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If an object the same mass as the Earth was to whiz past the Earth but miss by less than 500km, what would be the effects? Would there be massive tsunamis? Would people get pulled off the ground? Would there be no effect at all?
Mats Andersson
Mats Andersson, M Sc. Physics & Computer Science, Uppsala University (1991)
Updated May 1, 2017 · Upvoted by Andy Buckley, PhD in particle physics, visiting researcher at CERN, lecturer in physics and Mike Jarvis, Professional astrophysicist

The Roche limit for Earth is 16,000 km. This means that any object—provided it’s big enough so that it is held together by gravity—that comes closer to Earth than this, will be torn apart by tidal forces. Smaller objects hold together by other forces than gravity; spaceships and small meteors are fine.

If we have an object the same mass as Earth (and presumably a similar size), 500 km is well inside both objects’ Roche limits. Both bodies would be torn apart—on these scales, planets can be considered to be liquid—and presumably merge, after some complicated orbital shenanigans.

So no, “tsunamis” doesn’t even begin to describe it. There would be “tsunamis” consisting of the Earth’s entire mantle.

Now, you say “whiz past”. If the relative motion is fast enough, it is possible that only the parts closest to the other body would disintegrate like this, and that the molten mantle would eventually fall back on the original planets. I wouldn’t know exactly how fast “fast enough” is, and it’s still seriously bad news for the rest of the planet.

The best vantage point from which to watch this would be on Mars.

[EDIT: I am using a simple approximation of the Roche limit, such as the one used in this article. Victor T. Toth gives a more detailed explanation in his excellent comment below; all details considered, it doesn’t seem altogether certain that the bodies will be destroyed or merge. I think we’d have to do a detailed numerical simulation to find out. I still stand by the “mantle tsunamis”, though: I haven’t actually performed the calculations, but it seems overwhelmingly likely that the tidal forces would exceed the strength of the Earth’s crust.]