Back to hollowed-out asteroids:
Imagine finding a medium-sized asteroid---say 20m diameter. Break it up into eight 5m chunks. Then give each one a *small* delta-V to push it into an aerobraking trajectory.
It's a small delta-V only if the rock was going to come relatively near Earth in the first place, or you're very patient. If the asteroid isn't earth-crossing, then you'll simply have to change its orbit until it is, which will take a hefty delta-V unless you can do a slingshot off another planet.
That dumps most of the orbital speed, and another small delta-V gets you into whatever orbit you like.
"small" in this context may still be pretty hefty unless you're going for an orbit that has a very high apogee. If you're going to GEO, for example, the best you can do is put the asteroid into GTO, and it's still a hefty chunk of delta-V to go from GTO to GEO.
The GTO-like orbit is hazardous for this thing. If you don't do anything to the rock after the aerobraking, then about 12 hrs later (something like that, anyway) it'll re-enter the atmosphere a second time if you haven't imposed some new delta-V on it. Bear in mind that after aerobraking, the surface of the rock is still going to be very hot (molten?) for some time. With currently available technology, I'm really not sure how you'd put additional delta-V on it before it re-entered. Set off a bomb nearby?
Going to a higher transfer orbit buys your more time to get the 2nd delta-V maneuver. If you were accurate enough, I'd say you'd come out of the aerobraking into a lunar transfer orbit and then slingshot off the moon. That would give you weeks (or months, if you went into an inclined orbit after the lunar slingshot) to figure out how to put additional delta-V, but I suspect that the uncertainties are orders of magnitude too large for that kind of finesse.
1) There's some probability that you screw up the steering, and the rock ends up on a collision course for Manhattan rather than on a glancing path through the ionosphere. How small does it have to be to be guaranteed to burn up in the atmosphere?
Depends on the trajectory, composition, etc etc - but in general, a rock doesn't have to be all that big to make it to the ground.
In the satellite world, a "targeted re-entry" usually targets the Pacific, which is a pretty big target. In the case of the asteroid, we'd do something similar - time it so that if something went a few hundred km wrong, it would go into the middle of the Pacific. Evidently there aren't voters there.
2) Is it even possible to aerobrake a rock, or would it need a heat shield and/or mechanical support?
The question would be how well you could control it. With a potato-shaped rock, it would probably tumble chaotically, making it very difficult to accurately predict its path through the atmosphere. And unless your asteroid was very solid, there's a real chance that it would break up somewhere along the line, and none of the pieces would go where you'd want them to go. If you're adding a heat shield and mechanical bracing, then the heat shield will probably wind up being a capsule that completely enclosed the rock so that you could control the dynamics & aerodynamics. But at that point, we've gone from "aerobraking an asteroid" to "building a ship that can carry rocks and aerobrake."
3) Are #1 and #2 at odds? Maybe you could have some loose, rubbly asteroid ("safe" because it burns up easily) carrying an aerodynamic shell which doesn't get deployed until the orbit has been confirmed beyond all doubt.
I'll mostly agree with BenBurch on this. There was a movie where they blew up a huge asteroid just before it hit the Earth, and somehow that saved us - in reality, it wouldn't have been an improvement over the one big asteroid and might have been worse.
But with a rock only 5 meters across (maybe 50 tons?), on a grazing trajectory . . . let's see, maybe 1000 small rocks, each weighing 50 kg . . . they'd all make it to the ground, and might not make much of a crater but they'd destroy whatever they hit.
Maybe if you actually pulverized it down to sand. On a grazing trajectory, I think the sand would scatter pretty well. The heat shield would have to be light but rigid, so it would fall slowly if it came apart.
Interesting design problem.
ETA: R Mackey beat me to most of this.
