Re: Defending the Earth: Giant Asteroid
With a radius of 450 km, visible to the best Earth instruments but not detected before the warning---well, I'm taking a S.W.A.G. and saying it's 25 AU out. At that distance, Earth subtends 0.7 arcseconds. That's a tiny target.
If the planetoid is 25 AU out and will hit in 90 days, it's already travelling like a bat out of hell, and will accelerate as it approaches the Sun. Indeed, if it weren't for collisions, it would travel in a hyperbola--a very flat hyperbola. Thus, it will not take that much effort to get it off course, and it needs only a slight change of angle to miss the Earth.
It is probably wise to make sure it misses the Sun as well, since the effects of a collision are hard to predict, and might be very unfortunate.
It might be a good idea to see to it that it doesn't hit anything else, as the increase in "loose bits" could eventually lead to a (much smaller but still worrisome) later collision with the Earth.
In short, getting something or someone to the planetoid that can push sideways, even a rather small amount, would be sufficient. It would be wise to communicate with NASA or ESA or similar to find out how much pushing, and in which direction, is needed, especially to miss the Sun and all other sizable bodies.
Nuclear weapons are a foolish idea; instead of "dodging the bullet," Earth would need to "dodge a tight-patterned load of shot". Exploding a A-bomb or H-bomb near the planetoid is useless, as there's no air or other medium to transmit a shockwave.
OTOH, a few large rocket engines would work fine, so long as you could get the fuel out to them, if you can "strap them onto" some useful spot; if the planetoid is "chaotically tumbling" the problem gets much worse.
Still, it's not that big a problem so long as you have some way to get needed resources 25 or so AU out in a matter of a few days.
PS: Yes, CrosshairCollie is right; speeding up the planetoid is a way to deflect it. However, it would not be as effective a method for making sure it misses the Sun.