User:Tohline/Appendix/Mathematics/ToroidalSynopsis01
Synopsis of Toroidal Coordinate Approach
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Basics
Here we attempt to bring together — in as succinct a manner as possible — our approach and C.-Y. Wong's (1973) approach to determining the gravitational potential of an axisymmetric, uniform-density torus that has a major radius, <math>~R</math>, and a minor, cross-sectional radius, <math>~d</math>. The relevant toroidal coordinate system is one based on an anchor ring of major radius,
<math>~a^2 \equiv R^2 - d^2 \, .</math>
If the meridional-plane location of the anchor ring — as written in cylindrical coordinates — is, <math>~(\varpi, z) = (a,Z_0)</math>, then the preferred toroidal-coordinate system has meridional-plane coordinates, <math>~(\eta, \theta)</math>, defined such that,
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<math>~\eta</math> |
<math>~=</math> |
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math> |
and, |
<math>~\cos\theta</math> |
<math>~=</math> |
<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math> |
where,
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<math>~r_1^2 </math> |
<math>~\equiv</math> |
<math>~(\varpi + a)^2 + (z-Z_0)^2 \, ,</math> |
and, |
<math>~r_2^2 </math> |
<math>~\equiv</math> |
<math>~(\varpi - a)^2 + (z-Z_0)^2 \, ,</math> |
and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>. Mapping the other direction, we have,
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<math>~\varpi</math> |
<math>~=</math> |
<math>~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,</math> |
and, |
<math>~z-Z_0</math> |
<math>~=</math> |
<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math> |
The three-dimensional differential volume element is,
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<math>~d^3 r</math> |
<math>~=</math> |
<math>\varpi d\varpi ~dz ~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> |
Note that, if <math>~\eta_0</math> identifies the surface of the uniform-density torus, then,
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<math>~\cosh\eta_0</math> |
<math>~=</math> |
<math>~\frac{R}{d} \, ,</math> |
<math>~\sinh\eta_0</math> |
<math>~=</math> |
<math>~\frac{a}{d} \, ,</math> |
and, |
<math>~\coth\eta_0</math> |
<math>~=</math> |
<math>~\frac{R}{a} \, ;</math> |
and when the integral over the volume element is completed — that is, over all <math>~\psi</math>, over all <math>~\theta</math>, and over the "radial" interval, <math>~\eta_0 \le \eta \le \infty</math> — the resulting volume is,
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<math>~V</math> |
<math>~=</math> |
<math>~\frac{2\pi^2 \cosh\eta_0}{\sinh^3\eta_0}</math> |
<math>~=</math> |
<math>~2\pi^2 Rd^2 \, .</math> |
Also, given that,
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<math>~\cosh\eta</math> |
<math>~=</math> |
<math>~\frac{1}{2}\biggl[ e^\eta + e^{-\eta} \biggr]</math> |
and, |
<math>~\sinh\eta</math> |
<math>~=</math> |
<math>~\frac{1}{2}\biggl[ e^\eta - e^{-\eta} \biggr] \, ,</math> |
we have,
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<math>~\coth\eta</math> |
<math>~=</math> |
<math>~\biggl[ e^\eta + e^{-\eta} \biggr]\biggl[ e^\eta - e^{-\eta} \biggr]^{-1}</math> |
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<math>~=</math> |
<math>~\biggl[ \frac{r_1}{r_2} + \frac{r_2}{r_1} \biggr]\biggl[ \frac{r_1}{r_2} - \frac{r_2}{r_1} \biggr]^{-1}</math> |
Exploration
Want to explore argument of <math>~Q_{-1 / 2}(\Chi)</math>, namely,
<math> \Chi \equiv \frac{(\varpi^')^2 + \varpi^2 + (Z^' - Z)^2}{2\varpi^' \varpi} . </math>
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<math>~\Rightarrow~~~ 2\varpi^' \varpi \Chi</math> |
<math>~=</math> |
<math>~ (\varpi^')^2 + \varpi^2 + (Z^' - Z)^2 </math> |
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<math>~\Rightarrow~~~ 2\Chi \biggl[ \frac{a \sinh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr]\biggl[ \frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \biggr]</math> |
<math>~=</math> |
<math>~ \biggl[ \frac{a \sinh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr]^2 + \biggl[ \frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \biggr]^2 + \biggl[ \frac{a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} - \frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \biggr]^2 </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |