Nuclear won't do it either. When Nukes go off on earth, its the shock wave and heat that do the most damage. Out in space even hitting one side of a rather large asteroid may do a little melting but since a nuke in space would be mostly heat and light, there is no kinetic shock energy to break up the asteroid or to make it change course. Even if you could land on the asteroid to bury it or drill deep to place it, there is no air in the rock of said asteroid, just solid matter, so there is no guarantee that a nuke would do more than melt the rock on the inside....though melting the rock to vaporisation from the inside out may cause it to explode from the various gasses and pressures produced. This would create showers of meteors that would be very hard to predict in terms of impact. Smaller asteroids a mile or less may very well be destroyed easily this way but very large ones such as greater than one mile to hundreds of miles would need more more long range work.
Simply throwing missiles at them won't destroy them.
A few 1-ton objects guided to kinetic impact at orbital velocity on its way in, would likely break it up into a manageable cluster of rocks.
You don't need to obliterate it, just break it up enough that the pieces mostly burn up in the atmosphere
> Out in space even hitting one side of a rather large asteroid may do a little melting but since a nuke in space would be mostly heat and light, there is no kinetic shock energy to break up the asteroid or to make it change course.
You are incorrect. One thing you may want to look up: "Project Orion." It is something I have spent years researching, even to the point of getting quoted and some of my writing referenced in George Dyson's book of the same name. In short, Orion was the Coolest Propulsion System Ever: toss a nuke out the back of your spaceship, set it off, ride the blast, repeat. One 0.5 kiloton yield nuke every second or so would provide up to 8,000 to 16,000 *tons* of thrust.
In short, there are two ways to use a nuke to provide thrust:
1) Shock. Yes, there is no air in space. But the bomb itself is not converted to pure energy; it it for the most part converted to megadegree plasma. Orion pulse unit desings would be marvelous items for asteroid nudging. You build a bomb that has propellant on one side of it... propellant being something like water or polyethylene. (Orion used tungsten. even though the very high molecular weight of tungsten reduced the specific impulse available, the extreme density of the stuff made packaging *far* easier.)
When the bomb goes off, the radiation in the form of X-rays and neutrons can be modestly focussed into the propellant. As a result, the propellant is raised to several million degrees in microseconds, and explodes. Proper shaping of the propellant will give you a shaped charge effect... the propellant will turn into a jet of plasma moving at a good fraction of the speed of light. This jet, several *hundred* or *thousand* pounds, moving at maybe 5% lightspeed, will pack a massive wallop. Stand you bomb off from the asteroid however many miles required so that the jet spreads out as much as it can; this will spread the impulse over the largest possible surface and help reduce the chances of breaking the asteroid apart.
2) Ablation: build your nuke to emit as much energy as possible int the form of a narrow beam, again likely using shaped-charge principles. Put you bomb reasonable close to the surface and set it off. The result will be that one specific spot on the asteroid willbe converted to plasma, much as int the first case. This plasma explodes out of the asteroid, and serves as a brief but powerful rocket engine.
Total impulse per bomb can be far higher with 2) than with 1). However, by localizing the blombs effects on the asteroid, you stand a greater chances of simply shattering the asteroid. This is probably not what you want to do.
And then there's always the "Medusa* approach to moving an asteroid with nukes. But that's more complex.