Derivation of the day: collision rate part I

Background
There are several different regimes in which collisions can take place between a large body and a small body.  This time we consider a small body with random velocity \( u > v_{esc} \).  This is unaffected by the large body's Hill Sphere for the obvious reasons.

Quantities
\(\sigma\) = surface mass density
\(m\) = particle mass
\(u\) = random velocity
\(\Omega\) = angular speed
\(H\) = \(u/\Omega\) = scale height
\(R\) = radius of large body

Derivation
We first consider a "box" adjacent to the large body of length \(l\).  In three dimensions, this box has size \(l R^2\) since with \(u > v_{esc}\), the large body can only trap particles at its surface.  We then write the number of small bodies in this box, \(N\), as
\[N= lR^2 \times \text{number density of small bodies}\\
\frac{\sigma}{mH} = \text{number density of small bodies} \\
N = \frac{l\sigma R^2}{m u/\Omega}\]
We notice that \(l\) is simply \(u \Delta t\) so taking the derivative of \(N\) with respect to time, we find the rate of collisions;
\[\frac{dN}{dt} = \frac{\sigma \Omega}{m}R^2\]