Irregular behaviour and eccentricity increase in the trajectories of a dynamical system modelling asteroidal motions

Many trajectories of the third body are integrated numerically in a modified elliptical restricted three body problem (ERTBP), in which the eccentricity, e, of the orbit of the second primary varies sinusoidally with time. It is found that, in the case of the 2:1 resonance, the introduction of the time variability of e modifies significantly the behaviour of the trajectories of the third body. In particular their osculating eccentricity e, present the following two notable features: (a) In all cases it shows a definite chaotic variation, which appears at significantly shorter time-scales than the one found by Wisdom in the e = constant case. (b) In many cases it shows a significant increase, up and beyond the (critical) value e_crit = 0.52. As a result the third body approaches the first primary at distances smaller than 0.29 (where by we denote the semi-major axis of the trajectory of the second primary around the first), which in the actual Sun-Jupiter-asteroid problem corresponds to the semi-major axis of Mars. Our result might be of interest in the context of explaining the Kirkwood gaps at the resonances where the osculating eccentricity of asteroid trajectories calculated in the classical (e = constant) ERTBP does not reach Mars crosser values.