TROJAN COLLISION PROBABILITY - A STATISTICAL APPROACH

We study the long term evolution of the collision probability < P-i > and of the impact velocity v(imp) in the two Trojan asteroid swarms. T he new mathematical formalism by Dell'Oro and Paolicchi (1998) has bee n used since, in the calculation of the collision probability, it allo ws to account for the dynamical links among the Trojans and Jupiter or bital angles, due to the 1:1 resonance. This statistical method permit s to compute both < P-i > and v(imp) over a long timespan (we consider ed 1 Myr) without making use of heavy numerical integrations. Moreover , it allows to easily update the values of < P-i > and v(imp) anytime more complete samples of Trojan orbits are available. The values of < P-i > and v(imp) over a short timescale have been compared to those of Marzari et al. (1997) and a good agreement has been observed. Over a long timescale the influence of the secular frequency g5 - g6 is clear ly visible in < P-i >. The large oscillations due to the secular frequ ency are wider for L4 than for L5. We have considered two different in itial samples of orbits. The first is the same sample used by Marzari et al. (1997) and includes the orbits of 114 Trojans. The second, more complete, includes 223 objects. We observe an increase of < P-i > in both the swarms when the more complete sample of Trojan orbits (223) i s used. The v(imp), instead, slightly decreases compared to the v(imp) found by Marzari et al. (1997) from the sample of 114 Trojans.