Diffusion of Asteroids in Mean Motion Resonances

Abstract

If an asteroid is located in a mean motion resonance with Jupiter, its orbital elements, especially the eccentricity, e, can be transported to Jupiter-crossing values due to chaotic motion. For resonances closer to Jupiter, such as those placed in the outer asteroid belt (defined here by 3.45AU ≤ a ≤ 3.90AU), a large fraction of orbits is expected to be chaotic while, at the same time, the eccentricity value needed to cross the orbit of Jupiter is small (e ∼ 0.3). In the absence of mechanisms which can provide ‘shortcuts’ to high values of e (such as resonant periodic orbits; see Tsiganis et al., 2001), the ‘random walk’-like manner by which the eccentricity grows resembles a diffusion process. Murray & Holman (1997; hereafter MH97) constructed an analytical theory for this ‘slow chaos’ in outer-belt mean motion resonances, in the framework of the planar elliptic restricted three-body problem (hereafter ERTBP). Their calculations were based on an averaged Hamiltonian with 2 degrees of freedom. In this model, chaos is the result of the overlap among the terms of the resonance multiplet, which appear explicitly in the expansion of the disturbing function of the ERTBP. Their results indicate extended chaos in resonances of order q ≤ 5. Their estimate for the coefficient of diffusion inside a given resonance, D(I), in the action (e f =free eccentricity), reads D(I) ∼ I p , with p = q for e f ≈ e′ = 0.048 (primed elements refer to Jupiter). D(I) can be used to solve the associated transport equation and derive realistic estimates for the mean time that an asteroid takes to escape from the belt.