Direct-retrograde orbit flips at planetary close encounters

Abstract

Aims. We want to find the conditions under which planetary close encounters transform the orbits of small Solar System bodies from direct to retrograde, and vice versa. Methods. We derive analytical constraints on the orbital elements of the small body that allow direct-retrograde transitions at close encounters. We check the validity of the analytical constraints with numerical integrations of close encounters in the restricted, circular, three-dimensional three-body problem. Results. For bound orbits, inclination flips at close encounters are possible only for values of the Tisserand parameter, computed with respect to the planet actually encountered, which are within certain limits. We give an analytical expression for the probability per revolution of this transition, as function of the orbital parameters. We show how to identify, among the known asteroids and comets on direct orbits, those that can flip to retrograde motion due to an encounter with an outer planet. Conclusions. Inclination flips at planetary close encounters can be quantitatively characterized with the analytical theory of close encounters.