About averaging procedures in the problem of asteroid motion

  • I. V. Tupikova
Original Article


A new mathematically correct approach to construct an averaging procedure for the motion of a massless body around the central body perturbed by fully interacting planets is developed and the errors of the standard solution are discussed. The new technique allows to combine the advantages of the Hamiltonian representation with the usage of standard osculating elements in combination with all the standard expansions of the perturbing functions. The main idea is to introduce new additional variables conjugate to all the standard elements and to work in a corresponding super phase space. In this way, the number of variables is doubled at first, but one has to deal with only one Hamiltonian. The artificially introduced variables disappear from the final averaged equations as well as from the transformation formulae connecting the osculating and the mean elements.


Hamiltonian dynamics Averaging Lie series method Asteroid dynamics 


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  1. Beaugé, C., Ferraz-Mello, S., Michtchenko, T.A.: Planetary masses and orbital parameter from radial velocity measurements. In: Dvorak, R. (ed.) Extrasolar planets: formation, detection and dynamics, pp. 1–25, Wiley-VCH Verlag, Weinheim (2007)Google Scholar
  2. Beaugé C., Michtchenko T.A.: Modeling the high-eccentricity three-body problem. Application to the GJ876 planetary system. MNRAS 341, 760–770 (2003)CrossRefADSGoogle Scholar
  3. Beaugé C., Nesvorný D., Dones L.: A high-order analytical model for the secular dynamics of irregular satellites. Astron. J. 131, 2299–2313 (2006)CrossRefADSGoogle Scholar
  4. Bretagnon P.: Méthode iterative de construction d’une théorie générale planétaire. Astron. Astrophys. 231, 561–570 (1990)MATHADSMathSciNetGoogle Scholar
  5. Brouwer, D., van Woerkom, A.J.J.: The secular variations of the orbital elements of the principal planets. Astron. Pap. XIII(Part II), 81–107 (1950)Google Scholar
  6. Deprit A.: Canonical transformations depending on a small parameter. Celest. Mech. 1, 12–30 (1969)MATHCrossRefADSMathSciNetGoogle Scholar
  7. Ferraz-Mello S.: Do average Hamiltonians exist?. Celest. Mech. Dyn. Astron. 73, 243–248 (1999)MATHCrossRefADSMathSciNetGoogle Scholar
  8. Gronchi G.F., Milani A.: Proper elements for Earth-crossing asteroids. Icarus 152, 58–69 (2001)CrossRefADSGoogle Scholar
  9. Hori G.I.: Theory of general perturbations with unspecified canonical variables. Publ. Astron. Soc. Japan 18, 287–296 (1966)ADSGoogle Scholar
  10. Jacobi C.G.J.: Vorlesungen über Dynamik. Reimer Verlag, Berlin (1842)Google Scholar
  11. Kaula W.M.: Theory of Satellite Geodesy. Blaisdell Publ Co, MA (1966)Google Scholar
  12. Knežević, Z., Milani, A.: Higher order and iterative theories to compute asteroid mean elements. In: Proceedings of Colloquium IAU 172 impact of modern dynamics in astronomy, pp. 359–360. Namur (1998)Google Scholar
  13. Knežević Z., Milani A.: Synthetic proper elements for outer main belt asteroids. Celest. Mech. Dyn. Astron. 78, 17–46 (2000)MATHCrossRefADSGoogle Scholar
  14. Laskar J., Robutel P.: Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian. Celest. Mech. Dyn. Astron. 62, 193–217 (1995)MATHCrossRefADSMathSciNetGoogle Scholar
  15. Lemaitre A., Morbidelli A.: Proper elements for highly inclined asteroidal orbits. Celest. Mech. Dyn. Astron. 60, 29–56 (1994)MATHCrossRefADSGoogle Scholar
  16. LeVerrier U.-J.: Détermination des expressions analytique des coefficients du dévelopment de la fonction perturbatrice. Ann. Obs. Paris 1, 258–342 (1885)Google Scholar
  17. Michtchenko T.A., Ferraz-Mello S.: Modeling the 5:2 mean-motion resonance in the Jupiter-Saturn planetary system. Icarus 149, 357–374 (2001)CrossRefADSGoogle Scholar
  18. Milani A., Knežević Z.: Secular perturbation theory and computation of asteroid proper elements. Celest. Mech. 49, 347–411 (1990)MATHCrossRefADSGoogle Scholar
  19. Milani A., Knežević Z.: Asteroid proper elements and secular resonances. Icarus 98, 211–232 (1992)CrossRefADSGoogle Scholar
  20. Nesvorný, D., Ferraz-Mello, S., Holman, M., Morbidelli, A. : Regular and chaotic dynamics in the meanmotion resonances: Implications for the structure and evolution of the asteroid belt. In: Bottke, W.F., Paolicchi, P., Binzel, R.P., Cellino, A. (eds.) Asteroids III, pp. 379–394. University of Arizona Press, Tucson (2002)Google Scholar
  21. Poincarè H.: Sur une forme nouvelle des équations du probléme des trois corps. Bull. Astron. 14, 53 (1897)Google Scholar
  22. Tupikova, I.: Averaging in N-body problem with non-standard canonical transformation. In: Proceedings of Colloquium astrometry, geodynamics and celestial mechanics on the eve of XXI century, pp. 26–28. St.-Petersburg (2000)Google Scholar
  23. Tupikova, I.: Perturbation theory for asteroid motion in the gravitational field of a migrating planet. In: Proceedings of the conference analytical methods of celestial mechanics, St.Petersburg (2007a)Google Scholar
  24. Tupikova, I. Analytical theory for the motion of an asteroid in the gravitational field of migrating planet. Les Journées Systèmes de Reférence Spatio-temporels, pp. 121–123. Paris (2007b)Google Scholar
  25. Williams, J.G.: Secular perturbations in the solar system. Ph.D. Dissertation, University of California, Los Angeles (1969)Google Scholar
  26. Yuasa M.: Theory of secular perturbations of asteroids including terms of higher orders and higher degrees. Publ. Astron. Soc. Japan 25, 399–445 (1973)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Lohrmann ObservatoryInstitute of Planetary GeodesyDresdenGermany

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