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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.
We discuss this topic in a separate, accompanying chapter.
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.
Our Presentation of Wong's (1973) Result
Summary: First three terms in Wong's (1973) expression for the gravitational potential at any point, P(ϖ, z), outside of a uniformdensity torus.  
where, once the major ( R ) and minor ( d ) radii of the torus — as well as the vertical location of its equatorial plane (Z_{0}) — have been specified, we have,
NOTE: In evaluating these "leading coefficient expressions" for the case, , we have used the complete elliptic integral evaluations, K(k_{0}) = 1.854074677 and E(k_{0}) = 1.350643881. 
Setup
From our accompanying discussion of Wong's (1973) derivation, the exterior potential is given by the expression,



Wong (1973), §II.D, p. 294, Eqs. (2.59) & (2.61)
where,






Wong (1973), §II.D, p. 294, Eq. (2.63)
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
Part A
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,

Part B
Let's evaluate specifically for the case where , using the already separately evaluated values of the four relevant toroidal functions. We find,









Next, let's develop a consolidated expression for that replaces all the toroidal functions with complete elliptic integrals of the first and second kind.



















































Finally, let's evaluate this consolidated expression for the specific case of , remembering that in this specific case , , and . We find,









This matches the numerically evaluated expression, from above (6/30/2020). There is a tremendous amount of cancellation between the three key terms in this expression, so the match is only to three significant digits.</tr>
Part C
Next …
Useful Relations from Above

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









where, as above,
So we have,















Finally, inserting the expression for that we have derived, above, gives,






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. Finally, the third (n = 2) term in Wong's (1973) seriesexpression for the exterior gravitational potential is,









The Huré, et al (2020) Presentation
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
 Go Home 
Notation
In Huré, et al. (2020), the major and minor radii of the torus surface ("shell") are labeled, respectively, R_{c} and b, and their ratio is denoted,
Huré, et al. (2020), §2, p. 5826, Eq. (1)
The authors work in cylindrical coordinates, , whereas we refer to this same coordinatepair as, . The quantity,



Huré, et al. (2020), §2, p. 5826, Eqs. (5) & (7)
We have affixed the subscript "H" to their meridionalplane angle, θ, to clarify that it has a different coordinatebase definition from the meridionalplane angle, θ, that appears in our above discussion of Wong's (1973) work. In their paper, the subscript "0" is used in the case of an infinitesimally thin hoop , that is to say,



Huré, et al. (2020), §3, p. 5827, Eq. (13)
Generally, the argument (modulus) of the complete elliptic integral functions is,



Huré, et al. (2020), §2, p. 5826, Eq. (4)
and, as stated in the first sentence of their §3, reference may also be made to the complementary modulus,



(Again, we have affixed the subscript "H" in order to differentiate from the modulus employed by Wong (1973).) And in the case of an infinitesimally thin hoop ,



Huré, et al. (2020), §3, p. 5827, Eq. (12)
Key Finding
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)
Rewriting this last expression in a form that can more readily be compared with Wong's work, we obtain,






Hence, also,









Compare First Terms
Rewriting the first term in the Huré, et al. (2020) series expression for the potential, we have,



where,



For comparison, the first term in Wong's expression is,



where,


















This expression is correct for any value of the aspect ratio, . But let's set — as Huré, et al. (2020) have done — then see how the expression simplifies for an infinitesimally thin hoop, that is, if we let . First we note that,



so in this limit the modulus of the complete elliptic integral of the first kind becomes identical to the modulus used by Huré, et al. (2020), . Next, we note that,



As a result, we can write,



Now let's evaluate the coefficient, , in the limit of .
, in the limit of . First, drawing from our separate examination of the behavior of complete elliptic integral functions, we appreciate that,
Next, employing the binomial expansion, we find that,
and,
Hence, we have,

Given that,



we conclude that,



that is, we conclude that matches in the limit of, .
Go to Higher Order
Let's keep higher order terms in Wong's n = 0 component, and let's examine contributions to the same order that come from Wong's n = 1 and (if necessary) n = 2 components.
First, note that,



Keeping Higher Order in Wong's First Component









Add One Additional Term  

Next, employing the binomial expansion, we find that,
























Hence, we have,































































Hence,



or, more precisely,



Next Factors
Now,















Again, drawing from the binomial theorem, we have,












Now Work on Elliptic Integral Expressions
From a separate discussion we can draw the series expansion of , specifically,



where,






Also,



where,






What we want to do is write in terms of . Let's try …



where,




























































Let's subtract from the potential expression. But first, let's adopt the shorthand notation …
Given that,
let's define the variable, , such that,

We can therefore write,






where we should keep in mind that is . So, let's examine the piece,










































Now we have,






But, as we have just demonstrated,





















TEMPORARY BREAK HERE 
Hence,









Include Second Wong Term






Leading (Upsilon) Coefficient



















































































































Floating Comparison Summary  
As shown above, the first three terms of the Huré, et al. (2020) series expression may be written as,
Let's see how it compares to the first term of Wong's (1973) expression which, as shown separately above, can be written in the form,
First, as shown above,
Note that, in order to determine the functional form of the term in this expression, we will have to include terms in the various expressions for products of elliptic integrals. Second, we have shown that,
Hence,
and,
Second,

Geometric Factor
By definition,












Hence,



© 2014  2020 by Joel E. Tohline 