Transport in the outer asteroid belt: Fokker-Planck approach vs. numerical integration

In this paper we examine the statistical properties of transport, for orbits of fictitious asteroids initiated in three outer-belt resonances: the 7:4, 9:5 and 12:7 mean motion resonances of the 2-D elliptic restricted three-body problem. Two alternative approaches are used: (i) numerical integration of distributions of initial conditions and (ii) simulation of the diffusion of the eccentricity-related action, through the numerical solution of a 1-D Fokker-Planck equation. The diffusion coefficient, D, is determined numerically, using the definition of “local transport coefficients”. The statistical results of (i) and (ii) are compared. Qualitative agreement and even quantitative in some cases between the two approaches is found. For the improvement of the efficiency of (ii) proper modifications, concerning mostly the calculation of D(I), are needed. INTRODUCTION – MOTIVATION The possibility of formulating a statistical description of the motion of asteroids, at least in regions where chaotic motion dominates, has received little attention so far. It is known that chaotic motion results in a slow diffusion-like evolution of the action variables. Thus, the evolution of any initial distribution function of the actions could, in principle, be described by the solution of a Fokker-Planck-Kolmogorov (FPK) equation, provided that an appropriate calculation of the diffusion coefficient, D, could be achieved. Varvoglis & Anastasiadis (1996) managed to reproduce the statistical ‘law of escape’ of Lecar et al. (1992) by solving the FPK equation (Eq. 1) with D=aλ (λ=Lyapunov exponent; Konishi 1989). This result holds only in the resonance-overlap regime of the 2-D elliptic restricted three-body problem. Murray & Holman (1997, hereafter M&H), working on the planar elliptic three-body problem, derived an analytical approximation of the diffusion coefficient (in the quasi-linear approximation) for a single-resonance (of order q) domain. They used an approximate Hamiltonian derived by expanding the disturbing function and keeping only the resonant term and the lowest-order secular term. The coefficient then becomes action-