Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

Abstract

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. 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 ∼ − ln(µe 0 ) µ 3/2 e 0 . 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 Arnol'd diffusion, we also state a conjecture on its existence for a typical ε-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 Arnol'd diffusion on time scales − ln ε ε 2 .

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

“…In applications, using several scattering maps rather than a single one can be very advantageous. In astrodynamics, for example, the existence of multiple homoclinic intersections can be exploited to obtain diffusion [43,47] and to increase the versatility of space missions; see, e.g., [19,40].…”

Asymptotic Density of Collision Orbits in the Restricted Circular Planar 3 Body Problem

“…at each of the considered energy levels where the Hamiltonian H µ in (16) is different from the modified H µ in (17). They correspond to two disjoint intersections (see Fig.…”

“…The change of coordinates Ψ 2 is not fully explicit. Nevertheless, for some components it can be defined through successive changes of variables (for a more extensive explanation, one can see Appendix B.1 in [17]). Consider the set of all Diophantine numbers with constant type satisfying (20), which we have denoted by B γ .…”

Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification

“…A recent important development in the search for diffusion in the restricted three body problem is given in [21], where a different setting is considered. Instead of the Lyapunov orbits, the outer periodic orbits of the system are considered.…”

“…The diffusion mechanism follows from careful analysis of the properties of the circular problem, together with perturbation results, some of which (as is the case in this paper) are only verified numerically. We have heard that the authors of [21] are also considering validating their results to produce a computer assisted proof. One of the difficulties in such proof could be the fact that the considered manifolds intersect at a very small angle.…”