User:Tohline/Appendix/Ramblings/Bordeaux

Université de Bordeaux

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.

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.

Their Presentation

On an initial reading, it appears as though the most relevant section of the Huré, et al. (2020) paper is §8 titled, The Solid Torus. They write the gravitational potential in terms of the series expansion,

<math>~\Psi_\mathrm{grav}(\vec{r})</math>

<math>~\approx</math>

<math>~ \Psi_0 + \sum\limits_{n=1}^N \Psi_n \, , </math>

Huré, et al. (2020), §7, p. 5831, Eq. (42)

where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write,

<math>~\Psi_0 </math>

<math>~=</math>

<math>~ - \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}(k_0) \biggr] </math>

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

and,

<math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math>

<math>~=</math>

<math>~ - \frac{G\pi \rho_0 R_c b^2}{4 (k')^2 \Delta_0^3} \biggl\{ [\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k) - (k')^2 \Delta_0^2 \boldsymbol{K}(k) \biggr\} \, . </math>

Huré, et al. (2020), §8, p. 5832, Eq. (54)

Note that the argument of the elliptic integral functions is,

<math>~k</math>

<math>~\equiv</math>

<math>~ \frac{2\sqrt{\varpi R}}{\Delta} </math>

where,

<math>~\Delta</math>

<math>~\equiv</math>

<math>~ \biggl[ (R + \varpi)^2 + (Z-z)^2 \biggr]^{1 / 2} \, . </math>

Huré, et al. (2020), §2, p. 5826, Eqs. (4) & (5)

Our Presentation of Wong's (1973) Result

Setup

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

<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)</math>

<math>~=</math>

<math>~ -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) \, , </math>

where,

<math>~D_0 </math>	<math>~\equiv</math>	<math>~ \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] \, ,</math>
<math>~C_n(\cosh\eta_0)</math>	<math>~\equiv</math>	<math>~(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) \, </math>

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

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

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:

<math>~\varpi</math> <math>~=</math> <math>~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)}</math>      <math>~\Rightarrow ~</math>      <math>~\cos\theta</math> <math>~=</math> <math>~\cosh\eta - \frac{a\sinh\eta}{\varpi}</math> <math>~z - Z_0</math> <math>~=</math> <math>~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)}</math>      <math>~\Rightarrow ~</math>      <math>~\sin\theta</math> <math>~=</math> <math>~\frac{(z - Z_0)}{\varpi} \cdot \sinh\eta </math> Given that (sin2θ + cos2θ) = 1, we have, <math>~1</math> <math>~=</math> <math>~ \biggl[ \frac{(z - Z_0)}{\varpi} \cdot \sinh\eta \biggr]^2 + \biggl[\cosh\eta - \frac{a\sinh\eta}{\varpi}\biggr]^2 </math> <math>~\Rightarrow ~~~ \coth\eta</math> <math>~=</math> <math>~ \frac{1}{2a\varpi}\biggl[\varpi^2 + a^2 + (z - Z_0)^2 \biggr] \, . </math> We deduce as well that, <math>~\frac{2}{\coth\eta + 1}</math> <math>~=</math> <math>~ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \, , </math>        and, <math>~\sinh\eta + \cosh\eta</math> <math>~=</math> <math>~ \frac{\varpi^2 + a^2 + (z - Z_0)^2}{(\varpi + a)^2 + (z - Z_0)^2} \, . </math>

Leading (n = 0) Term

Wong's Expression

Now, from our separate derivation we have,

<math>~P_{-1 / 2}(\cosh\eta)</math>

<math>~=</math>

<math>~ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, . </math>

And if we make the function-argument substitution, <math>~z \rightarrow \coth\eta</math>, in the "Key Equation,"

	<math>~Q_{-\frac{1}{2}}(z)</math>	<math>~=</math>	<math>~ \sqrt{ \frac{2}{z+1} } ~K\biggl( \sqrt{ \frac{2}{z+1}} \biggr) </math>
Abramowitz & Stegun (1995), p. 337, eq. (8.13.3)

we can write,

<math>~P_{-1 / 2}(\cosh\eta)</math>

<math>~=</math>

<math>~ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) \, , </math>

where, from above, we recognize that,

<math>~ 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} \, . </math>

So, the leading (n = 0) term gives,

<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)</math>	<math>~=</math>	<math>~ -D_0 (\cosh\eta - \cos\theta)^{1 / 2} ~C_0(\cosh\eta_0)P_{-\frac{1}{2}}(\cosh\eta) </math>
	<math>~=</math>	<math>~ -D_0~C_0(\cosh\eta_0) \biggl[ \frac{a \sinh\eta}{\varpi} \biggr]^{1 / 2} ~\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) </math>
	<math>~=</math>	<math>~ -\frac{D_0~C_0(\cosh\eta_0)}{\pi} \biggl[ \frac{2a }{\varpi} \biggr]^{1 / 2} ~ k \boldsymbol{K}(k) </math>
	<math>~=</math>	<math>~ - C_0(\cosh\eta_0) \cdot \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot \boldsymbol{K}(k) \, . </math>

Thin-Ring Evaluation of C₀

In an accompanying discussion of the thin-ring approximation, we showed that as <math>~\cosh\eta_0 \rightarrow \infty</math>

<math>~C_0(x)\biggr|_{x\rightarrow \infty}</math>

<math>~=</math>

<math>~\biggl( \frac{3 \pi^2}{2^2} \biggr) \frac{1}{\cosh^2\eta_0} \, . </math>

Hence, in this limit we can write,

<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{thin-ring}</math>

<math>~=</math>

<math>~ - \frac{2 }{\pi} \cancelto{1}{\biggl[\frac{\sinh\eta_0}{\cosh\eta_0}\biggr]^3 } \frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot \boldsymbol{K}(k) \, . </math>

More General Evaluation of C₀

NOTE of CAUTION: In our above evaluation of the toroidal function, <math>~Q_{-\frac{1}{2}}(z)</math>, we appropriately associated the function argument, <math>~z</math>, with the hyperbolic-cotangent of <math>~\eta</math>; that is, we made the substitution, <math>~z \rightarrow \coth\eta</math>. Here, as we assess the behavior of, and evaluate, the leading coefficient, <math>~C_0</math>, an alternate substitution is appropriate, namely, <math>~z_0 \rightarrow \cosh\eta_0</math>; we affix the subscript zero to this function argument in an effort to minimize possible confusion.

More generally, though, drawing from our accompanying tabulation of Toroidal Function Evaluations, we have,

<math>~2C_0(\cosh\eta_0)</math>	<math>~=</math>	<math>~ \biggl[ Q_{+\frac{1}{2}}(\cosh \eta_0) \biggr] \biggl[ Q_{ - \frac{1}{2}}^2(\cosh \eta_0) \biggr] + 3 \biggl[ Q_{ - \frac{1}{2}}(\cosh \eta_0) \biggr] \biggl[ Q^2_{ + \frac{1}{2}}(\cosh \eta_0) \biggr] </math>
	<math>~=</math>	<math>~ \biggl[ \cosh\eta_0 ~k_0~K(k_0) ~-~ [2(\cosh\eta_0+1)]^{1 / 2} E(k_0) \biggr] \times \biggl\{ \frac{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) }{ [2^{3} (\cosh\eta_0+1) (\cosh\eta_0-1)^{2} ]^{1 / 2}} \biggr\} </math>
		<math>~ - \frac{3}{2^2} \biggl[ k_0 ~K ( k_0) \biggr] \times \biggl\{ \cosh\eta_0~ k_0~K ( k_0 ) ~-~(\cosh^2\eta_0+3) \biggl[ \frac{2}{(\cosh\eta_0 - 1)(\cosh^2\eta_0 -1)} \biggr]^{1 / 2} E(k_0) \biggr\} \, , </math>

where,

<math>~k_0</math>

<math>~\equiv</math>

<math>~\biggl[ \frac{2}{\cosh\eta_0+1}\biggr]^{1 / 2} ~~~\Rightarrow ~~~ (\cosh\eta_0 + 1) = \frac{2}{k_0^2} \, .</math>

Looking back at our previous numerical evaluation of <math>~C_0(\cosh\eta_0)</math> when <math>~z_0 = \cosh\eta_0 = 3 ~~\Rightarrow ~~~ k_0 = 2^{-1 / 2}</math>, we see that, Appendix Expression: <math>~Q_{-\tfrac{1}{2}}(z_0)</math> <math>~=</math> <math>~k_0 K(k_0)</math> Hence MF53 value, <math>~Q_{-\tfrac{1}{2}}(3)</math> <math>~=</math> <math>~1.311028777 ~~~\Rightarrow ~~~ K(k_0) = 1.854074677</math> Appendix Expression: <math>~Q_{+\tfrac{1}{2}}(z_0)</math> <math>~=</math> <math>~z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0)</math> Hence MF53 value, <math>~Q_{+\tfrac{1}{2}}(3)</math> <math>~=</math> <math>~0.1128885424 ~~~\Rightarrow~~~ E(k_0) = 1.350643881</math> Appendix Expression: <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math> <math>~=</math> <math>~[2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)]</math> Hence, <math>~Q^2_{-\tfrac{1}{2}}(3)</math> <math>~=</math> <math>~1.104816977</math>, which matches MF53 value Additional derivation: <math>~Q^2_{+\tfrac{1}{2}}(z_0)</math> <math>~=</math> <math>~ -~\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\} </math> Hence, <math>~Q^2_{+\tfrac{1}{2}}(3)</math> <math>~=</math> <math>~0.449302588</math> <math>~\Rightarrow ~~~ C_0(3)</math> <math>~=</math> <math>~ \frac{1}{2}~Q_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) + \frac{3}{2}~ Q_{- \frac{1}{2}}(3)\cdot Q^2_{+ \frac{1}{2}}(3) = 0.945933522 \, . </math>

Attempting to simplify this expression, we have,

<math>~2C_0(\cosh\eta_0)</math>	<math>~=</math>	<math>~ \biggl\{ \cosh\eta_0 ~k_0~K(k_0) ~-~ \biggl(\frac{2}{k_0}\biggr) E(k_0) \biggr\} \times \biggl\{ \frac{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) }{ [2^{2} k_0^{-1} (\cosh\eta_0-1) ]} \biggr\} </math>
		<math>~ - \frac{3}{2^2} \biggl[ k_0 ~K ( k_0) \biggr] \times \biggl\{ \cosh\eta_0~ k_0~K ( k_0 ) ~-~(\cosh^2\eta_0+3) \biggl[ \frac{k_0^2}{(\cosh\eta_0 - 1)^2} \biggr]^{1 / 2} E(k_0) \biggr\} </math>
<math>~\Rightarrow ~~~ 2^3(\cosh\eta_0 - 1)C_0(\cosh\eta_0)</math>	<math>~=</math>	<math>~ \biggl\{ \cosh\eta_0 ~k_0^2~K(k_0) ~-~ 2 E(k_0) \biggr\} \times \biggl\{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) \biggr\} </math>
		<math>~ - 3 k_0 ~K ( k_0) \times \biggl\{ \cosh\eta_0(\cosh\eta_0 - 1)~ k_0~K ( k_0 ) ~-~(\cosh^2\eta_0+3) k_0 E(k_0) \biggr\} </math>
	<math>~=</math>	<math>~ -~K(k_0)\cdot K(k_0) \biggl[ (\cosh\eta_0-1) \cdot \cosh\eta_0 ~k_0^2 + 3\cosh\eta_0~ (\cosh\eta_0~-1)k_0^2\biggr] </math>
		<math>~ + K(k_0)\cdot E(k_0) \biggl[ 2^2 \cosh^2\eta_0 ~k_0^2 + 2(\cosh\eta_0 ~-1) + 3k_0^2 (\cosh^2\eta_0 ~ + 3)\biggr] - E(k_0)\cdot E(k_0) \biggl[2^3\cosh\eta_0 \biggr] </math>
<math>~\Rightarrow ~~~ \biggl[ \frac{ 2^3(\cosh\eta_0 - 1)}{k_0^2} \biggr] C_0(\cosh\eta_0)</math>	<math>~=</math>	<math>~ -~K(k_0)\cdot K(k_0) \biggl[ (\cosh\eta_0-1) \cdot \cosh\eta_0 + 3\cosh\eta_0~ (\cosh\eta_0~-1) \biggr] </math>
		<math>~ + K(k_0)\cdot E(k_0) \biggl[ 2^2 \cosh^2\eta_0 + \frac{2}{k_0^2}(\cosh\eta_0 ~-1) + 3 (\cosh^2\eta_0 ~ + 3)\biggr] - E(k_0)\cdot E(k_0) \biggl[\frac{2^3\cosh\eta_0}{k_0^2} \biggr] </math>
<math>~\Rightarrow ~~~ (\cosh^2\eta_0 - 1) C_0(\cosh\eta_0)</math>	<math>~=</math>	<math>~ K(k_0)\cdot K(k_0) \biggl[ \cosh\eta_0(1 - \cosh\eta_0) \biggr] + 2K(k_0)\cdot E(k_0) \biggl[ \cosh^2\eta_0 + 1\biggr] - E(k_0)\cdot E(k_0) \biggl[ \cosh\eta_0(1 + \cosh\eta_0) \biggr] </math>

This last, simplifed expression gives, as above, <math>~C_0(3) = 0.945933523</math>. TERRIFIC!

Finally then, for any choice of <math>~\eta_0</math>,

<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{exterior}</math>	<math>~=</math>	<math>~ - \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot \boldsymbol{K}(k) </math>
		<math>~ \times \biggl\{ K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] + 2K(k_0)\cdot E(k_0) l[ \cosh^2\eta_0 + 1 ] - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] \biggr\} \, . </math>

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