(2|k|+1)(1-y)y \frac{d\Upsilon}{dy} - 4 n y^{2} \frac{d\Upsilon}{dy} </math> |
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<math>~\Rightarrow~~~~ - \frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math> |
<math>~=</math> |
<math>~ (1-y)y \frac{d^2\Upsilon}{dy^2} + \frac{1}{2}(1-y) \frac{d\Upsilon}{dy} + \frac{1}{2}(2|k|+1)(1-y)\frac{d\Upsilon}{dy} - n y \frac{d\Upsilon}{dy} </math> |
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<math>~=</math> |
<math>~ (1-y)y \frac{d^2\Upsilon}{dy^2} + (|k|+1)(1-y)\frac{d\Upsilon}{dy} - n y \frac{d\Upsilon}{dy} </math> |
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<math>~=</math> |
<math>~ (1-y)y \frac{d^2\Upsilon}{dy^2} + (|k|+1)\frac{d\Upsilon}{dy} -y (|k|+1+n)\frac{d\Upsilon}{dy} \, . </math> |
This matches equation (3.9) of Blaes85. According to Blaes (1985), this equation "… is a standard eigenvalue problem whose only solutions are the Jacobi polynomials …"
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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Solution
My own background training and experience has not previously exposed me to the general class of Jacobi polynomials. In my effort to understand this class of polynomials and, specifically, their relationship to the Sturm-Liouville Equation, I have found the following references to be useful:
- PDF-formatted class notes presented by Professor Haijun Yu
- PDF-formatted class notes presented by Professor Russell Herman
- A couple of scanned pages from §22 of Abramowitz & Stegun (1972) posted online by Colin Macdonald
Drawing on Theorem 3.16 from Yu's class notes, we find that each one of the set of "n" Jacobi polynomials, <math>~J_n^{\alpha,\beta}(x)</math>, is an eigenfunction of the singular Sturm-Liouville problem whose mathematical definition is provided by the 2nd-order ODE,
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<math>~\mathcal{L}_{\alpha,\beta}J_n^{\alpha,\beta}(x)</math> |
<math>~=</math> |
<math>~\lambda_n^{\alpha,\beta}J_n^{\alpha,\beta}(x) \, ,</math> |
where the differential operator,
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<math>~\mathcal{L}_{\alpha,\beta}</math> |
<math>~\equiv</math> |
<math>~ -(1-x)^{-\alpha}(1+x)^{-\beta} \cdot \frac{d}{dx} \biggl[ (1-x)^{\alpha+1}(1+x)^{\beta+1} \cdot \frac{d}{dx} \biggr] </math> |
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<math>~=</math> |
<math>~ (x^2-1)\cdot \frac{d^2}{dx^2} + [\alpha - \beta + (\alpha+\beta+2)x]\cdot \frac{d}{dx} \, ,</math> |
and the corresponding nth eigenvalue is,
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<math>~\lambda_n^{\alpha,\beta}</math> |
<math>~=</math> |
<math>~n(n+\alpha+\beta + 1) \, .</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |