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SpheroidRing Systems
Through a research collaboration at the Université de Bordeaux, B. Basillais & J. M. Huré (2019), MNRAS, 487, 45044509 have published a paper titled, Rigidly Rotating, Incompressible SpheroidRing Systems: New Bifurcations, Critical Rotations, and Degenerate States.
Exterior Gravitational Potential of Toroids
J. M. Huré, B. Basillais, V. Karas, A. Trova, & O. Semerák (2020), MNRAS, 494, 58255838 have published a paper titled, The Exterior Gravitational Potential of Toroids. Here we examine how their work relates to the published work by C.Y. Wong (1973, Annals of Physics, 77, 279), which we have separately discussed in detail.
Their Presentation
On an initial reading, it appears as though the most relevant section of the Huré, et al. (2020) paper is §8 titled, The Solid Torus. They write the gravitational potential in terms of the series expansion,



Huré, et al. (2020), §7, p. 5831, Eq. (42)
where, after setting and acknowledging that we can write,



Huré, et al. (2020), §8, p. 5832, Eqs. (52) & (53)
and,



Huré, et al. (2020), §8, p. 5832, Eq. (54)
Note that the argument of the elliptic integral functions is,



where, 



Huré, et al. (2020), §2, p. 5826, Eqs. (4) & (5)
Our Presentation of Wong's (1973) Result
Setup
From our accompanying discussion of Wong's (1973) derivation, the exterior potential is given by the expression,



where,






and where, in terms of the major ( R ) and minor ( d ) radii of the torus — or their ratio, ε ≡ d/R,






These expressions incorporate a number of basic elements of a toroidal coordinate system. In what follows, we will also make use of the following relations:
Once the primary scale factor, , has been specified, the illustration shown at the bottom of this inset box — see also our accompanying set of similar figures used by other researchers — helps in explaining how transformations can be made between any two of the referenced coordinate pairs: , , .
Given that (sin^{2}θ + cos^{2}θ) = 1, we have,
We deduce as well that,
Given the definitions,
we can use the transformations,
Or we can use the transformations,
Additional potentially useful relations can be found in an accompanying chapter wherein we present a variety of basic elements of a toroidal coordinate system. 
Leading (n = 0) Term
Wong's Expression
Now, from our separate derivation we have,



And if we make the functionargument substitution, , in the "Key Equation,"




Abramowitz & Stegun (1995), p. 337, eq. (8.13.3) 
we can write,



where, from above, we recognize that,
So, the leading (n = 0) term gives,












ThinRing Evaluation of C_{0}
In an accompanying discussion of the thinring approximation, we showed that as



Hence, in this limit we can write,



More General Evaluation of C_{0}
NOTE of CAUTION: In our above evaluation of the toroidal function, , we appropriately associated the function argument, , with the hyperboliccotangent of ; that is, we made the substitution, . Here, as we assess the behavior of, and evaluate, the leading coefficient, , an alternate substitution is appropriate, namely, ; we affix the subscript zero to this function argument in an effort to minimize possible confusion with the argument, . 
Drawing from our accompanying tabulation of Toroidal Function Evaluations, we have more generally,









where,



Looking back at our previous numerical evaluation of when , we see that,

Attempting to simplify this expression, we have,



























This last, simplifed expression gives, as above, . TERRIFIC!
Finally then, for any choice of ,






Second (n = 1) Term
The second (n = 1) term in Wong's (1973) expression for the exterior potential is,



where, is the same as above, and,



Now, from our accompanying table of "Toroidal Function Evaluations", it appears as though,



where, as above,



Hence, we have,


















From the above function tabulations & evaluations — for example, and — and a separate listing of Example Recurrence Relations, we have,
Then, letting and, for all m ≥ 2, letting in the "Key Equation,"
we have,
Therefore, specifically for m = 1, we obtain the recurrence relation,

While keeping in mind that,



and, 



let's attempt to express this leading coefficient, , entirely in terms of the pair of complete elliptic integral functions.
























Hence,


















Hence, we have,



Third (n = 2) Term
The third (n = 2) term in Wong's (1973) expression for the exterior potential is,



where, is the same as above, and,



In order to evaluate , we will need the following pair of expressions in addition to the ones already used:
And, setting m = 2 in the above recurrence relation for gives,

Next …
Useful Relations from Above

Now, from our tabulation of example recurrence relations, we see that,









where, as above,
So we have,















Summary
Once the major ( R ) and minor ( d ) radii of the torus have been specified, two key model parameters can be immediately determined, namely,
and,
in which case also, Once the massdensity ( ρ_{0} ) of the torus has been specified, the torus mass is given by the expression,
In addition to the principal pair of meridionalplane coordinates, , it is useful to define the pair of distances,






where, the equatorial plane of the torus is located at . As we have shown above, the leading (n = 0) term in Wong's (1973) seriesexpression for the gravitational potential anywhere outside the torus is,






where, the two distinctly different arguments — one with, and one without a zero subscript — of the complete ellipticintegral functions are,






As we also have shown above, the second (n = 1) term in Wong's (1973) seriesexpression for the exterior gravitational potential is,






Note that a transformation from the coordinate pair to the toroidalcoordinate pair includes the expression,



So this (n = 1) term's explicit dependence on "cos(nθ)" is clear.
© 2014  2020 by Joel E. Tohline 