If it can be broke in 3 or even 2 pieces, the net energy in each part is going to be on a separate vector - diverging by definition - from the original course.
Why is impacting a nuke on the surface of an asteroid - regardless of position, rotation, axis, time, or depth - not 1000 times easier than trying to create an imaginary rocket/thruster,sail,ion jet or anything else that must be developed, built, flown to the intercept, maneuvered into place remotely - with a 8-15 minutes delay ?????? between signal and response - and somehow mounted exactly on the axis at the right azimuth to drive the asteroid?
We can’t even land satellites on Mars successfully in routine ops yet.
The old fire cracker anomaly...hold a fire cracker on its extreme end with just your finger nails and it will explode with little damage...grasp it with an enclosed fist and you run the risk of blowing off your fingers.
The concern is you do little damage and do little to change a large asteroid’s course by exploding it on its surface. With smaller asteroids and comets say small sledding hill size or less, you might melt a good part of it and the ejecta might change its momentum enough to miss earth on the first pass but larger asteroids are going to need something more drastic!
The thing is, nukes are all we have at this point in the short term to counter smaller threats say 500 feet or less as close up nuclear releases might simply melt them into vapor. You are talking about an instantaneous temp of greater than 100 million degrees centigrade(higher for h- bombs) and a smaller object’s elements might literally catch fire even with out oxygen(causing secondary high temp chemical reactions) if the object was sufficiently was close enough to the nuke.
A 1000 foot to 2000(6300’ circumference) foot object might be lit for a short time on one end like a candle and hopefully change the object’s direction if you caught the object far enough out sufficiently. The key there is if you had sufficient warning, time for prep, and could catch the object far enough out!
The real bad scenario would be a mile wide asteroid or more spotted less than 6 months, perhaps as little as a month or sooner out of its collision point with Earth. Depending on its composition a nuke would have little impact especially iron/nickle core objects.
A mixture of frozen water/ methane and rock/metal might be dealt with successfully as the nuke’s instant high heat at an impact or near impact explosion point would vaporize the elements with oxygen being stipped from hydrogen because of the super thermal heat...one could see a series of secondary explsions and chemical reactions until they gave out due to the remaining bulk of a super cold asteroid/comet.
If a shaped nuclear charge could be introduced into the body of a icy comet/asteroid, then the ejecta could thrown out in a more funneled/tuned fashion like a jet flame which, if far enough out would definitely change the direction/speed of the object so that it would miss earth.
The point is we would dependant at this point of the heat/radiation of the nuke to do any thing useful as atmospherics wouldn’t be available for concussive factors. The flash of the nuke is only for an instant in space so it would have to be in contact or very close vicinity to do anything reasonable.
Now, one could conceive in the next 30 to fifty years if we could get enough international cooperation, in building a more robust planetary defense. There is no atmosphere in space for nukes to do much good on large mile wide or more objects...but what if a good 100 mega ton nuke could take some atmosphere with it? A series of interplanetary grenades(a mile in diameter or less depending on the job you needed it for) if you will, with a heavy iron/nickel lead base topped with a sturdy shell that was studded with large rock or metal projectiles...this shell would be designed to fracture in key places. In side of this shell would be highly compressed or frozen oxygen and hydrogen or filled with highly oxygenated,dueterium/tritium laden water. At the center of it would lay your nuclear device. These devices could be kept at various lagrange points between Earth and moon or even between Earth or sun or various planets and the sun. If an object was spotted coming at Earth, say a 1000 foot diameter(3149’ circumference)...a juniour sized grenade say 100’ or two with an appropriate sized nuke could be launched out to meet it, using planetary gravity assist to help sling shot it to at least 20 to 30 miles a second. The idea of the bomb would be to amplify the effect of the nuke by creating a 1-2 kinetic plus heat effect weapon. The heavey iron metal base would survive long enough to shape the charge of the explosion forward, the forward velocity of the weapon cancels out reverse action of the explosion. The outer shell would fracture forward at high velocity propelling the iron/stone objects into the asteroid helping shread it...especially if the asteroid were of mixed consistency or loose objects. Then of course there is the matter of a nuke being exploded with tons of water or liquid hydrogen and oxygen surrounding it all of it suddenly flashing into a shaped superheated plasma crashing into an oncoming asteroid. If the nukes were set in a deuterium/tritium heavily enhanced water shell with a relatively compressed air/water shell around that, the resulting explosion and aftermath would probably be visible at least through telescopes from millions of miles away. The technology is feasable and most of it would be off the shelf albeit expensive. We would have decide as a world to put our differences aside and decide to lauch into space in a big way in order to put such a system into operation