Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem
Féjoz, Jacques; Guardia, Marcel; Kaloshin, Vadim; Roldán, Pablo (2016), Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem, Journal of the European Mathematical Society, 18, 10, p. 2313-2401. 10.4171/JEMS/642
TypeArticle accepté pour publication ou publié
Journal nameJournal of the European Mathematical Society
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Institut de Mécanique Céleste et de Calcul des Ephémérides [IMCCE]
Department of Mathematics, University of Maryland
Escola Tècnica Superior d'Enginyeria Industrial de Barcelona [Barcelona] [ETSEIB]
Abstract (EN)We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun--Jupiter--asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio μ=10−3 and a small eccentricity e0>0. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. We verify certain non-degeneracy conditions numerically. Based on the work of Treschev, it is natural to conjecture that diffusion time for this problem is ∼−ln(μe0)μ3/2e0. We expect our instability mechanism to apply to realistic values of e0 and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable. It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnold diffusion, we state a conjecture on its existence for a typical \eps-perturbation of the product of a pendulum and a rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnold diffusion on time scales −ln\eps\eps2.
Subjects / KeywordsThree-body problem; instability; resonance; hyperbolicity; Mather mechanism; Arnol’d diffusion; Solar System; Asteroid Belt; Kirkwood gap
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