What is the rate of collisons?

Question

Lets assume that planet Earth is the only massive body around here, and moves at speed 1km/s towards a huge field of almost motionless small rocks. The concenteration of rocks is 1 rock per 1,000,000 km³ (n = 1e-6/km³).

http://i16.tinypic.com/4vefxjr.jpg

What will be the rate of bombardment of the entire planet (rocks per second) when it enters the field?

Answer 1

We have to consider the problem of gravitational capture. The ref. says the capture area = planet area * (1+[(Vescape^2)/(v^2)]), where v is the velocity at infinity. This fits our situation. Given Vescape = 11.2 km/s, v = 1 km/s and R = 6378 km, the capture area = 1.616E10km^2.
Capture rate = area * velocity * density = 1.616E10 * 1 *1E-6 = 1.616E4 rocks/s. We have increased the effective area by a factor of 126.44 and the effective radius by 11.245.

Answer 2

Assuming the rocks aren't gravitationally pulled toward earth, you attack this problem as follows:

The earth has a radius a so the cross sectional area perpendicular to the velocity vector is:

A = pi*a^2

The earth moves at a speed v so it sweeps out a volume of

V = vA = v*pi*a^2 every second

The earth passes through the rocks with number density n so the collision rate is

C = n*V = n*v*pi*a^2 now a = 6378 km so you get

C = 128 rocks/s (rounding up)