# Template:LSU CT99CommonTheme3B

Beginning with his integral expression for $~\Phi(\eta,\theta)|_\mathrm{axisym}$, Wong (1973) was able to complete the integrals in both coordinate directions and obtain an analytic expression for the potential both inside and outside of a uniformly charged (equivalently, uniform-density), circular torus. This is a remarkable result that has been largely unnoticed and unappreciated by the astrophysics community. We detail how he accomplished this task in an accompanying chapter titled, Wong's (1973) Analytic Potential.

If a torus has a major radius, $~R$, and cross-sectional radius, $~d$, Wong realized that every point on the surface of the torus will have the same toroidal-coordinate radius, $~\eta_0 = \cosh^{-1}(R/d)$, if the anchor ring of the selected toroidal coordinate system has a radius, $~a = \sqrt{R^2 - d^2}$. His derived expressions for the potential — one, outside, and the other, inside the torus — are:

Exterior Solution:  $~\eta_0 \ge \eta$
 $~\Phi_\mathrm{W}(\eta,\theta)$ $~=$ $~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta)$ $~ \times P_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) - (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh \eta_0) ~ Q_{n - \frac{1}{2}}(\cosh \eta_0) \biggr] \, .$
Interior Solution:  $~\eta \ge \eta_0$
 $~\Phi_\mathrm{W}(\eta,\theta)$ $~=$ $~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta)$ $~ \times \biggl\{ Q_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0})  - (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0}) \biggr] - Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggr\} \, .$

The following pair of animated images result from our numerical evaluation of this pair of expressions for $~\Phi_\mathrm{W}$ (including the first four, and most dominant, terms in the series summation) for the case of: (left) $~R/d = 3$, which is the aspect ratio Wong chose to illustrate in his publication; and (right) tori having a variety of different aspect ratios over the range, $~1.8 \le R/d \le 8$.

Does this expression for the potential behave as we expect in the "thin ring" approximation? On p. 295 of Wong (1973), we find the following statement:

 "For the case of a very thin ring (i.e., $~\eta_0 \rightarrow \infty$), the exterior solution has contributions mostly from the first term in the expansion of the series …"

Using the notation, $~\Phi_\mathrm{W0}$, to represent the leading-order term in the expression for the exterior potential, we have (see the accompanying chapter for details),

 $~\Phi_{\mathrm{W}0} (\eta,\theta)$ $~=$ $~-\biggl( \frac{GM}{a} \biggr) F(\cosh\eta_0)\cdot [\cosh\eta - \cos\theta]^{1 / 2} P_{-\frac{1}{2}}(\cosh\eta)$ $~=$ $~- \frac{2}{\pi}\biggl( \frac{GM}{a} \biggr) F(\cosh\eta_0)\cdot \biggl[ \frac{2a^2}{(\varpi + a)^2 + z^2} \biggr]^{1 / 2} K(k) \, ,$

where,

 $~k$ $~\equiv$ $~ \biggl[ \frac{2}{1+\coth\eta}\biggr]^{1 / 2} = \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + z^2} \biggr]^{1 / 2} \, ,$       and, $~F(\cosh\eta_0)$ $~\equiv$ $~\frac{2^{1 / 2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \biggl[ Q_{+\frac{1}{2}}(\cosh \eta_0) Q_{- \frac{1}{2}}^2(\cosh \eta_0) + 3 Q_{- \frac{1}{2}}(\cosh \eta_0)~Q^2_{+ \frac{1}{2}}(\cosh \eta_0)\biggr] \, .$

In our accompanying discussion we show that,

 $~F(\cosh\eta_0)\biggr|_{\eta_0\rightarrow \infty}$ $~=$ $~\biggl\{ \frac{2^{1 / 2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \biggl[ \biggl( \frac{3 \pi^2}{2} \biggr) \frac{1}{\cosh^2\eta_0} \biggr] \biggr\}_{\eta_0\rightarrow \infty} = \frac{1}{\sqrt{2}} \, .$

Hence, we have,

 $~\Phi_{\mathrm{W}0} (\eta,\theta)\biggr|_{\eta_0 \rightarrow \infty}$ $~=$ $~- \biggl[ \frac{2GM}{\pi} \biggr] \frac{K(k)}{\sqrt{(\varpi + a)^2 + z^2}} \, ,$

which precisely matches the above-referenced Gravitational Potential in the Thin Ring (TR) Approximation.