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Asteroid Intercept

Actually, it was also very much about momentum. The amount of deflection of the asteroid isn't just due to the momentum transferred from the spacecraft, but also due to the equal and opposite reaction to the force that blasted debris from the impact crater into space. NASA will be studying the new orbit to calculate how much material was thrown out of the crater and how fast it escaped. The recoil is a significant factor in how effective this deflection method is, and it had to be determined by experiment, not just calculated from Newtonian mechanics.
This. Since we don't have detailed data on what the target is like it's impossible to accurately model what will happen. We need to shoot (both kinetic and nuclear) at various targets and see what sort of results we actually get. (The only other proposal for which the nature of the target matters is blowing it apart with a high-yield bomb in it's core.)
 
If you wanted to pulverize a large asteroid it might take multiple atomic bombs drilled into ateroid expaloding simualteneously.

Conventional explosives might work. That would probably require humans on the asteroid.

There is an idea to keep a probe at a combatant distance form the asteroid. Use gravitational attraction.

Why is this even in consideration?

The only thing a gravity tractor does is allow you to pull the target without landing on it--it doesn't matter if it's a pile of dust, it will still move. However, it requires more thrust than it would take if the engine simply landed on the asteroid and does nothing to provide such thrust. It also has the problem that the thrust has to be applied off-axis because on-axis means your rocket hits the target and wastes much of your thrust--in an extreme case it could actually push the asteroid the other way.
But how do we get it alongside the object? That sounds like a lot of energy would be needed in the probe.
 
If you wanted to pulverize a large asteroid it might take multiple atomic bombs drilled into ateroid expaloding simualteneously.

Conventional explosives might work. That would probably require humans on the asteroid.

There is an idea to keep a probe at a combatant distance form the asteroid. Use gravitational attraction.

Why is this even in consideration?

The only thing a gravity tractor does is allow you to pull the target without landing on it--it doesn't matter if it's a pile of dust, it will still move. However, it requires more thrust than it would take if the engine simply landed on the asteroid and does nothing to provide such thrust. It also has the problem that the thrust has to be applied off-axis because on-axis means your rocket hits the target and wastes much of your thrust--in an extreme case it could actually push the asteroid the other way.
But how do we get it alongside the object? That sounds like a lot of energy would be needed in the probe.
Yes, a tractor requires rendezvous which itself is only viable for NEOs that are going to hit on a future orbit. Then your acceleration is limited to how much thrust you can apply to the tractor without simply pushing it away from the asteroid. It's not a practical system short of some sort of torch drive that can be throttled to a whisper.

Against comets you're going to have to do it with a flyby and that means kinetic impactor or Orion.
 
There is a possibility of an asteroid heading for Earth to get missed.


CNN —

Astronomers have spotted three near-Earth asteroids that were lurking undetected within the glare of the sun. One of the asteroids is the largest potentially hazardous object posing a risk to Earth to be discovered in the last eight years.


The asteroids belong to a group found within the orbits of Earth and Venus, but they’re incredibly difficult to observe because the brightness of the sun shields them from telescope observations.


To avoid the sun’s glare, astronomers leaped at the chance to conduct their observations during the brief window of twilight. An international team spied the space rocks while using the Dark Energy Camera located on the Víctor M. Blanco 4-meter Telescope located at the Cerro Tololo Inter-American Observatory in Chile.

“Our twilight survey is scouring the area within the orbits of Earth and Venus for asteroids,” said lead study author Scott S. Sheppard, an astronomer at the Earth & Planets Laboratory of the Carnegie Institution for Science in Washington, DC, in a statement.


“So far we have found two large near-Earth asteroids that are about 1 kilometer across, a size that we call planet killers.”


Scientists determined that the asteroid crosses Earth’s orbit, but it occurs when Earth is on the opposite side of the sun – this pattern will continue for centuries since it takes the asteroid five years to complete an orbit around the sun. But over time, the asteroid’s orbital movement will be more in sync with Earth’s. Scientists don’t know the asteroid’s orbit with enough precision to say how dangerous it could become in the future, but for now, it “will stay well away from Earth,” Sheppard said.
 
I remember attending a talk many years ago in which it was stated that the greatest risk comes from those earth crossing asteroids with orbits mostly interior to the earth’s because they may remain undetected until possibly too late.
 
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