Celestial Mechanics and Dynamical Astronomy

, Volume 99, Issue 4, pp 261–292

Long-term evolution of orbits about a precessing oblate planet: 3. A semianalytical and a purely numerical approach

Original Article


Construction of an accurate theory of orbits about a precessing and nutating oblate planet, in terms of osculating elements defined in a frame associated with the equator of date, was started in Efroimsky and Goldreich (2004) and Efroimsky (2004, 2005, 2006a, b). Here we continue this line of research by combining that analytical machinery with numerical tools. Our model includes three factors: the J2 of the planet, its nonuniform equinoctial precession described by the Colombo formalism, and the gravitational pull of the Sun. This semianalytical and seminumerical theory, based on the Lagrange planetary equations for the Keplerian elements, is then applied to Deimos on very long time scales (up to 1 billion years). In parallel with the said semianalytical theory for the Keplerian elements defined in the co-precessing equatorial frame, we have also carried out a completely independent, purely numerical, integration in a quasi-inertial Cartesian frame. The results agree to within fractions of a percent, thus demonstrating the applicability of our semianalytical model over long timescales. Another goal of this work was to make an independent check of whether the equinoctial-precession variations predicted for a rigid Mars by the Colombo model could have been sufficient to repel its moons away from the equator. An answer to this question, in combination with our knowledge of the current position of Phobos and Deimos, will help us to understand whether the Martian obliquity could have undergone the large changes ensuing from the said model (Ward 1973; Touma and Wisdom 1993, 1994; Laskar and Robutel 1993), or whether the changes ought to have been less intensive (Bills 2006; Paige et al. 2007). It has turned out that, for low initial inclinations, the orbit inclination reckoned from the precessing equator of date is subject only to small variations. This is an extension, to non-uniform equinoctial precession given by the Colombo model, of an old result obtained by Goldreich (1965) for the case of uniform precession and a low initial inclination. However, near-polar initial inclinations may exhibit considerable variations for up to ±10 deg in magnitude. This result is accentuated when the obliquity is large. Nevertheless, the analysis confirms that an oblate planet can, indeed, afford large variations of the equinoctial precession over hundreds of millions of years, without repelling its near-equatorial satellites away from the equator of date: the satellite inclination oscillates but does not show a secular increase. Nor does it show secular decrease, a fact that is relevant to the discussion of the possibility of high-inclination capture of Phobos and Deimos.


Orbital elements Osculating elements Mars Natural satellites Natural satellites’ orbits Deimos Equinoctial precession The Goldreich lock 


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  1. Bills, B.G.: Non-chaotic obliquity variations of Mars. The 37th Annual Lunar and Planetary Science Conference, pp. 13–17, March 2006, League City, TX (2006)Google Scholar
  2. Brouwer D. (1959). Solution of the problem of artificial satellite theory without drag. Astron. J. 64: 378–397 CrossRefADSMathSciNetGoogle Scholar
  3. Brouwer, D., van Woerkom, A.J.J.: The secular variations of the orbital elements of the principal planets. Astronomical papers prepared for the use of the American Ephemeris and Nautical Almanac, vol. 13, Part 2, pp. 81–107. US Government Printing Office, Washington, DC (1950)Google Scholar
  4. Brumberg V.A., Evdokimova L.S. and Kochina N.G. (1971). Analytical methods for the orbits of artificial satellites of the moon. Celestial Mech. 3: 197–221 MATHCrossRefADSGoogle Scholar
  5. Burns J. (1972). Dynamical characteristics of phobos and deimos. Rev. Geophys. Space Phys. 6: 463–483 ADSGoogle Scholar
  6. Burns J. (1978). The dynamical evolution and origin of the Martian moons. Vistas Astron. 22: 193–210 CrossRefADSGoogle Scholar
  7. Colombo G. (1966). Cassini’s second and third laws. Astron. J. 71: 891–896 CrossRefADSGoogle Scholar
  8. Cook G.E. (1962). Luni-solar perturbations of the orbit of an earth satellite. Geophys. J. 6(3): 271–291 MATHADSGoogle Scholar
  9. Efroimsky M. and Goldreich P. (2004). Gauge freedom in the N-body problem of celestial mechanics. Astron. Astrophys. 415: 1187–1199, astro-ph/0307130CrossRefADSMATHGoogle Scholar
  10. Efroimsky, M.: Long-term evolution of orbits about a precessing oblate planet. 1. The case of uniform precession. astro-ph/0408168 (2004) [This preprint is a very extended version of Efroimsky (2005)]Google Scholar
  11. Efroimsky M. (2005). Long-term evolution of orbits about a precessing oblate planet: 1. The case of uniform precession. Celestial Mech. Dynam. Astron. 91: 75–108 MATHCrossRefADSMathSciNetGoogle Scholar
  12. Efroimsky M. (2006). Long-term evolution of orbits about a precessing oblate planet: 2. The case of variable precession. Celestial Mech. Dynam. Astron. 96: 259–288 MATHCrossRefADSMathSciNetGoogle Scholar
  13. Efroimsky, M.: Long-term evolution of orbits about a precessing oblate planet: 2. The case of variable precession. astro-ph/0607522 (2006b) [This preprint is a very extended version of Efroimsky (2006a)]Google Scholar
  14. Efroimsky M. (2006). Gauge freedom in orbital mechanics. Ann. N. Y. Academy Sci. 1065: 346–374, astro-ph/0603092CrossRefADSGoogle Scholar
  15. Efroimsky, M., Lainey, V.: The physics of bodily tides in terrestrial planets, and the appropriate scales of dynamical evolution. J. Geophys. Res.—Planets (2007, in press)Google Scholar
  16. Everhart, E.: An efficient integrator that uses Gauss-Radau spacings. Dynamics of comets: their origin and evolution. In: Carusi, A., Valsecchi, G.B. (eds.) Proceedings of IAU Colloquium 83 held in Rome on 11–15 June 1984. vol. 115, p. 185. Astrophysics and Space Science Library, Dordrecht, Reidel (1985)Google Scholar
  17. Goldberg D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA MATHGoogle Scholar
  18. Goldreich P. (1965). Inclination of satellite orbits about an oblate precessing planet. Astron. J. 70: 5–9 CrossRefADSGoogle Scholar
  19. Gurfil P., Kasdin N.J., Arrell R.J., Seager S. and Nissanke S. (2002). Infrared space observatories: How to mitigate zodiacal dust interference. Astrophys. J. 567: 1250–1261 CrossRefADSGoogle Scholar
  20. Gurfil P. and Kasdin N.J. (2002a). Characterization and design of out-of-ecliptic trajectories using deterministic crowding genetic algorithms. Comput. Methods Appl. Mech. Eng. 191: 2169–2186 MATHCrossRefGoogle Scholar
  21. Gurfil P. and Kasdin N.J. (2002b). Niching genetic algorithms-based characterization of geocentric orbits in the 3D elliptic restricted three-body problem. Comput. Methods Appl. Mech. Eng. 191: 5673–5696 Google Scholar
  22. Hartmann W.K. (2007). Martian cratering 9: toward resolution of the controversy about small craters. Icarus 189: 274–278 CrossRefADSGoogle Scholar
  23. Innanen K.A., Zheng J.Q., Mikkola S. and Valtonen M.J. (1997). The Kozai Mechanism and the stability of planetary orbits in binary star systems. Astron. J. 113: 1915–1919 CrossRefADSGoogle Scholar
  24. Kilgore T.R., Burns J.A. and Pollack J.B. (1978). Orbital evolution of “Phobos” following its “capture”. Bull. Am. Astron. Soc. 10: 593 ADSGoogle Scholar
  25. Kozai Y. (1959). On the effects of the Sun and the Moon upon the motion of a close Earth satellite. SAO Special Report 22: 7–10 ADSGoogle Scholar
  26. Kozai Y. (1960). Effect of precession and nutation on the orbital elements of a close earth satellite. Astron. J. 65: 621–623 CrossRefADSMathSciNetGoogle Scholar
  27. Lainey V., Duriez L. and Vienne A. (2004). New accurate ephemerides for the Galilean satellites of Jupiter. I. Numerical integration of elaborated equations of motion. Astron. Astrophys. 420: 1171–1183 CrossRefADSGoogle Scholar
  28. Lainey, V., Gurfil, P., Efroimsky, M.: Long-term evolution of orbits about a precessing oblate planet: 4. A comprehensive model (2008 in preparation)Google Scholar
  29. Laskar J. (1988). Secular evolution of the solar system over 10 million years. Astron. Astrophys. 198: 341–362 ADSGoogle Scholar
  30. Laskar J. and Robutel J. (1993). The chaotic obliquity of the planets. Nature 361: 608–612 CrossRefADSGoogle Scholar
  31. Murison, M.: Satellite Capture and the Restricted Three-Body Problem. Ph.D. Thesis, University of Wisconsin, Madison (1988)Google Scholar
  32. Nesvorný D. and Vokrouhlický D. (2007). Analytic theory of the YORP effect for near-spherical objects. Astron. J. 134: 1750–1768 ADSGoogle Scholar
  33. Paige, D.A., Golombek, M.P., Maki, J.N., Parker, T.J., Crumpler, L.S., Grant, J.A., Williams, J.P.: MER small-crater statistics: evidence against recent quasi-periodic climate variations. Seventh Int. Conference Mars, 9–13 July 2007, Caltech, Pasadena, CA (2007)Google Scholar
  34. Pang K.D., Pollack J.B., Veverka J., Lane A.L. and Ajello J.M. (1978). The composition of phobos: evidence for carbonateous chondrite surface from spectral analysis. Science 199: 64 CrossRefADSGoogle Scholar
  35. Pollack J.B., Burns J.A. and Tauber M.E. (1979). Gas drag in primordial circumplanetary envelopes. a mechanism for satellite capture. Icarus 37: 587 CrossRefADSGoogle Scholar
  36. Proskurin V.F. and Batrakov Y.V. (1960). Perturbations of the motion of artificial satellites, caused by the earth oblateness. Bull. Inst. Theor. Astro. 7: 537–548 Google Scholar
  37. Smith D.E., Lemoine F.G. and Zuber M.T. (1995). Simultaneous estimation of the masses of mars, phobos, and deimos using spacecraft distant encounters. Geophys. Res. Lett. 22: 2171–2174 CrossRefADSGoogle Scholar
  38. Szebehely V. (1967). Theory of Orbits. Academic Press, NY Google Scholar
  39. Tolson, R.H., 15, collaborators.: Viking first encounter of phobos. Preliminary results. Science 199, 61 (1978) Google Scholar
  40. Touma J. and Wisdom J. (1993). The chaotic obliquity of Mars. Science 259: 1294–1297 CrossRefADSGoogle Scholar
  41. Touma J. and Wisdom J. (1994). Lie-Poisson integrators for rigid body dynamics in the solar system. Astron. J. 107: 1189–1202 CrossRefADSGoogle Scholar
  42. Veverka J. (1977). Phobos and deimos. Sci. Am. 236: 30 ADSCrossRefGoogle Scholar
  43. Ward W. (1973). Large-scale variations in the obliquity of Mars. Science 181: 260–262 CrossRefADSGoogle Scholar
  44. Ward W. (1974). Climatic variations of Mars. Astronomical theory of insolation. J. Geophys. Res. 79: 3375–3386 ADSCrossRefGoogle Scholar
  45. Ward W. (1979). Present obliquity oscillations of Mars—Fourth-order accuracy in orbital e and i. J. Geophys. Res. 84: 237–241 ADSGoogle Scholar
  46. Ward W. (1982). Comments on the long-term stability of the earth’s obliquity. Icarus 50: 444–448 CrossRefADSGoogle Scholar
  47. Zhang K., Hamilton, D.P.: Dynamics of inner neptunian satellites. Abstracts of the 37th DPS Meeting of the Americal Astronomical Society. In: AAS Bulletin,37, 667–668 (2005)Google Scholar
  48. Waz P. (2004). Analytical theory of the motion of phobos: a comparison with numerical integration. Astron. Astrophys. 416: 1187–1192 CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V 2007

Authors and Affiliations

  • Pini Gurfil
    • 1
  • Valéry Lainey
    • 2
    • 3
  • Michael Efroimsky
    • 4
  1. 1.Faculty of Aerospace EngineeringTechnionHaifaIsrael
  2. 2.IMCCE/Observatoire de Paris, UMR 8028 du CNRSParisFrance
  3. 3.Observatoire Royal de BelgiqueBrusselsBelgium
  4. 4.US Naval ObservatoryWashingtonUSA

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