It seems like it should be quadratic in that double the satellites should get 4x the number of collisions, but satellite orbits don’t start out random. Without stationkeeping initially clear orbits become chaotic in that slight differences in initial position and velocity result in wildly different orbits. That transition results in an exponential increase in risk.

This is most obvious with geostationary orbit. Over a single year you could have say 10,000 satellites in geostationary orbit with essentially zero risk of collision, but without station keeping risk continues to grow every year those 10,000 satellites drift around.

I see your point that it’s not random but an exponential increase wouldn’t result from chaotic changes in orbit. That randomizes, making it quadratic.

It looks like inactive geostationary satellites will drift to the same longitudes, and tilt their axis away from the pole. If you have a set of geostationary satellites in that situation, the risk of collision is still quadratic in growth.

The risk is the sum of the pairwise probabilities of collision. So the only way you get super quadratic is if incrementally added satellites have increased pairwise risk with other satellites. It is possible that could be the case, because if you have to pack satellites more tightly, then maybe each satellite has higher pairwise risk with its direct neighbors.

That would still be sub-exponential, unless the collision risk between two satellites got exponentially higher as they get more closely packed. For example, suppose you have a ring of satellites, and you double the number. Then immediately neighboring satellites are twice as close. By basic inequalities you can show that considering the non-neighbor pairs of satellites, they have less than quadrupled the total risk of a collision. So by the master theorem we know the risk of collision would only grow exponentially if there were an exponentially higher risk of collision between two neighboring satellites in terms of the reciprocal of their distance. But that is not the case, is it? You’d expect it to be polynomial. Fewer orbits before the satellite drifts past its neighbor, a smaller expected distance to the neighbor as it drifts past… as with most physics problems these are linear components, multiplied together.

Thinking about it more, it could probably be modeled by the intersection of random walks that are initially unable to come into contact. I was thinking that was exponential but it’s presumably some polynomial function with a large exponent that’s eventually bound by a different quadratic function for sufficient values of time.

It would seem to be exponential after the first collision, at least.