There is some worry about an asteroid striking Earth and wiping out humanity. Asteroids have struck before, such as the one that caused extermination of the dinosaurs. The rate of these events is about once every hundred million years, so this isn’t a personal problem. However, it is a humanity-level problem and thus it should be approached with a tiny bit of support from everyone. A penny per person per year would be $70 million, which is a reasonable amount to spend searching for potential Earth-crossing asteroids. Once one was discovered, which is very unlikely, a greater input would be needed; but major monetary input can be put off until something serious is discovered.
When a Doomsday asteroid is discovered there comes the problem of how to prevent the collision, and those considerations can be developed while we have time to think about our options. Blasting it with a hydrogen bomb would probably work, even if the collision were only a few days away, but there would still be big pieces if it got that close. If it were years away from collision the debris from an H-bomb hitting it would be scattered enough to give only occasional falling pieces. Most of these would be small and no problem; the few larger pieces would be tracked after the H-bomb explosion, and any pieces still on a collision course with Earth could be hit again.
The H-bomb is the ugly brute-force method and is a last-resort solution. There are other ways using rocket propulsion to gently push the asteroid into a different course. These are incredibly expensive because it requires getting heavy fuel to very distant places and burning it precisely in expensive equipment. The problem of weight comes with the fueling and refueling of the rocket engines. Even ion thrusters would require a lot of mass to make them big enough to give the needed push. This would probably work, but to be effective it would require precision at the limits of possibility to push soon enough, and in the right way.
There is a way to use the asteroid itself and the energy from the sun to create thrust. Place a very large membrane mirror near the asteroid and focus the reflected rays of the sun onto a point on the asteroid. This would create a reactive jet of superheated material that would gently push the asteroid away from the heated point. That reactive point would be determined by the path calculated to guide the asteroid away from a collision course. Once again this would require years of precision guidance and monitored pressure to guide the asteroid away from collision. The bigger the asteroid, the earlier the push must begin, and the course projected accurately.
An ion thruster could be powered by an atomic battery, and it wouldn’t require as much weight to be put on the surface, but it would still be a lot, and be technically sophisticated. Perhaps a solar panel could be used to power the ion thruster. All the same it would take years of steady push to make enough difference with a large asteroid. On the other hand it will be smaller ones, with the power of the Arizona Meteor Crater, that are far more numerous, and these might be guided with a more easily created system. These have an Earth striking rate of one every ten thousand years. I have better plans for such an object. It would become a useful object if it could be steered in such a way as to return to a near-Earth position and possibly even put into orbit around the Earth. If that could be done with a rubble asteroid it would be a huge boon for humanity, because it could be mined for creating materials to be used in manufacturing huge space stations. Another application would be to use its inertia to lift objects off the Earth by dropping a line and at just the right instant hooking a payload to the line.
A Doomsday asteroid might become a huge asset to humanity.