Theoretical and computational models for Saturn's co-orbiting moons, Janus and Epimetheus

Two moons of Saturn, Janus and Epimetheus, are in co-orbital motion, exchanging orbits approximately every four Earth years as the inner moon approaches the outer moon and they gravitationally interact. The orbital radii of these moons differ by only 50 km (less than the moons' mean physical radii), and it is this slight difference in their orbits that enables their periodic exchanges. Numerical n-body simulations can accurately model these exchanges using only Newtonian physics acting upon three objects: Saturn, Janus, and Epimetheus. Here we present analytical approaches and solutions, and corresponding computer simulations, designed to explore the effects of the initial orbital radius difference on otherwise similar co-orbital systems. Comparison with our simulation results illustrates that our analytic expressions provide very accurate predictions for the moon separations at closest approach and simulated post-exchange orbital radii. Our analytic estimates of the exchange period also match the simulated value for Janus and Epimetheus to within a few percent, although systems with smaller differences in their orbital radii are less well-modeled by our simple approach, suggesting that either full simulations or more sophisticated analytic approaches would be required to estimate exchange periods in those cases.