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 the 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, within eq. (10.3.75) of MF53; in §14.19 of NIST's Digital Library of Mathematical Functions
See Also
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© 2014 - 2021 by Joel E. Tohline |