If an extinction-level asteroid were to be detected on an impact course with Earth, could we avert it effectively with existing technology and systems, such as course adjustment via nuclear missiles? (How much advance warning would we need in order to do so?)
It seems like the ideal answer would take the form of an equation relating the mass and velocity of the asteroid to the warning time and detonative tonnage needed to avert impact.
(But it requires a bit of dishonesty.)
Simply because an "extinction level" asteroid does sufficiently limit the possible nature of the object.
To be extinction-level, it would need to be quite large. More than 3km in diameter.
To be asteroid, it would need to be an inner-solar-system object. This pretty much limits it to being a large asteroid, perturbed from its previously stable orbit by the gravity of Jupiter or one of the solar system bodies inward from Jupiter. Even at the absolute latest, we would detect it directly after such an encounter and have half an orbit to prepare for it. This means six years of warning, if the encounter event is such that it is directed directly to an Earth encounter, which is ludicrously unlikely.
The thing is, we have already mapped all, or very nearly all, asteroids of that size range. And we certainly will notice if one passes close enough to Jupiter to be deviated into an Earth collision in one gravitational event.
We have the ability to build a defense against a very large asteroid, although with only six years warning the result will be very messy indeed. We should be able to fragment it, and divert much of the asteroid mass from collision, but the resultant buckshot effect from the fragments will be extremely nasty. Still, we could.
What is much more dangerous is those classes of objects not eliminated by our clever wordplay analysis at the beginning.
A not-quite-extinction-level but still civilization-ending asteroid of 500m to 1km in the exact same scenario might not be noticed in time.
And worst a visitor from the outer solar system, or even from interstellar space (thus technically a comet not an asteroid), does not need to be perturbed by Jupiter to enter an Earth-targeting orbit. It might already be on such a path. It would potentially require a lot less than six years to cross the same distance. And it would come from somewhere out of the sky, not from the very highly observed space near Jupiter.
It would depend massively on the size of the asteroid, it's composition and how far out it was detected.
For the true planet killers it is extremely likely that the asteroid would be detected years in advance. Asteroids "planet killer" like Apophis are simply too big (Apophis is over 300m) to be missed. This gives us a significant advantage as it both gives us time to put together a mission and vastly reduces the amount by which the asteroid has to be re-directed. In this circumstances there are three main solutions:
Gravitational tractor - the good
Fly a spacecraft next to the asteroid, stop it stationary to the asteroid (using it's thrusters to maintain distance) and allow the gravitational attraction between the asteroid and the spacecraft to act as a tow rope, towing the asteroid along. This has the advantages of having very little which can go wrong and creating no orbital debris. However, the forces involved are simply too small to redirect a "planet killer" size asteroid in a reasonable timeframe
Nuke thruster - the cool
Fly a spacecraft packed with a (for a this kind of asteroid, really really big) nuke next to one side of the asteroid and blow it up. The energy released by the nuke will vaporise one side of the asteroid creating thrust. If a planet killer was discovered today this would probably be our best bet, as (thanks to the cold war) we have the technology to make some extremely powerful nukes as well as the rockets to get them there. This does, however, have some major downsides. While there is no threat of nukes going off in the event of a launch failure (setting off a nuclear bomb is an extremely precise process - the US has accidentally set fire to them and dropped them out of aeroplanes on multiple occasions and they have never gone off) it would scatter a lot of highly radioactive material. Additionally, it is likely that this would create a vast amount of orbital debris.
Smaller asteroids
Our current system - ATLAS https://atlas.fallingstar.com/home.php - gives one day's warning for a 30 kiloton asteroid, a week for a 5 megaton and three weeks for a 100-megaton. For context the Chelyabinsk meteor had a mass of about 12000 tons (note at 11km/s - escape velocity and therefore the lowest velocity at which an asteroid can hit Earth - each ton of matter has about the same energy as 15 tons of TNT). Depending on the accuracy with which the asteroid's trajectory can be calculated this warning it may allow the impact area to be evacuated (though considering that even a 5 megaton asteroid "only" has a TNT equivalent of about 3 Tsar Bomba's there are still a lot of places on earth where this could hit and do almost no damage). Therefore while this class certainly isn't good news their yields are low enough and the warning time long enough that they don't pose an existential threat.
However, the 100kt+ range is the real worry, with our current detection capabilities we'd need something ready to go - which we don't have. And their yield (equivalent to hundreds of Tsar Bomba's) is sufficient to obliterate the area for hundreds of kilometres and cause something similar to a nuclear winter.
ICBM's
Before someone inevitably suggest ICBM's, no they wouldn't work - while they look good on paper. They lack the delta V to intercept an asteroid at sufficient distance to redirect it, they also lack the yield to vaporise all but the smallest asteroids (it is also doubtful they they would have the yield for either). To destroy a large asteroid after a late detection would require an extremely large rocket carrying an extremely large nuke and guided very precisely to it's target.
So we should first be clear about how big an object we are talking about. The dinosaur killer was thought to be about 10 km across, so that is the size range we should be considering, if we are serious about "extinction level." (The lower end may be below that, but it's a good round number and we know mammals and birds did survive.)
To my understanding, surveys have already concluded that there are no threats of that size in the inner solar system, at least as far into the future as orbits can be projected.
The Wikipedia article Asteroid Impact Avoidance gives a good overview of the many approaches. Apparently it has been proposed to develop a 1 gigaton nuclear explosive, which could deflect a 10 km asteroid with only perhaps six months warning. That would perhaps be sufficient for one detected coming inward from the outer solar system, but only if a bomb was kept prepared at all times.
With decades of notice, a 10 km asteroid could apparently be deflected with just a long series of small impacts, with no nuclear explosives involved, according to this article in Space.com. In general, if you read a lot about research on asteroid deflection, you will find that people feel more comfortable proposing methods that don't involve nuclear explosives. I think a lot of the reason for this is that nuclear devices are so tied up in military secrecy, international treaty rules, and security (falling-into-the-wrong-hands) concerns that other approaches just seem safer. For instance, building a 1 gigaton bomb would be politically problematic. Better, rather, to improve surveillance to catch outer-solar-system threats sooner, and then deal with them with greater advance notice if necessary.
