Proper Elements and Stable Chaos

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.