Abstract
The long term evolution of the orbits of the asteroids is studied by means of proper elements, which are quasi-integrals of the motion. After a short review of the classical theories for secular perturbations, this paper presents the state of the art for the computation of proper elements. The recent theories have been extended to higher degree in the eccentricities and inclinations, and to the second order in the perturbing masses; they use new iterative algorithms to compute secular perturbations with fixed initial conditions but variable frequencies. This allows to compute proper elements stable over time spans of several million years, within a range of oscillations small enough to allow the identification of asteroid families; the same iterative algorithm can also be used to automatically detect secular resonances, that is to map the dynamical structure of the main asteroid belt. However the proper element theories approximate the true solution of the N-body problem with a conditionally periodic solution of a truncated problem, while the orbits of most asteroids are not conditionally periodic, but chaotic; positive Lyapounov exponents have been detected for a large number of real asteroids. The phenomenon of stable chaos occurs whenever the range of oscillations of the proper elements, as computed by state of the art theories, remains small for time spans of millions of years, while the Lyapounov time (in which the orbits diverge by a factor exp(1)) is much shorter, e.g. a few thousand years. This can be explained only by a theory which accounts correctly for the degeneracy of the unperturbed 2-body problem used as a first approximation. The two stages of computation of mean and proper elements are each subject to the phenomena of resonance and chaos; stable chaos occurs when a weak resonance affects the computation of mean elements, but the solution of the secular perturbation equations is regular.
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References
Arnold, V., 1976, “Méthodes Mathématiques de la Mécanique Classique”, MIR, Moscou.
Baccili, S. and Cattaneo, L., 1993, Exam paper in “Advanced Mechanics”, University of Pisa.
Benettin, G., Galgani, L. and Giorgilli, A., 1984, A proof of Kolmogorov theorem on invariant tori using canonical transformations without inversion, Nuovo Cimento, 79b, 201–223.
Bretagnon, P., 1974, Termes à longues périodes dans le système solaire, Astron. Astrophys. 30, 141–154.
Brouwer, D. 1951, Secular variations of the orbital elements of minor planets, Astron. J. 56, 9–32.
Carpino, M., Milani, A. and Nobili, A.M., 1987, Long—term numerical integrations and synthetic theories for the motion of the outer planets, Astron. Astrophys., 181, 182–194.
Duriez, L., 1979, Approche d’une théorie générale planétaire en variables elliptiques héliocentriques, Ph.D. Thesis, Univ. Lille.
Duriez, L., 1990, Le dévelopment de la fonction perturbatrice, in Modern methods in celestial mechanics, Benest, D. and Froeschlé, C. eds., Editions Frontières, Gif-sur Yvette, pp. 35–62.
Farinella, P., Gonczi, R. and Froeschlé, Ch.,1993, Meteorites from the asteroid 6 Hebe, Celest. Mech., 56, 287–305.
Ferraz-Mello, S., 1994, The convergence domain of the laplacian expansion of the disturbing function, Cel. Mech., 58.
Froeschlé, C., 1984, The Lyapounov characteristic exponents and applications to the dimension of the invariant manifolds of a chaotic attractor, in Stability of the solar system and its minor natural and artificial bodies, Szebehely, V. ed., Reidel, Dordrecht, 265–282.
Henrard, J., 1990, A semi-numerical perturbation method for separable Hamiltonian systems, Celest. Mech. 49, 43–67.
Henrard, J., and Lemaitre, A. 1983, A second fundamental model for resonance, Celest. Mech. 30, 197–218.
Hori, G., 1966, Theory of general perturbations with unspecified canonical variables, Publ. Astron. Soc. Japan 18, 287–296.
Kneievié, Z., 1989, Asteroid long-periodic perturbations: the second order Hamiltonian, Celes. Mech. 46, 147–158.
Kneievié, Z., 1991, Asteroid long periodic perturbations; derivation of proper elements and assessment of their accuracy, Astron. Astrophys. 241, 267–288.
Kneievié, Z., Carpino, M., Farinella, P., Froeschlé, C., Froeschlé, Ch., Gonczi, R., Jovanovie, B., Paolicchi, P. and Zappalà, V., 1988, Asteroid short-periodic perturbations and the accuracy of mean orbital elements, Astron. Astrophys. 192, 360–369.
Kneievié, Z., Milani, A., Farinella, P., Froeschlè, Ch. and Froeschlè, C., 1991, Secular resonances from 2 to 50 AU, Icarus, 93, 316–330.
Kneievié, Z., Froeschlé, Ch., Lemaitre, A., Milani, A., and Morbidelli, A., 1994, Comparison of two theories for calculation of asteroid proper elements, in preparation.
Kozai, Y. 1962, Secular perturbations of asteroids with high inclinations and eccentricities, Astron. J. 67, 591–598.
Kozai,Y., 1979, The dynamical evolution of the Hirayama families, in Asteroids, Gehrels, T. ed. (Univ. of Arizona Press), pp. 334–358.
Laskar, J., 1986, Secular terms of classical planetary theories using the results of general theory, Astron. Astrophys., 157, 59–70.
Lemaitre, A., and A. Morbidelli, 1994, Calculation of proper elements for high inclined asteroidal orbits, Celest. Mech., in press.
Milani, A., 1988, Secular perturbations of planetary orbits and their representation as series, in Long Term Behaviour of Natural and Artificial N-Body Systems, Roy, A.E., editor, Kluwer, Dordrecht, 73–108
Milani, A., 1990, Perturbation methods in Celestial Mechanics, in Modern methods in celestial mechanics, Benest, D. and Froeschlé, C. eds., Editions Frontière, Gif-sur Yvette, pp. 109–150.
Milani, A., 1991, Chaos in the three body problem, in Predictability, stability, and chaos in N-body dynamical systems, Roy, A.E. ed., Plenum, New York, pp. 11–33.
Milani, A. and Farinella, P., 1994, Chaos as a clock: the age of the Veritas asteroid family, submitted for publication.
Milani, A., and Kneievié, Z., 1990, Secular perturbation theory and computation of asteroid proper elements, Celest. Mech. 49, 347–411.
Milani, A., and Kneievie, Z., 1992, Asteroid proper elements and secular resonances, Icarus 98, 211–232.
Milani, A., and Kneievié, Z., 1994, Asteroid proper elements and the dynamical structure of the asteroid belt, Icarus,in press.
Milani, A. and Labianca, A., 1994, The radius of convergence of fixed frequency perturbation theories for the motion of the asteroids, in preparation.
Milani, A., and A. M. Nobili 1992. An example of stable chaos in the solar system. Nature 357, 569–571.
Milani, A., Nobili, A.M. and Carpino, M. 1987, Secular variations of the semimajor axes: theory and experiments, Astron. Astrophys. 172, 265–279.
Milani, A., Carpino, M., Hahn, G. and Nobili, A.M., 1989, Dynamics of planet-crossing asteroids: classes of orbital behaviour, Project SPACEGUARD, Icarus 78, 212–269.
Milani, A., Farinella, P. and Kneievié, Z., 1992, On the search for asteroid families, in Interrelations between Physics and Dynamics for Minor Bodies in the Solar System, D. Benest and C. Froeschlé eds., Editions Frontières, Gif-sur-Yvette, pp. 85–132.
Milani, A., Bowell, E., Kneievié, Z., Lemaitre, A., Morbidelli,. A. and Muinonen, K., 1994a, A composite catalogue of asteroid proper elements, in Asteroids, Comets, Meteors 1993, Milani, A., Di Martino, M. and Cellino, A. eds., Kluwer, Dordrecht, in press.
Milani, A., Nobili, A.M., and Kneievié, Z., 1994b, Stable chaos in the asteroid belt, in preparation.
Morbidelli, A., and Henrard, J., 1991, Secular resonances in the asteroid belt: theoretical perturbation approach and the problem of their location, Celestial Mechanics, 51, 131–168.
Poincaré, H., 1884, Sur une généralisation des fractions continues, Comp. Rend. Acad. Sci. Paris 99, 1014–1016.
Poincaré H. 1892–1899, Les Méthodes Nouvelles de la Méchanique Céleste,Vol. I, 1892; Vol. II, 1893; Vol. III, 1899; Gauthier-Villars, Paris (reprinted by Blanchard, Paris, 1987).
Roy, A.E., 1982, The stability of N-body hierarchical dynamical systems, in Applications of modern dynamics to celestial mechanics and astrodynamics, Szebehely, V. ed., Kluwer, Dordrecht, pp. 103–130.
Williams, J.G., 1969, Secular perturbations in the Solar System, Ph.D. Thesis, Univ. California, Los Angeles.
Williams, J.G., 1973, Meteorites from the asteroid belt? Eos, 54, 233.
Williams, J.G., 1979, Proper orbital elements and family memberships of the asteroids, in Asteroids, Gehrels,T. ed., Univ.Arizona Press, pp. 1040–1063.
Williams, J.G., 1989, Asteroid family identifications and proper elements, in Asteroids II, Binzel, R.P, Gehrels, T. and Matthews, M.S. eds., Univ.Arizona Press, Tucson, pp. 1034–1072.
Williams, J.G., and Faulkner, J., 1981, The position of secular resonance surfaces, Icarus 46, 390–399.
Yuasa, M., 1973, Theory of secular perturbations of asteroids including terms of higher order and higher degree, Publ. Astron. Soc. Japan 25, 399–445.
Zappalà, V., Cellino, A., Farinella, P. and Kneievié, Z., 1990, Asteroid families I: identification by hierarchical clustering and reliability assessment, Astron. J. 100, 2030–2046.
Zappalà, V., Cellino, A., Farinella, P. and Milani, A., 1994, Asteroid families. II. Extension to unnumbered multi-opposition asteroids, Astron. J. 107, 772–801.
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Milani, A. (1995). Proper Elements and Stable Chaos. In: Roy, A.E., Steves, B.A. (eds) From Newton to Chaos. NATO ASI Series, vol 336. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1085-1_5
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