The long-term evolution of orbits in the solar system: A mapping approach

Abstract

We derive a mapping that approximates the restricted circulat three-body problem (Sun, planet, test particle) when the eccentricity ofr test particle is small and its semimajor axis is close to that of the planet, | a − a p|/ a p ⪡ 1. The mapping is based on the approximation that perturbations to the test particle orbit are localized near conjunction with the planet. We use the mapping to confirm Wisdom's result that initially circular test particle orbits are chaotic when |a − 1 p |/a p ≲ μ 2 3 , where μ is the planet/Sun mass ratio. The map can be generalized to include perturbations from two nearby planets, in the two-planet map to model the evolution of initially circular test particle orbits in the Solar System. We find that many orbits between Uranus and Neptune and most orbits between Jupiter and Saturn become planet-crossing and hence unstable within the age of the Solar System (4.5 Gyr), and that most orbits between Saturn and Uranus, between Venus and Earth, and in the asteroid belt are stable over this time scale. (Between Venus and Earth, 4.5 Gyr corresponds to over 2 × 10 9 iterations of the map.) These results suggest that there may still be stable bands in the outer Solar System containing residual planetesimals from the protoplanetary disk. In addition, they show that interesting dynamical evolution of Solar System orbits is likely to occur on Gyr times scales.