On orbit satellite collision: What are the odds?

Here's a quick back-of-the-envelope:

The collision happened at 750 km altitude, which is the most populated
part of earth orbit:

http://www.armscontrolwonk.com/2185/the-future-is-now

....from that histogram, it looks like there are about N_satellites =
300 satellites between 700 and 800 km altitude. For a collision to
happen, two satellites have to be in the same place at the same time.

By "in the same place", let's say the two satellites would have to be
in a cubical that is 2 m x 2 m x 2 m (satellite bus is usually smaller
than this, but solar panels bigger, so this is a compromise) for a
collision to happen.

Between 700 and 800 km, how many of these "cubes" are the

N_cubes = 4*pi*R^2*t_layer / V_cube (where t_layer is the
"thickness" of the layer of LEO we are examining: 100 km)

= 4 * 3.14 * ((6380+700)*10^3 m)^2 * 100*10^3 / (2 m)^3 =
7.9*10^18 cubes

By "at the same time", the residence time of a satellite moving
through a 2 m cube at 7.5 km/s is

t_residence = 2 m / (7500 m/s) = 2.7*10^-4 s (or 270
microseconds)

In a given year, the number of these "timeslots" is:

N_timeslots = 365*24*60*60 seconds/t_residence = 1.2*10^11
timeslots

If satellites are more or less randomly bouncing around this volume of
space (think of a gas where satellites are individual molecules), then
the odds of any satellite being in the same cube as a particular
satellite at a given instant (i.e., in a given time slot) is:

P_1sat = N_satellites / N_cubes = 3.8*10^-17

Very unlikely. But over the course of a year, the odds become

P_1satYear = N_timeslots * P_1sat = 4.57*10^-6

or "one in 200,000". But, there are N_satellites, so the odds of *any*
satellite colliding with *any* *other* satellite over a year are

P_anySatYear = N_satellites * P_1satYear = 1.37*10^-3

or "one in 700".

Now, over the past 30 years of having a large number of satellites on
orbit, the odds of satellite-satellite collision drop to:

P_SpaceAge = P_anySatYear * 30 years = 0.04

....or "one in 25". Entirely reasonable.

The only mystery is why this collision wasn't predicted and the orbit
of the Iridium satellite tweaked well in advance.
--
Andrew J. Higgins Mechanical Engineering Dept.
Associate Professor McGill University
Shock Wave Physics Group Montreal, Quebec CANADA
http://people.mcgill.ca/andrew.higgins/