Epicycle in a Keplerian Potential

In the case of a Keplerian potential, the epicycle has aspect ratio 2:1 with frequency equal to the orbital frequency. Hence the path traced out is closed in a stationary frame, and approximates an ellipse with focus at the gravitational center. This, and the harmonic potential, are the only to central force fields for which the orbit is closed.

Don't confuse this with Ptolemy's epicycles which were prograde circles with period equal to the synodic period.

Figure from BT-I Fig 3.6