I know that the reason why they stay in orbit but why do some move away from the earth(the moon) or come closer and eventually fall? And why does the moon move away from earth?
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Objects can be in a stable orbit around the earth if the only force acting on them would be earth's gravity. Often other factors influence the objects and they either fall or use small rocket thruster to stay in a particular orbit.
What happens for example with satellites is that they experience atmospheric drag which slows them down. The atmosphere hundreds of kilometers above ground contains only a few molecules per cubic meter but this is already enough to have an effect. Other possible causes for falling or leaving the orbit is gravity from other sources that are nearby, magnetic fields, particles from the sun, etc.
Alexander
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@RudyGruse: The moon has no atmosphere but the earth is nearby, so while objects can be in a stable orbit they will be influenced by the earths gravitational field and the other effects I mentioned. – Alexander Nov 15 '11 at 23:09
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@RudyGruse: This is due to a tiny acceleration of the moon from the the gravity of the tides on earth. The explanation is nicely written here cornell.edu – Alexander Nov 15 '11 at 23:30