User:Tohline/Appendix/Mathematics/ToroidalSynopsis01

Synopsis of Toroidal Coordinate Approach

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>

where,

<math>~C_0 \equiv \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2Z_0 (Z_0 - z^' ) \biggr]</math>

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation