Make a few assumptions (months to impact/intersection; mass of incoming object; number of km it needs to move to miss (number of degrees of deflection).
Figure out what impact/impulse is needed to create that much movement.
Use a chemical rocket? Probably can’t get that much mass that far out and still keep accelerating it: the Apollo program (and we have no large rockets now!) could only get a little bitty command capsule and LEM + eqpt module to the moon - and that by coasting (not accelerating!) .
But, that creates a starting point for discussion.
Lets start with twelve months and we know a thousand foot(3140’ circumference) asteroid has a 99 percent probability of hitting Earth. Its a mixed class got a heavey iron/rock core and some dirty icy water and frozen methane on it. It’s mass would be that of say a couple of super aircraft carriers appox 200000 tons.
Because it is coming in towards the sun and us it is actually accelerating so that when it reaches us it will be traveling at roughly 10 miles a second and it is currently 310 million mile from us leaving roughly a year.
So we need to know at what point to do we need to know when an asteroid is going to hit us so that we can meet it a 6 months to a year out so that we can make use of its slower velocity farther out so that the feeble forces we do have can make the most good....lets say 1 year out.
Now this is where my astro-physics fails me since I don’t know the gravitational accelration constant of the sun. I know on earth its roughly 9.75 meters per second squared and I’ve read that most object hit the earth between 7 to 20 miles a second depending on direction and other factors. My reading suggests that such an object hitting our earth at 36000 miles per hour would strike us with a 500 megaton eqivalent of energy released, a real killer.
So the parameters are to strike or effect the asteroid at 310 million miles out while the object is traveling at about 7 miles a second but accellerating as it comes toward the sun. We would have to affect it enough so that it did not strike us on its way about the sun and on its return outward again. We would need it to miss us by 30000 miles so as to not affect our geo-synchronous satelites. We know also is that its got movement along it three axis(tumbling slowly)!