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Figures of equilibrium of an inhomogeneous self-gravitating fluid

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Abstract

This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.

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Notes

  1. Relevant calculations can be easily performed using the formulae of Sect. 3.3 and astronomical data available from the Internet.

  2. A. Clairaut took part in the first expeditions which confirmed I. Newton’s viewpoint that the Earth is compressed from the poles.

  3. The proof of convergence was not published by Lyapunov. Problems of convergence for the Maclaurin and Huygens–Roche figures are solved in Kholshevnikov and Elkin (2002), Kholshevnikov (2003) and Kholshevnikov and Kurdubov (2004).

  4. This work was not published during the life-time of S. A. Chaplygin and appeared for the first time in his posthumous collected works prepared by L. N. Sretenskii.

  5. In the case of piecewise constant distribution the function \(\rho ^{\prime }(\mu )\) is a combination of Dirac delta functions.

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Acknowledgments

The authors thank A. Albouy for useful advice and invaluable assistance in the course of work. The work of Alexey V. Borisov was carried out within the framework of the state assignment to the Udmurt State University “Regular and Chaotic Dynamics”. The work of Ivan S. Mamaev was supported by the RFBR Grants 14-01-00395-a.

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Authors and Affiliations

  1. Udmurt State University, Universitetskaya 1, Izhevsk, 426034, Russia

    Ivan A. Bizyaev & Ivan S. Mamaev

  2. Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja Str. 16, Ekaterinburg, 620990, Russia

    Alexey V. Borisov

  3. National Research Nuclear University “MEPhI”, Kashirskoye Shosse 31, Moscow, 115409, Russia

    Alexey V. Borisov

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  1. Ivan A. Bizyaev
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  2. Alexey V. Borisov
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  3. Ivan S. Mamaev
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Correspondence to Alexey V. Borisov.

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This is a revised version of the paper (Bizyaev et al. 2014), previously published in Russian.

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Bizyaev, I.A., Borisov, A.V. & Mamaev, I.S. Figures of equilibrium of an inhomogeneous self-gravitating fluid. Celest Mech Dyn Astr 122, 1–26 (2015). https://doi.org/10.1007/s10569-015-9608-5

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