Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession



It was believed until very recently that a near-equatorial satellite would always keep up with the planet’s equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich [Astronomy & Astrophysics (2004) Vol. 415, pp. 1187–1199], this misconception originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet’s obliquity change. Some of these terms will be non-trivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet’s obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations will resonate with the satellite’s orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in a subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics.


near-equatorial satellites of oblate planets contact orbital elements 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arnold, V. I. 1989Mathematical Methods of Classical Mechanics.Springer-VerlagNew YorkGoogle Scholar
  2. Ashby, N., Allison, T. 1993‘Canonical planetary perturbation equations for velocity- dependent forces and the lense-Thirring precession’Celest. Mech. Dyn. Astr57537585CrossRefMATHADSMathSciNetGoogle Scholar
  3. Brumberg, V. A., Evdokimova, L. S., Kochina, N. G. 1971‘Analytical methods for the orbits of artificial satellites of the Moon’Celestial Mech.3197221CrossRefMATHADSGoogle Scholar
  4. Brumberg, V. A. 1992Essential Relativistic Celestial MechanicsAdam HilgerBristolGoogle Scholar
  5. Brouwer, D. 1959‘Solution of the problem of artificial satellite theory without drag’The Astronomical J.64378397CrossRefADSMathSciNetGoogle Scholar
  6. Chernoivan, V. A., Mamaev, I. S. 1999‘The restricted two-body problem and the Kepler problem in the constant-curvature spaces’Reg. Chaot. Dynam4112124CrossRefMATHMathSciNetGoogle Scholar
  7. Defraigne, P., Rivoldini, A., Van Hoolst, T., Dehant, V. 2003Mars nutation resonance due to free inner core nutation’J. Geophys. Res.1085128CrossRefGoogle Scholar
  8. Dehant, V., van Hoolst, T., Defraigne, P. 2000‘Comparison between the nutations of the planet Mars and the nutations of the Earth’Surv. Geophys2189110CrossRefADSGoogle Scholar
  9. Efroimsky, M.: 2002a, ‘Equations for the orbital elements. Hidden symmetry’, Preprint No 1844 of the Institute of Mathematics and its Applications, University of Minnesota http://www.ima.umn.edu/preprints/feb02/feb02.html. Google Scholar
  10. Efroimsky, M.: 2002b, ‘The implicit gauge symmetry emerging in the N-body problem of celestial mechanics’, astro-ph/0212245.Google Scholar
  11. Efroimsky, M. and Goldreich, P.: 2003, ‘Gauge symmetry of the N-body problem in the Hamilton-Jacobi approach’, J. Math. Phys. 44, 5958–5977 astro-ph/0305344. Google Scholar
  12. Efroimsky, M. and Goldreich, P.: 2004, ‘Gauge freedom in the N-body problem of celestial mechanics’, Astron. Astrophys. 415, 1187–1199; astro-ph/0307130. Google Scholar
  13. Efroimsky, M.: 2004, ‘Long-term evolution of orbits about a precessing oblate planet, The case of uniform precession’, astro-ph/0408168 Google Scholar
  14. Eubanks, T. M. 1993‘Variation in the orientation of the Earth’ In: Contributions of Space Geodesy to Geodynamics: Earth DynamicsAmer. Geophys. UnionWashington154Google Scholar
  15. Goldreich, P. 1965‘Inclination of satellite orbits about an oblate precessing planet’Astron. J7059CrossRefADSGoogle Scholar
  16. Kinoshita, T. 1993‘Motion of the orbital plane of a satellite due to a secular change of the obliquity of its mother planet’Celest. Mech. Dyn. Astr.57359368CrossRefMATHADSGoogle Scholar
  17. Kozai, Y. 1960‘Effect of precession and nutation on the orbital elements of a close earth satellite’The Astronomical J.,65621623ADSMathSciNetGoogle Scholar
  18. Laskar, J. 2004‘A comment on ‘Accurate Spin Axes and Solar System Dynamics: Climatic Variations for the Earth and Mars’Astron. Astrophys416799800CrossRefADSGoogle Scholar
  19. Laskar, J., Robutel, J. 1993‘The chaotic obliquity of the planets’Nature361608612CrossRefADSGoogle Scholar
  20. Laskar, J. 1990‘The chaotic motion of the solar system A numerical estimate of the size of the chaotic zones’Icarus88266291CrossRefADSGoogle Scholar
  21. Marsden, J., Ratiu, T 2003Introduction to Mechanics and Symmetry.SpringerNYGoogle Scholar
  22. Morbidelli, A. 2002Modern Celestial Mechanics: Dynamics in the Solar System.Taylor & Francis LondonGoogle Scholar
  23. Murison, M. 1988‘Satellite Capture and the Restricted Three-Body Problem’Ph.D. thesis, University of WisconsinMadisonGoogle Scholar
  24. Newman, W., Efroimsky, M. 2003‘The method of variation of constants and multiple time scales in orbital mechanics’Chaos13476485CrossRefPubMedMATHADSMathSciNetGoogle Scholar
  25. Proskurin, V. F., Batrakov, Y. V. 1960‘Perturbations of the Motion of Artificial Satellites, caused by the Earth Oblateness’The Bulletin of the Institute of Theoretical Astronomy7537548Google Scholar
  26. Richardson, D. L., Kelly, T. J. 1988‘Two-body motion in the post-Newtonian approximation’Celestial Mech43193210CrossRefMATHADSGoogle Scholar
  27. Touma, J., Wisdom, J. 1994‘Lie-Poisson integrators for rigid body dynamics in the solar system’Astron J.10711891202CrossRefADSGoogle Scholar
  28. Van Hoolst, T., Dehant, V., Defraigne, P. 2000‘Chandler wobble and free core nutation for Mars’Planet. Space Sci4811451151CrossRefADSGoogle Scholar
  29. Ward, W. 1973‘Large-scale variations in the obliquity of Mars’Science181260262ADSGoogle Scholar
  30. Ward, W. 1974‘Climatic variations of Mars. Astronomical theory of insolation’J. Geophys. Res7933753386CrossRefADSGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.US Naval ObservatoryWashingtonUSA

Personalised recommendations