Hmmmm.... Well, FFS sez you need to focus your laser down to a square cm and get at least 1 MW in that to have any hope of punching through armor. However, I have it from other sources that penetrating armor is not required to make a kill.
I would guess that the beam is not nearly so well focused as to allow a 1 sq cm spot, nor so well aimed to stay on that spot for 20 seconds. Probably has either/both of a small focal diameter and a low wavelength. Someone suggested it was IR, which would make sense. I don't have the formula handy, but at a range of 600km, we're talking some beam spreading.
All the laser really needs to do is impart its energy to the target, warming it up a lot faster than it can cool itself. 20 seconds is a long time; a real long time. Some missiles can move 20 km in 20 seconds, but we're talking about a ballistic missile in its launch phase, so we probably have to track it for about 5km or less.
Let's say it's a 5 MW laser. I'm sure if it was 10, they'd've said something to that effect, and 5 is half way between 1 and 10. That laser is going to impart 5 Mj a second, for 20 seconds, or 100 Mj, onto its target. Remember, it's not focused, it's a frickin spotlight, but thankfully it's not mounted on a frickin shark.
Anyway, just what can 100 Mj do to something? We can be pretty sure that 100 Mj isn't the actual point at which the missile blows up; it's going to be cooling; maybe half that is all it's going to get. So 50 Mj. What's the volume of the target? I don't know how big a rocket we're talking here, but you can be sure that 50 Mj is going to have to heat the whole bleeping thing.
Now if someone has their handy joule-to-calorie converter handy, we can find out the number of calories we're imparting, and divide that by the volume of the target. The result should be the number of degrees celcius we are adding to the target. btw, 1 calorie is the energy required to raise 1 ml of water 1 degree; since a missile is not water, this is only an approximation. Metal doesn't require a lot of energy to increase its temp, water is a pretty good heat sink. I would GUESS that you could safely multiply the result by 10 and be closer to the correct answer then you were before you did so.
I will let a math-monger give us some numbers.
