As has already been said, the further out it is the less force will be required to deflect it sufficiently.
There are many schemes for slightly changing it’s vector and causing a miss, but what I see as a problem is the tendency to always think of speeding one up to widen it’s orbit or slow it to shrink it’s orbit rather than deflecting it out of the plane of the ecliptic where it won’t become a problem ever again.
But there is an even easier solution, but I can hear conservationists howl in outrage before I even make the following suggestion…
Use the moon as a catcher’s mitt.
or to be more technical, use INTENTIONAL, but harmless collisions
to rid ourselves of those troublesome space rocks.
Once an asteroid has impacted the moon we need never worry about that asteroid again, and in terms of interplanetary distances
it’s easier to make something miss a particular house than a particular continent.
Another possibility is using just enough deflection to intentionally steer a “near miss” CLOSER to intentionally “slingshot” it out of consideration as a future threat using the earth’s gravity to either fling it into a highly inclined orbit.
another advantage to highly inclined orbits is that once an object is in such an orbit it won’t encounter any major planets that can disturb that orbit
a last advantage to deflecting asteroids out of the plane of the ecliptic is that absolutely brutal methods can be employed.
Personally I’d love to do something really strange with a large
Insert into earth orbit.
Ponder the potential uses…