User:Tohline/2DStructure/ToroidalGreenFunction
Using Toroidal Coordinates to Determine the Gravitational Potential
NOTE: An earlier version of this chapter has been shifted to our "Ramblings" Appendix.
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Here we build upon our accompanying review of the types of numerical techniques that various astrophysics research groups have developed to solve for the Newtonian gravitational potential, <math>~\Phi(\vec{x})</math>, given a specified, three-dimensional mass distribution, <math>~\rho(\vec{x})</math>. Our focus is on the use of toroidal coordinates to solve the integral formulation of the Poisson equation, namely,
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<math>~ \Phi(\vec{x})</math> |
<math>~=</math> |
<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math> |
For the most part, we will adopt the notation used by C.-Y. Wong (1973, Annals of Physics, 77, 279); in an accompanying discussion, we review additional results from this insightful 1973 paper, as well as a paper of his that was published the following year in The Astrophysical Journal, namely, Wong (1974).
Basic Elements of the Toroidal Coordinate System
Given the meridional-plane coordinate location of a toroidal-coordinate system's axisymmetric anchor ring, <math>~(\varpi,z) = (a,Z_0)</math>, the relationship between toroidal coordinates and Cartesian coordinates is,
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<math>~x</math> |
<math>~=</math> |
<math>~\frac{a \sinh\eta \cos\psi}{(\cosh\eta - \cos\theta)} \, ,</math> |
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<math>~y</math> |
<math>~=</math> |
<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math> |
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<math>~z - Z_0</math> |
<math>~=</math> |
<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math> |
This set of coordinate relations appear as equations 2.1 - 2.3 in Wong (1973). They may also be found, for example, on p. 1301 within eq. (10.3.75) of [MF53]; in §14.19 of NIST's Digital Library of Mathematical Functions; or even within Wikipedia. (In most cases the implicit assumption is that <math>~Z_0 = 0</math>.)
Mapping the other direction [see equations 2.13 - 2.15 of Wong (1973) ], we have,
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<math>~\eta</math> |
<math>~=</math> |
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math> |
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<math>~\cos\theta</math> |
<math>~=</math> |
<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math> |
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<math>~\tan\psi</math> |
<math>~=</math> |
<math>~\frac{y}{x} \, ,</math> |
where,
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<math>~r_1^2 </math> |
<math>~\equiv</math> |
<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + (z-Z_0)^2 \, ,</math> |
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<math>~r_2^2 </math> |
<math>~\equiv</math> |
<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + (z-Z_0)^2 \, ,</math> |
and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.
According to p. 1301, eq. (10.3.75) of [MF53] — or, for example, as found in Wikipedia — the differential volume element is,
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<math>~d^3x</math> |
<math>~=</math> |
<math>~h_\eta h_\theta h_\psi d\eta d\theta d\psi</math> |
<math>~=</math> |
<math>~\biggl[ \frac{a^3 \sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .</math> |
Green's Function Expression
As presented by Wong (1973)
Referencing [MF53], Wong (1973) states that, in toroidal coordinates, the Green's function is,
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<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits^\infty_{m,n=0} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math> |
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<math>~ \times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, , </math> |
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Wong (1973), p. 293, Eq. (2.53) |
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where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)," and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>. This Green's function expression can indeed be found as eq. (10.3.81) on p. 1304 of [MF53], but it should be noted that the MF53 expression contains two (presumably type-setting) errors: First, the factor, <math>~(-1)^m</math>, appears as <math>~(-i)^m</math> in MF53; and, second, in the term that is composed of a ratio of gamma functions, the denominator appears in MF53 as <math>~\Gamma(n - m + \tfrac{1}{2})</math>, whereas it should be <math>~\Gamma(n + m + \tfrac{1}{2})</math>, as presented here.
Rearranging Terms
Let's focus on the situation when <math>~\eta^' > \eta</math>, and begin rearranging or substituting terms.
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<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')] </math> |
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<math>~ \times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} ~P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') </math> |
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<math>~=</math> |
<math>~ \frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} } \sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')] </math> |
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<math>~ \times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~ \sqrt{ \frac{\pi}{2} }~\Gamma(n-m+\tfrac{1}{2}) \sqrt{\sinh\eta}~P^m_{n-1 / 2}(\cosh\eta) \biggl\}\biggr\{ ~\sqrt{ \frac{2}{\pi} }~\frac{\sqrt{\sinh\eta^'}}{\Gamma(n + m + \tfrac{1}{2})} Q^m_{n-1 / 2}(\cosh\eta^') \biggr\} </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |