Skip to main content
SpringerLink
Log in

Motion of deformable planets and stability of their stationary rotations

  • Published:
Cosmic Research Aims and scope Submit manuscript

Abstract

A mechanical system consisting from N deformable spheres interacting according to the law of gravity is considered as a model of planetary system. Deformations of the viscoelastic spheres are described according to the model of the theory of elasticity of small deformations, the Kelvin-Voigt model of viscous forces, and occur under the action of gravitational fields and fields of centrifugal forces. Approximate equations describing motions of the centers of mass of the spheres and their rotations relative to the centers of mass are constructed by the method of separation of motions on the basis of solving quasistatic problems of the theory of viscoelasticity with allowance made for smallness of sphere deformations. Using the first integral of conservation of the angular momentum of the system relative to its center of mass, the expression for the changed potential energy is obtained with the use of the Routh method. An investigation of stationary rotations is carried out, and it is shown that all of them are unstable, if the number of planets is more than two.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Spravochnoe rukovodstvo po nebesnoi mekhanike i astrodinamike (Handbook on Celestial Mechanics and Astrodynamics), Duboshin, G.K., Ed., Moscow: Nauka, 1976.

    Google Scholar 

  2. Vil’ke, V.G., Motion of a Viscoelastic Sphere in a Central Newtonian Field of Forces, Prikl. Mat. Mekh., 1980, vol. 44, no. 3, pp. 395–402.

    MathSciNet  Google Scholar 

  3. Vil’ke, V.G., Analiticheskaya mekhanika sistem s beskonechnym chislom stepenei svobody (Analytical Mechanics for Systems with Infinite Number of Degrees of Freedom), Moscow: Izd. Mekh.-Mat. Fak., Mos. Gos. Univ., 1997.

    Google Scholar 

  4. Vil’ke, V.G. and Shatina, A.V., Evolution of Motion of a Binary Planet, Kosm. Issled., 2001, vol. 39, no. 3, pp. 316–323. [Cosmic Research, pp. 295–305].

    Google Scholar 

  5. Vil’ke, V.G., Teoreticheskaya mekhanika (Theoretical Mechanics), St. Petersburg: Lan’, 2003.

    Google Scholar 

  6. Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge Univ. Press, 1927. Translated under the title Matematicheskaya teoriya uprugosti, Moscow: ONTI, 1935.

  7. Vil’ke, V.G. and Shatina, A.V., Translational-Rotational Motion of a Viscoelastic Sphere in Gravitational Field of an Attracting Center and a Satellite, Kosm. Issled., 2004, vol. 42, no. 1, pp. 95–106. [Cosmic Research, pp. 91–102].

    Google Scholar 

  8. Karapetyan, A.V. and Rumyantsev, V.V., Stability of Conservative and Dissipative Systems, Itogi Nauki Tekh., Ser.: Obshch. Mekh., vol. 6, Moscow: VINITI, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Vorob’evy gory, Moscow, 119899, Russia

    V. G. Vil’ke

Authors
  1. V. G. Vil’ke
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.G. Vil’ke, 2010, published in Kosmicheskie Issledovaniya, 2010, Vol. 48, No. 3, pp. 279–288.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vil’ke, V.G. Motion of deformable planets and stability of their stationary rotations. Cosmic Res 48, 273–282 (2010). https://doi.org/10.1134/S0010952510030093

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0010952510030093

Keywords

Search

Navigation