Direct-retrograde orbit flips at planetary close encounters

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