Figures of equilibrium of an inhomogeneous self-gravitating fluid

  • Ivan A. Bizyaev
  • Alexey V. Borisov
  • Ivan S. Mamaev
Original Article


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.


Self-gravitating fluid Confocal stratification Homothetic stratification Chaplygin problem Axisymmetric equilibrium figures Space of constant curvature 

Mathematics Subject Classification




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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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
  • Alexey V. Borisov
    • 2
    • 3
  • Ivan S. Mamaev
    • 1
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia
  3. 3.National Research Nuclear University “MEPhI”MoscowRussia

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