On the origin of the planetary spin by accretion of planetesimals: II. Collisional orbits at the hill surface

Abstract

The structure of the totality of orbits starting at the Hill surface is investigated numerically and systematically in the framework of the planar-restricted three-body problem. We have obtained the following results. (1) The qualitative features of the totality of orbits at the Hill surface are clarified. (2) A general formula for the collisional probability as a function of the mass ratio μ, radius R, and Jacobi constant C is obtained for μ ≤ 10 −4, 2 × 10 −5 ≤ R < 2 × 10 −4, and the range of C corresponding to Γ ≤ 2, where Γ is the Jacobi constant for the Hill problem of the restricted three-body problem. The collisional probability is also given as a function of the velocity at the Hill surface. (3) The relative collisional probability as a function of ( a, e) is obtained for 0.86 ≤ a ≤ 1.14 and 0.0 ≤ e ≤ 0.1.