If the mass of the satellite is much smaller than the moon, the earth, the sun, we don't need to deal with n-body problem. The gravitational force is a time-varying potential field that is not affected by the satellite itself.

N-body problem comes in when you try to calculate the motion of the moon, the earth, the other planets and the sun. They are big enough to perturb each other's orbit. Still they can be calculated pretty precisely before modern computers come into play. For example Einstein's general relativity is tested by giving a lightly different Mercury's orbit from Newton's theory.