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 \int \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> |
See Also
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© 2014 - 2021 by Joel E. Tohline |