An h-bomb’s destructive force on earth is dependent on atmospherics to conduct the concussive forces of the blast...the concussive shock and massive heating of the air do the most damage. Closer in, of course the heat and radiation does a lot of direct damage but with out atmosphere the damage radius is fairly limited.
So you have to bury the bomb in any large asteroid for it to do any good. Smaller asteroids or icy commets may be melted by a sufficiently close atomic release. Yet, in space an atomic flash would be brief and intense like a large flashbulb with a spherical out put levels of thermal and multispectral radiation which would decline with the square of the distance as the pulse expanded out ward.
And that is the problem...transporting a suitable bomb to a large asteroid, then getting it buried deep enough in a fissure that it would split the asteroid or shaping the charge so that a huge amount of ejecta is thrown out into space, the force of which acts as a chemical
rocket which alters the asteroids course.
That is why nukes are poo poohed. I think when it came to “saving the planet”, even the most PC types would “grudgingly” approve the use of nukes if the science supported their use.
Still, I’d bet desperate governments might still authorize nukes as a “hail Mary” last ditch effort to stop a mountain size asteroid from hitting the Earth
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.