Abstract
A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e ∼ 0.3).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ferraz-Mello, S., and Klafke, J. C.: 1992, ‘A model for the study of very-high-eccentricity asteroidal motion. The 3:1 resonance’, to appear inChaos, resonance, and collective dynamical phenomena in the solar system (IAU Symp. 152), ed. S. Ferraz-Mello.
Kinoshita, H., Yoshida, H., Nakai, H.: 1991, ‘Symplectic integrators and their application to dynamical astronomy’,Celest. Mech. 50, 59–71.
Murray, C. D.: 1986, ‘Structure of the 2:1 and 3:2 Jovian resonances’,Icarus,65, 70–82.
Neri, F.: 1988, ‘Lie algebras and canonical integration’, Dept. of Physics, University of Maryland, preprint.
Šidlichovský, M.: 1988, ‘The origin of 5/2 Kirkwood gap’,The Few Body Problem (ed. M. Valtonen) p.117–121, Kluwer Acad. Publishers.
Šidlichovský, M.: 1991, ‘Tables of the disturbing function for resonant asteroids’,Bull. Astron. Inst. Czechosl.,42, 116–123.
Šidlichovský, M.: 1992, ‘Mapping for asteroidal resonances’, accepted in Astron. Astrophys.
Šidlichovský, M., Melendo, B.: 1986, ‘Mapping for 5/2 asteroidal commensurability’,Bull. Astron. Inst. Czechosl.,37, 65–80.
Wisdom, J.: 1982, ‘The origin of Kirkwood gaps: A mapping for asteroidal motion near the 3/1 commensurability’,Astron. J.,87, 577–593.
Wisdom, J.: 1983, ‘Chaotic behavior and the origin of the 3/1 Kirkwood gap’,Icarus,56, 51–74.
Wisdom, J.: 1985, ‘A perturbative treatment of motion near the 3/1 commensurability’,Icarus,63, 272–289.
Wisdom, J., Holman, M.: 1991, ‘Symplectic maps for theN-body problem’,Astron. J,102, 1528–1538.
Yoshida, H.: 1990, ‘Construction of higher order symplectic integrators’,Phys. Let. A 150, 262–268.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Šidlichovský, M. Chaotic behaviour of trajectories for the asteroidal resonances. Celestial Mech Dyn Astr 56, 143–152 (1993). https://doi.org/10.1007/BF00699727
Issue Date:
DOI: https://doi.org/10.1007/BF00699727