Collisional parameters of planetesimal belts, precursor of debris disks, perturbed by a nearby giant planet

Planetesimal belts are invoked to explain the prolonged existence of debris disks. Important parameters to model their collisional evolution and to compute the dust production rate are the intrinsic probability of collision $P_i$ and the mean impact velocity $U_c$. If a planet orbits close to the belt, the values of both these parameters are affected by its secular perturbations yielding a strong correlation between eccentricity $e$ and pericentre longitude $\varpi$. We adopt a new algorithm to compute both $P_i$ and $U_c$ in presence of various levels of secular correlation due to different ratios between proper and forced eccentricity. We tested this algorithm in a standard case with a Jupiter--sized planet orbiting inside a putative planetesimal belt finding that it is less collisionally active compared to a self--stirred belt because of the $e - \varpi$ coupling. The eccentricity of the planet is an important parameter in determining the amount of dust production since the erosion rate is 10 times faster when the planet eccentricity increases from 0.1 to 0.6. Also the initial conditions of the belt (either warm or cold) and its average inclination strongly affects $P_i$ and $U_c$ and then its long term collisional evolution in presence of the planet. We finally apply our method to the planetesimal belts supposedly refilling the dust disks around HD 38529 and $\epsilon$ Eridani. In the most collisionally active configurations, only a small fraction of bodies smaller than 100 km are expected to be fragmented over a time--span of 4 Gyr.