On the Lyapunov exponents of the asteroidal motion subject to resonances and encounters

Abstract

AbstractIn theoretical as well as practical issues of the asteroidal hazard problem, it is important to be able to assess the degree of predictability of the orbital motion of asteroids. Some asteroids move in a virtually predictable way, others do not. The characteristic time of predictability of any motion is nothing but the Lyapunov time (the reciprocal of the maximum Lyapunov exponent) of the motion. In this report, a method of analytical estimation of the maximum Lyapunov exponents of the orbital motion of asteroids is described in application for two settings of the problem. Namely, the following two types of the motion are considered: (1) the motion close to the ordinary or three-body mean motion resonances with planets, and (2) the motion in highly eccentric orbits subject to moderately close encounters with planets. Whatever different these settings may look, the analytical treatment is universal: it is performed within a single framework of the general separatrix map theory. (Recall that the separatrix maps describe the motion near the separatrices of a nonlinear resonance.) The analytical estimates of the Lyapunov times are compared to known numerical ones, i.e., to known estimates obtained by means of numerical integration of the orbits.