Difference between revisions of "User:Tohline/Appendix/Ramblings/Bordeaux"
Line 280: | Line 280: | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\frac{2}{ | <math>~\frac{2}{\coth\eta + 1}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 323: | Line 323: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, | \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, . | ||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
And if we make the function-argument substitution, <math>~z \rightarrow \coth\eta</math>, in the "Key Equation," | |||
{{ User:Tohline/Math/EQ_QminusHalf01 }} | {{ User:Tohline/Math/EQ_QminusHalf01 }} | ||
we can write, | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
Line 343: | Line 343: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} k | \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) \, , | ||
</math> | </math> | ||
</td> | </td> | ||
Line 351: | Line 349: | ||
</table> | </table> | ||
where, | where, from above, we recognize that, | ||
<div align="center"> | <div align="center"> | ||
<math>~ | <math>~ | ||
k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} | k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} | ||
= | = | ||
\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + z^2} \biggr]^{1 / 2} \, . | \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} \, . | ||
</math> | </math> | ||
</div> | </div> |
Revision as of 21:46, 17 June 2020
Université de Bordeaux
![]() |
---|
| Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
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
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:
Given that (sin2θ + cos2θ) = 1, we have,
We deduce as well that,
|
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> |
for example … |
<math>~Q_{-\frac{1}{2}}(\cosh\eta)</math> |
<math>~=</math> |
<math>~ 2 e^{-\eta/2} K(e^{-\eta}) </math> |
|
Abramowitz & Stegun (1995), p. 337, eq. (8.13.3) |
Abramowitz & Stegun (1995), p. 337, eq. (8.13.4) |
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>
![]() |
---|
© 2014 - 2021 by Joel E. Tohline |