  The rate of collision and the relative velocities of the colliding particles
in turbulent flows is a crucial part of several natural phenomena, e.g., rain
formation in warm clouds and planetesimal formation in a protoplanetary disks.
The particles are often modeled as passive, but heavy and inertial. Within this
model, large relative velocities emerge due to formation of singularities
(caustics) of in the gradient matrix of the velocities of the particles. Using
extensive direct numerical simulations of heavy particles in both two (direct
and inverse cascade) and three dimensional turbulent flows we calculate the
rate of formation of caustics, $J$ as a function of the Stokes number (${\rm
St}$).The best approximation to our data is $J \sim \exp(-C/{\rm St})$, in the
limit ${\rm St} \to 0 $ where $C$ is a non-universal constant.
