  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.
