*Average probability of some (any) operational satellite at a give altitude being hit by something, either a piece of space debris, rocket body, or another satellite each year. This was calculated using a flux of space objects with diameters greater than 1 cm from NASA’s model that is valid for the equatorial region. It predicts a collision with an operational satellite once every 37 years.*

Is anyone else bothered about the odds of two satellites actually hitting each other? Even if one is from the substantially larger group of dead satellites? I’ve been telling reporters that I’m not surprised that something (a piece of space junk) hit a satellite, but that I am surprised that it was another satellite. Of course, space junk has hit satellites before, as the excellent fact sheet written by David Wright for UCS catalogs. In fact, eight satellites have been hit by space debris (only three were active satellites) since 1991, the earliest recorded date of such a strike. But, if you start from basics, the first answer you get is that few, if any, ordinary satellites should be hit by space debris. Its only when you realize that danger lurks at the poles do you begin to realize the full danger satellites face.

Let’s do the wrong calculation first, because it’s very informative. If all space around the Earth was the same, you could just take the “flux” of space debris through any given point for a specified altitude, multiply it by the area of each satellite and the number of such satellites at that altitude and the time you want to add up all the chances of a collision to get a total probability. In fact, NASA spends a great deal of time determining that flux so it can use it to design debris mitigation for satellites and the Space Station. (See the NASA report on the “Orbital Debris Engineering Model.”) You can see the flux of space debris with diameters greater than 1 cm here . (Note that this is pre-2007.) It was created using the Haystack radar to survey the space junk density. I used this flux to calculate the “average” collision rate shown above. Note that it is independent of errors on orbital elements etc. that plague calculations of near misses between two specific objects. It predicts that an operational satellite should be hit by a piece of space debris greater than 1 cm in diameter about once every 37 years. That’s wrong, so don’t get too excited about it. Instead, let’s see why it is wrong.

Space around the Earth is not uniform: it is “shaped” by the uses humanity puts it to. Near Earth space is dominated by satellites (and space junk) in near-polar orbits. You can see two plots I made, one of the inclination vs. altitude of operational satellites (as defined by the UCS database of satellites) and the same type of plot for all cataloged space objects . Because the near-polar orbits all converge near the poles, the spatial density near those two spots increases dramatically. The NASA plot below, for a specific orbital altitude range, demonstrate this. It shows that the flux near the polar regions are almost 10 times what it is in the equatorial regions. Of course, satellites spend only a fraction of their time near the poles. Because of that, it doesn’t seem to be quite enough to account for the number of collisions (that’s based on a very qualitative assessment and not a calculation) but it illustrates the problem. I also have to emphasize that this is a “back of the envelop calculation.”

*click on the image for a larger version*

This rather casual calculation might account for the number of collisions between satellites (operational and nonoperational) and rocket bodies that David Wright has cataloged, but it still seems hard to understand the likelihood of a collision between the Iridium and Cosmos satellites. That still seems to be an unfortunate “outlier.”

After writing this piece, David Wright and colleague graciously shared with me a paper that also points out the special role the poles play in space debris. I haven’t done a literature search, but I suspect that my calculation is far from unique and that many others have also discussed this point.

update (25 Feb 2009): For those who like to keep track of these things, the current debris count is:

Cosmos 2251 — 244

Iridium 33 — 108

_________________

Total so far = 352

Geoff,

very nice — could you make a plot of the number of satellites vs. inclination in a given altitude range: eg. 700 to 900km? …Or else, a 3D plot that has the number of satellites along the 3rd axis to lift the degeneracy? From the 2D plots you have it is hard to get a sense of the population of satellites at a given altitude as I believe many of the data points may be falling on the almost exact same value. i.e. what looks like one datapoint may conceal several satellites (?).