Continuing the discussion from Date of the kickstarter?:
I got tired of the rough estimates, so I went and plotted this up properly. I made a few assumptions in doing so:
The planet being orbited is, for all intents and purposes, Earth. It has Earth’s mass and radius. It is an idealized, perfectly spherical Earth.
The station is orbiting in a circular orbit at an altitude of 2000 km.
There is no atmospheric drag.
The explosion is radially symmetric, and, for our purposes, constrained to the plane of its orbit.
The large chunks of the station are launched away from the explosion with a Gaussian (normal) velocity distribution. I had to make up some numbers here for the mean velocity and standard deviation, so I chose a mean of 981 m/s and a standard deviation of 327 m/s (981/3). 981 was selected from a list of velocities on Wikipedia because A) it was faster than the muzzle velocity of an M16 (so that it’s, literally, faster than a speeding bullet), and B) listed as the velocity of an actual aircraft (the SR-71 Blackbird, not, for some reason, called the “Superman”).
The space station explodes into 20 large pieces. 20, because it’s a nice round number, and didn’t take too long to run the calculations. These 20 pieces are not assumed to be distributed symmetrically around the centre of the explosion, however.
Here’s the colossal mess that was spit out:
In this run, 8/20 (40%) of the space station chunks collided with the planet, while the remaining 12 (60%) were launched into a cascade of orbits. None achieved escape velocity.
I ran the calculations a few more times, to get a reasonable sample (and also to restrict the colour options of the orbits, because dear god):
13/20 (65%) collide, while 7/20 (35%) achieve stable orbits. None escape.
9/20 (45%) collide, and the remaining 11 pieces get distributed over a wide range of orbits.
Again, 9/20 (45%) collide with the planet.
14/20 (a whopping 70%) of the fragments hit the planet!
In total, I ran 10 simulations, with the number of collisions in each simulation given below:
The mean average of these is 100/200 (50%), so, on average, half of the space station will collide with the planet. Strangely enough, it’s not a front half/back half split (although, that does contribute the majority (~90%) of the collisions). The forward half can collide if it gets a suitably large push toward the planet from the explosion (it ends up in an orbit with a higher apoapsis, but with a periapsis beneath the planet’s surface).
Mind you, while these numbers are independent of the number of space station chunks, they are not independent of the station’s orbit. A lower orbit will result in more chunks hitting the planet, while a higher orbit will result in not only fewer chunks colliding, but more chunks escaping the planet’s hold altogether. And, naturally, these numbers are very much dependent on the mean velocity chosen for the explosion.