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.

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,

<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 a toroidal-coordinate system's axisymmetric anchor ring, <math>~(\varpi,z) = (a,Z_0)</math>, the relationship between toroidal coordinates and Cartesian coordinates is,

<math>~x</math>	<math>~=</math>	<math>~\frac{a \sinh\eta \cos\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
<math>~y</math>	<math>~=</math>	<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
<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, on p. 1301 within eq. (10.3.75) of [MF53]; in §14.19 of NIST's Digital Library of Mathematical Functions; or even within Wikipedia. (In most cases the implicit assumption is that <math>~Z_0 = 0</math>.)

Mapping the other direction [see equations 2.13 - 2.15 of Wong (1973) ], we have,

<math>~\eta</math>	<math>~=</math>	<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
<math>~\cos\theta</math>	<math>~=</math>	<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
<math>~\tan\psi</math>	<math>~=</math>	<math>~\frac{y}{x} \, ,</math>

where,

<math>~r_1^2 </math>	<math>~\equiv</math>	<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + (z-Z_0)^2 \, ,</math>
<math>~r_2^2 </math>	<math>~\equiv</math>	<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + (z-Z_0)^2 \, ,</math>

and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.

According to p. 1301, eq. (10.3.75) of [MF53] — or, for example, as found in Wikipedia — the differential volume element is,

<math>~d^3x</math>

<math>~=</math>

<math>~h_\eta h_\theta h_\psi d\eta d\theta 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>

Green's Function Expression

On p. 293, Wong (1973) states that, in toroidal coordinates, the Green's function is,

<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </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>.

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