Volume 8, Issue 2, June 2020, Page: 30-34
A Differential Rotation Law for Stars and Fluid Planets
Joel Uriel Cisneros-Parra, Science Faculty, Autonomous University of San Luis Potosí, San Luis Potosí, México; Physics Institute, Autonomous University of San Luis Potosí, San Luis Potosí, México
Francisco Javier Martínez-Herrera, Physics Institute, Autonomous University of San Luis Potosí, San Luis Potosí, México
Daniel Montalvo-Castro, Physics Institute, Autonomous University of San Luis Potosí, San Luis Potosí, México
Received: Mar. 31, 2020;       Accepted: May 3, 2020;       Published: May 15, 2020
DOI: 10.11648/j.ajaa.20200802.13      View  18      Downloads  6
Abstract
We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number l of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let l tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed ad initio. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.
Keywords
Gravitation, Hydrodynamics, Planets and Satellites, Stars, Rotation
To cite this article
Joel Uriel Cisneros-Parra, Francisco Javier Martínez-Herrera, Daniel Montalvo-Castro, A Differential Rotation Law for Stars and Fluid Planets, American Journal of Astronomy and Astrophysics. Vol. 8, No. 2, 2020, pp. 30-34. doi: 10.11648/j.ajaa.20200802.13
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Cisneros, J. U., F. J. Martínez, & D. Montalvo, RMx, AA, 2016, 52, 375.
[2]
Cisneros, J. U., F. J. Martínez, & D. Montalvo, 2017, ApJ, 848, 109.
[3]
Kitchatinov, L. L., 2005, Physics-Uspekhi, 48 (5), 449.
[4]
Cisneros, J. U., F. J. Martínez, & D. Montalvo, American Journal of Astronomy and Astrophysics. Vol. 8, No. 1, 2020, pp. 8-14.
[5]
Jeans, J. H. 1919, Phil Trans. R. Soc., (Cambridge, England Cambridge University Press).
[6]
Poincare, H., Theorie des Turbillons, (1893), Gauthier-Villars, Paris, 212.
[7]
Landau, L.\ D.\ \&Lifshitz E.\ M., 1987, Course of Theoretical Physics: Fluid Mechanics, New York: Pergamon Press.
[8]
Tassoul, J. L., Stellar Rotation, (2000), Cambridge University Press, 256.
[9]
Hubbard, W. B., 2012, ApJL, 756: L15.
[10]
Jacobi, C. G. J., Uber die Figur des Gleichgewichts, Poggendorff Annallen der Physik und Chemie, Vol. 33, 1834, pp. 229-238; reprinted in Gesammelte Werke, Vol. 2, pp. 17-72, G. Reimer, Berlin, 1882.
[11]
Lyttleton, R. A. 1953, The Stability of Rotating LiquidMasses (Cambridge: Cambridge University Press).
[12]
Chandrasekhar, S. 1969 Ellipsoidal Figures Of Equilibrium (Yale: University Press).
[13]
Dedekind, R., Zusatz zu der vorstehenden Abhandlung, J. Reine Angew. Math., Vol. 58, 1860.
[14]
Ostriker, J. P., J. W-K. Mark, 1968, ApJ, 151.
[15]
Tassoul, J. L., J. P. Ostriker ApJ, 1968, 154, 613.
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