Celestial Mechanics and Dynamical Astronomy

, Volume 60, Issue 1, pp 29–56 | Cite as

Proper elements for highly inclined asteroidal orbits

  • Anne Lemaitre
  • Alessandro Morbidelli


Based on Williams' work and rewritten in action angle variables, a method for the calculation of proper elements is here presented. The averaging over the long periodic terms is performed by the semi numerical method developed by Henrard (1990); no series expansion in eccentricity or inclination of the asteroid is used which allows calculating proper elements for highly inclined orbits. Conversely, the theory is truncated at the first degree in the eccentricity and the inclination of the perturbing planets. A few tests about accuracy and consistency are presented.

Key words

asteroids proper elements averaging method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Brouwer, D., Van Woerkom, A.J.: 1950, “The secular variations of the orbital elements of the principal planets”,Astr. Papers U.S. Naval Obs.,13, 85–107Google Scholar
  2. Everhart, E.: 1985, “An efficient integrator that uses Gauss-Radau spacings”,Dynamics of Comets: their origin and evolution,Eds A. Carusi and G.B. Valsecchi, D. Reidel, Dordrecht, 185-Google Scholar
  3. Froeschlé, Ch., Morbidelli, A., Scholl, H.: 1991, “Complex dynamical behaviour of the asteroid 2335 James associated with the secular resonancesv 5 andv 16: numerical studies and theoretical interpretation”,Astron. Astroph.,24, 553Google Scholar
  4. Henrard, J.: 1990, “A Semi- Numerical Perturbation Method for Separable Hamiltonian Systems”,Celest. Mech. and Dyn. Astr.,49, 43–67Google Scholar
  5. Hirayama, K.: 1918, “Groups of asteroids probably of common origin”,Astron. J.,31, 185–188Google Scholar
  6. Knežević, Z.: 1988, “Asteroid Mean Orbital Elements”,Bull. Obs. Astron. Belgrade,139, 1–6Google Scholar
  7. Knežević: 1989, “Asteroid long-periodic perturbations: the second order Hamiltonian”,Celest. Mech. and Dyn. Astr.,46, 147–158Google Scholar
  8. Kozai, Y.: 1962, “Secular Perturbations of asteroids with high inclinations and eccentricities”,Astron. J.,67, 591–598Google Scholar
  9. Milani, A., Knežević, Z.: 1990, “Secular Perturbation Theory and Calculation of Asteroid Proper Elements”,Celest. Mech. and Dyn. Astr.,49, 347–411Google Scholar
  10. Milani, A., Knežević, Z.: 1992, “Asteroid proper elements and secular resonances”,Icarus,98, 211–232Google Scholar
  11. Milani, A., Knežević, Z.: 1994, “Asteroid proper elements and the dynamical structure of the asteroid belt”,Icarus,, in pressGoogle Scholar
  12. Morbidelli A, Henrard J.: 1991, “Secular resonances in the asteroid belt: theoretical perturbation approach and the problem of their location”,Celest. Mech. and Dyn. Astr.,51, 131–167Google Scholar
  13. Nobili, A., Carpino, M., Milani, A.: 1989, “Fundamental frequencies and small divisors in the orbits of the outer planets”,Astron. Astroph.,210, 313–336Google Scholar
  14. Williams, J.G.: 1969, “Secular Perturbations in the Solar System”,Ph.D. Dissertation, University of California, Los AngelesGoogle Scholar
  15. Williams, J.G.: 1979, “Proper Elements and Families Memberships of the Asteroids”,Asteroids,Ed. Gehrels T., Ch. III pp 1040–1063Google Scholar
  16. Williams, J.G.: 1989, “Proper Elements and Families Memberships of the Asteroids”,Asteroids II,Eds. Binzel R.P., Gehrels T., Matthews, M.S., Ch. VI pp 1034–1072Google Scholar
  17. Yuasa, M.: 1973, “Theory of Secular Perturbations of asteroids Including Terms of Higher Orders and Higher Degrees”,Publ. Astron. Soc. Japan,25, 399–445Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Anne Lemaitre
    • 1
  • Alessandro Morbidelli
    • 1
  1. 1.Department of Mathematics - FUNDPNamurBelgium

Personalised recommendations