Depletion of the Asteroid Belt at Resonances

  • S. Ferraz-Mello
  • R. Dvorak
  • T. A. Michtchenko
Part of the NATO ASI Series book series (NSSB, volume 336)


The existence of gaps and groups in the distribution of outer belt asteroid orbits, at resonances with Jupiter’s orbit, is explained by different rates of destruction of the flow of regular motions by Saturn perturbations. This result completes the explanation given for the gaps of the inner belt where extended chaotic regions exist without the need of taking into account the action of Saturn. The 2/1 and 3/2 resonances are studied and it is shown how they are affected by Saturn. Lyapunov times are estimated from simulations over 10 Myr.


Close Approach Chaotic Region Chaotic Orbit Asteroidal Belt Invariant Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Cl. Froeschlé and R. Greenberg, Mean motion resonances, in “Asteroids”, R.P.Binzel et al. eds., Univ. Arizona Press, Tucson, pp. 827–844 (1989).Google Scholar
  2. [2]
    S. Ferraz-Mello, Kirkwood gaps and groups, in Proc. IAU Symposium 160, A.Milani et al. eds., Kluwer, Dordrecht (in press).Google Scholar
  3. [3]
    J. Wisdom, The origin of Kirkwood gaps: A mapping for asteroidal motion near the 3/1 commensurability, Astron. J. 85, 1122–1133 (1982).MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J. Wisdom, Chaotic behaviour and the origin of the 3/1 Kirkwood gap, Icarus 56, 51–74 (1983).ADSCrossRefGoogle Scholar
  5. [5]
    J. Wisdom, A perturbative treatment of motion near the 3/1 commensurability, Icarus 63, 272–289 (1985).ADSCrossRefGoogle Scholar
  6. [6]
    Efemeridi Malikh Planet na 1993 god“, Inst. Teoret. Astron., St. Petersburg, Russia (1992).Google Scholar
  7. [7]
    S. Ferraz-Mello and J.C. Klafke, A Model for the study of very-high-eccentricity asteroidal motion. The 3:1 resonance, in “Predictability, Stability and Chaos in N body Dynamical Systems”, A.E.Roy, ed., Plenum Press, New York, pp. 177–184 (1991).CrossRefGoogle Scholar
  8. [8]
    Saha, P. “Simulating the 3:1 Kirkwood gap”, Icarus 100, 434–439 (1992).ADSCrossRefGoogle Scholar
  9. [9]
    M.Sidlichovskÿ and B. Melendo, Mapping for the 5/2 asteroidal commensurability, Bull. Astron. Inst. Czechoslov. 37, 65–80 (1986).ADSGoogle Scholar
  10. [10]
    C.D. Murray, Structure of the 2:1 and 3:2 jovian resonances, Icarus 65, 70–82 (1986).ADSCrossRefGoogle Scholar
  11. [11]
    J. Henrard and A. Lemaitre, A perturbative treatment of the 2/1 jovian resonance, Icarus 69, 266–279 (1987).ADSCrossRefGoogle Scholar
  12. [12]
    T. Yokoyama and J.M. Balthazar, Application of Wisdom’s perturbative method for 5:2 and 7:3 resonances, Icarus 99, 175–190 (1992).ADSCrossRefGoogle Scholar
  13. [13]
    M. Sidlichovskÿ, Chaotic behaviour of trajectories for the asteroidal resonances, Celest. Mech. Dyn. Astron. 56, 143–152 (1993).ADSMATHCrossRefGoogle Scholar
  14. [14]
    K.F. Sundman, Sur les conditions nécessaires et suffisantes pour la convergence du développement de la fonction perturbatrice dans le mouvement plan, Ofversigt Finska Vetenskaps-Soc. Fórh. 58 A No. 24 (1916).Google Scholar
  15. [15]
    S. Ferraz-Mello, The convergence domain of the Laplacian expansion of the disturbing function, Celest. Mech. Dyn. Astron. 58 (1994).Google Scholar
  16. [16]
    S. Ferraz-Mello and M. Sato, A very-high-eccentricity asymmetric expansion of the disturbing function near resonances of any order, Astron. Astrophys. 225, 541–547 (1989).ADSGoogle Scholar
  17. [17]
    J.C. Klafke, S. Ferraz-Mello and T. Michtchenko, Very-high-eccentricity librations at some higher-order resonances, in Proc. IAU Symposium 152, S.Ferraz-Mello, ed., Kluwer, Dordrecht, pp. 153–158 (1992).Google Scholar
  18. [18]
    M. Moons and A. Morbidelli, The main mean-motion commensurabilities in the planar circular and elliptic problem, Celest. Mech. Dyn. Astron. 57, 99–108 (1993).MathSciNetADSMATHCrossRefGoogle Scholar
  19. [19]
    A. Morbidelli and M. Moons, Secular resonances in mean-motion commensurabilities. The 2/1 and 3/2 cases, Icarus 102, 316–332 (1993).ADSCrossRefGoogle Scholar
  20. [20]
    R. Giffen, A study of commensurable motion in the asteroidal belt, Astron. Astrophys. 23, 387–403 (1973).ADSGoogle Scholar
  21. [21]
    Cl. Froeschlé and H. Scholl, On the dynamical topology of the Kirkwood gaps, Astron. Astrophys. 48, 389–393 (1976).ADSGoogle Scholar
  22. [22]
    J. Wisdom, Chaotic dynamics in the Solar System, Icarus 72, 241–275 (1987).ADSCrossRefGoogle Scholar
  23. [23]
    S.P. Ipatov, Evolution of asteroidal orbits at the 5:2 resonance, Icarus 95, 100–114 (1992).ADSCrossRefGoogle Scholar
  24. [24]
    F. Franklin, M. Lecar and M. Murison, Chaotic orbits and long-term stability: an example from asteroids of the Hilda group, Astron. J. 105, 2336–2343 (1993).ADSCrossRefGoogle Scholar
  25. [25]
    A. Hanslemeier and R. Dvorak, Numerical integration with Lie series, Astron. Astrophys. 132, 203–207 (1983).ADSGoogle Scholar
  26. [26]
    S. Ferraz-Mello, The high-eccentricity libration of the Hildas, Astron. J. 96, 400–408 (1988).ADSCrossRefGoogle Scholar
  27. [27]
    J. Hadjidemetriou, Resonant families of periodic orbits in the restricted 3-body problem at the 3:1 resonance and their relation to the averaged model. Celest. Mech. Dyn. Astron. 53, 151–183 (1992).MathSciNetADSMATHCrossRefGoogle Scholar
  28. [28]
    E.W. Brown and C.A. Shook, “Planetary Theory”, (University Press, Cambridge), p. 232 (1933).Google Scholar
  29. [29]
    D. Brouwer, The problem of the Kirkwood gaps in the asteroidal belt, Astron. J. 68, 152–159 (1963).ADSCrossRefGoogle Scholar
  30. [30]
    E. Everhart, An efficient integrator that uses Gauss-Radau spacings, in “Dynamics of Comets, their origin and evolution”, A. Carusi and G.B. Valsecchi, eds., Reidel, Dordrecht, 185–202 (1985).CrossRefGoogle Scholar
  31. [31]
    A. Milani, A. Nobilli and M. Carpino, Secular variations of the semimajor axes: theory and experiment, Astron. Astrophys. 172, 265–279. (1987).ADSMATHGoogle Scholar
  32. [32]
    T.A. Michtchenko and S. Ferraz-Mello, The high-eccentricity libration of the Hildas. II. Synthetic-theory approach, Celest. Mech. Dynan. Astron. 56, 121–129 (1993).ADSCrossRefGoogle Scholar
  33. [33]
    G. Benettin, L. Galgani and J.M. Strelcyn, Kolmogorov entropy and numerical experiments, Phys Rev. A, 14, 2338–2345 (1976).ADSCrossRefGoogle Scholar
  34. [34]
    Cl. Froeschlé and H. Scholl, The stochasticity of peculiar orbits in the 2/1 Kirkwood gap, Astron. Astrophys. 93, 62–66 (1981).ADSGoogle Scholar
  35. [35]
    M. Lecar, F. Franklin and M. Munson, i On predicting long-term orbital instability: A relation between the Lyapunov time and sudden orbital transitions, Astron. J. 104, 1230–1236 (1992).ADSCrossRefGoogle Scholar
  36. [36]
    H.A. Levison and M.J. Duncan, The gravitational sculpting of the Kuiper belt, Astrophys. J. Lett. 406, L35 - L38 (1993).ADSCrossRefGoogle Scholar
  37. [37]
    M. Holman and J. Wisdom, Dynamical stability of the outer Solar System and the delivery of short-period comets, Astron. J. 105, 1987–19990 (1993).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • S. Ferraz-Mello
    • 1
  • R. Dvorak
    • 2
  • T. A. Michtchenko
    • 1
  1. 1.Instituto Astronômico e GeofísicoUniversidade de São PauloSão PauloBrasil
  2. 2.Institut für AstronomieUniversität WienWienAustria

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