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We present a diffusion mechanism for time-dependent perturbations ofautonomous Hamiltonian systems introduced in [25]. This mechanism is based onshadowing of pseudo-orbits generated by two dynamics: an `outer dynamics',given by homoclinic trajectories to a normally hyperbolic invariant manifold,and an `inner dynamics', given by the restriction to that manifold. On theinner dynamics the only assumption is that it preserves area. Unlike otherapproaches, [25] does not rely on the KAM theory and/or Aubry-Mather theory toestablish the existence of diffusion. Moreover, it does not require to checktwist conditions or non-degeneracy conditions near resonances. The conditionsare explicit and can be checked by finite precision calculations in concretesystems.  As an application, we study the planar elliptic restricted three-bodyproblem. We present a rigorous theorem that shows that if some concretecalculations yield a non zero value, then for any sufficiently small, positivevalue of the eccentricity of the orbits of the main bodies, there are orbits ofthe infinitesimal body that exhibit a change of energy that is bigger than somefixed number, which is independent of the eccentricity.  We verify numerically these calculations for values of the masses close tothat of the Jupiter/Sun system. The numerical calculations are not completelyrigorous, because we ignore issues of round-off error and do not estimate thetruncations, but they are not delicate at all by the standard of numericalanalysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where1 would be enough). The code of this verifications is available. We hope thatsome full computer assisted proofs will be obtained in a near future sincethere are packages (CAPD) designed for problems of this type.

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