# User:Tohline/Appendix/Ramblings/Bordeaux

(Difference between revisions)
 Revision as of 15:29, 26 June 2020 (view source)Tohline (Talk | contribs) (→Second (n = 1) Term)← Older edit Current revision as of 17:05, 16 December 2020 (view source)Tohline (Talk | contribs) (205 intermediate revisions not shown) Line 1: Line 1: - =Université de Bordeaux= + =Université de Bordeaux (Part 1)= {{LSU_HBook_header}} {{LSU_HBook_header}} Line 8: Line 8: Through a research collaboration at the [https://www.u-bordeaux.com Université de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract 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.'' Through a research collaboration at the [https://www.u-bordeaux.com Université de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract 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 [[User:Tohline/Appendix/Ramblings/BordeauxSequences#Spheroid-Ring_Systems|separate, accompanying chapter]]. ==Exterior Gravitational Potential of Toroids== ==Exterior Gravitational Potential of Toroids== [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract 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 [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]]. [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract 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 [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]]. - ===Their Presentation=== + ===Our Presentation of Wong's (1973) Result=== - On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)] paper is §8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion, + -
+ - + + +
'''Summary:'''  First three terms in [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] expression for the gravitational potential at any point, P(ϖ, z), outside of a uniform-density torus.
+ + [[File:WongTorusIllustration02.png|500px|center|Wong diagram]] + + ---- + + -
- $~\Psi_\mathrm{grav}(\vec{r})$ + $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}$ - $~\approx$ + $~=$ $~ [itex]~ - \Psi_0 + \sum\limits_{n=1}^N \Psi_n \, , + - \biggl( \frac{2^{3} }{3\pi^3} \biggr) + \Upsilon_{W0}(\eta_0) \biggl\{ + \frac{a}{ r_1 } \cdot \boldsymbol{K}(k) \biggr\}\, ,$ [/itex]
- - [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §7, p. 5831, Eq. (42) - - where, after setting $~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c$ and acknowledging that $~V_{0,0} = 1 \, ,$ we can write, -
- -
- $~\Psi_0$ + $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}$ Line 47: Line 49: $~ [itex]~ - - \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}(k_0) \biggr] + - \biggl( \frac{2^{3} }{3\pi^3} \biggr) + \Upsilon_{W1}(\eta_0) \times \cos\theta + \biggl\{ \frac{a}{r_2} \cdot + \boldsymbol{E}(k) \biggr\} \, ,$ [/itex]
- - [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §8, p. 5832, Eqs. (52) & (53) -
- and, -
- Line 76: Line 78:
- $~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]$ + $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\varpi, z)\biggr|_\mathrm{exterior}$ Line 68: Line 66: $~ [itex]~ - - \frac{G\pi \rho_0 R_c b^2}{4 (k')^2 \Delta_0^3} \biggl\{ + - \biggl( \frac{2^{3} }{3\pi^3} \biggr)\Upsilon_{W2}(\eta_0) - [\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k) - (k')^2 \Delta_0^2 \boldsymbol{K}(k) + \times \cos(2\theta) - \biggr\} \, . + \biggl\{ + \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) + - + \frac{a}{r_1} \cdot \boldsymbol{K}(k) + \biggr\} \, ,$ [/itex]
- [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §8, p. 5832, Eq. (54) + 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 that the argument of the elliptic integral functions is, + -
+
- + + + + + + + + + + + + + + + + + + + + + Line 109: Line 152:
- $~k$ + $~a^2$ Line 91: Line 90: $~ [itex]~ - \frac{2\sqrt{\varpi R}}{\Delta} + R^2 - d^2$ +       and,       + $~\cosh\eta_0 \equiv \frac{R}{d} ~~~~\Rightarrow ~~~\sinh\eta_0 = \frac{a}{d} + \, ,$ [/itex]     where,
- $~\Delta$ + $~r_1^2$ + + $~\equiv$ + + $~(\varpi + a)^2 + (z - Z_0)^2 \, ,$ +
+ $~r_2^2$ + + $~\equiv$ + + $~(\varpi - a)^2 + (z - Z_0)^2 \, ,$ +
+ $~\cos\theta$ + + $~\equiv$ + + $~\biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \, ,$ +
+ $~k$ Line 103: Line 143: $~ [itex]~ - \biggl[ (R + \varpi)^2 + (Z-z)^2 \biggr]^{1 / 2} \, . + \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} + = \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} + = \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} + \, .$ [/itex]
- [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §2, p. 5826, Eqs. (4) & (5) + ---- -
+ - ===Our Presentation of Wong's (1973) Result=== + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Leading Coefficient Expressions …… evaluated for:   $~\frac{R}{d} = \cosh\eta_0 = 3$ +
+ $~\Upsilon_{W0}(\eta_0)$ + + $~\equiv$ + + $~ + \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ + K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] + + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] + - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] + \biggr\} \, , +$ + 7.134677
+ $~\Upsilon_{W1}(\eta_0)$ + + $~\equiv$ + + $~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] + \biggl\{ + K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] + +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] + -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] + \biggr\} + \, , +$ + 0.130324
+ $~\Upsilon_{W2}(\eta_0)$ + + $~\equiv$ + + $~ + \frac{2^{3 / 2}}{3^2} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ + K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13] + + 2 K ( k_0 ) \cdot E(k_0) [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ] +$ +
+   + +   + + $~ + -~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] + \biggr\} \, , +$ + 0.003153
where,
+ $~k_0$ + + $~\equiv$ + + $~ + \biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, . +$ + 0.707106781
+ NOTE:  In evaluating these "leading coefficient expressions" for the case, $~R/d = 3$, we have used the complete elliptic integral evaluations, '''K'''(k0) = 1.854074677 and  '''E'''(k0) = 1.350643881. +
+ + ====Setup==== ====Setup==== From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression, From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression, +
Line 134: Line 274: - + [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], §II.D, p. 294, Eqs. (2.59) & (2.61) +
where, where, +
Line 168: Line 310: + [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W 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, and where, in terms of the major ( R ) and minor ( d ) radii of the torus — or their ratio, ε ≡ d/R, Line 1,148: Line 1,292: - +
Line 1,252: Line 1,396: $~ [itex]~ - -~\frac{1}{2^2}\biggl\{ z k_0~K ( k_0 ) + -~\frac{1}{2^2} + \biggl\{ z k_0~K ( k_0 ) ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\}$ [/itex] Line 1,294: Line 1,439: $~ [itex]~ - 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) \, , + 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) +$ +
+   + + $~=$ + + $~ + 5 \biggl\{ [2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)] \biggr\} + + z \biggl\{ z k_0~K ( k_0 ) + ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} +$ +
+   + + $~=$ + + $~ + 2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} \biggl\{ [5 z] + ~-~z (z^2+3) \biggr\} E(k_0) + + \biggl\{ z^2 k_0~ + - [(z-1)(z^2-1)]^{-1 / 2} [ 2^{-3 / 2} \cdot 5 (z-1)] \biggr\}K(k_0) +$ +
+   + + $~=$ + + $~ + 2^{-3 / 2}(z+1)^{-1 / 2} [4 z^2 - 5 ]K(k_0) + -~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z E(k_0)$ [/itex] - $~\frac{2}{\coth\eta_0 + 1} + [itex]~\frac{2}{\cosh\eta_0 + 1} = = - \frac{2(z_0^2-1)^{1 / 2}}{z_0 + (z_0^2 - 1)^{1 / 2}} + \frac{2}{z_0 + 1} \, ,$ \, ,[/itex]
+ + + + + + + + + + + + + + + + + + Line 1,359: Line 1,552: Line 1,494: Line 1,687: + 5[2^3(z-1)(z^2-1)]^{-1 / 2} \cdot 4z^2k_0 + 5[2^3(z-1)(z^2-1)]^{-1 / 2} \cdot 4z^2k_0 + 5[2^3(z-1)(z^2-1)]^{-1 / 2}\cdot [2(z+1)]^{1 / 2} \cdot (z-1) + 5[2^3(z-1)(z^2-1)]^{-1 / 2}\cdot [2(z+1)]^{1 / 2} \cdot (z-1) - \biggr\} + \biggr\} \, . [/itex] [/itex] + + + Hence, +
- $~\Rightarrow ~~~ 2[(z-1)(z^2-1)]^{1 / 2} C_1(z_0)$ + $~2[(z-1)(z^2-1)]^{1 / 2} C_1(z_0)$ Line 1,509: Line 1,706: $~ [itex]~ z k_0 \cdot K(k_0)\cdot K(k_0) \biggl\{ \frac{k_0}{2^2} \biggl[(z-1)(z^2-1) \biggr]^{1 / 2} - \frac{5(z-1)}{2^{3/2}} \biggr\} z k_0 \cdot K(k_0)\cdot K(k_0) \biggl\{ \frac{k_0}{2^2} \biggl[(z-1)(z^2-1) \biggr]^{1 / 2} - \frac{5(z-1)}{2^{3/2}} \biggr\} - + -~10 z(z+1)^{1 / 2} \cdot E(k_0)\cdot E(k_0) -~10 z(z+1)^{1 / 2} \cdot E(k_0)\cdot E(k_0)$ [/itex] Line 1,524: Line 1,720: $~ [itex]~ - +~K(k_0)\cdot E(k_0) \biggl\{ + +~2^{-3/2} K(k_0)\cdot E(k_0) \biggl\{ - -~\frac{1}{2^2} \cdot k_0(z^2+3) \biggl[ 2 \biggr]^{1 / 2} + k_0[19z^2 - 3 ] - + 5[2^3]^{-1 / 2} \cdot 4z^2k_0 + + 5(z-1) [2(z+1)]^{1 / 2} - + 5[2^3]^{-1 / 2}\cdot [2(z+1)]^{1 / 2} \cdot (z-1) + \biggr\} \biggr\} +$ +
+ $~\Rightarrow ~~~2^{3/2}\biggl[ \frac{(z-1)}{k_0} \biggr] C_1(z_0)$ + + $~=$ + + $~ + z k_0 \cdot K(k_0)\cdot K(k_0) \biggl\{ \frac{k_0}{2^2} \biggl[\frac{2^{1 / 2}(z-1)}{k_0} \biggr] - \frac{5(z-1)}{2^{3/2}} \biggr\} + -~\biggl[ \frac{2^{3 / 2} \cdot 5z}{k_0} \biggr] E(k_0)\cdot E(k_0) +$ +
+   + +   + + $~ + +~2^{-3/2} K(k_0)\cdot E(k_0) \biggl\{ + k_0[19z^2 - 3 ] + + \frac{10 (z-1)}{k_0} + \biggr\} +$ +
+ $~\Rightarrow ~~~C_1(z_0)$ + + $~=$ + + $~ + \biggl[ \frac{2(3z^2 - 1)}{(z^2-1)} \biggr]K(k_0)\cdot E(k_0) + -~\biggl[ \frac{z}{(z+1)} \biggr] K(k_0)\cdot K(k_0) + -~\biggl[ \frac{ 5z}{(z-1)} \biggr] E(k_0)\cdot E(k_0) +$ +
+ $~\Rightarrow ~~~(z_0^2-1)C_1(z_0)$ + + $~=$ + + $~ + 2(3z^2 - 1) K(k_0)\cdot E(k_0) + -~z_0(z_0-1) K(k_0)\cdot K(k_0) + -~5z_0(z_0+1) E(k_0)\cdot E(k_0) \, .$ [/itex]
+ + + + + + + + + + + + + + + + + + + + + + + + + + Hence, we have, + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\eta,\theta)$ + + $~=$ + + $~ + - \frac{2^{3} a}{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2^2} \biggr] + \boldsymbol{E}(k) \, . +$ +
+ + + + + + + ====Third (n = 2) Term==== + + =====Part A===== + The third (n = 2) term in [[User:Tohline/Apps/Wong1973Potential#D0andCn|Wong's (1973) expression for the exterior potential]] is, + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)$ + + $~=$ + + $~ + -D_0 + (\cosh\eta - \cos\theta)^{1 / 2} \cdot 2 \cos(2\theta) \cdot C_2(\cosh\eta_0)P_{+\frac{3}{2}}(\cosh\eta) \, , +$ +
+ + + + + + where, $~D_0$ is the same as [[#Setup|above]], and, + + +
+ $~C_2(\cosh\eta_0)$ + + $~\equiv$ + + $~\tfrac{5}{2}Q_{+\frac{5}{2}}(\cosh \eta_0) Q_{+\frac{3}{2}}^2(\cosh \eta_0) + - \tfrac{1}{2} Q_{+\frac{3}{2}}(\cosh \eta_0)~Q^2_{+ \frac{5}{2}}(\cosh \eta_0) \, . +$ +
+ + + + + + + +
+ In order to evaluate $~C_2(z)$, we will need the following pair of expressions in addition to the ones already used: + + + + + + + +
+ $~Q^0_{m-\tfrac{1}{2}}$ [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|recurrence]] with m = 3, gives:      + $~Q_{+\tfrac{5}{2}}(z_0)$ + + $~=$ + + $~\frac{8}{5} z~Q_{+\tfrac{3}{2}}(z_0) - \frac{3}{5} Q_{+\tfrac{1}{2}}(z_0)$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\Rightarrow~~~15Q_{+\tfrac{5}{2}}(z_0)$ + + $~=$ + + $~8 z~\biggl[ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0) \biggr] + - + 9 \biggl[ z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0) \biggr]$ +
+   + + $~=$ + + $~ + z~k_0 K(k_0) \biggl[ 8(4z^2 - 1 ) - 9 \biggr] + + + [2(z+1)]^{1 / 2} E(k_0) \biggl[-32z^2 + 9 \biggr] +$ +
+   + + $~=$ + + $~ + z~k_0 K(k_0) [ 32z^2 - 17 ] + + + [2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ] \, . +$ +
+ Hence,     $~Q_{+\frac{5}{2}}(3)$ + + $~=$ + + $~0.002080867 \, .$ +
+ + And, setting m = 2 in the [[#Qrecurrence|above recurrence relation for]] $~Q^2_{m+\frac{1}{2}}(z)$ gives, + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\biggl[ (m - \tfrac{3}{2})Q^{2}_{m+\tfrac{1}{2}} (z) \biggr]_{m=2}$ + + $~=$ + + $~ + \biggl[ (2m)z Q_{m-\tfrac{1}{2}}^{2}(z) - (m + \tfrac{3}{2})Q^{2}_{m - \tfrac{3}{2}}(z) \biggr]_{m=2} +$ +
+ $~\Rightarrow ~~~ Q^{2}_{+\tfrac{5}{2}} (z)$ + + $~=$ + + $~ + 8z Q_{+\tfrac{3}{2}}^{2}(z) - 7 Q^{2}_{+ \tfrac{1}{2}}(z) +$ +
+   + + $~=$ + + $~ + 8z \biggl[ 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) \biggr] - 7 Q^{2}_{+ \tfrac{1}{2}}(z) +$ +
+   + + $~=$ + + $~ + 40z Q^{2}_{- \tfrac{1}{2}}(z_0) + - [32z^2 +7]Q_{+\tfrac{1}{2}}^{2}(z_0) +$ +
+   + + $~=$ + + $~ + 40z \biggl\{ + [2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)] + \biggr\} +$ +
+   + +   + + $~ + + \frac{[32z^2 +7]}{4} \biggl\{ + z k_0~K ( k_0 ) + ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0) + \biggr\} +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\Rightarrow ~~~ 4Q^{2}_{+\tfrac{5}{2}} (z)$  + $~=$ + + $~ + 2^5\cdot 5z \biggl\{ 2^{1 / 2} + [(z-1)(z^2-1)]^{-1 / 2} [zE(k_0) ] + - + 2^{-3 / 2}[(z-1)(z^2-1)]^{-1 / 2} [(z-1)K(k_0)] + \biggr\} +$ +
+   + +   + + $~ + + [32z^2 +7] \biggl\{ + z k_0~K ( k_0 ) + ~-~2^{1 / 2}(z^2+3) [ (z-1)(z^2-1) ]^{-1 / 2} E(k_0) + \biggr\} +$ +
+   + + $~=$ + + $~ + \biggl\{ + 2^{11 / 2}\cdot 5 [z^2 ] - 2^{1 / 2} [32z^2 +7] (z^2+3) + \biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0) +$ +
+   + +   + + $~ + -~2^{7 / 2}\cdot 5 \biggl\{ [(z-1)(z^2-1)]^{-1 / 2} (z-1) \biggr\} z K(k_0) + + [32z^2 +7] \biggl\{ 2^{1 / 2}[z + 1]^{-1 / 2} \biggr\} z K ( k_0 ) +$ +
+   + + $~=$ + + $~ + 2^{1 / 2}\biggl\{ 32z^2 - 33 \biggr\} z [z + 1]^{-1 / 2} K ( k_0 ) + -~2^{1 / 2} + \biggl\{ + 32z^4 - 57 z^2 + 21 + \biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0) +$ +
+   + + $~=$ + + $~ + 2^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33) z (z-1) K ( k_0 ) + -~(32z^4 - 57 z^2 + 21)E(k_0) \biggr] + \, . +$ +
+ Hence,     $~Q^2_{+\frac{5}{2}}(3)$ + + $~=$ + + $~0.03377378 \, .$ +
+ +
+ + =====Part B===== + + Let's evaluate $~C_2(z)$ specifically for the case where $~z = \cosh\eta_0 = 3$, using the already separately evaluated values of the four relevant toroidal functions.  We find, + + +
+ $~2C_2(3)$ + + $~=$ + + $~5Q_{+\frac{5}{2}}(3) Q_{+\frac{3}{2}}^2(3) + - Q_{+\frac{3}{2}}(3)~Q^2_{+ \frac{5}{2}}(3) +$ +
+   + + $~=$ + + $~ + 5\cdot ( 0.002080867 ) \times ( 0.132453829 ) - ( 0.014544576 ) \times (0.03377378 ) +$ +
+   + + $~=$ + + $~ + 8.868687\times 10^{-4} \, . +$ +
+ + + + + + + + + + + + + + + + + + + Next, let's develop a consolidated expression for $~C_2(z_0)$ that replaces all the toroidal functions with complete elliptic integrals of the first and second kind. + + +
+ $~2C_2(z_0)$ + + $~=$ + + $~5Q_{+\frac{5}{2}}(z_0) Q_{+\frac{3}{2}}^2(z_0) + - Q_{+\frac{3}{2}}(z_0)~Q^2_{+ \frac{5}{2}}(z_0) +$ +
+   + + $~=$ + + $~ + \frac{1}{3}\biggl\{ + z~k_0 K(k_0) [ 32z^2 - 17 ] + + + [2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ] + \biggr\} + \times \biggl\{ + 2^{-3 / 2}(z+1)^{-1 / 2} [4 z^2 - 5 ]K(k_0) + -~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z E(k_0) + \biggr\} +$ +
+   + +   + + $~ + - \frac{1}{2^2\cdot 3} + \biggl\{ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0) \biggr\} + \times \biggl\{ + 2^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33) z (z-1) K ( k_0 ) + -~(32z^4 - 57 z^2 + 21)E(k_0) \biggr] + \biggr\} +$ +
+ + + + + + + + + + + + + + + + + + + + +
+ $~\Rightarrow ~~~ 2^{2} \cdot 3 (z^2-1) C_2(z_0)$ + + $~=$ + + $~ + \biggl\{ + K(k_0) z[ 32z^2 - 17 ] + + + (z+1) E(k_0) [9 -32z^2 ] + \biggr\} + \times \biggl\{ + (z-1) [4 z^2 - 5 ]K(k_0) + -~4 (z^2 - 2)z E(k_0) + \biggr\} +$ +
+   + +   + + $~ + - ~ + \biggl\{ (4z^2 - 1 ) K(k_0) - 4 z(z+1) E(k_0) \biggr\} + \times \biggl\{ + (32z^2 - 33) z (z-1) K ( k_0 ) + -~(32z^4 - 57 z^2 + 21)E(k_0) + \biggr\} +$ +
+   + + $~=$ + + $~ + \biggl\{ + (z-1)[ 32z^2 - 17 ] [4 z^2 - 5 ]z K(k_0) \cdot K(k_0) + -~4 (z^2 - 2)z^2 [ 32z^2 - 17 ] K(k_0) \cdot E(k_0) + \biggr\} +$ +
+   + +   + + $~ + + \biggl\{ + (z-1) (z+1) [9 -32z^2 ] [4 z^2 - 5 ]K(k_0) \cdot E(k_0) + -~4 (z^2 - 2)z (z+1) [9 -32z^2 ] E(k_0) \cdot E(k_0) + \biggr\} +$ +
+   + +   + + $~ + + ~ + \biggl\{ + (32z^4 - 57 z^2 + 21)(4z^2 - 1 ) K(k_0) \cdot E(k_0) + -~(32z^2 - 33) z (z-1)(4z^2 - 1 ) K ( k_0 ) \cdot K(k_0) + \biggr\} +$ +
+   + +   + + $~ + + ~ + \biggl\{ + 4 z(z+1)(32z^2 - 33) z (z-1) K ( k_0 ) \cdot E(k_0) + -~4 z(z+1)(32z^4 - 57 z^2 + 21)E(k_0) \cdot E(k_0) + \biggr\} +$ +
+   + + $~=$ + + $~(z-1)\biggl\{ + \biggl[ + ( 32z^2 - 17 ) (4 z^2 - 5 )z \biggr] + -~\biggl[ (32z^2 - 33) z (4z^2 - 1 ) \biggr] \biggr\} K ( k_0 ) \cdot K(k_0) +$ +
+   + +   + + $~ + + \biggl\{ \biggl[ + (z-1) (z+1) (9 -32z^2 ) (4 z^2 - 5 )\biggr] + -~\biggl[ 4 (z^2 - 2)z^2 ( 32z^2 - 17 ) \biggr] +$ +
+   + +   + + $~ + + ~ + \biggl[ + (32z^4 - 57 z^2 + 21)(4z^2 - 1 ) \biggr] + + ~ \biggl[ 4 z(z+1)(32z^2 - 33) z (z-1)\biggr]\biggr\} K ( k_0 ) \cdot E(k_0) +$ +
+   + +   + + $~ + -~2z(z+1) \biggl\{ \biggl[ 2 (32z^4 - 57 z^2 + 21) \biggr] + +~2\biggl[ (z^2 - 2) (9 -32z^2 )\biggr] \biggr\} E(k_0) \cdot E(k_0) +$ +
+   + + $~=$ + + $~ + z(z-1)\biggl\{[ 52 - 64z^2 ] \biggr\} K ( k_0 ) \cdot K(k_0) + -~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0) +$ +
+   + +   + + $~ + + \biggl\{ + \biggl[ 5(32z^4 - 41z^2 + 9 ) \biggr] + - \biggl[ (32z^4 - 57 z^2 + 21)\biggr] +$ +
+   + +   + + $~ + + ~ + 4z^2\biggl[ (32z^4 - 57 z^2 + 21) + + (32z^4 - 65z^2 + 33) + (-32z^4 + 41z^2 -9 ) +~( -32z^4 + 81z^2 - 34 ) + \biggr]\biggr\} K ( k_0 ) \cdot E(k_0) +$ +
+   + + $~=$ + + $~ + 4z(z-1)\biggl\{ 13 - 16z^2 \biggr\} K ( k_0 ) \cdot K(k_0) + -~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0) + + 8\biggl\{ + 16z^4 -13z^2 + 3 \biggr\} K ( k_0 ) \cdot E(k_0) \, . +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Finally, let's evaluate this consolidated expression for the specific case of $~z_0 = \cosh\eta_0 = 3$, remembering that in this specific case $~k_0 = 2^{-1 / 2}$, $~K(k_0) = 1.854074677$, and $~E(k_0) = 1.350643881$.  We find, + + +
+ $~2C_2(z_0)$ + + $~=$ + + $~ + [2 \cdot 3 (z^2-1) ]^{-1} \biggl\{ + 4z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0) + -~4z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0) + + 8[ + 16z^4 -13z^2 + 3 ] K ( k_0 ) \cdot E(k_0) + \biggr\} +$ +
+   + + $~=$ + + $~ + [48 ]^{-1} \biggl\{ + -24[ 131 ] K ( k_0 ) \cdot K(k_0) + -~48 [ 147] E(k_0) \cdot E(k_0) + + 8[ 1182 ] K ( k_0 ) \cdot E(k_0) + \biggr\} +$ +
+   + + $~=$ + + $~ + 8.8708 \times 10^{-4} \, . +$ +
+ + + + + + + + + + + + + + + + + + + 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. + + =====Part C===== + + Next … + + +
+
'''Useful Relations from Above'''
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\cosh\eta$ + + $~=$ + + $~\frac{r_1^2 + r_2^2}{2r_1 r_2} \, ;$ +
+ $~\sinh\eta$ + + $~=$ + + $~\frac{r_1^2 - r_2^2}{2r_1 r_2} \, ;$ +
+ $~\varpi$ + + $~=$ + + $~\frac{r_1^2 - r_2^2}{2a} \, ;$ +
+ $~\cosh\eta - \cos\theta$ + + $~=$ + + $~\frac{2a^2}{r_1 r_2} \, ;$ +
+ $~ \cos\theta$ + + $~=$ + + $~\frac{r_1^2 + r_2^2-4a^2}{2r_1 r_2} \, ;$ +
+ $~\frac{2}{\coth\eta + 1}$ + + $~=$ + + $~\frac{4a\varpi}{r_1^2} \, .$ +
+ +
+ + + + Now, from our tabulation of [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|example recurrence relations]], we see that, + + +
+ $~ P_{+\frac{3}{2}}(\cosh\eta)$ + + $~=$ + + $~\frac{4}{3} \cdot \cosh\eta ~P_{+\frac{1}{2}}(\cosh\eta) - \frac{1}{3} ~ P_{-\frac{1}{2}}(\cosh\eta)$ +
+   + + $~=$ + + $~\frac{4}{3} \cdot \cosh\eta \biggl[ \frac{\sqrt{2}}{\pi} (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) \biggr] + - + \frac{1}{3} ~ \biggl[ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr]$ +
+   + + $~=$ + + $~\frac{2^{1 / 2}}{3\pi} + \biggl[ 4 \cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) + - + (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr] + \, ,$ +
+ + + + + + + + + + + + + + + + + + + where, as above, +
+ $~ + k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} + = + \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} + = + \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} \, . +$ +
+ So we have, + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)$ + + $~=$ + + $~ + -\frac{2^{5/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] + C_2(\cosh\eta_0)\cos(2\theta) + \biggl\{ (\cosh\eta - \cos\theta)^{1 / 2}P_{+\frac{3}{2}}(\cosh\eta) \biggr\} +$ +
+   + + $~=$ + + $~ + -\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] + C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2} + \biggl\{ + 4 \cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) + - + (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) + \biggr\} +$ +
+   + + $~=$ + + $~ + -\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] + C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2} +$ +
+   + +   + + $~ + \times + \biggl\{ + 4 \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \biggl[\frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^{+1 / 2} \biggl[ \frac{4a}{r_1^2} \biggl( \frac{r_1^2 - r_2^2}{2a} \biggr)\biggr]^{-1 / 2} \boldsymbol{E}(k) + - + \biggl[ \frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^{-1 / 2} ~\biggl[ \frac{4a}{r_1^2}\biggl( \frac{r_1^2 - r_2^2}{2a} \biggr) \biggr]^{1 / 2} \boldsymbol{K}(k) + \biggr\} +$ +
+   + + $~=$ + + $~ + -\frac{2^{9/2} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] + C_2(\cosh\eta_0)\cos(2\theta) + \times + \biggl\{ + \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) + - + \frac{a}{r_1} \cdot \boldsymbol{K}(k) + \biggr\} \, . +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Finally, inserting the expression for $~\sinh^2\eta_0~ C_2(\cosh\eta_0) = (z_0^2-1)C_2(z_0)$ that we have derived, above, gives, + + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)$ + + $~=$ + + $~ + -\frac{2^{9/2} }{3^3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] + \times \cos(2\theta) + \biggl\{ + \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) + - + \frac{a}{r_1} \cdot \boldsymbol{K}(k) + \biggr\} +$ +
+   + +   + + $~ + \times \biggl\{ + z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0) + -~z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0) + + 2 [ 16z^4 -13z^2 + 3 ] K ( k_0 ) \cdot E(k_0) + \biggr\} \, . +$ +
+ + + + + + + + + + + + + ====Summary==== + + Once the major ( R ) and minor ( d ) radii of the torus have been specified, two key model parameters can be immediately determined, namely, +
+ $~a^2 \equiv R^2 - d^2\, ,$       and,       $~\cosh\eta_0 \equiv \frac{R}{d} \, ,$ +
+ in which case also, $~\sinh\eta_0 = a/d \, .$  Once the mass-density ( ρ0 ) of the torus has been specified, the torus mass is given by the expression, +
+ $~M = 2\pi^2 \rho_0 d^2 R \, .$ +
+ In addition to the principal pair of meridional-plane coordinates, $~(\varpi, z)$, it is useful to define the pair of distances, + + +
+ $~r_1^2$ + + $~\equiv$ + + $~(\varpi + a)^2 + (z - Z_0)^2 \, ,$ +
+ $~r_2^2$ + + $~\equiv$ + + $~(\varpi - a)^2 + (z - Z_0)^2 \, ,$ +
+ + + + + + + + + + + + where, the equatorial plane of the torus is located at $~z = Z_0$.  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, + + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}$ + + $~=$ + + $~ + - \frac{2^{3} }{3\pi^3} + \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] + \frac{a}{ r_1 } \cdot \boldsymbol{K}(k) +$ +
+   + +   + + $~ + \times \biggl\{ + K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] + + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] + - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] + \biggr\} \, . +$ +
+ + + + + + + + + + + + + where, the two distinctly different arguments — one with, and one without a zero subscript — of the complete elliptic-integral functions are, + + +
+ $~k$ + + $~\equiv$ + + $~ + \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} + = + \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} + = \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} + = \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} \, , +$ +
+ $~k_0$ + + $~\equiv$ + + $~ + \biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, . +$ +
+ + + + + + + + + + + + + As we also have shown above, the second (n = 1) term in Wong's (1973) series-expression for the exterior gravitational potential is, + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}$ + + $~=$ + + $~ + - \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot + \boldsymbol{E}(k) +$ +
+   + +   + + $~\times + \biggl\{ + K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] + +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] + -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] + \biggr\} + \, . +$ +
+ + + + + + + + + + + + Note that a transformation from the $~(r_1, r_2)$ coordinate pair to the toroidal-coordinate pair $~(\eta, \theta)$ includes the expression, + + +
+ $~\cos\theta$ + + $~=$ + + $~ + \frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, . +$ +
+ + + + + + 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, + + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)$ + + $~=$ + + $~ + -\frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] + \times \cos(2\theta) + \biggl\{ + \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) + - + \frac{a}{r_1} \cdot \boldsymbol{K}(k) + \biggr\} +$ +
+   + +   + + $~ + \times \frac{2^{3 / 2}}{3^2}\biggl\{ + K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13] + + 2 K ( k_0 ) \cdot E(k_0) [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ] +$ +
+   + +   + + $~ + -~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] + \biggr\} \, . +$ +
+ + + + + + + + + + + + + + + + + + + ===The Huré, ''et al'' (2020) Presentation=== + + {{LSU_WorkInProgress}} + + ====Notation==== + + In [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract 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, +
+ $~e \equiv \frac{b}{R_c} \, .$ + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §2, p. 5826, Eq. (1) +
+ The authors work in cylindrical coordinates, $~(R, Z)$, whereas we refer to this same coordinate-pair as, $~(\varpi_W, z_W)$.  The quantity, +
+ + +
+ $~\Delta^2$ + + $~\equiv$ + + $~ + [R + (R_c + b\cos\theta_H)]^2 + [Z - b\sin\theta_H]^2 \, . +$ +
+ + + + + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract 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 [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] work. In their paper, the subscript "0" is used in the case of an infinitesimally thin hoop $~(b \rightarrow 0)$, that is to say, +
+ + +
+ $~\Delta_0^2$ + + $~=$ + + $~ + [R + R_c]^2 + Z^2 \, . +$ +
+ + + + + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §3, p. 5827, Eq. (13) + + + Generally, the argument (modulus) of the complete elliptic integral functions is, +
+ + +
+ $~k_H$ + + $~=$ + + $~ + \frac{2}{\Delta}\biggl[ R (R_c + b\cos\theta_H) \biggr]^{1 / 2} \, , +$ +
+ + + + + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract 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'', + + +
+ $~k'_H$ + + $~\equiv$ + + $~[1 - k_H^2]^{1 / 2} \, .$ +
+ + + + + + + (Again, we have affixed the subscript "H" in order to differentiate from the modulus employed by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].)  And in the case of an infinitesimally thin hoop $~(b\rightarrow 0)$, +
+ + +
+ $~[k^2_H]_0$ + + $~=$ + + $~ + \frac{4R R_c}{\Delta_0^2} \, . +$ +
+ + + + + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract 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 [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)] paper is §8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion, +
+ + +
+ $~\Psi_\mathrm{grav}(\vec{r})$ + + $~\approx$ + + $~ + \Psi_0 + \sum\limits_{n=1}^N \Psi_n \, , +$ +
+ + + + + + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §7, p. 5831, Eq. (42) + + where, after setting $~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c$ and acknowledging that $~V_{0,0} = 1 \, ,$ we can write, +
+ + +
+ $~\Psi_0$ + + $~=$ + + $~ + - \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}([k_H]_0) \biggr] +$ +
+ + + + + + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §8, p. 5832, Eqs. (52) & (53) + + and, +
+ + +
+ $~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]$ + + $~=$ + + $~ + - \frac{G\pi \rho_0 R_c b^2}{4 (k'_H)^2 \Delta_0^3} \biggl\{ + [\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H) + \biggr\} \, . +$ +
+ + + + + + + [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract 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, + + +
+ $~\frac{2^3\pi}{e^2} \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]$ + + $~=$ + + $~ + - \frac{ 1 }{(k'_H)^2 \Delta_0^3}\biggl\{ + [\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H) + \biggr\} +$ +
+   + + $~=$ + + $~ + - \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} + \frac{\boldsymbol{K}(k_H)}{\Delta_0} \, . +$ +
+ + + + + + + + + + + + + Hence, also, + + +
+ $~ \frac{ \Psi_0 }{GM} + \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]$ + + $~=$ + + $~- \frac{2}{\pi}\biggl\{ + \frac{\boldsymbol{K}([k_H]_0)}{\Delta_0} + \biggr\} + + \frac{e^2}{2^3\pi}\biggl\{ + \frac{\boldsymbol{K}(k_H)}{\Delta_0} + - \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} + \biggr\} +$ +
+   + + $~=$ + + $~- \frac{2}{\pi \Delta_0}\biggl\{ + \boldsymbol{K}([k_H]_0) + - + \frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) \biggr\} + - \frac{2}{\pi\Delta_0} \cdot \frac{e^2}{2^4}\biggl\{\biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] + \biggr\} \boldsymbol{E}(k_H) +$ +
+ $~\Rightarrow ~~~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr]$ + + $~=$ + + $~ + \boldsymbol{K}([k_H]_0) + - + \frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) + + \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \boldsymbol{E}(k_H) \, . +$ +
+ + + + + + + + + + + + + + + + + + + ===Compare First Terms=== + + Rewriting the first term in the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)] series expression for the potential, we have, + + +
+ $~\frac{\Psi_0}{GM}$ + + $~=$ + + $~ + - \frac{2}{\pi} \biggl\{ \frac{\boldsymbol{K}([k_H]_0) }{[ (\varpi_W + R_c)^2 + z_W^2]^{1 / 2}} \biggr\} \, , +$ +
+ + + + + + where, + + +
+ $~[k_H]_0$ + + $~=$ + + $~ + \biggl[ \frac{4\varpi_W R_c}{\Delta_0^2} \biggr]^{1 / 2} + = + \biggl\{ \frac{4\varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2} \biggr\}^{1 / 2} \, . +$ +
+ + + + + + + For comparison, the first term in Wong's expression is, + +
+ $~\frac{\Phi_\mathrm{W0}}{GM}$ + + $~=$ + + $~ + - \biggl( \frac{2^{3} }{3\pi^3} \biggr) + \Upsilon_{W0}(\eta_0) \biggl\{ + \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\, , +$ +
+ + + + + + where, + + +
+ $~a^2$ + + $~\equiv$ + + $~ + R^2 - d^2 ~~~\Rightarrow ~~~ a = R_c(1 - e^2)^{1 / 2} \, , +$ +
+ $~r_1^2$ + + $~\equiv$ + + $~\biggl[ \varpi + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + \biggl[z - Z_0 \biggr]^2 \, ,$ +
+ $~k$ + + $~\equiv$ + + $~ + \biggl\{ \frac{4\varpi R_c(1-e^2)^{1 / 2}}{[\varpi + R_c(1-e^2)^{1 / 2}]^2 + [z - Z_0]^2} \biggr\}^{1 / 2} + \, , +$ +
+ $~\Upsilon_{W0}(\eta_0)$ + + $~=$ + + $~ + \frac{\sinh\eta_0}{\cosh\eta_0}\biggl\{ + K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] + + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] + - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] + \biggr\} \, , +$ +
+   + + $~=$ + + $~ + \frac{(1-e^2)^{1 / 2}}{e^2} \biggl\{ + - K(k_0)\cdot K(k_0) (1-e) + + 2K(k_0)\cdot E(k_0) (1+e^2) + - E(k_0)\cdot E(k_0) (1+e) + \biggr\} \, , +$ +
+ $~k_0$ + + $~=$ + + $~ + \biggl[ \frac{2}{1+1/e} \biggr]^{1 / 2} + = + \biggl[ \frac{2e}{1+e} \biggr]^{1 / 2} \, . +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + This expression is correct for any value of the aspect ratio, $~e$.  But let's set $~Z_0 = 0$ — as Huré, et al. (2020) have done — then see how the expression simplifies for an infinitesimally thin hoop, that is, if we let $~e \rightarrow 0$.  First we note that, + + + +
+ $~k\biggr|_{e\rightarrow 0}$ + + $~=$ + + $~ + \biggl\{ \frac{4\varpi R_c}{[\varpi + R_c]^2 + z^2} \biggr\}^{1 / 2} + \, , +$ +
+ + + + + + 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), $~[k_H]_0$.  Next, we note that, + + +
+ $~r_1\biggr|_{e\rightarrow 0}$ + + $~=$ + + $~[( \varpi + R_c )^2 + z^2]^{1 / 2} \, .$ +
+ + + + + + As a result, we can write, + +
+ $~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}$ + + $~=$ + + $~ + \frac{\Psi_0}{GM} \cdot + \biggl[ + \biggl( \frac{2^{2} }{3\pi^2} \biggr) + \Upsilon_{W0}(\eta_0) + \biggr]_{e\rightarrow 0} \, . +$ +
+ + + + + + Now let's evaluate the coefficient, $~\Upsilon_{W0}$, in the limit of $~e \rightarrow 0$. + + +
+ +
+ $~\Upsilon_{W0}$, in the limit of $~e \rightarrow 0$. +
+ First, drawing from our [[User:Tohline/Apps/Wong1973Potential#Phase_0C|separate examination of the behavior of complete elliptic integral functions]], we appreciate that, + + + + + + + + + + + + + + + + + + + + +
+ $~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot E(k_0) \biggr]$ + + $~=$ + + $~ + 1 ~+~\frac{1}{2^5} ~k_0^4 + ~+~\frac{1}{2^5} ~ k_0^6 + + \mathcal{O}(k_0^{8}) \, , +$ +
+ $~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot K(k_0) \biggr]$ + + $~=$ + + $~ + 1 + \frac{1}{2} k_0^2 + + \frac{11}{2^5} ~k_0^4 + + \frac{17}{2^6} ~ k_0^6 + + \mathcal{O}(k_0^{8}) \, , +$ +
+ $~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot E(k_0) \biggr]$ + + $~=$ + + $~ + 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~-~\frac{1}{2^6} ~ k_0^6 + + \mathcal{O}(k_0^{8}) \, . +$ +
+ Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that, + + + + + + + + + + + + + +
+ $~k_0^2$ + + $~=$ + + $~2e(1+e)^{-1}$ +
+   + + $~=$ + + $~2e( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;$ +
+ and, + + + + + + + + + + + + + + +
+ $~k_0^4$ + + $~=$ + + $~4e^2(1+e)^{-2}$ +
+   + + $~=$ + + $~4e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, .$ +
+ Hence, we have, + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)$ + + $~=$ + + $~ + - (1-e) \biggl[ 1 + \frac{1}{2} k_0^2 + + \frac{11}{2^5} ~k_0^4 + + \mathcal{O}(k_0^{6}) + \biggr] +$ +
+   + +   + + $~ + + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4 + + \mathcal{O}(k_0^{6}) + \biggr] +$ +
+   + +   + + $~ + - (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~ + + \mathcal{O}(k_0^{6}) + \biggr] +$ +
+   + + $~=$ + + $~ + - (1-e) \biggl[ 1 + e ( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) + + \frac{11}{2^3} \cdot ~e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) + + \mathcal{O}(e^{3}) + \biggr] +$ +
+   + +   + + $~ + + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) + + \mathcal{O}(e^{3}) + \biggr] +$ +
+   + +   + + $~ + - (1+e) \biggl[ 1 - e ~( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) ~ + -~ \frac{1}{2^3} \cdot~ e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) ~ + + \mathcal{O}(e^{3}) + \biggr] +$ +
+   + + $~=$ + + $~ + - (1-e) \biggl[ 1 + e ( 1 - e ) + + \frac{11}{2^3} \cdot ~e^2 + \biggr] + + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2 + \biggr] + - (1+e) \biggl[ 1 - e ~( 1 - e ) ~ + -~ \frac{1}{2^3} \cdot~ e^2 ~ + \biggr] + + \mathcal{O}(e^{3}) +$ +
+   + + $~=$ + + $~ + - \biggl[ 1 + e ( 1 - e ) + + \frac{11}{2^3} \cdot ~e^2 + \biggr] + +e \biggl[ 1 + e + \biggr] + + 2 \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2 + \biggr] + + 2 e^2 + - \biggl[ 1 - e ~( 1 - e ) ~ + -~ \frac{1}{2^3} \cdot~ e^2 ~ + \biggr] + - e \biggl[ 1 - e + \biggr] + + \mathcal{O}(e^{3}) +$ +
+   + + $~=$ + + $~ + -1 - e + e^2 + - \frac{11}{2^3} \cdot ~e^2 + + e + e^2 + + 2 ~+~\frac{1}{2^2} \cdot~e^2 + + 2 e^2 + -1 + e - e^2 ~ + +~ \frac{1}{2^3} \cdot~ e^2 ~ + - e +e^2 + + \mathcal{O}(e^{3}) +$ +
+   + + $~=$ + + $~\frac{e^2}{2^3} + \biggl[ 2^5 + - 11~ + ~+~3 \biggr]~ + + \mathcal{O}(e^{3}) +$ +
+   + + $~=$ + + $~ + 3e^2 + + \mathcal{O}(e^{3}) +$ +
+ $~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)$ + + $~=$ + + $~ + [1 + \mathcal{O}(e^{1})]\cdot (1 - e^2)^{1 / 2} +$ +
+ $~\Rightarrow ~~~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}$ + + $~=$ + + $~ + 1 \, .$ +
+ +
+ + Given that, + + +
+ $~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}$ + + $~=$ + + $~ + 1 \, ,$ +
+ + + + + + we conclude that, + + +
+ $~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}$ + + $~=$ + + $~ + \frac{\Psi_0}{GM} \, , +$ +
+ + + + + + that is, we conclude that $~\Psi_0$ matches $~\Phi_{W0}$ in the limit of, $~e\rightarrow 0$. + + ===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, + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~=$ + + $~ + \Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr) + \Upsilon_{W0}(\eta_0) \biggl\{ + \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\, . +$ +
+ + + + + + + + ====Keeping Higher Order in Wong's First Component==== + + + + +
+ $~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot E(k_0) \biggr]$ + + $~=$ + + $~ + 1 ~+~\frac{1}{2^5} ~k_0^4 + ~+~\frac{1}{2^5} ~ k_0^6 + ~+~\frac{231}{2^{13}} ~ k_0^8 + + \mathcal{O}(k_0^{10}) \, , +$ +
+ $~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot K(k_0) \biggr]$ + + $~=$ + + $~ + 1 + \frac{1}{2} k_0^2 + + \frac{11}{2^5} ~k_0^4 + + \frac{17}{2^6} ~ k_0^6 + + \frac{1787}{2^{13}} ~k^8 + + \mathcal{O}(k_0^{10}) + \, , +$ +
+ $~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot E(k_0) \biggr]$ + + $~=$ + + $~ + 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 + ~-~\frac{1}{2^6} ~ k_0^6 + ~-~\frac{77}{2^{13}} ~ k_0^8 + + \mathcal{O}(k_0^{10}) \, . +$ +
+ + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\frac{2^2}{\pi^2} \biggl[K(k) \cdot E(k) \biggr]$ + + $~=$ + + $~ + \biggl\{ + 1 + \biggl( \frac{1}{2} \biggr)^2k^2 + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 + + \cdots + + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} + + \cdots + \biggr\} +$ +
+   + +   + + $~ + \times~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 + - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} + - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} + ~-~ \cdots + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} + ~-~ \cdots + \biggr\} +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 + - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} + - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} + \biggr\} + ~+~ + \biggl\{ + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggr\} + \times~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 + - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} + \biggr\} +$ +
+   + +   + + $~+~ + \biggl\{ + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + \biggr\} + \times~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 + \biggr\} + ~+~ + \biggl\{ + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + \biggr\}\times + \biggl\{1 - \frac{1}{2^2} ~k^2 + \biggr\} + + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{3}{2^6}~ k^4 + - \biggl(\frac{5}{2^8}\biggr)~k^6 + - \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)~k^8 + \biggr\} + ~+~ + \biggl\{ + \biggl( \frac{1}{2^2} \biggr)k^2 + - \frac{1}{2^4} ~k^4 + - \frac{3}{2^8}~ k^6 + -\frac{5}{2^{10}} ~k^8 + \biggr\} +$ +
+   + +   + + $~ + ~+~ + \biggl( \frac{3^2}{2^6}\biggr) k^4 + ~-~ + \biggl( \frac{3^2}{2^8}\biggr) k^6 + ~-~ \biggl(\frac{3^3}{2^{12}}\biggr) ~k^8 + ~+~ + \biggl( \frac{5^2}{2^8}\biggr) k^6 + ~-~\biggl(\frac{5^2}{2^{10}}\biggr) ~k^8 + ~+~\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) ~k^8 + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + 1 + ~+~\biggl[ \frac{1}{2^2} + - \frac{1}{2^2} \biggr] ~k^2 + ~+~\biggl[ \frac{3^2}{2^6} + - \frac{3}{2^6} + - \frac{1}{2^4} \biggr]~k^4 + ~+~\biggl[ \frac{5^2}{2^8} + - \frac{5}{2^8} + - \frac{3}{2^8} + ~-~\frac{3^2}{2^8} \biggr]~ k^6 + + \biggl[ + \biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) - \biggl(\frac{5^2}{2^{10}}\biggr) - \biggl(\frac{3^3}{2^{12}}\biggr) - \frac{5}{2^{10}} - \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr) + \biggr]~k^8 + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + 1 ~+~\frac{1}{2^5} ~k^4 + ~+~\frac{1}{2^5} ~ k^6 + ~+~\frac{231}{2^{13}} ~ k^8 + + \mathcal{O}(k^{10}) +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\frac{2^2}{\pi^2} \biggl[K(k) \cdot K(k) \biggr]$ + + $~=$ + + $~ + \biggl\{ + 1 + \biggl( \frac{1}{2} \biggr)^2k^2 + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 + + \cdots + + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} + + \cdots + \biggr\} +$ +
+   + +   + + $~ + \times~ + \biggl\{ + 1 + \biggl( \frac{1}{2} \biggr)^2k^2 + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 + + \cdots + + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} + + \cdots + \biggr\} +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 + + \frac{1}{2^2} k^2 + + \frac{3^2}{2^6} k^4 + + \frac{5^2}{2^8} k^6 + + \frac{5^2 \cdot 7^2}{2^{14}} k^8 + \biggr\} + \times~ + \biggl\{ + 1 + + \frac{1}{2^2} k^2 + + \frac{3^2}{2^6} k^4 + + \frac{5^2}{2^8} k^6 + + \frac{5^2 \cdot 7^2}{2^{14}} k^8 + \biggr\} + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 + + \frac{1}{2^2} k^2 + + \frac{3^2}{2^6} k^4 + + \frac{5^2}{2^8} k^6 + + \frac{5^2 \cdot 7^2}{2^{14}} k^8 + \biggr\} + ~+~\frac{1}{2^2} k^2 + \biggl\{ + 1 + + \frac{1}{2^2} k^2 + + \frac{3^2}{2^6} k^4 + + \frac{5^2}{2^8} k^6 + \biggr\} + ~+~\frac{3^2}{2^6} k^4 + \biggl\{ + 1 + + \frac{1}{2^2} k^2 + + \frac{3^2}{2^6} k^4 + \biggr\} + ~+~ + \biggl\{ + \frac{5^2}{2^8} k^6 + \biggr\} + \biggl\{ + 1 + + \frac{1}{2^2} k^2 + \biggr\} + ~+~ + \biggl\{ + \frac{5^2 \cdot 7^2}{2^{14}} k^8 + \biggr\} + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 + \frac{1}{2^2} k^2 + + \frac{3^2}{2^6} k^4 + + \frac{5^2}{2^8} k^6 + + \frac{5^2 \cdot 7^2}{2^{14}} k^8 + \biggr\} + ~+~ + \biggl\{ + \frac{1}{2^2} k^2 + + \frac{1}{2^4} k^4 + + \frac{3^2}{2^8} k^6 + + \frac{5^2}{2^{10}} k^8 + \biggr\} + ~+~ + \biggl\{ + \frac{3^2}{2^6} k^4 + ~+~\frac{3^2}{2^8} k^6 + ~+~\frac{3^4}{2^{12}} k^8 + \biggr\} + ~+~ + \biggl\{ + \frac{5^2}{2^8} k^6 + ~+~\frac{5^2}{2^{10}} k^8 + \biggr\} + ~+~ + \biggl\{ + \frac{5^2 \cdot 7^2}{2^{14}} k^8 + \biggr\} + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + 1 + \biggl[ \frac{1}{2^2} + ~+~ \frac{1}{2^2} \biggr] ~k^2 + + \biggl[ \frac{3^2}{2^6} + + \frac{1}{2^4} + ~+~\frac{3^2}{2^6} \biggr]~k^4 + + \biggl[ \frac{5^2}{2^8} + + \frac{3^2}{2^8} + ~+~\frac{3^2}{2^8} + ~+~\frac{5^2}{2^8} \biggr]~ k^6 + ~+~\biggl[ + \frac{5^2 \cdot 7^2}{2^{14}}+ \frac{5^2}{2^{10}} ~+~\frac{3^4}{2^{12}} ~+~\frac{5^2}{2^{10}} ~+~\frac{5^2 \cdot 7^2}{2^{14}} + \biggr]k^8 + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + 1 + \frac{1}{2} k^2 + + \frac{11}{2^5} ~k^4 + + \frac{17}{2^6} ~ k^6 + + \frac{1787}{2^{13}} ~k^{8} + + \mathcal{O}(k^{10}) +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~\frac{2^2}{\pi^2} \biggl[E(k) \cdot E(k) \biggr]$ + + $~=$ + + $~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 + - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} + - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} + ~-~ \cdots + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} + ~-~ \cdots + \biggr\} +$ +
+   + +   + + $~ + \times~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 + - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} + - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} + ~-~ \cdots + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} + ~-~ \cdots + \biggr\} +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{3}{2^6}~ k^4 + - \frac{5}{2^8}~ k^6 + - \frac{5^2\cdot 7}{2^{14}} ~k^8 + \biggr\} + \times + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{3}{2^6}~ k^4 + - \frac{5}{2^8}~ k^6 + - \frac{5^2\cdot 7}{2^{14}} ~k^8 + \biggr\} + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{3}{2^6}~ k^4 + - \frac{5}{2^8}~ k^6 + - \frac{5^2\cdot 7}{2^{14}} ~k^8 + \biggr\} + ~+~ + \biggl\{- \frac{1}{2^2} ~k^2 + \biggr\} + \times + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{3}{2^6}~ k^4 + - \frac{5}{2^8}~ k^6 + \biggr\} +$ +
+   + +   + + $~ + ~+~ + \biggl\{ + - \frac{3}{2^6}~ k^4 + \biggr\} + \times + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{3}{2^6}~ k^4 + \biggr\} + ~+~ + \biggl\{ + - \frac{5}{2^8}~ k^6 + \biggr\} + \times + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + \biggr\} + ~+~ + \biggl\{ + - \frac{5^2\cdot 7}{2^{14}}k^8 + \biggr\} + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + \biggl\{ + 1 - \frac{1}{2^2} ~k^2 + - \frac{3}{2^6}~ k^4 + - \frac{5}{2^8}~ k^6 + - \frac{5^2\cdot 7}{2^{14}} ~k^8 + \biggr\} + ~+~ + \biggl\{ + - \frac{1}{2^2} ~k^2 + \frac{1}{2^4} ~k^4 + + \frac{3}{2^8}~ k^6 + + \frac{5}{2^{10}}~k^8 + \biggr\} +$ +
+   + +   + + $~ + ~+~ + \biggl\{ + ~-~ \frac{3}{2^6}~ k^4 + ~+~ \frac{3}{2^8}~ k^6 + + \frac{3^2}{2^{12}}~k^8 + \biggr\} + ~+~ + \biggl\{ + ~-~\frac{5}{2^8}~ k^6 + + \frac{5}{2^{10}}~k^8 + \biggr\} + ~+~ + \biggl\{ + ~-~ \frac{5^2\cdot 7}{2^{14}}k^8 + \biggr\} + + \mathcal{O}(k^{10}) +$ +
+ $~$ + + $~=$ + + $~ + 1 + \biggl[ - \frac{1}{2^2}- \frac{1}{2^2} \biggr]k^2 + + \biggl[ - \frac{3}{2^6}+ \frac{1}{2^4}~-~ \frac{3}{2^6} \biggr] k^4 + + \biggl[ - \frac{5}{2^8}+ \frac{3}{2^8}~+~ \frac{3}{2^8}~-~\frac{5}{2^8} \biggr] k^6 + + \biggl[ - \frac{5^2\cdot 7}{2^{14}} + \frac{5}{2^{10}}+ \frac{3^2}{2^{12}}+ \frac{5}{2^{10}}~-~ \frac{5^2\cdot 7}{2^{14}} \biggr] k^8 + + \mathcal{O}(k^{10}) +$ +
+ $~$ + + $~=$ + + $~ + 1 + \biggl[ - \frac{2}{2^2} \biggr]k^2 + + \biggl[ - \frac{3}{2^5}+ \frac{2}{2^5} \biggr] k^4 + + \biggl[ - \frac{5}{2^7}+ \frac{3}{2^7} \biggr] k^6 + + \biggl[ - \frac{5^2\cdot 7}{2^{13}} + \frac{5\cdot 2^4}{2^{13}}+ \frac{2 \cdot 3^2}{2^{13}}\biggr] k^8 + + \mathcal{O}(k^{10}) +$ +
+   + + $~=$ + + $~ + 1 - \frac{1}{2} ~k^2 ~-~ \frac{1}{2^5} ~ k^4 ~-~\frac{1}{2^6} ~ k^6 ~-~\frac{77}{2^{13}} ~ k^8 + + \mathcal{O}(k^{10}) +$ +
+ +
+ + + + + Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that, + + +
+ $~k_0^2$ + + $~=$ + + $~2e(1+e)^{-1}$ +
+   + + $~=$ + + $~2e( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;$ +
+ $~k_0^4$ + + $~=$ + + $~4e^2(1+e)^{-2}$ +
+   + + $~=$ + + $~4e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, ;$ +
+ $~k_0^6$ + + $~=$ + + $~2^3e^3(1+e)^{-3}$ +
+   + + $~=$ + + $~2^3e^3( 1 - 3e + 6e^2 - 10e^3 + 15e^4 - 21e^5 + \cdots ) \, ;$ +
+ $~k_0^8$ + + $~=$ + + $~2^4e^4(1+e)^{-4}$ +
+   + + $~=$ + + $~2^4e^4(1 - 4e + 10 e^2 - 20e^3 + 35e^4 - 56e^5 + \cdots) \, .$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Hence, we have, + + + +
+ $~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)$ + + $~=$ + + $~ + - (1-e) \biggl[ 1 + \frac{1}{2} k_0^2 + + \frac{11}{2^5} ~k_0^4 + + \frac{17}{2^6} ~ k_0^6 + + \frac{1787}{2^{13}} ~k_0^8 + + \mathcal{O}(k_0^{10}) + \biggr] +$ +
+   + +   + + $~ + + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4 + ~+~\frac{1}{2^5} ~ k_0^6 + ~+~ \frac{231}{2^{13}}~k_0^8 + + \mathcal{O}(k_0^{10}) + \biggr] +$ +
+   + +   + + $~ + - (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~ + ~-~\frac{1}{2^6} ~ k_0^6 + ~-~\frac{77}{2^{13}}~k_0^8 + + \mathcal{O}(k_0^{10}) + \biggr] +$ +
+   + + $~=$ + + $~ + - (1-e) \biggl[ 1 + e ( 1 - e +e^2 -e^3 ) + + \frac{11}{2^3} \cdot ~e^2( 1 - 2e + 3e^2) + + \frac{17}{2^6} ~\cdot 2^3 e^3 ( 1 - 3e) + + \frac{1787}{2^{13}} \cdot 2^4e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} \cdot~4e^2( 1 - 2e + 3e^2) + ~+~\frac{1}{2^5} ~ 2^3e^3 (1-3e) + ~+~\frac{231}{2^{13}} \cdot 2^4e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + - (1+e) \biggl[ 1 - e ~( 1 - e +e^2 -e^3) ~ + -~ \frac{1}{2^5} \cdot~ 4e^2( 1 - 2e +3e^2) ~ + ~-~\frac{1}{2^6} ~ 2^3e^3(1 - 3e) + ~-~\frac{77}{2^{13}}~2^4e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + + $~=$ + + $~ + - 2^{-9}(1-e) \biggl[ 512 (1+ e - e^2 +e^3 -e^4 ) + + 704 \cdot ~( e^2 - 2e^3 + 3e^4) + + 1088 ~\cdot ( e^3 - 3e^4) + + 1787 \cdot e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + + (1+e^2) 2^{-9}\biggl[ 1024 + ~+~128 \cdot~( e^2 - 2e^3 + 3e^4) + ~+~256 ~ (e^3-3e^4) + ~+~462 \cdot e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + - 2^{-9}(1+e) \biggl[ 512 ~(1 -e + e^2 -e^3 + e^4) ~ + -~ 64 \cdot~ ( e^2 - 2e^3 +3e^4) ~ + ~-~64 ~ (e^3 - 3e^4) + ~-~77~e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + + $~=$ + + $~ + - 2^{-9}(1-e) \biggl[ + 512 + 512e + 192e^2 + e^3(512 - 1408 + 1088) + e^4(704-512 - 3264 + 1787) + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + + 2^{-9}(1+e^2) \biggl[ + 1024 + 128e^2 + e^3(-256 + 256 ) + e^4(384 -768 + 462) + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + - 2^{-9}(1+e) \biggl[ + 512 - 512e + e^2(512 - 64 ) + e^3(-512 +128 -64 ) + e^4(512 - 192 - 192 - 77) + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + + $~=$ + + $~ + - 2^{-9}(1-e) \biggl[ + 512 + 512e + 192e^2 + 192e^3 - 1285 e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + + 2^{-9}(1+e^2) \biggl[ + 1024 + 128e^2 + 78e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + - 2^{-9}(1+e) \biggl[ + 512 - 512e + 448e^2 - 448 e^3 + 51e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + + $~=$ + + $~ + 2^{-9}\biggl[ + (- 192+ 128 - 448)e^2 + + (- 192 + 448) e^3 + + (1285 + 78- 51)e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + + 2^{-9} e \biggl[ + 1024e + (192- 448)e^2 + + (192+ 448) e^3 + + \mathcal{O}(e^{4}) + \biggr] + + 2^{-9} e^2 \biggl[ + 1024 + 128e^2 + + \mathcal{O}(e^{3}) + \biggr] +$ +
+   + + $~=$ + + $~ + 2^{-9} \biggl[ + (- 192+ 128 - 448)e^2 + 2048 e^2 + + (1285 + 78- 51)e^4 + + (192+ 448) e^4 + + 128e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + + $~=$ + + $~ + 2^{-9} \biggl[ + 1536e^2 + + 2080e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + + $~=$ + + $~ + 3e^2 + + \biggl[ \frac{5\cdot 13}{2^4}\biggr] e^4 + + \mathcal{O}(e^{5}) +$ +
+ $~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)$ + + $~=$ + + $~ + \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, . +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Hence, + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~=$ + + $~ + [1 + \mathcal{O}(e^{2})]\cdot (1 - e^2)^{1 / 2} + \biggl\{ + \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\Delta_0 \, , +$ +
+ + + + + + or, more precisely, + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~=$ + + $~ + \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} + \biggl\{ + \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\Delta_0 \, . +$ +
+ + + + + + + ====Next Factors==== + + + Now, + + +
+ $~\Delta_0^2$ + + $~=$ + + $~ + (\varpi_W + R_c)^2 + z_W^2 \, , +$ +
+ $~r_1^2$ + + $~=$ + + $~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 \, ,$ +
+ $~\Rightarrow ~~~ r_1^2 - \Delta_0^2$ + + $~=$ + + $~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 - \biggl[ (\varpi_W + R_c)^2 + z_W^2 \biggr]$ +
+   + + $~=$ + + $~\varpi_W^2 + 2\varpi_W R_c (1 - e^2 )^{1 / 2} + R_c^2 (1 - e^2 ) + z_W^2 - [\varpi_W^2 + 2\varpi_W R_c + R_c^2 + z_W^2 ]$ +
+   + + $~=$ + + $~2\varpi_W R_c [(1 - e^2 )^{1 / 2} - 1] -e^2 R_c^2 \, .$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ---- + + Again, drawing from the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we have, + + +
+ $~(1 -e^2)^{1 / 2}$ + + $~=$ + + $~ + 1 - \frac{1}{2}e^2 + \biggl[\frac{ \tfrac{1}{2}(-\tfrac{1}{2}) }{ 2 } \biggr]e^4 - \biggl[ \frac{ \tfrac{1}{2}(-\tfrac{1}{2} )(-\tfrac{3}{2} ) }{ 3! } \biggr]e^6 + + \biggl[ \frac{ \tfrac{1}{2} (-\tfrac{1}{2})(-\tfrac{3}{2})(-\tfrac{5}{2}) }{ 4! } \biggr]e^8 + \cdots +$ +
+   + + $~=$ + + $~ + 1 - \frac{1}{2}e^2 - \frac{1}{2^3} e^4 - \frac{1}{2^4}e^6 - \frac{5}{2^7} e^8 - \mathcal{O}(e^{10}) \, . +$ +
+ + + + + + + + + + + + + ---- + +
+ + +
+ $~\Rightarrow ~~~ r_1^2 - \Delta_0^2$ + + $~=$ + + $~\varpi_W R_c \biggl[- e^2 - \frac{1}{2^2} e^4 - \frac{1}{2^3}e^6 - \frac{5}{2^6} e^8 - \mathcal{O}(e^{10}) \biggr] -e^2 R_c^2$ +
+ $~\Rightarrow ~~~\frac{ r_1^2}{\Delta_0^2}$ + + $~=$ + + $~1 + -e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr] + - \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10}) \biggr]$ +
+ + + + + + + + + + + + + + + + ====Now Work on Elliptic Integral Expressions==== + + + From a [[User:Tohline/2DStructure/ToroidalGreenFunction#Series_Expansions|separate discussion]] we can draw the series expansion of $~\boldsymbol{K}(k)$, specifically, + + +
+ $~\frac{2K(k)}{\pi}$ + + $~=$ + + $~ + 1 + \biggl( \frac{1}{2} \biggr)^2k^2 + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + + \cdots +$ +
+ + + + + + where, + + + +
+ $~\frac{k^2}{4}$ + + $~\equiv$ + + $~ + \frac{1}{4} \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr] + = \biggl[ \frac{a\varpi}{r_1^2} \biggr] + = \biggl[ \frac{a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr] +$ +
+   + + $~=$ + + $~ + \frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2} \, . +$ +
+ + + + + + + + + + + + + Also, + + +
+ $~\frac{2K(k_H)}{\pi}$ + + $~=$ + + $~ + 1 + \biggl( \frac{1}{2} \biggr)^2k_H^2 + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 + + \cdots +$ +
+ + + + + + where, + + +
+ $~\frac{k_H^2}{4}$ + + $~=$ + + $~ + \frac{1}{\Delta^2}\biggl[ R (R_c + b\cos\theta_H) \biggr] +$ +
+   + + $~=$ + + $~ + \frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2} \, . +$ +
+ + + + + + + + + + + + What we want to do is write $~K(k)$ in terms of $~K(k_H)$.  Let's try … + + +
+ $~\frac{2K(k)}{\pi}$ + + $~=$ + + $~\frac{2K(k_H)}{\pi} + \delta_K \, ,$ +
+ + + + + + + + where, + + +
+ $~\delta_K$ + + $~\equiv$ + + $~\frac{2K(k)}{\pi} - \frac{2K(k_H)}{\pi}$ +
+   + + $~=$ + + $~\biggl\{1 + \frac{k^2}{4} + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + + \cdots + \biggr\} + - \biggl\{1 + \frac{k_H^2}{4} + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 + + \cdots + \biggr\} +$ +
+   + + $~\approx$ + + $~\biggl\{1 + \frac{k^2}{4} + \biggr\} + - \biggl\{1 + \frac{k_H^2}{4} + \biggr\} + = \frac{k^2}{4} - \frac{k_H^2}{4} +$ +
+   + + $~=$ + + $~ + \frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2} + - + \frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2} +$ +
+   + + $~=$ + + $~ + \biggl\{ \varpi_W R_c(1-e^2)^{1 / 2} \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2 + 2\varpi_W R_c(1-e^2)^{1 / 2} - R_c^2 e^2 \biggr\}^{-1} +$ +
+   + +   + + $~ + - + \biggl\{ \varpi_W R_c (1 + e\cos\theta_H) \biggr\} \biggl\{[\varpi_W + R_c(1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1} +$ +
+   + + $~\approx$ + + $~ + \biggl\{ \varpi_W R_c \biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2 + 2\varpi_W R_c\biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] - R_c^2 e^2 \biggr\}^{-1} +$ +
+   + +   + + $~ + - + \biggl\{ \varpi_W R_c \biggl[ 1 + e\cos\theta_H \biggr] \biggr\} + \biggl\{\varpi_W^2 + 2\varpi_W R_c(1+e\cos\theta_H) + + R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H) + z_W^2 - 2 z_W R_c e\sin\theta_H + R_c^2 e^2 \sin\theta^2_H \biggr\}^{-1} +$ +
+   + + $~\approx$ + + $~ + \biggl\{\varpi_W R_c - e^2 \biggl[ \frac{\varpi_W R_c}{2} \biggr] + \mathcal{O}(e^4) \biggr\} + \biggl\{ [ (\varpi_W +R_c)^2 + z_W^2 ] - e^2 \biggl[ \varpi_W R_c + R_c^2 \biggr] + \mathcal{O}(e^4) \biggr\}^{-1} +$ +
+   + +   + + $~ + - + \biggl\{ \varpi_W R_c + e \biggl[ \varpi_W R_c \cos\theta_H \biggr]\biggr\} + \biggl\{ [ (\varpi_W^2 + R_c)^2 + z_W^2 ] + + 2R_c e(R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) + R_c^2 e^2 \biggr\}^{-1} +$ +
+   + + $~\approx$ + + $~ + \biggl\{ \frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr] \biggr\} + \biggl\{ 1 - e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2 }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\}^{-1} +$ +
+   + +   + + $~ + -~ + \biggl\{ \frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] } \biggl[ 1 + e\cos\theta_H \biggr] \biggr\} + \biggl\{ 1 + + e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] + + e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr] \biggr\}^{-1} +$ +
+   + + $~\approx$ + + $~ + \frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr] + \biggl\{ 1 + e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2 }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\} +$ +
+   + +   + + $~ + -~\frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] } \biggl[ 1 + e\cos\theta_H \biggr] + \biggl\{ 1 + - e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] + - e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr] \biggr\} +$ +
+   + + $~\approx$ + + $~ + -~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] } + \biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] +$ +
+   + +   + + $~ + +~\frac{e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } + \biggl\{ \biggl[ \frac{ \varpi_W R_c + R_c^2 }{ (\varpi_W +R_c)^2 + z_W^2 } - \frac{1}{2} \biggr] + - \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr] + - \cos\theta_H \biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] + \biggr\} +$ +
+   + + $~\approx$ + + $~ + -~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] [ (\varpi_W^2 + R_c)^2 + z_W^2 ] } + \biggl[ \cos\theta_H [(\varpi_W^2 + R_c)^2 + z_W^2 ] -2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) \biggr] +$ +
+   + +   + + $~ + +~\frac{\tfrac{1}{2}e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ][ (\varpi_W^2 + R_c)^2 + z_W^2 ] } + \biggl\{ 2 (\varpi_W R_c + R_c^2 ) - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] + - 2R_c^2 + - 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) + \biggr\} +$ +
+   + + $~\approx$ + + $~ + -~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 } + \biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr] +$ +
+   + +   + + $~ + +~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 } + \biggl\{ 2 (\varpi_W R_c + R_c^2 ) - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] + - 2R_c^2 + - 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) + \biggr\} \, . +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Let's subtract $~K([k_H]_0)$ from the potential expression.  But first, let's adopt the shorthand notation … + +
+ Given that, + + + + + + + + + + + + + + + + + + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~=$ + + $~ + \biggl\{ + \boldsymbol{K}(k) \biggr\} \frac{\Delta_0}{r_1}~\biggl( \frac{2^2}{3\pi^2} \biggr) \Upsilon_{W0}(\eta_0) +$ +
+   + + $~=$ + + $~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl[\frac{r_1^2 }{ \Delta_0^2 } \biggr]^{-1 / 2} + \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} +$ +
+   + + $~=$ + + $~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\} + \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2 - \mathcal{O}(e^{4}) \biggr] +$ +
+ + let's define the variable, $~\mathcal{A}$, such that, + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~=$ + + $~ + \biggl\{\boldsymbol{K}(k)\biggr\} \{1 + e^2\mathcal{A}\} +$ +
+ $~\Rightarrow ~~~ \{ 1 + e^2 \cdot \mathcal{A} \}$ + + $~=$ + + $~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\} + \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2 - \mathcal{O}(e^{4}) \biggr]$ +
+   + + $~\approx$ + + $~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] \biggr\} + \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 \biggr] \biggl[ 1 - \frac{1}{2}e^2 \biggr]$ +
+   + + $~\approx$ + + $~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 - \frac{1}{2}e^2 \biggr\}$ +
+ $~\Rightarrow ~~~ 2\mathcal{A}$ + + $~\approx$ + + $~\biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{41}{2^3\cdot 3}\biggr) \, .$ +
+ +
+ + + We can therefore write, + + + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)$ + + $~\approx$ + + $~- K([k_H]_0) + + \biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\} + \{ 1 + e^2 \cdot \mathcal{A} \} +$ +
+   + + $~\approx$ + + $~- K([k_H]_0) + + + K(k_H) + \{ 1 + e^2 \cdot \mathcal{A} \} + + + \frac{\pi}{2} \cdot \delta_K \, , +$ +
+ + + + + + + + + + + + where we should keep in mind that $~\delta_k$ is $~\mathcal{O}(e^1)$.  So, let's examine the piece, + + +
+ $~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]$ + + $~=$ + + $~ + \biggl\{1 + \frac{k_H^2}{4} + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 + + \cdots + \biggr\} - + \biggl\{1 + \frac{k_H^2}{4} + + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 + + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 + + \cdots + \biggr\}_{e\rightarrow 0} +$ +
+   + + $~\approx$ + + $~ + \frac{k_H^2}{4} - \biggl[ \frac{k_H^2}{4} \biggr]_{e\rightarrow 0} +$ +
+   + + $~=$ + + $~ + \biggl[ \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr] + - + \biggl[ \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr]_{e\rightarrow 0} +$ +
+   + + $~=$ + + $~\varpi_W R_c (1 + e\cos\theta_H) + \biggl\{ [\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1} + - + \biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] +$ +
+   + + $~=$ + + $~\varpi_W R_c (1 + e\cos\theta_H) + \biggl\{ + \varpi_W^2 + 2\varpi_W R_c(1 + e\cos\theta_H) + R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H) + z_W^2 -2z_W R_c e\sin\theta_H + R_c^2 e^2\sin^2\theta_H + \biggr\}^{-1} +$ +
+   + +   + + $~ + - + \biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] +$ +
+   + + $~=$ + + $~ + -\biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] + + \biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] + \biggl\{1 + + \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} + + \frac{e^2 [R_c^2\cos^2\theta_H + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2} + \biggr\}^{-1} +$ +
+   + + $~\approx$ + + $~ + -\biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] + +\biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] + \biggl\{1 + - \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} +$ +
+   + +   + + $~ + - \frac{e^2 [R_c^2\cos^2\theta_H + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2} + + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ [(\varpi_W + R_c)^2 + z_W^2]^2} + \biggr\} +$ +
+   + + $~\approx$ + + $~ + \biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] + \biggl\{ + - \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} +$ +
+   + +   + + $~ + - \frac{e^2 [R_c^2\cos^2\theta_H + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2} + + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ [(\varpi_W + R_c)^2 + z_W^2]^2} + \biggr\} +$ +
+   + +   + + $~ + +\biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] + \biggl\{ \frac{e\cos\theta_H [(\varpi_W + R_c)^2 + z_W^2] -e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} + \biggr\} +$ +
+   + + $~\approx$ + + $~ + \biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr] + \biggl\{ + e\cos\theta_H + - \frac{ eR_c\cos\theta_H [2\varpi_W + 2 R_c - 2z_W \tan\theta_H] }{ \Delta_0^2} + - \frac{e^2 R_c^2 }{ \Delta_0^2} +$ +
+   + +   + + $~ + - \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{ \Delta_0^2} + + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^4} + \biggr\} \, . +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Now we have, + + + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)$ + + $~\approx$ + + $~- K([k_H]_0) + + + K(k_H) + \{ 1 + e^2 \cdot \mathcal{A} \} + + + \frac{\pi}{2} \cdot \delta_K +$ +
+ $~\Rightarrow ~~~ - \Delta_0 \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~\approx$ + + $~ + \frac{2}{\pi} K([k_H]_0) + + \frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr] + + + \delta_K + + + \frac{2}{\pi} K(k_H) e^2 \cdot \mathcal{A} \, . +$ +
+ + + + + + + + + + + + + But, as we have just demonstrated, + + +
+ $~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]+ \delta_K +$ + + $~\approx$ + + $~ + \biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr] + \biggl\{ + e\cos\theta_H + - \frac{ eR_c\cos\theta_H [2\varpi_W + 2 R_c - 2z_W \tan\theta_H] }{ \Delta_0^2} + - \frac{e^2 R_c^2 }{ \Delta_0^2} +$ +
+   + +   + + $~ + - \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{ \Delta_0^2} + + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^4} + \biggr\} +$ +
+   + +   + + $~ + -~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 } + \biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr] +$ +
+   + +   + + $~ + +~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 } + \biggl\{ 2 (\varpi_W R_c + R_c^2 ) - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] + - 2R_c^2 + - 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) + \biggr\} \, . +$ +
+   + + $~\approx$ + + $~ + \biggl[ \frac{e^2 \cdot \varpi_W R_c}{ \Delta_0^4 } \biggr] + \biggl\{ + - R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] +$ +
+   + +   + + $~ + +~(\varpi_W R_c + R_c^2 ) - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] + - R_c^2 + - 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) +$ +
+   + +   + + $~ + + \frac{ [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^2} + \biggr\} +$ +
TEMPORARY BREAK HERE
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Hence, + + +
+ $~ - \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~\approx$ + + $~ + K([k_H]_0) + + + K(k_H) e^2 \cdot \mathcal{A} + +~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot \varpi_W R_c}{ \Delta_0^4 } \biggr] + \biggl\{ + - R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] +$ +
+   + +   + + $~ + +~(\varpi_W R_c + R_c^2 ) - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] + - R_c^2 + - 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) +$ +
+   + +   + + $~ + + \frac{ [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^2} + \biggr\} +$ +
+ + + + + + + + + + + + + + + + + + + ===Include Second Wong Term=== + + + +
+ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}$ + + $~=$ + + $~ + - \biggl( \frac{2^{3} }{3\pi^3} \biggr) + \Upsilon_{W1}(\eta_0) \times \cos\theta + \biggl\{ \frac{a}{r_2} \cdot + \boldsymbol{E}(k) \biggr\} +$ +
+ $~\Rightarrow ~~~ - \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr]$ + + $~=$ + + $~ + \biggl( \frac{2^{2} }{3\pi^2} \biggr) + \Upsilon_{W1}(\eta_0) \times \cos\theta + \biggl\{ \frac{\Delta_0}{r_2} \cdot + \boldsymbol{E}(k) \biggr\} \, ; +$ +
+ + + + + + + + + + + + + ====Leading (Upsilon) Coefficient==== + + + +
+ $~\Upsilon_{W1}(\eta_0)$ + + $~\equiv$ + + $~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] + \biggl\{ + K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] + +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] + -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] + \biggr\} +$ +
+ $~\Rightarrow~~~ \biggl[ \frac{e^2}{ (1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)$ + + $~=$ + + $~ + - (1-e)K(k_0)\cdot K(k_0) + +~2(3-e^2)K(k_0)\cdot E(k_0) + -~5(1+e) E(k_0)\cdot E(k_0) \, . +$ +
+ $~\Rightarrow ~~~ \frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)$ + + $~=$ + + $~ + - (1-e) \biggl[ 1 + \frac{1}{2} k_0^2 + + \frac{11}{2^5} ~k_0^4 + + \frac{17}{2^6} ~ k_0^6 + + \frac{1787}{2^{13}} ~k_0^8 + + \mathcal{O}(k_0^{10}) + \biggr] +$ +
+   + +   + + $~ + + 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4 + ~+~\frac{1}{2^5} ~ k_0^6 + ~+~ \frac{231}{2^{13}}~k_0^8 + + \mathcal{O}(k_0^{10}) + \biggr] +$ +
+   + +   + + $~ + - 5(1+e) \biggl[ 1 + - ~\frac{1}{2} ~k_0^2 ~ + -~ \frac{1}{2^5} ~ k_0^4 ~ + ~-~\frac{1}{2^6} ~ k_0^6 + ~-~\frac{77}{2^{13}}~k_0^8 + + \mathcal{O}(k_0^{10}) + \biggr] +$ +
+   + + $~=$ + + $~ + - (1-e) \biggl[ 1 + \frac{1}{2} \cdot 2e( 1 - e +e^2 - e^3 ) + + \frac{11}{2^5} ~\cdot 4e^2( 1 - 2e + 3e^2 ) + + \frac{17}{2^6} ~ \cdot 2^3e^3( 1 - 3e ) + + \frac{1787}{2^{13}} ~\cdot 2^4e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + + 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~\cdot 4e^2( 1 - 2e + 3e^2 ) + ~+~\frac{1}{2^5} ~ \cdot 2^3e^3( 1 - 3e ) + ~+~ \frac{231}{2^{13}}~\cdot 2^4e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + - 5(1+e) \biggl[ 1 + - ~\frac{1}{2} ~\cdot 2e( 1 - e +e^2 - e^3 ) ~ + -~ \frac{1}{2^5} ~ \cdot 4e^2( 1 - 2e + 3e^2 ) + ~-~\frac{1}{2^6} ~ \cdot 2^3e^3( 1 - 3e ) + ~-~\frac{77}{2^{13}}~\cdot 2^4e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + + $~=$ + + $~ + - 2^{-9}(1-e) \biggl[ 2^9 + 2^9e( 1 - e +e^2 - e^3 ) + + 2^6 \cdot 11 ~\cdot e^2( 1 - 2e + 3e^2 ) + + 2^6\cdot 17 ~ \cdot e^3( 1 - 3e ) + + 1787 ~\cdot e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + + 2^{-9} (6- 2e^2) \biggl[ 2^9 ~+~2^6~\cdot e^2( 1 - 2e + 3e^2 ) + ~+~2^7 \cdot e^3( 1 - 3e ) + ~+~ 231~\cdot e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+   + +   + + $~ + - 2^{-9}(5+ 5e) \biggl[ 2^9 + - ~2^9 \cdot e( 1 - e +e^2 - e^3 ) ~ + -~ 2^6 \cdot e^2( 1 - 2e + 3e^2 ) + ~-~2^6 \cdot e^3( 1 - 3e ) + ~-~77~\cdot e^4 + + \mathcal{O}(e^{5}) + \biggr] +$ +
+ $~\Rightarrow ~~~\frac{2^{11}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)$ + + $~=$ + + $~ + - 2^9 - 2^9e( 1 - e +e^2 - e^3 ) + - 2^6 \cdot 11 ~\cdot e^2( 1 - 2e + 3e^2 ) + - 2^6\cdot 17 ~ \cdot e^3( 1 - 3e ) + - 1787 ~\cdot e^4 +$ +
+   + +   + + $~ + + 2^9e + 2^9e^2( 1 - e +e^2 ) + + 2^6 \cdot 11 ~\cdot e^3( 1 - 2e ) + + 2^6\cdot 17 ~ \cdot e^4 +$ +
+   + +   + + $~ + + 6\biggl[ 2^9 ~+~2^6~\cdot e^2( 1 - 2e + 3e^2 ) + ~+~2^7 \cdot e^3( 1 - 3e ) + ~+~ 231~\cdot e^4 + \biggr] + -2^{10}e^2 ~-~2^7~\cdot e^4 +$ +
+   + +   + + $~ + + 5 \biggl[ -2^9 + + ~2^9 \cdot e( 1 - e +e^2 - e^3 ) ~ + +~ 2^6 \cdot e^2( 1 - 2e + 3e^2 ) + ~+~2^6 \cdot e^3( 1 - 3e ) + ~+~77~\cdot e^4 + \biggr] +$ +
+   + +   + + $~ + +~5 \biggl[ -2^9e + + ~2^9 \cdot e^2( 1 - e +e^2 ) ~ + +~ 2^6 \cdot e^3( 1 - 2e ) + ~+~2^6 \cdot e^4 + \biggr] + + \mathcal{O}(e^{5}) +$ +
+   + + $~=$ + + $~ + 2^9( e^2 - e^3 + e^4 ) + - 2^6 \cdot 11 ~( e^2 - 2e^3 + 3e^4 ) + - 2^6\cdot 17 ~( e^3 - 3e^4 ) + - 1787 ~e^4 +$ +
+   + +   + + $~ + +2^9 ( e^2 - e^3 +e^4 ) + + 2^6 \cdot 11 ( e^3 - 2e^4 ) + + 2^6\cdot 17 ~ e^4 +$ +
+   + +   + + $~ + +~ 3\cdot 2^7~( e^2 - 2e^3 + 3e^4 ) + ~+~3\cdot 2^8 ( e^3 - 3e^4 ) + ~+~ 2\cdot 3^2 \cdot 7\cdot 11~\cdot e^4 + -2^{10}e^2 ~-~2^7~\cdot e^4 +$ +
+   + +   + + $~ + + 5\cdot 2^9 ( - e^2 +e^3 - e^4 ) ~ + +~ 5\cdot 2^6 ( e^2 - 2e^3 + 3e^4 ) + ~+~5\cdot 2^6 ( e^3 - 3e^4 ) + ~+~5\cdot 77~e^4 +$ +
+   + +   + + $~ + +5\cdot 2^9 ( e^2 - e^3 +e^4 ) ~ + +~ 5\cdot 2^6 ( e^3 - 2e^4 ) + ~+~5\cdot 2^6 e^4 + + \mathcal{O}(e^{5}) +$ +
+   + + $~=$ + + $~ + e^2 [ 2^9 - 2^6\cdot 11 + 2^9 + 3\cdot 2^7 -2^{10} -5\cdot 2^9 + 5\cdot 2^6 + 5\cdot 2^9 ] +$ +
+   + +   + + $~ + + e^3 [ -2^9 - 2^7\cdot 11 - 2^6\cdot 17 - 2^9 +2^6\cdot 11 - 3\cdot 2^8+3\cdot 2^8 + 5\cdot 2^9-5\cdot 2^7 +5\cdot 2^6 -5\cdot 2^9 + 5\cdot 2^6] +$ +
+   + +   + + $~ + + e^4 [ 2^9 - 3\cdot 11\cdot 2^6+3\cdot 17\cdot 2^6 - 1787 + 2^9 - 11\cdot 2^7 + 17\cdot 2^6 + 3^2\cdot 2^7 - 3^2\cdot 2^8+ 2\cdot 3^2\cdot 7\cdot 11 - 2^7 + - 5\cdot 2^9 + 3\cdot 5\cdot 2^6 - 3\cdot 5\cdot 2^6 + 5\cdot 7\cdot 11 + 5\cdot 2^9 - 5\cdot 2^7 +5\cdot 2^6] + + \mathcal{O}(e^{5}) +$ +
+   + + $~=$ + + $~ + e^2 [ - 2^6\cdot 11 + 3\cdot 2^7 + 5\cdot 2^6 ] + + e^3 [ -2^{10} - 2^7\cdot 11 - 2^6\cdot 17 +2^6\cdot 11 ] +$ +
+   + +   + + $~ + + e^4 [ 2^{10}~-~ 1787~+~ 2\cdot 3^2\cdot 7\cdot 11~+~ 5\cdot 7\cdot 11 ~+~ 2^6\cdot (3\cdot 17 - 3\cdot 11 - 22 + 17 + 18 - 36 - 2 - 10 + 5) ] + + \mathcal{O}(e^{5}) +$ +
+   + + $~=$ + + $~ + 2^6 e^2 [ 0 ] ~- 2^8 \cdot 11 e^3 + e^4 [ 2^{10}~-~ 1787~+~ 7\cdot 11\cdot 23 ~-~ 2^8\cdot 3 ] + + \mathcal{O}(e^{5}) +$ +
+   + + $~=$ + + $~ + - 2^8 \cdot 11 e^3 + 2^4\cdot 3\cdot 5e^4 + + \mathcal{O}(e^{5}) +$ +
+ $~\Rightarrow ~~~\frac{2^{2}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)$ + + $~=$ + + $~ + - \frac{11}{2} e^3 + \frac{3\cdot 5}{2^5} e^4 + + \mathcal{O}(e^{5}) \, . +$ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
'''Floating Comparison Summary'''
+ As [[#Step01|shown above]], the first three terms of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)] series expression may be written as, + + + + + + + +
+ $~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr]$ + + $~=$ + + $~ + \boldsymbol{K}([k_H]_0) + - + \frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) + + \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \boldsymbol{E}(k_H) \, . +$ +
+ + Let's see how it compares to the first term of Wong's (1973) expression which, as [[#Step02|shown separately above]], can be written in the form, + + + + + + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~=$ + + $~ + \Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr) + \Upsilon_{W0}(\eta_0) \biggl\{ + \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\, . +$ +
+ + ---- + First, as [[#Step03|shown above]], + + + + + + + +
+ $~\frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)$ + + $~=$ + + $~ + \biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, . +$ +
+ Note that, in order to determine the functional form of the $~\mathcal{O}(e^{2})$ term in this expression, we will have to include $~k_0^8$ terms in the various expressions for products of elliptic integrals.  Second, [[#Step04|we have shown that]], + + + + + + + + + + + + + + + + + + +
+ $~\frac{ r_1^2}{\Delta_0^2}$ + + $~=$ + + $~1 + -e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr] + - \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10}) \biggr]$ +
+ $~\Rightarrow ~~~ \frac{\Delta_0}{r_1}$ + + $~\approx$ + + $~ + 1 + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr] + \, ,$       and we are defining $~\delta_K$ such that, +
+ $~K(k)$ + + $~=$ + + $~K(k_H) + \frac{\pi}{2} \cdot \delta_K \, .$ +
+ + ---- + + Hence, + + + + + + + + + + + + + + + + + + + + + + + + +
+ $~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr]$ + + $~\approx$ + + $~ + \biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\} + \biggl\{ 1 + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr] \biggr\}\biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} +$ +
+   + + $~\approx$ + + $~ + \boldsymbol{K}([k_H]_0) + + + \boldsymbol{K}(k_H) e^2 \cdot \mathcal{A} + +~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot \varpi_W R_c}{ \Delta_0^4 } \biggr] + \biggl\{ + - R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] +$ +
+   + +   + + $~ + +~(\varpi_W R_c + R_c^2 ) - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] + - R_c^2 + - 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) +$ +
+   + +   + + $~ + + \frac{ [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^2} + \biggr\} \, , +$ +
+ and, + + + + + + +
+ $~\mathcal{A}$ + + $~\approx$ + + $~\biggl[ \frac{R_c(R_c + \varpi_W)}{2\Delta_0^2}\biggr] + \biggl( \frac{41}{2^4\cdot 3}\biggr) \, .$ +
+ + ---- + + Second, + + + + + + + + + + + + + +
+ $~- \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr]$ + + $~=$ + + $~ + \boldsymbol{E}(k) + \biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\} + \biggl[ \biggl( \frac{2^{2} }{3\pi^2} \biggr) \Upsilon_{W1}(\eta_0) \biggr] +$ +
+   + + $~=$ + + $~ + \biggl\{ \boldsymbol{E}(k_H) + \frac{\pi}{2}\cdot \delta_E \biggr\} + \biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\} + \biggl[- \biggl(\frac{11}{2\cdot 3} \biggr) e + \biggl(\frac{5}{2^5}\biggr) e^2 + \mathcal{O}(e^{5}) \biggr]\cdot (1 - e^2)^{1 / 2} \, . +$ +
+ +
+ + + + + + ====Geometric Factor==== + + By definition, + + + +
+ $~\Delta_0^2$ + + $~=$ + + $~ + (\varpi_W + R_c)^2 + z_W^2 \, , +$ +
+ $~r_1^2$ + + $~=$ + + $~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 \, ,$ +
+ $~r_2^2$ + + $~=$ + + $~\biggl[ \varpi_W - R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 \, ,$ +
+ $~\cos^2\theta$ + + $~=$ + + $~\biggl[ \frac{r_1^2 + r_2^2 - 4R_c^2(1-e^2)}{2r_1 r_2} \biggr]^2 \, .$ +
+ + + + + + + + + + + + + + + + + + + + + + + + Hence, + + +
+ $~r_2^2 - \Delta_0^2 \cdot \cos^2\theta$ + + $~=$ + + $~$ +
+
• Université de Bordeaux (Part 2):  [[User:Tohline/Appendix/Ramblings/BordeauxSequences|Spheroid-Ring Sequences]]
• +
• Université de Bordeaux (Part 3):  [[User:Tohline/Appendix/Ramblings/BordeauxPostDefense|Discussions Following Dissertation Defense]]
• +
{{LSU_HBook_footer}} {{LSU_HBook_footer}}

# Université de Bordeaux (Part 1)

## 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.

 $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}$ $~=$ $~ - \biggl( \frac{2^{3} }{3\pi^3} \biggr) \Upsilon_{W0}(\eta_0) \biggl\{ \frac{a}{ r_1 } \cdot \boldsymbol{K}(k) \biggr\}\, ,$ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}$ $~=$ $~ - \biggl( \frac{2^{3} }{3\pi^3} \biggr) \Upsilon_{W1}(\eta_0) \times \cos\theta \biggl\{ \frac{a}{r_2} \cdot \boldsymbol{E}(k) \biggr\} \, ,$ $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\varpi, z)\biggr|_\mathrm{exterior}$ $~=$ $~ - \biggl( \frac{2^{3} }{3\pi^3} \biggr)\Upsilon_{W2}(\eta_0) \times \cos(2\theta) \biggl\{ \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) - \frac{a}{r_1} \cdot \boldsymbol{K}(k) \biggr\} \, ,$

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,

 $~a^2$ $~\equiv$ $~ R^2 - d^2$       and,       $~\cosh\eta_0 \equiv \frac{R}{d} ~~~~\Rightarrow ~~~\sinh\eta_0 = \frac{a}{d} \, ,$ $~r_1^2$ $~\equiv$ $~(\varpi + a)^2 + (z - Z_0)^2 \, ,$ $~r_2^2$ $~\equiv$ $~(\varpi - a)^2 + (z - Z_0)^2 \, ,$ $~\cos\theta$ $~\equiv$ $~\biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \, ,$ $~k$ $~\equiv$ $~ \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} = \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} = \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} \, .$

 Leading Coefficient Expressions … … evaluated for: $~\frac{R}{d} = \cosh\eta_0 = 3$ $~\Upsilon_{W0}(\eta_0)$ $~\equiv$ $~ \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] \biggr\} \, ,$ 7.134677 $~\Upsilon_{W1}(\eta_0)$ $~\equiv$ $~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] \biggr\} \, ,$ 0.130324 $~\Upsilon_{W2}(\eta_0)$ $~\equiv$ $~ \frac{2^{3 / 2}}{3^2} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13] + 2 K ( k_0 ) \cdot E(k_0) [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]$ $~ -~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] \biggr\} \, ,$ 0.003153 where, $~k_0$ $~\equiv$ $~ \biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, .$ 0.707106781

NOTE: In evaluating these "leading coefficient expressions" for the case, $~R/d = 3$, we have used the complete elliptic integral evaluations, K(k0) = 1.854074677 and E(k0) = 1.350643881.

#### Setup

From our accompanying discussion of Wong's (1973) derivation, the exterior potential is given by the expression,

 $~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)$ $~=$ $~ -D_0 (\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, ,$

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

where,

 $~D_0$ $~\equiv$ $~ \frac{2^{3/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] = \frac{2^{3/2} }{3\pi^2} \biggl[\frac{(R^2 - d^2)^{3 / 2}}{d^2 R} \biggr] \, ,$ $~C_n(\cosh\eta_0)$ $~\equiv$ $~(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) - (n - \tfrac{3}{2}) Q_{n - \frac{1}{2}}(\cosh \eta_0)~Q^2_{n + \frac{1}{2}}(\cosh \eta_0) \,$

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,

 $~\cosh\eta_0$ $~=$ $~\frac{R}{d} = \frac{1}{\epsilon} \, ,$ $~\sinh\eta_0$ $~=$ $~\frac{a}{d} = \frac{1}{d}\biggl[ R^2 - d^2 \biggr]^{1 / 2} = \frac{1}{\epsilon} \biggl[1 - \epsilon^2 \biggr]^{1 / 2} \, .$

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, $~a$, 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: $~(\varpi, z)$, $~(\eta, \theta)$, $~(r_1, r_2)$.

 $~\varpi$ $~=$ $~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)}$ $~\Rightarrow ~$ $~\cos\theta$ $~=$ $~\cosh\eta - \frac{a\sinh\eta}{\varpi}$ $~z - Z_0$ $~=$ $~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)}$ $~\Rightarrow ~$ $~\sin\theta$