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 look

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.