Abstract
In this paper a theoretical perturbation approach to the problem of the dynamics in secular resonance is exposed. This approach avoids any expansion of the main term of the Hamiltonian (linear term in the masses) with respect to the eccentricity or the inclination of the asteroid, in order to achieve results valid for any value of these variables. Moreover suitable action-angle variables are introduced to take properly into account the dynamics related to the motion of the argument of perihelion of the asteroid, which is relevant at high inclination. A class of secular resonances wider than that usually considered is found. An explicit computation of the location of the main secular resonances, estimating also the contribution of the quadratic term in the masses by means of classical series expansion, is reported in the last sections. The accuracy of computations obtained by series expansion is discussed in the paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
13 April 2024
Author update of Alessandro Morbidelli.
References
Bretagnon, P.: 1974, ‘Termes à longues périodes dans le système solaire.’, Astron. Astrophys., 30, 141–154.
Brouwer, D. and Van Woerkom, A. J. J.: 1950, ‘The secular variations of the orbital elements of the principal planets.’, Astr. Papers. U.S. Naval Obs., 13, 85–107.
Colombo, R. and Giorgilli, A.: 1990, ‘A quantitative algebraic approach to Lie transforms in Hamiltonian mechanics.’, preprint.
Ferraz-Mello, S.: 1988, ‘The high eccentric libration of the Hildas’, Astron. J., 96, 400–408.
Ferraz-Mello, S.: 1989, ‘A semi-numerical expansion of the averaged disturbing function for some very-high eccentric orbits’, Celest. Mech., 45, 65–68.
Froeschlé, Ch., and Scholl, H.: 1986a, ‘The secular resonance ν6 in the asteroidal belt.rs, Astron. Astrophys., 166, 326–332.
Froeschlé, Ch., and Scholl, H.: 1986b, ‘The effects of the secular resonances ν16 and φ5 on the asteroidal orbits.’, Astron. Astrophys., 170, 138–144.
Froeschlé, Ch., and Scholl, H.: 1987a, ‘Chaotic motion in secular resonances.’, in Proc. 10th ERAM of the IA U, 3.
Froeschlé, Ch., and Scholl, H.: 1987b, ‘Orbital evolution of asteroids near the secular resonance ν6.’, Astron. Astrophys., 179, 294–303.
Froeschlé, Ch., and Scholl, H.: 1988a, ‘A possible source for highly inclined Apollo-Amor asteroids: the secular resonance ν16.’, in The few body problem, M.J.Valtonen (ed.), 123–127.
Froeschlé, Ch., and Scholl, H.: 1988b, ‘Secular resonances: new results.’, Celest. Mech., 43, 113–117.
Froeschlé, Ch., and Scholl, H.: 1989, ‘The three principal resonances ν5, ν6 and ν16 in the asteroidal belt.’, Celest. Mech., 46, 231–251.
Froeschlé, Ch., and Scholl, H.: 1990, ‘Orbital evolution of known asteroids in the ν5 resonance region.’, Astron. Astrophys., 227, 255–263.
Giorgilli, A. and Galgani, L.: 1978, ‘Formal integrals for an autonomous Hamiltonian system near an equilibrium point.’, Celest. Mech., 17, 267–280.
Giorgilli, A. and Galgani, L.: 1985, ‘Rigorous estimates for the series expansions of Hamiltonian perturbation theory.’, Celest. Mech., 37, 95–112.
Henrard, J., and Lemaître, A.: 1983, ‘A second fundamental model for resonance.’, Celest. Mech., 30, 197–218.
Henrard, J.: 1990, ‘A semi-numerical perturbation method for separable hamiltonian systems.’, Celest. Mech., 49, 43–67.
Henrard, J.: 1991, ‘The adiabatic invariant in celestial mechanics.’, to be published in Dynamics reported.
Heppenheimer, T., A.: 1980,‘Secular resonances and the origin of eccentricities of Mars and the asteroids.’, Icarus, 41, 76–88.
Knežević, Z., Milani, A., Farinella, P, Froeschlé, Ch., and Froeschlé, Cl.: 1991,‘Secular resonances from 2 to 50 A.U.’ In preparation.
Kozai, Y.: 1962, ‘Secular perturbations of asteroids with high inclination and eccentricities.’, The Astronomical Journal, 67, 591–598.
Laskar, J.: 1988, ‘Secular evolution of the solar system over 10 million years.’, Astron. Astrophys., 198, 341–362.
Laskar, J.: 1990, ‘The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones’, Icarus, 88, 266–291.
Lemaître, A., and Dubru, P.: 1990, ‘The secular resonances in the primitive solar nebula.’ Celest. Mech., in press.
Lemaître, A., and Henrard, J.: 1990, ‘Origin of the chaotic behaviour in the 2/1 Kirkwood gap.’, Icarus, 83, 391–409.
Milani, A., and Knežević, Z.: 1990, ‘Asteroid proper elements.’, Celest. Mech., 49, 347–411.
Morbidelli, A., and Giorgilli, A.: 1990, ‘On the dynamics in the asteroid belt’, Celest. Mech., 47, 145–204.
Nakai, H., and Kinoshita, H.: 1985, ‘Secular perturbations of asteroids in secular resonances.’, Celest. Mech., 36, 391–407.
Nobili, A., Milani, A., and Carpino, M.: 1989, ‘Fundamental frequencies and small divisors in the orbits of the outer planets.’, Astron. Astrophys., 210, 313–336.
Poincaré, H.: 1892, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris.
Šidlichovský, M.: 1989, ‘The existence of a chaotic region due to the overlap of secular resonances ν5 and ν6.’, Celest. Mech., in press.
Tisserand, M. F.: 1882, Ann. Obs. Paris, 16, E1.
Wetherill, G. W. and Williams, J. G.: 1979, ‘Origin of differentiated meteorites.’, in Origin and distribution of the elements, L. H. Aherens (ed.), 19–31.
Williams, J. G.: 1969, ‘Secular perturbations in the solar system.’, Ph.D. dissertation, University of California, Los Angeles.
Williams, J. G., and Faulkner, J.: 1981, ‘The position of secular resonance surfaces.’, Icarus, 46, 390–399.
Yoshikawa, M.: 1987, ‘A simple analytical model for the ν6 resonance.’, Celest. Mech., 40, 233–272.
Yuasa, M.: 1973, ‘Theory of secular perturbations of asteroids including terms of higher order and higher degree.’, Publ. Astr. Soc. Japan, 25, 399–445.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Morbidelli, A., Henrard, J. Secular resonances in the asteroid belt: Theoretical perturbation approach and the problem of their location. Celestial Mech Dyn Astr 51, 131–167 (1991). https://doi.org/10.1007/BF00048606
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00048606