A STUDY OF ORBITS NEAR SATURNS TRIANGULAR LAGRANGIAN POINTS

We revisit the question whether orbits near Saturn's triangular Lagran gian points (L4 and L5) may be stable for the age of the Solar System, In this paper, asteroids potentially on these orbits are named ''Brui ns'' for short. We numerically integrated orbits around the L4 and L5 Lagrangian paints of Jupiter (Trojans) and Saturn: 40 Trojans and 350 Bruins, all of inclination less than 12 degrees, Trojan orbits were nu merically integrated along with Bruin orbits, so that by comparing the results, we might better understand Bruin orbital dynamics, Four Brui ns were stable when the numerical integration was stopped at 412 Myrs. Properties of these stable orbits were: (1) proper eccentricities les s than 0.028; (2) longitudes of perihelion that librate about a point 45 degrees from Saturn's perihelion, such that the perihelia are never close when the Bruin's eccentricity is near maximum; (3) maximum ecce ntricities that do not occur when perihelia are near Jupiter's aphelio n or when Jupiter is near its maximum eccentricity; and (4) libration angle about L4 or L5 of more than 80 degrees (a measure of tadpole len gth), Orbits with libration angles less than 80 degrees were unstable, the time to instability being correlated with libration angle, In con trast, long-lived Trojans may have very small tadpole orbits and longi tudes of perihelion that either circulate or librate with respect to J upiter's. We numerically integrated various Bruin orbits using differe nt Solar System models to develop a Hamiltonian perturbation theory fo r low-inclination Bruin orbits, Although only at the beginning stages of development, the theory already identifies three separatrices of Br uin motion due in part to the Great Inequality (GI) between Jupiter an d Saturn, These GI separatrices are a major contributor to the unstabl e region near Saturn's L4 and L5 points. We found a secular resonance between the perihelion precession rates of Saturn and a Bruin in the e lliptic, restricted problem of three bodies with imposed motion of Sat urn's perihelion, This resonance creates a separatrix of Bruin motion, which may cause low-inclination Bruins with circulating longitudes of perihelion to go unstable when Jupiter is added to the model. Althoug h we still cannot say whether all Bruin orbits eventually go unstable, we can predict candidate stable orbits based on this work. (C) 1996 A cademic Press, Inc.