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The Sciences

# Kablam! Satellite collision math, and a correction

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:

KE = 0.5 x 700,000 grams x (800,000 cm/sec)^2 = 2 x 10^17 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 10^16 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.

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