Dispersion velocity revisited

The aim of this paper is to provide an analytical tool, which might improve models in which the particle-in-a-box approach has been applied and that may be also used when the thin disk approximation could not be longer appropriate. The dispersion velocity is the root-mean-square planetesimal, asteroid, or Kuiper belt object velocity with respect to the local mean circular orbit. This velocity is a function of the object orbital eccentricity and inclination. We calculate a general expression of the dispersion velocity for the planar case in which the object’s orbit has no inclination with respect to the local mean circular orbit and for the spatial case in which it has an inclined orbit. Our general expression of the square of the dispersion velocity may be expanded around any value of e for the planar and spatial cases, being in space an exact solution of the orbital inclination i. We expanded our expression around e= 0 with i= e/ 2 to study solid accretion rates and collision probabilities. We find that in the whole range of eccentricities and inclinations, our results are lower than solid accretion rates and collision probabilities computed by using the expressions of the dispersion velocity usually adopted in the literature. We apply our expressions of the square of the dispersion velocity expanded around e=0 and up to sixth order in e in our numerical model of planetary formation with planetesimal fragmentation and in our model of the collisional frequency on large asteroids. Our formalism, although generally giving lower values than previous approximations, validates the formerly used estimates for the applications presented here. In addition, we calculate the statistical velocity dispersion obtaining a straightforward expression as a function of the eccentricity.