Dynamical behaviour of planetesimals temporarily captured by a planet from heliocentric orbits: basic formulation and the case of low random velocity

Planetesimals encountering with a planet cannot be captured permanently unless energy dissipation is taken into account, but some of them can be temporarily captured in the vicinity of the planet for an extended period of time. Such a process would be important for the origin and dynamical evolution of irregular satellites, short-period comets, and Kuiper-belt binaries. In this paper, we describe the basic formulation for the study of temporary capture of planetesimals from heliocentric orbits using three-body orbital integration, such as the definition of the duration and rate of temporary capture, and present results in the case of low random velocity of planetesimals. In the case of planetesimals initially on circular orbits, we find that planetesimals undergo a close encounter with the planet before they become temporarily captured. When planetesimals are scattered by the planet into the vicinity of one of periodic orbits around the planet, the duration of temporary capture tends to be extended. Typically, these capture orbits are in the retrograde direction around the planet. We evaluate the rate of temporary capture of planetesimals, and find that the ratio of this rate to their collision rate on to the planet increases with increasing semimajor axis of the planet. Similar results are obtained for planetesimals with non-zero but small random velocities, as long as Kepler shear dominates the relative velocity between the planet and planetesimals. For larger initial random velocities of planetesimals, temporary capture in both prograde and retrograde directions with much longer duration becomes possible.