Collisional velocities and rates in resonant planetesimal belts

We consider a belt of small bodies (planetesimals, asteroids, dust particles) around a star, captured in one of the external or 1:1 mean-motion resonances with a massive perturber (protoplanet, planet). The objects in the belt collide with each other. Combining methods of celestial mechanics and statistical physics, we calculate mean collisional velocities and mean collisional rates, averaged over the belt. The results are compared to collisional velocities and rates in a similar, but non-resonant belt, as predicted by the particle-in-a-box method. It is found that the effect of the resonant lock on the velocities is rather small, while on the rates more substantial. At low to moderate eccentricities and libration amplitudes of tens of degrees, which are typical of many astrophysical applications, the collisional rates between objects in an external resonance are by about a factor of two higher than those in a similar belt of objects not locked in a resonance. For Trojans under the same conditions, the collisional rates may be enhanced by up to an order of magnitude. The collisional rates increase with the decreasing libration amplitude of the resonant argument, depend on the eccentricity distribution of objects, and vary from one resonance to another. Our results imply, in particular, shorter collisional lifetimes of resonant Kuiper belt objects in the solar system and higher efficiency of dust production by resonant planetesimals in debris disks around other stars.