 11129 Jacques Fejoz, Marcel Guardia, Vadim Kaloshin, Pablo Roldan
 Diffusion along mean motion resonance in the restricted planar threebody problem
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Sep 12, 11

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Abstract. We study the dynamics of the restricted planar threebody problem
near a mean motion resonance, 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
SunJupiterasteroid systems. For the primaries (Sun and Jupiter), we pick a
realistic mass ratio $\mu=10^{3}$ and a small eccentricity $e_0>0$.
The main result is a construction of a variety of diffusing orbits
which show a drastic change of the osculating 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. In the proof we verify certain
nondegeneracy conditions numerically.
Based on the work of Treschev, it is natural to conjecture that the
time of diffusion for this problem is at least $\sim \ln (\mu e_0)/
(\mu^{3/2} e_0)$. We expect our instability mechanism to apply to
realistic values of $e_0$ and we give heuristic arguments in its
favor. If so, the applicability of Nekhoroshev theory to the
threebody 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
socalled {\it 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 also state a conjecture on its existence for a typical
$\eps$perturbation of the product of the pendulum and the 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 / \eps^{2}$.
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