The Keplerian map for the planar restricted three-body problem as a model of comet evolution

We examine the evolution of highly eccentric, planet-crossing orbits in the planar restricted three-body problem (Sun, planet, comet), using a simple Keplerian map in which the comet energy changes instantaneously at periheli on by an amount depending only on the azimuthal angle between the planet an d the comet at the time of perihelion passage. This approximate but very fa st mapping allows us to explore the evolution of large ensembles of long-pe riod comets. We compare our results on comet evolution with those given by the diffusion approximation and by direct integration of comet orbits. Our mapping suggests that at long times the number of surviving comets on highl y eccentric, planet-crossing orbits may be determined by resonance sticking rather than a random walk. (C) 1999 Academic Press.