Evidences of diffusion related to the center manifold of L3 of the SRTBP

Resumo

Copyright © 2014 by the authors.In this contribution we present evidences of diffusion of trajectories in the framework of the Spatial Restricted Three-Body Problem (SRTBP) and introduce a methodology to quantify and to examine the diffusion process. In the circular SRTBP, the center manifold of the equilibrium L3,W L3C is four-dimensional and contains vertical Lyapunov orbits, planar Lyapunov orbits, two-dimensional invariant tori, other periodic orbits associated to resonances, and small chaotic zones. We compute the invariant structures inside W L3C, in particular, we obtain Fourier representations of invariant curves in a four-dimensional space, corresponding to Poincare sections of bidimensional invariant tori inside W L3C. Then, we report on the diffusion of trajectories with initial conditions very close to these invariant curves. We introduce a methodology for diffusion analysis which provides statistics of the process and shows that the diffusion rate of trajectories is not constant in the phase space. The diffusion rate increases as trajectories go away from the vertical periodic orbit and then decreases when they approach the planar periodic orbit, wandering across the hyperbolic manifolds of distinct tori. Besides the analysis of a large ensemble of trajectories, specific cases are also studied to illustrate the process. In some cases, stickiness to tridimensional tori is observed. Eventually, the trajectories escape due to approximation to the secondary. Finally, we show that the diffusion mechanism is associated to the existence of transition chains of heteroclinic connections and relate the rate of diffusion with the splitting of the stable and unstable hyperbolic manifolds of the two-dimensional invariant tori. Our analysis corresponds to the investigation for the mass ratio of the Saturn-Titan system, but our methodology can be extended to many other concrete applications in orbital dynamics in the Solar system.