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>

Arguments of Q and K

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>
	<math>~=</math>	<math>~ \biggl[ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] \, . </math>

Also,

<math>~ \Chi +1 </math>	<math>~=</math>	<math>~ \biggl[ \frac{\sinh\eta \cdot \sinh\eta^' + \cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] </math>
	<math>~=</math>	<math>~ \biggl[ \frac{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] </math>
<math>~ \Rightarrow ~~~\mu^2 \equiv \frac{ 2 }{\Chi +1 }</math>	<math>~=</math>	<math>~ \biggl[ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } \biggr] \, . </math>

NOTE by Tohline: On 5 June 2018, I used Excel to test the validity of the toroidal-coordinate-based expressions that have been derived here, and summarized in the following table.

Summary Table
Quantity	Raw Expression in Cylindrical Coordinates	Expression in Terms of Toroidal Coordinates
<math>~\Chi</math>	<math> \frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi} </math>	<math>~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } </math>
<math>~\mu^2 \equiv \frac{2}{\Chi + 1}</math>	<math> \frac{4\varpi^' \varpi}{(\varpi^' + \varpi)^2 + (z^' - z)^2} </math>	<math>~ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } </math>

Potential

The potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary mass distribution, <math>~\rho({\vec{r}})</math>, is,

<math>~U({\vec{r}}~')</math>

<math>~=</math>

<math>~-G \iiint \frac{\rho(\vec{r}) d^3r}{|~\vec{r} - {\vec{r}}^{~'} ~|} \, .</math>

Volume Element

See above.

Green's Function

Wong (1973) points out that in toroidal coordinates the Green's function is,

<math>~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} </math>	<math>~=</math>	<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math>
		<math>~ \times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, , </math>
Wong (1973), Eq. (2.53)

where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)," and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>. According to CT99, the Green's function written in toroidal coordinates is,

<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>	<math>~=</math>	<math>~ \frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) </math>
	<math>~=</math>	<math>~ \frac{1}{a\pi} \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) \, . </math>

Things to note:

1. The argument of <math>~Q_{m - 1 / 2}</math> in the CT99 expression is very different from the argument of <math>~Q^m_{n - 1 / 2}</math> (or <math>~P^m_{n - 1 / 2}</math>) in Wong's expression.
2. In both expressions, <math>~m</math> is the integer multiplying the azimuthal angle, <math>~\psi</math>, but in the CT99 expression this index serves as the subscript index of the function, <math>~Q</math>, whereas in Wong's expression it serves as the superscript index of both functions, <math>~Q</math> and <math>~P</math>. In this context, note that,

   <math>~Q^m_{n-\frac{1}{2}}(\cosh\eta)</math>

   <math>~=</math>

   <math>~(-1)^m \sqrt{\frac{\pi}{2}} ~\Gamma(m-n+\tfrac{1}{2}) \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{n}_{m - \frac{1}{2}} (\coth\eta) \, . </math>

3. Wong's expression contains not only a summation over the index, <math>~m</math>, but also an explicit summation over the index, <math>~n</math>, which multiplies the "polar" angle, <math>~\theta</math>; no such additional summation appears in the CT99 expression, indicating that the summation over <math>~n</math> has implicitly already been completed. In this context, note that the summation expression gives,

   <math>~Q^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)</math>

   <math>~=</math>

   <math>~ e^{\mu\pi i} \Gamma\left(\mu+ \tfrac{1}{2} \right) \biggl[ \dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu }}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}}\biggr] - 2\sum_{n=1}^{\infty} Q^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left(n \phi\right) \, ; </math>

   or, specifically for the case of <math>~\mu = 0</math>,

   <math>~ \sum_{n=0}^{\infty} \epsilon_n Q_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left(n \phi\right) </math>

   <math>~=</math>

   <math>~ \biggl[ \dfrac{ \pi/\sqrt{2} }{\left(\cosh\xi-\cos\phi\right)^{\frac{1}{2}} }\biggr] \, . </math>

4. Next thought …

See Also

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