Epicycle Arising from Coriolis Force

Non-rotating (inertial) Frame (Left: A) and Rotating Frame (Right: B)

In a frame B that rotates with constant angular velocity relative to an inertial frame A, the accelerations (forces) that appear in frame B are given by:

In our case, frame B rotates with the galaxy: so aA is the gravitational force towards the galactic center, and the centrifugal pseudo-force term just balances this (so in the absence of perturbation, the particle would remain stationary in the rotating frame, and it is this balance that determines ). The third term is the coriolis pseudo-force. Notice it is always perpendicular to the velocity seen in the rotating frame, vB. Hence any perturbed motion is subject to an apparent force that keeps that motion moving in a closed loop - an epicycle.

The animation shows two views of a system in which there is a centripetal force (in this example, the force is harmonic, i.e. aA -rA). The object executes "orbital motion" around the center at a mean rate shown by the rotation of the periphery (in this case, the orbit is the sum of two SHM components out of phase by /2).

The right hand image is the same motion but seen in a frame rotating at the mean orbital frequency (so the periphery now looks stationary). From this frame, the particle seems to execute a retrograde circular epicycle. Notice the acceleration is always perpendicular to the velocity in this rotating frame.

This particular animation (from the wikipedia) matches a well-known illustration of the Coriolis effect: a concave paraboloid bowl gives the harmonic centripetal force, and a dry-ice-puck moves without friction across the surface, in an "orbit". One can spin the bowl at just the right rate so that the puck can remain stationary (moving in a circle in the inertial frame). Perturbing this sends the puck in a non-circular "orbit". A camera rotating with the bowl now sees just the epicycle.