Actually I did make an error in the yield required (see below). The reason I went with nukes is the mass of the system to be delivered is small, and the thrust is instantaneous...since the base problem is detection at a range useful, in order to keep the radars reasonable, you need a big push. (I was trying to see if all aspects could be dome with current tech). The issue with continuous thrust is the system is heavy (roughly 3-5kg/kw for advanced nuclear...my design holds the record) yielding about 10-20mN of thrust...so you would need far more warning (5-10 years) to make it work...with heavy lift launchers you could reduce that to under a year, but then you would still need to detect the rock 6+ AU out.
Note, I assumed this was an attack rock, so it was coming in on a highly elliptical orbit from the kuiper/Oort clouds. For a NEA or something else (eg non-weapon) the problem is much simpler, as the orbits are predictable, and the rock is nearby, simplifying tracking.
Optical telescopes would be better than radar, and since it would take decades to reach earth from even the kuiper belt, you would have more than enough time to detect and affect it. Also, the yield required decreases if you go nuke as you get more and more warning--new horizons could have carried the required payload if we were able to engage it more than a year out (a few hundred kT).
Now, if it was truly a hypervelocity (e.g. Near c) impactor (see the space marines novels as an example)...the effects are much worse...even a few hundred tons of gravel at 0.99c would sterilize earth (an effective yield of a few 10's of TERATONS of TNT equivalent). And there is virtually no defense, as you couldn't see it coming in time (if you detected it at Pluto, you would have maybe minutes of warning). However, such weapons are WAY beyond current tech...though represent the ultimate in destructive power.
Also, I went with a near surface detonation to ensure the whole rock is moved--subsurface could result in substantial chunks on the original course--also, if the density were a lot lower (cometary for example), the ablation method would be even more effective and not pulverize the object.
Ideal would be (for the original problem discussed size) several 10's of 10MT class weapons detonated about 400-500m down...that's enough yield to pulverize the entire thing to dust/gravel ensuring burn up, AMD thus could be done a lot later...but would require drilling, etc.
Oh, for the ablation calc--the impulse efficiency of vaporization is typically about 1MJ of delivered energy equals about 20-100Ns of impulse (for microsecond pulses). This can be increased by lengthening the pulse and really improved by using two or more nukes, one to preheat/pre vaporize the surface, the second to heat that larger mass to stretch out the pulse--possibly yielding a 10-20x or more improvement. A megaton is about 5 billion MJ, or a delivered impulse of (assuming the multi-stage method) 100-500 billion Ns and 1-10 trillion Ns per megaton--so you would need 20-200 megatons, hence my 10-20 requirement (as I was pretty conservative on the mass calcs--a cometary body would probably only need 10)
The best possible method would again be the deep pulverization method, but that requires drilling and such--I'd much rather send multiple bomb-craft that are lighter...that way if one or more fail, you can still get the mission done, and more only helps. Anyway, a full discussion is the stuff of theses...and there have been quite a few.
