Without any external forces a moving object will continue to move in a straight line. The gravitational force between the two objects will provide the centripetal force to keep the objects moving around one another.
1. satellite in orbit around the earth (motion of earth is negligible)
2. moon in orbit around the earth (center of motion several thousand miles
from center of earth)
3. earth in orbit around sun (center of rotation close to center of sun)
4. binary stars (if masses of stars are equal center of rotation is in middle)
