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Up to now ~ 400 asteroids are known which move close to the Lagrangian equilibrium points L 4 (246) and L 5 (167) of Jupiter. In this investigation the orbits of all known Trojans were integrated numerically for 10 million years using the Lie Series integrator with adaptive stepsize (Hanslmeier and Dvorak, 1984) in the dynamical model of the outer planetary system. The goal of the study was to extend the computation of the proper elements for all known Trojans for a longer time interval; the respective results are compared to already existing ones (e.g. Bien & Schubart, 1987; Milani, 1993).
The determination of the characteristic quantities proper eccentricity ep , proper inclination ip and libration D was done by numerical filtering techniques using the method of Labrouste (Burger, 1998) and by a very precise frequency analysis (Chapront, 1997). Our results differ from the ones calculated for a shorter time interval by Milani (1993) between 5% and 10%. In Fig.1 we plotted the nomogram for the three proper elements mentioned above for L 4 and the ep for L 5. As a new interesting result we found Trojans with 0.15 < ep < 0.22 for L 4 (upper left) and L 5 (upper right) which answers an open question of an existing gap in this range of the ep . For L 4 Trojans there is a well defined maximum in the nomogram for ep ∼ 0.06; for the L 5 Trojans the maximum is flat and shifted versus ep ∼ 0.07. The proper elements sin(ip ) for L 4 (bottom left graph) show a maximum for Trojans moving in the plane of Jupiter’s orbit and a decrease versus larger inclinations, although a plateau for 14° < i < 27° is visible; this confirms that the Trojan orbits are more inclined that the orbits of the main belt asteroids.
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