Difference between revisions of "User:Tohline/Appendix/Ramblings/Bordeaux"
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[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]]. | ||
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 state that the gravitational potential outside of an homogeneous (circular cross-section) torus is, | 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 state that the gravitational potential outside of an homogeneous (circular cross-section) torus with (major, minor) radii (R<sub>C</sub>, b) is, | ||
<div align="center"> | <div align="center"> | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
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<math>~ | <math>~ | ||
-4G\rho_0 \int_0^b \int_0^{2\pi} a\kappa b' d\theta db' | -4G\rho_0 \int_0^b \int_0^{2\pi} a\kappa b' d\theta db' \, , | ||
</math> | </math> | ||
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[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §8, p. 5831, Eq. (48) | |||
</div> | |||
where, the function, | |||
<div align="center"> | |||
<math>~\kappa \equiv \frac{K(k)}{\Delta} \, ,</math> | |||
</div> | |||
<font color="orange">can be replaced by its Taylor expansion, namely</font> … | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
Revision as of 23:11, 16 June 2020
Université de Bordeaux
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Spheroid-Ring Systems
Through a research collaboration at the Université de Bordeaux, B. Basillais & J. -M. Huré (2019), MNRAS, 487, 4504-4509 have published a paper titled, Rigidly Rotating, Incompressible Spheroid-Ring Systems: New Bifurcations, Critical Rotations, and Degenerate States.
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.
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 state that the gravitational potential outside of an homogeneous (circular cross-section) torus with (major, minor) radii (RC, b) is,
|
<math>~\Psi_\mathrm{grav}(\vec{r})</math> |
<math>~=</math> |
<math>~ -4G\rho_0 \int_0^b \int_0^{2\pi} a\kappa b' d\theta db' \, , </math> |
Huré, et al. (2020), §8, p. 5831, Eq. (48)
where, the function,
<math>~\kappa \equiv \frac{K(k)}{\Delta} \, ,</math>
can be replaced by its Taylor expansion, namely …
|
<math>~</math> |
<math>~=</math> |
<math>~</math> |
|
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