The Hill entry rate for u < vH

The relative velocity of the small planetesimal and big planetesimal as the small one enters the big one's hill sphere is approximately vH. We can imagine the big planetesimal as moving at the hill velocity vH through an infinitely thin disk of small planetesimals, which have a surface mass density sigma and individual mass m. It encounters objects that enter into its hill spill, of radius RH. So the number of small planetesimals that enter its hill sphere in a unit time delta t is:
N = number small bodies per area * encounter area
number of small bodies per area = mass per area/mass of small body = sigma/m
encounter area = RH * length it has traveled = vH delta_t

So N = sigma/m rH * vH delta_t
The hill entry rate is:
dN/dt = sigma/m rH * vH
We can rewrite vH = Omega rH, where Omega is the orbital frequency

Thus:
dN/ dt = sigma/m rH^2 Omega