# Translationally invariant speed distribution

During impact events, asteroids et al strike Earth at high relative speed, in part because objects in space move in random directions with no preferred reference frame with respect to which objects could tend to be slow. This raises the question of what probability distribution the velocity of objects might have. Presumably, the result has no preferred direction or reference frame. If $f\left(\mathbf{v}\right)$ is a spherically symmetric pdf of velocity, the pdf of speed is $v^2f\left(v\right)$. Can we choose $f$ so that $\mathbf{v}\to \mathbf{v}-\mathbf{v_0}$ preserves the pdf of speed?

Edit: for bodies we can expect to be confined to the plane of planetary orbits, we'd have a different symmetry law, giving a pdf $vf\left( v\right)$.

• When talking about motion in the solar system I think the reference frame of the sun is very much a preferred frame. – By Symmetry Sep 20 '16 at 20:13
• @BySymmetry Bodies' velocities might be drawn from a distribution that averages to the sun's velocity, but if we're concerned with the relative velocities of two small bodies we want the distribution to transform appropriately under a change in the velocity of one. Perhaps I need to convolute a spherically symmetric $f$ with itself. – J.G. Sep 20 '16 at 21:01
• Are you thinking about Liouville's theorem being applied to the velocity distribution function of asteroids? – honeste_vivere Sep 21 '16 at 12:52