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Université de Bordeaux (Part 1)
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Spheroid-Ring Systems
Through a research collaboration at the Université de Bordeaux, B. Basillais & J. -M. Huré (2019), MNRAS, 487, 4504-4509 have published a paper titled, Rigidly Rotating, Incompressible Spheroid-Ring 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, 5825-5838 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 uniform-density torus. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
where, once the major ( R ) and minor ( d ) radii of the torus — as well as the vertical location of its equatorial plane (Z0) — have been specified, we have,
NOTE: In evaluating these "leading coefficient expressions" for the case, |
Setup
From our accompanying discussion of Wong's (1973) derivation, the exterior potential is given by the expression,
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Wong (1973), §II.D, p. 294, Eqs. (2.59) & (2.61)
where,
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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,
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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,
Given that (sin2θ + cos2θ) = 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,
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And if we make the function-argument substitution, , in the "Key Equation,"
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Abramowitz & Stegun (1995), p. 337, eq. (8.13.3) |
we can write,
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where, from above, we recognize that,
So, the leading (n = 0) term gives,
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Thin-Ring Evaluation of C0
In an accompanying discussion of the thin-ring approximation, we showed that as
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Hence, in this limit we can write,
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More General Evaluation of C0
NOTE of CAUTION: In our above evaluation of the toroidal function, |
Drawing from our accompanying tabulation of Toroidal Function Evaluations, we have more generally,
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where,
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Looking back at our previous numerical evaluation of
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Attempting to simplify this expression, we have,
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This last, simplifed expression gives, as above, . TERRIFIC!
Finally then, for any choice of ,
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Second (n = 1) Term
The second (n = 1) term in Wong's (1973) expression for the exterior potential is,
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where, is the same as above, and,
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Now, from our accompanying table of "Toroidal Function Evaluations", it appears as though,
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where, as above,
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Hence, we have,
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From the above function tabulations & evaluations — for example,
Then, letting
we have,
Therefore, specifically for m = 1, we obtain the recurrence relation,
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While keeping in mind that,
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and, |
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let's attempt to express this leading coefficient, , entirely in terms of the pair of complete elliptic integral functions.
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Hence,
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Hence, we have,
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Third (n = 2) Term
Part A
The third (n = 2) term in Wong's (1973) expression for the exterior potential is,
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where, is the same as above, and,
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In order to evaluate
And, setting m = 2 in the above recurrence relation for
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Part B
Let's evaluate specifically for the case where
, using the already separately evaluated values of the four relevant toroidal functions. We find,
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Next, let's develop a consolidated expression for that replaces all the toroidal functions with complete elliptic integrals of the first and second kind.
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Finally, let's evaluate this consolidated expression for the specific case of , remembering that in this specific case
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, and
. We find,
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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
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Now, from our tabulation of example recurrence relations, we see that,
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where, as above,
So we have,
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Finally, inserting the expression for that we have derived, above, gives,
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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 mass-density ( ρ0 ) of the torus has been specified, the torus mass is given by the expression,
In addition to the principal pair of meridional-plane coordinates, , it is useful to define the pair of distances,
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where, the equatorial plane of the torus is located at . As we have shown above, the leading (n = 0) term in Wong's (1973) series-expression for the gravitational potential anywhere outside the torus is,
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where, the two distinctly different arguments — one with, and one without a zero subscript — of the complete elliptic-integral functions are,
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As we also have shown above, the second (n = 1) term in Wong's (1973) series-expression for the exterior gravitational potential is,
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Note that a transformation from the coordinate pair to the toroidal-coordinate pair
includes the expression,
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So this (n = 1) term's explicit dependence on "cos(nθ)" is clear. Finally, the third (n = 2) term in Wong's (1973) series-expression for the exterior gravitational potential is,
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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, Rc 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 coordinate-pair as,
. The quantity,
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Huré, et al. (2020), §2, p. 5826, Eqs. (5) & (7)
We have affixed the subscript "H" to their meridional-plane angle, θ, to clarify that it has a different coordinate-base definition from the meridional-plane 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,
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Huré, et al. (2020), §3, p. 5827, Eq. (13)
Generally, the argument (modulus) of the complete elliptic integral functions is,
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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,
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(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 ,
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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,
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Huré, et al. (2020), §7, p. 5831, Eq. (42)
where, after setting and acknowledging that
we can write,
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Huré, et al. (2020), §8, p. 5832, Eqs. (52) & (53)
and,
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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,
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Hence, also,
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Compare First Terms
Rewriting the first term in the Huré, et al. (2020) series expression for the potential, we have,
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where,
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For comparison, the first term in Wong's expression is,
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where,
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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,
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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,
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As a result, we can write,
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Now let's evaluate the coefficient, , in the limit of
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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,
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Given that,
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we conclude that,
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that is, we conclude that matches
in the limit of,
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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,
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Keeping Higher Order in Wong's First Component
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Add One Additional Term | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Next, employing the binomial expansion, we find that,
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Hence, we have,
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Hence,
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or, more precisely,
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Next Factors
Now,
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Again, drawing from the binomial theorem, we have,
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Now Work on Elliptic Integral Expressions
From a separate discussion we can draw the series expansion of , specifically,
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where,
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Also,
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where,
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What we want to do is write in terms of
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where,
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Let's subtract from the potential expression. But first, let's adopt the shorthand notation …
Given that,
let's define the variable,
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We can therefore write,
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where we should keep in mind that is
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Now we have,
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But, as we have just demonstrated,
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TEMPORARY BREAK HERE |
Hence,
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Include Second Wong Term
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Leading (Upsilon) Coefficient
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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
Hence,
and,
Second,
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Geometric Factor
By definition,
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Hence,
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See Also
- Université de Bordeaux (Part 2): Spheroid-Ring Sequences
- Université de Bordeaux (Part 3): Discussions Following Dissertation Defense
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© 2014 - 2020 by Joel E. Tohline |