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First, to clarify: I'm not asking if perhaps there's a moon that we haven't found yet. The question is, theoretically, would the earth be able to have another stable moon in addition to the current one? Or, if the orbit couldn't be stable, why not? How large/small of a moon would it be able to have? And how do we know all this?

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While earth does only have 1 full satellite, the Moon, it does have a number of objects which orbit the sun at about the same interval. Cruithne, for example. These are called quasi-satellites. –  Nate Kerkhofs Apr 23 at 9:14
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Jupiter has over 60 moons, and the dozens of man-made satellites (along with the thousands of pieces of space-trash) orbiting the Earth could be considered tiny moons.

Earth's main Moon would disrupt the orbit of anything smaller at certain radii. The particular disrupted orbits are called resonances, and occur where the orbital period in question divided by the period of the Moon reduces to a small fraction, like 1/2, 2/3, etc. It comes from the Moon giving a predictable tug on objects at those resonance orbits, similar to pushing a child on a swing higher and higher so that you eventually pitch him right out onto the ground. (Do not try this at home.)

The principle is spectacularly illustrated by Saturn's rings. (Some of*) the dozens of dark tracks indicate where one of the moons has a resonance. http://upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Saturn%27s_rings_dark_side_mosaic.jpg/2200px-Saturn%27s_rings_dark_side_mosaic.jpg

[EDIT: To answer more of your questions, there are lots of stable orbits, and the Moon can also act to stabilize, not just destabilize orbits- see http://en.wikipedia.org/wiki/Trojan_(astronomy) . Any stable orbit would have to be above the top of the atmosphere, about 100 km, or else friction with atmospheric gas would drag it down to Earth. Moons could range from specks of dust up to potentially extremely large, even as large as the first Moon, but at great size the options for stability become much more limited. See http://en.wikipedia.org/wiki/Euler%27s_three-body_problem. All of this arises from classical, Newtonian mechanics. To greatly simplify history, Tycho Brahe took a lot of data on planet positions in the sky, Kepler took those observations and developed his purely empirical laws of orbital motion, and Newton took those empirical laws and developed a theoretical framework to account for them.]

Further reading-

http://en.wikipedia.org/wiki/Orbital_resonance

http://en.wikipedia.org/wiki/Rings_of_Saturn

*Some gaps are due to other or unknown factors, but for the most part...

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But what about the question of how much mass would it take to move the earth out of orbit of the sun? –  Cameron Jun 14 '11 at 17:09
    
If you gave the new moon sufficient angular momentum to move along properly with the Earth, then it will have no effect on the Earth's bulk orbit, though it will introduce wobbles. If you want the Earth, Moon, and new moon all to share the same amount of angular momentum as the Earth and Moon have right now, then you would need to add a huge moon indeed to crash the whole enchilada into the Sun. That's a sophomore-level Newtonian celestial mechanics problem. I don't have time to work it out, but by intuition, I'd guess probably much more mass than the Earth itself, which wouldn't be a "moon". –  Andrew Jun 14 '11 at 17:16
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Thanks for the info. However, the ISS is usually between 350 and 400 km above the Earth's surface, yet it still needs periodic boosts every few months because of atmospheric drag. 100 km is nowhere near high enough to escape the Earth's atmosphere completely. –  voithos Jun 14 '11 at 17:26
    
Oh yeah. I was recalling the 100 km figure from the coverage of the X Prize, but I think they just set the bar there to be above most of the atmosphere. –  Andrew Jun 14 '11 at 17:39
    
There is also the roche limit to consider for moons of significant size –  Carson Myers Jun 14 '11 at 19:33
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