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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.

3D Depiction of Wong's Toroidal Potential Well
3D Depiction of Wong's Toroidal Potential Well

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.

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