I don't understand the magnitudes and masses?<br /><br /><i>"It's not like accelerating a spacecraft. It's like accelerating 1,000,000,000,000,000,000,000 spacecraft all at once."</i> <br /><br />Your figure exceeds the mass of the Moon (assuming the spacecraft you're talking about is more than 70 kg). All Earth-crossing asteroids are significantly less massive than this. For example, Apophis is about 20,000,000,000 kg.<br /><br />Apophis' surface gravity is only 0.00001 that of Earth's surface gravity (assuming 125 meter radius and 2e10 kg mass). You couldn't even walk on its surface as even your most gentle step would accelerate you well beyond its escape velocity. The Castle Bravo nuclear test (15 megatons or 6e16 Joules of released energy) produced a crater 6500 feet in diameter, roughly the size of Apophis, and 250 feet deep. Why would something that can produce an Apophis-sized crater on Earth barely blow dust off an asteroid with negligable surface gravity? And the crust of Earth, at 2.7 grams per cubic centimeter, is more dense than an asteroid. Apophis is estimated to be about 2.4 g/cc which is why it is theorized to be a loose pile of rubble, rather than solid rock.<br /><br /><i>The best chance is to change the speed by a millimeter a second far enough in advance that it will make the asteroid miss us. A nuke MIGHT be large enough to do that if placed in the right spot<br /></i><br />The strategy behind a nuke is not to nudge it by a mm/s, but to break it into fragments, each carrying off momentum. When the spacing between the fragments exceeds 1 Earth diameter, Earth will simply pass through a debris field possibly getting hit by a few negligable fragments, but much more likely not getting hit at all.<br /><br />Another advantage of nuking it is that we don't have to worry about the error bars on our orbit calculations. Since the nudging methods work best a few decades in advance, and we might not know to a precision of a few thousand kilometers where this