Orbits Viewed in a Rotating Frame

Normally, the orbit and its epicycle do not have commensurate frequencies, and so the orbit doesn't close when viewed in a stationary frame. However, if it is viewed from a frame which is rotating at an angular frequency of g - n / m where n,m are integers, the orbit will appear closed.

The figure shows this situation for three pairs of n,m. For n = 0, the frame rotates with the orbit and so we just see the epicycle motion. If the frame rotates more slowly by then we see the orbit go once around in one epicyclic period. The orbit closes, but is elongated away from being centered. If the frame rotates more slowly by / 2 then the epicycle has completed one cycle when the orbit appears to have completed half a cycle, so it is at its extreme radius on the opposite side. After another epicycle period, we return to our starting point, and the orbit closes --- it is an ellipse centered on the galaxy center.

More complicated combinations of n and m are shown.