User:Tohline/Appendix/Mathematics/ToroidalSynopsis01
Synopsis of Toroidal Coordinate Approach
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
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,
|
<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,
|
<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,
|
<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,
|
<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,
|
<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,
|
<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,
|
<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,
|
<math>~\coth\eta</math> |
<math>~=</math> |
<math>~\biggl[ e^\eta + e^{-\eta} \biggr]\biggl[ e^\eta - e^{-\eta} \biggr]^{-1}</math> |
<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> |
|
|
<math>~=</math> |
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1 r_2} \biggr]\biggl[ \frac{r_1^2 - r_2^2}{r_1 r_2} \biggr]^{-1}</math> |
<math>~=</math> |
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1^2 - r_2^2} \biggr]</math> |
|
|
<math>~=</math> |
<math>~ \frac{ \varpi^2 + a^2 + (z - Z_0)^2 }{ 2a\varpi } \, . </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>
Therefore,
|
<math>~2\varpi \biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math> |
<math>~=</math> |
<math>~ (\varpi^')^2 + \varpi^2 + (z^' - z)^2 - [\varpi^2 + a^2 + (z - Z_0)^2 ] </math> |
|
|
<math>~=</math> |
<math>~ (\varpi^')^2 - a^2 + [ (z^')^2 - 2z^' z + z^2]- [z^2 - 2zZ_0 + Z_0^2] </math> |
|
|
<math>~=</math> |
<math>~ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2z(Z_0 - z^' ) </math> |
|
<math>~\Rightarrow ~~~2a\biggl[ \frac{\sinh\eta }{(\cosh\eta - \cos\theta)} \biggr]\biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math> |
<math>~=</math> |
<math>~ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2(Z_0 - z^' )\biggl[ Z_0 + \frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \biggr] </math> |
|
|
<math>~=</math> |
<math>~ 2aC_0 +2a(Z_0 - z^' )\biggl[ \frac{\sin\theta}{(\cosh\eta - \cos\theta)} \biggr] </math> |
|
<math>~\Rightarrow ~~~ \sinh\eta \biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math> |
<math>~=</math> |
<math>~ C_0 (\cosh\eta - \cos\theta) + (Z_0 - z^' ) \sin\theta </math> |
|
<math>~\Rightarrow ~~~ \varpi^' \Chi </math> |
<math>~=</math> |
<math>~ \frac{1}{\sinh\eta} \biggl[ C_0 (\cosh\eta - \cos\theta) + (Z_0 - z^' ) \sin\theta + a\cosh\eta\biggr] </math> |
|
<math>~\Rightarrow ~~~ \Chi </math> |
<math>~=</math> |
<math>~ \frac{1}{\varpi^' \sinh\eta} \biggl[ (C_0 + a)\cosh\eta + (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr] </math> |
where,
<math>~ C_0 \equiv \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2Z_0 (Z_0 - z^' ) \biggr] = \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^')^2 +Z_0^2 - 2Z_0 z^' \biggr] = \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^' - Z_0)^2 \biggr] \, . </math>
Now, notice that,
|
<math>~ ( \varpi^')^2 + a^2 + (z^' - Z_0)^2 </math> |
<math>~=</math> |
<math>~ 2a\varpi^'~\coth\eta^' </math> |
||
|
<math>~\Rightarrow ~~~ ( \varpi^')^2 - a^2 + (z^' - Z_0)^2 </math> |
<math>~=</math> |
<math>~ 2a\varpi^'~\coth\eta^' - 2a^2 </math> |
||
|
<math>~\Rightarrow ~~~ C_0 </math> |
<math>~=</math> |
<math>~ \varpi^'~\coth\eta^' - a </math> |
||
|
|
<math>~=</math> |
<math>~ \biggl[ \frac{a \sinh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] ~\coth\eta^' - a </math> |
||
|
|
<math>~=</math> |
<math>~ \biggl[ \frac{a \cosh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] - a \, . </math> |
||
Hence,
|
<math>~ \Chi </math> |
<math>~=</math> |
<math>~ \frac{\cosh\eta}{\varpi^' \sinh\eta} \biggl[ \varpi^' \coth\eta^' \biggr] + \frac{1}{\sinh\eta} \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta^' } \biggr] \biggl[ (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr] </math> |
|
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' + \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta \cdot \sinh\eta^' } \biggr] \biggl[ (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr]
</math> |
|
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' - \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta \cdot \sinh\eta^' } \biggr] \biggl\{ \biggl[ \frac{a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \biggr] \sin\theta + \biggl[ \frac{a \cosh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] \cos\theta - a\cos\theta\biggr\}
</math> |
|
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' - \biggl[ \frac{1 }{ \sinh\eta \cdot \sinh\eta^' } \biggr] \biggl\{ \sin\theta^' \sin\theta + \cosh\eta^' \cos\theta - (\cosh\eta^' - \cos\theta^')\cos\theta\biggr\}
</math> |
|
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' - \biggl[ \frac{\sin\theta^' \sin\theta +\cos\theta^'\cos\theta }{ \sinh\eta \cdot \sinh\eta^' } \biggr] \, .
</math> |
See Also
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |