Kablam! Satellite collision math, and a correction

Contributed by
Feb 12, 2009
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OK, so last night I posted about satellites that collided in orbit. I mentioned that the energy created in the collision was about the same as detonating a ton of TNT. I got asked how I did that math. That's no problem (well, a little one), but a bigger problem is that I screwed up the physics of the orbits. As commenter Marco Langbroek pointed out, the angle of impact I used was wrong. I forgot about the angle of the nodes.

Let me explain.

Both satellites were in polar orbits, more or less. One had an inclination (tilt) of about 86 degrees to the Equator -- in other words, it passed 4 degrees (90 - 86 = 4) from being directly over the Earth's poles, and the other had an inclination of 74 degrees from the Equator. I assumed that meant that the angle of approach was 12 degrees. FYI, a head on collision would be an angle of approach of 180 degrees (they are moving in opposite directions, toward each other), one catching up to the other would be 0 degrees, and a broadside "T-bone" collision is an angle of 90 degrees.

So where did I screw this up? The inclination is not the only important angle. What's also important is what's called the node of an orbit, or the angle around the Earth. Here's an illustration:

Earth with two possible satellite orbits

The yellow and red tracks represent two polar satellite orbits. You can see that a satellite in either orbit will pass very close over the Earth's pole (the south pole is seen here), so the inclinations of both orbits is high, near 90 degrees. But you can see they are rotated with respect to each other, in this case by about 60 degrees. That means that where they intersect, over the Earth's poles, the angle between them is about 60 degrees.

That has a huge impact (har har) on the collision speed. If they had the same node and the same inclination, the collision speed would be zero; they'd be on the same orbit. But if the nodes are rotated by 90 degrees, the collision would be a broadside, one slamming directly into the side of the other.

That's what happened with the Cosmos and Iridium satellites. The inclination difference was about 12 degrees, but the node angle difference was about 83 degrees (according to Mr. Langbroek, whose word I'll accept here since he's an amateur satellite tracker). So the impact angle was almost a total broadside.

How does that affect the energy of the impact? Well, it's possible to get a pretty rough idea. What follows is basically a back-of-the-envelope calculation, meant to be pretty loose. BotE calculations aren't supposed to be truly accurate; they're meant to give you an idea of the resulting number. So I wouldn't be surprised if the true conclusion I reach here as far as explosive yield is concerned is off by a factor of 2 or more, but the thing is I don't care. We're just trying to grasp the magnitude of the forces involved, not their exact measurements.

The collision energy depends on the relative velocities of the satellites. Imagine two cars approaching an intersection. One car is in the middle when the other slams into it at a 90 degree angle. The only important number here is the velocity of the impacting car; the velocity of the other one doesn't matter. It could be sitting there in the intersection, or moving at 100 kph; the velocity of the collision really only depends on how fast the other car was moving when it hit.

In reality, with satellites, it's more complicated. The actual three dimensional trigonometry of the event is a little fierce, but it turns out that an impact at 83 degrees is mathematically very close to a simple 90 degree collision (the difference in angles results in just a small percentage difference in velocity). In other words, a car hitting you at an 83 degree angle has almost the same velocity as if it were hitting you exactly broadside.

Assuming the satellites were both moving at the same speed, the impact velocity then is simply the velocity of one of the satellites, or about 8 kilometers per second.

The energy of impact depends on the mass of the satellites, too. The Iridium satellite was about 700 kg, and the Cosmos was probably about the same. I couldn't find a good figure for Cosmos, just estimates... but it's close enough. Remember, I'm trying to be very rough here; I don't care if I'm off in my numbers by a factor or two or not; I just want an approximation.

The kinetic energy of an object is equal to 1/2 x its mass x its velocity2. Again, because I'm being really rough here, it doesn't matter which satellite hits the other. We can assume they have about the same mass, so the kinetic energy (in ergs, which I'll explain in a second) is

KE = 0.5 x 700,000 grams x (800,000 cm/sec)2 = 2 x 1017 ergs

An erg is a small unit of energy, but 200,000,000,000,000,000 is a lot of them. Blowing up a ton of TNT releases about 4 x 1016 ergs, so this collision was roughly the equivalent of lobbing 5 tons of TNT at the satellites.

Boom. Yikes.

[FWIW, my mistake in the previous post was assuming the collision angle was 12 degrees, and the velocity of collision depends on the sine of the angle between the objects. Sin(12) = .20, so my number was 1/5 as big as it should be.]

That's why there's a large cloud of expanding debris; each piece of shattered satellite carried away a piece of that violent energy release. The energy of collision changed the orbits of all those shards, so they are now orbiting the Earth on new paths that take them higher or lower over the surface, right into the traffic lanes of other satellites.

So eventually, some may once again find an object in their way. And because of the high velocity, the kinetic energy of impact even from a low-mass piece can be pretty fierce. A rifle bullet does a huge amount of damage when it hits something, and it has a mass of about 10 grams and moves at a paltry 1 km/sec. Now think of the damage inflicted by a small satellite chunk that masses about 1 kg (100 times as much as the bullet) and moving 8 times faster... the energy of impact is 6400 times that of the bullet. Imagine getting hit by six thousand rifle bullets, and you start to get an idea of why satellite collisions are not just catastrophic for the two birds involved, but also a danger to other objects in orbit as well.