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
Singular Sturm-Liouville Problem
Drawing on Theorem 3.16 from Yu's class notes, we find that each one of the set of <math>j=0 \rightarrow \infty</math> Jacobi polynomials, <math>~J_j^{\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_j^{\alpha,\beta}(x)</math> |
<math>~=</math> |
<math>~\lambda_j^{\alpha,\beta}J_j^{\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 jth eigenvalue is,
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<math>~\lambda_j^{\alpha,\beta}</math> |
<math>~=</math> |
<math>~j(j+\alpha+\beta + 1) \, .</math> |
(Note that we have used "j" instead of the more traditional use of "n" to identify the specific Jacobi polynomial, because we are already using "n" to denote the polytropic index.) According to Theorem 3.17 from Yu's class notes, for each specified value of the index, <math>~j</math>, the eigenfunction solution to this eigenvalue problem — that is, the relevant Jacobi polynomial — is,
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<math>~J_j^{\alpha,\beta}(x)</math> |
<math>~=</math> |
<math>~(1-x)^{-\alpha}(1+x)^{-\beta} \biggl\{ \frac{(-1)^j}{2^j j!} \cdot \frac{d^j}{dx^j}\biggl[ (1-x)^{j+\alpha}(1+x)^{j+\beta} \biggr] \biggr\} \, .</math> |
Table 1 provides a few examples.
| Table 1: Example Jacobi Polynomials | ||
|---|---|---|
| <math>~j</math> | <math>~J_j^{\alpha,\beta}(x)</math> | <math>~\lambda_j^{\alpha,\beta}</math> |
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<math>~0</math> |
<math>~1</math> |
<math>~0</math> |
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<math>~1</math> |
<math>~\tfrac{1}{2}(\alpha+\beta+2)x + \tfrac{1}{2}(\alpha-\beta)</math> |
<math>~(\alpha+\beta+2)</math> |
| See also, eqs. (35)-(37) of Wolfram MathWorld. | ||
Note: When setting <math>~j=1</math> then performing the differentiation specified in the above definition of <math>~J_j^{\alpha,\beta}</math>, I don't actually obtain the expression provided by eq. (36) of Wolfram MathWorld or (which is the same) eq. (3.110) in Yu's class notes. This, combined with the recognition that the Blaes85 ODE does not have precisely the same form as the singular Sturm-Liouville equation makes me uncomfortable pursuing this specific line of investigation.
Jacobi Differential Equation
Instead, let's pursue the following lead. Wolfram MathWorld provides a discussion of the Jacobi Differential Equation. Equation (1) of this reference appears to be identical to the singular Sturm-Liouville equation presented above. However, in addition, this MathWorld chapter points out that Zwillinger (1997, p. 123) gives a related differential equation that is referred to as "Jacobi's equation," namely,
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<math>~0</math> |
<math>~=</math> |
<math>~x(1-x)y + [\gamma - (\alpha+1)x]y' + j(\alpha+j)y \, .</math> |
If we rewrite our above-derived ODE — which is identical to eq. (3.9) of Blaes85 — as,
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<math>~ 0 </math> |
<math>~=</math> |
<math>~ (1-y)y \frac{d^2\Upsilon}{dy^2} +\biggl[ (|k|+1) -y (|k|+1+n)\biggr] \frac{d\Upsilon}{dy} + \frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math> |
we see that its form is identical to this so-called "Jacobi's equation." The association is complete if we make the following parameter associations:
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<math>~\gamma</math> |
<math>~\leftrightarrow</math> |
<math>~|k|+1 \, ,</math> |
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<math>~\alpha</math> |
<math>~\leftrightarrow</math> |
<math>~|k|+n \, ,</math> |
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<math>~j(|k|+n + j)</math> |
<math>~\leftrightarrow</math> |
<math>~\frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \, .</math> |
We can infer from this third association that, for the jth eigenfunction solution of the governing eigenvalue problem, the relevant oscillation frequency is,
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<math>~\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2</math> |
<math>~=</math> |
<math>~|k| + \frac{2j}{n}(|k|+n + j)</math> |
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<math>~=</math> |
<math>~\frac{1}{n}\biggl[n|k| + 2j(|k|+n + j)\biggr]</math> |
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<math>~=</math> |
<math>~\frac{1}{n}\biggl[2j^2 + 2jn + 2j|k| + n|k| \biggr] \, .</math> |
This is identical to the eigenfrequency identified in eq. (3.12) of Blaes85. So, we must be on the right track!
See Also
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