Chaotic Transport in the Solar System

This thesis is dedicated to the study of chaos and diffusion in the mean mo- tion resonances (MMRs) of the circular restricted three-body problem (CT3BP), focusing on the orbits that undergo close encounters with the secondary body. First we implement a semi-analytical averaging method to compute the averaged Hamiltonian for a specific MMR, and we verify its effectiveness to approximate the motions whose orbits cross the orbit of the secondary body, as well as the situations in which such an approximation is not acceptable. This study requires a careful examination of the representation of the collision singularities and the as- sociated Fourier expansions of the perturbing function, in the specific action-angle variables used for the representation of the averaged Hamiltonian. Afterwards, we investigate two aspects of the problem: the boundaries of application of Hamil- tonian perturbation theory (i.e. the conjugation of the averaged Hamiltonian to the original one) and the chaotic diffusion that can result from the accumulation of the effects of multiple close encounters with a planet. The planar and spatial CR3BP have been studied separately. Indeed, the different dimensionality of the planar and spatial cases impacts both the analytic representation of the afore- mentioned singularities as well as on the mechanisms that determine the chaotic nature and instability of the resonant motions. Both problems have been studied by leveraging: the computation of approximate first integrals, obtained from a canonical transformation of the Hamiltonian of the CR3BP; the computation of Fast Lyapunov Indicators, obtained from numerical integrations of the regularized equations of motion; the analytic representation of the gravitational singularities and of the singularities of the Fourier expansion of the perturbing function, in the resonant action-angle variables; the computation of the adiabatic invariants, which are the main tool to study the chaos generated by the crossing of a separatrix in the spatial CR3BP. Our results indicate that the efficacy of the averaging method is confirmed also for the orbits which cross the orbit of the secondary body, except when they enter a neighborhood of the set where the Fourier transform of the perturbation with respect to the fast angle of the resonant is singular. Quite unexpectedly, the averaging method proves to be effective also when large parts of the resonant phase- plane are affected by chaotic diffusion, as it has been revealed by the computation of the regularized FLI. In these regimes, already one step of the perturbation theory is sufficient to reveal the properties of these chaotic diffusions which occur with small random almost-stepwise variations of the quasi-integrals of motion similarly to the Arnold diffusion. Chaotic diffusion in particular is effective in removing phase-protection mechanisms, leading to possible deep close encounters. For the spatial problem, we also provide examples of chaotic diffusion which, in addition to the crossings of a separatrix, are due to the accumulation of effects of weak close encounters. In particular, we will provide orbits having a deeper close encounter which determines the extraction from a main resonance, and the subsequent cascade of close encounters determines a slower chaotic diffusion be- tween the nearby resonances. The numerical examples of Chapter 3 and 4 have been provided for the 1:2 external mean motion resonances, for the value of the mass parameter which represents the Sun-Neptune system.