Trojan collision probability: a statistical approach

We study the long term evolution of the collision probability <P_i> and of the impact velocity vimp in the two Trojan asteroid swarms. The new mathematical formalism by Dell'Oro and Paolicchi (1998) has been used since, in the calculation of the collision probability, it allows to account for the dynamical links among the Trojans and Jupiter orbital angles, due to the 1:1 resonance. This statistical method permits to compute both <P_i> and vimp over a long timespan (we considered 1 Myr) without making use of heavy numerical integrations. Moreover, it allows to easily update the values of <P_i> and vimp anytime more complete samples of Trojan orbits are available. The values of <P_i> and vimp 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 clearly visible in <P_i>. The large oscillations due to the secular frequency are wider for L4 than for L5. We have considered two different initial 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) is used. The vimp, instead, slightly decreases compared to the vimp found by Marzari et al. (1997) from the sample of 114 Trojans.