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I am looking for a theory that shows whether or not two passing celestial bodies will go into orbit.

I assume there would have to be a critical point where the gravitational attraction is stronger than the inertia of the moving bodies.

The bodies would then orbit until they collide, or be affected by a force to fall behind the critical point and escape orbit.

I am looking for a theory that states that objects past this critical point of gravitational attraction must either collide immediately, or go into orbit after falling within a certain distance.

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  • $\begingroup$ This is called "capturing", by the way. For example, sending a probe to Mars involves insuring that the probe is in some way captured by Mars' gravity; usually this involves retrorockets or aerobraking, but occasionally you can get away with nothing but the gravitational capturing mentioned in pela's answer. Natural bodies do not capture each other very often at all, although it can happen. $\endgroup$ – Nathan Tuggy Apr 23 '15 at 16:30
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If there are only two bodies, then they will never enter a mutual orbit. For two objects initially gravitationally unbound, in order to become gravitationally bound you must remove energy from the system. With only two bodies (that don't collide), this does not happen. They will accelerate toward each other, change directions according to how close they get, and then leave each other again with exactly the same total energy and momentum as before, but in general shared in some other ratio (for instance, if a small body encounters a large body, the smaller will gain energy and leave with a larger velocity).

On the other hand, if you have three (or more) bodies, one may get slung out with high velocity, thus extracting energy from the two others, which can then go in orbit. But alas, there's no equation for this; the so-called N-body problem has no analytic solution, and must in general be solved numerically.

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  • $\begingroup$ Just to clarify: if the bodies aren't point masses and their minimal distance is less than the sum of their radii, they will collide. $\endgroup$ – barrycarter Apr 23 '15 at 16:54
  • $\begingroup$ What does "gravitationally unbound" mean in this case? If I just looked at the Earth and Sun as two bodies passing each other, my continued observation would show one IS in orbit around the other. Is it possible for two bodies to be gravitationally unbound since gravity presumably can affect objects at any distance? I believe the correct answer is that one will orbit the other elliptically, approach and then veer away like a parabola, or approach to a minimum distance like a hyperbola. $\endgroup$ – barrycarter Apr 23 '15 at 16:57
  • $\begingroup$ Bound would mean that U + KE < 0. Hyperbolic approaches would be an example of an interaction between objects that are not bound. $\endgroup$ – BowlOfRed Apr 23 '15 at 20:37
  • $\begingroup$ How can 2 bodies (only) redistribute energy and momentum while concerving both? Postulate a very distant observer to allow determination of which is moving and by how much. The only solutions that concerve momentum and energy rotate the momentum vectors of both objects around the center of mass of the system. You cannot change either's magnitude. The "smaller will gain energy" is only possible with 3 bodies. $\endgroup$ – JDługosz Apr 26 '15 at 8:49
  • $\begingroup$ Maybe "gravitationally unbound" is a bad term, but yes, I meant like BowlOfRed explains. Concerning the "smaller will gain energy", this is for instance what happens in gravitational slingshots. $\endgroup$ – pela Apr 26 '15 at 20:52

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