Motion near the 3/1 resonance of the planar elliptic restricted three body problem

  • Jacques Henrard
  • N.D. Caranicolas


The global semi-numerical perturbation method proposed by Henrard and Lemaître (1986) for the 2/1 resonance of the planar elliptic restricted three body problem is applied to the 3/1 resonance and is compared with Wisdom's perturbative treatment (1985) of the same problem. It appears that the two methods are comparable in their ability to reproduce the results of numerical integration especially in what concerns the shape and area of chaotic domains. As the global semi-numerical perturbation method is easily adapted to more general types of perturbations, it is hoped that it can serve as the basis for the analysis of more refined models of asteroidal motion. We point out in our analysis that Wisdom's uncertainty zone mechanism for generating chaotic domains (also analysed by Escande 1985 under the name of slow Hamiltonian chaotic layer) is not the only one at work in this problem. The secondary resonance ω p = 0 plays also its role which is qualitatively (if not quantitatively) important as it is closely associated with the random jumps between a high eccentricity mode and a low eccentricity mode.


resonance Kirkwood gaps perturbation method chaotic motion secondary resonance 


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  1. Borderies, N., and Goldreich, P.: 1984, “A simple derivation of capture probabilities”, Celest. Mech., 32, 127–136.Google Scholar
  2. Escande, D.F.: 1985, “Change of adiabatic invariant at separatrix crossing : Application to slow Hamiltonian chaos”. In “Advances in Nonlinear Dynamics and Stochastic Processes” (R. Livi and A. Politi eds.), World Scientific, Singapore, 67–69.Google Scholar
  3. Froeschlé, C., and Scholl, H.: 1976, “On the dynamical topology of the Kirkwood gaps”, Astron. Astrophys., 48, 389–393.Google Scholar
  4. Froeschlé, C., and Scholl, H.: 1977, “A qualitative comparison between the circular and elliptic Sun-Jupiter-Asteroid problem at commesurabilities”, Astron. Astrophys., 57, 33–39.Google Scholar
  5. Froeschlé, C., and Scholl, H.: 1981, “The stochasticity of peculiar orbits in the 2/1 Kirkwood gap”, Astron. Astrophys., 93, 62–66.Google Scholar
  6. Giffen, R.: 1973, “A study of commensurable motion in the asteroid belt”, Astron. Astrophys., 23, 387–403.Google Scholar
  7. Henrard, J. : 1988, “Resonances in the Planar Elliptic Restricted Problem”. In Long Term Behaviour of Natural and Artificial N-Body Systems, (A. Roy ed.).Google Scholar
  8. Henrard, J. : 1989, “The adiabatic invariant in Classical Mechanics”, Dynamics Reported, to be published.Google Scholar
  9. Henrard, J., and Lemaître, A.: 1986, “A perturbation method for problems with two critical arguments”, Celest. Mech., 39, 213–238.Google Scholar
  10. Henrard, J., and Lema#x00EE;tre, A.: 1987, “A perturbative treatment of the 2/1 Jovian resonance”, Icarus, 69, 266–279.Google Scholar
  11. Henrard, J., Lema#x00EE;tre, A., Milani, A. and Murray, C.D.: 1986, “The reducing transformation and apocentric librators”, Celest. Mech., 38, 335–344.Google Scholar
  12. Koiller, J., Balthazar, J.M. and Yokoyama, T.: 1987, “Relaxation-chaos phenomena in Celestial Mechanics”, Physica, 26D, 85–122.Google Scholar
  13. Lema#x00EE;tre, A.: 1984, “High order resonance in the restricted three body problem”, Celest. Mech., 32, 109–126.Google Scholar
  14. Lema#x00EE;tre, A., and Henrard, J. : 1989, “Chaotic motion in the 2/1 resonance”, Icarus in print.Google Scholar
  15. Message, J.P.: 1966, “On nearly commensurable periods in the restricted problem of three bodies”, Proc. LA. U. Symp. No. 25, 197–222.Google Scholar
  16. Murray, C.D.: 1986, “Structure of the 2/1 and 3/2 Jovian resonances”, Icarus, 65, 70–82.Google Scholar
  17. Murray, C.D., and Fox, K.: 1984, “Structure of the 3/2 Jovian resonance : A comparison of numerical methods”, Icarus, 59, 221–233.Google Scholar
  18. Peale, S.J. : 1986, “Orbital resonance, unusual configurations and exotic rotation states”, in Satellites (J. Burns and M. Matthews eds.), Univ. of Arizona Press, 159–223.Google Scholar
  19. Poincaré, H. : 1899, Les méthodes nouvelles de la Mécanique Céleste, Gauthier Villars (Tome II, p. 43).Google Scholar
  20. Poincare, H.: 1902, “Sur les planètes du type d'Hécube”, Bull. Astron., 19, 289–310.Google Scholar
  21. Scholl, H., and Froeschlé, C.: 1974, “Asteroidal motion at the 3/1 commensurability”, Astron. Astrophys., 33, 455–458.Google Scholar
  22. Scholl, H., and Froeschlé, C.: 1975, “Asteroidal motion at the 5/2, 7/3 and 2/1 resonances”, Astron. Astrophys., 42, 457–463.Google Scholar
  23. Schubart, J.: 1966, “Special cases of the restricted problem of three bodies”, Proc. of the LA. U. Symp. No. 25, 187–193.Google Scholar
  24. Sessin, W., and Ferraz-Mello, S.: 1984, “Motion of two planets with period commensurable in the ratio 2/1”, Celest. Mech., 32, 307–332.Google Scholar
  25. Szebehely, V. : 1967, Theory of Orbits, Academic Press.Google Scholar
  26. Wisdom, J.: 1982, “The origin of the Kirkwood's gaps : A mapping for asteroidal motion near the 3/1 commensurability”, Astron. J.,85, 1122–1133.Google Scholar
  27. Wisdom, J.: 1983, “Chaotic behaviour and the origin of the 3/1 Kirkwood gap”, Icarus, 56, 51–74.Google Scholar
  28. Wisdom, J.: 1985, “A perturbative treatment of motion near the 3/1 commensurability”, Icarus, 63, 272–289.Google Scholar
  29. Wisdom, J.: 1986, “Canonical solution of the two critical argument problem”, Celest. Mech.,38, 175–180.Google Scholar
  30. Wisdom, J.: 1987, “Urey prize lecture : Chaotic dynamics in the Solar System”, Icarus, 72, 241–275.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Jacques Henrard
    • 1
  • N.D. Caranicolas
    • 2
  1. 1.Département de mathématique, FUNDPNamurBelgium
  2. 2.Department of AstronomyUniversity of ThessalonikiThessalonikiGreece

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