A Semianalytical Model for the Motion of the Trojan Asteroids: Proper Elements and Families

Abstract

In this paper we develop a semianalytical model to describe the long-term motion of Trojan asteroids located in tadpole orbits around the L 4 and L 5 jovian Lagrangian points. The dynamical model is based on the spatial elliptic three-body problem, including the main secular variations of Jupiter's orbit and the direct perturbations of the remaining outer planets. Based on ideas introduced by A. H. Jupp (1969, Astron. J. 74, 35–43), we develop a canonical transformation which allows the transformation of the tadpole librating orbits into circulating orbits. The disturbing function is then explicitly expanded around each libration point by means of a Taylor–Fourier asymmetric expansion. Making use of the property in which the different degrees of freedom in the Trojan problem are well separated with regard to their periods of oscillation, we are able to find approximate action-angle variables combining Hori's method with the theory of adiabatic invariants. This procedure is applied to estimate proper elements for the sample of 533 Trojans with well determined orbits at December 2000. The errors of our semianalytical estimates are about 2–3 times larger than those previously obtained with numerical approaches by other authors. Finally, we use these results to search for asteroidal families among the Trojan swarms. We are able to identify and confirm the existence of most of the families previously detected by Milani (1993, Celest. Mech. Dynam. Astron. 57, 59–94). The families of Menelaus and Epeios, both around L 4, are the most robust candidates to be the by-product of catastrophic disruption of larger asteroids. On the other hand, no significant family is detected around L 5.