The best case performance of the thing would require that the engine thrust equals the amount of force to be exerted on the asteroid, but unless the distance between the craft and the asteroid is large or the craft uses a pair of engines firing at extremely shallow angles relative to the asteroid (which would be extremely inefficient) I would think the ejecta from the engines would push on the asteroid in such fashion as to work against the desired motion.
Why not just have a spacecraft slowly land on the asteroid, fasten itself, and then use thrusters to push the thing directly?
There are several factors that could make this extremely difficult to accomplish. The asteroid may be tumbling in a random fashion at a high rotational velocity. Even if one could manage to land, using thrusters effectively on a tumbling object is problematic. The surface of the asteroid might be extremely irregular and rocky making the landing treacherous. Furthermore, the composition and properties of the asteroid are likely not well known which makes it difficult to design anchors to plant into the surface.
The best case performance of the thing would require that the engine thrust equals the amount of force to be exerted on the asteroid, but unless the distance between the craft and the asteroid is large or the craft uses a pair of engines firing at extremely shallow angles relative to the asteroid (which would be extremely inefficient) I would think the ejecta from the engines would push on the asteroid in such fashion as to work against the desired motion.
Agreed, but how about this?:
Replace the spacecraft with an equivalent inert mass. Tether the mass to a platform via a very long (~1000 kilometer) cable. Mount four, or better yet, six engines symmetrically on the platform so as to exert the necessary force keeping the inert mass at a constant distance from the asteroid. The angling of the engines can now easily be made shallow enough that their ejecta will miss the asteroid while providing a reasonably efficient thrust.
All of this technology is doable now (at admittedly enormous expense). In at most a few decades, it will be cheap enough for serious consideration.