Question

Suppose two bodies are orbiting about their common center of mass, and the distance between the bodies is a fixed distance d. Surely one possible configuration is the following: each body moves in a circle, the circles are concentric (centered at the center of mass of the system), and they have the same angular speed. Does it hold that this is the only possible configuration?

Note: I believe the answer is yes but I am clueless about how to prove it.

Answer 1

Yes, you are correct. Here is an easy way to see this:

Suppose you have to bodies (A and B), which do not necessarily have the same mass. Now suppose these bodies exist far from any other disturbances and that we are sitting in a reference frame stationary with respect to their center of mass. Then in our reference frame, the center of mass cannot move, and our reference frame is inertial because there are no external forces (we are far from disturbances). Thus the center of mass is fixed.

Now draw an imaginary line segment between the two masses. This line should always have length d, since we specified that the bodies are always at a fixed distance d from each other. Furthermore, the center of mass of the system is located on this line, and in particular, its location is given by the ratio of the two masses (i.e. if they are equal then it will be located at the midpoint of this line). Since the center of mass does not move in our reference frame and the bodies are always at a fixed d from each other, we can thus imagine the bodies as actually being connected to a rod that is fixed at the center of mass and can pivot around it. In this case it is obvious that the bodies must move in concentric circles and that this is the only possible motion. Any other motion will result in one of our requirements breaking down (either the center of mass will move, in which case there must be some external force, or distance between the bodies will change)