Tidal evolution of the Galilean satellites - A linearized theory

The Laplace resonance among the Galilean satellites Io, Europa, and Ganymede is traditionally reduced to a pendulum-like dynamical problem by neglecting short-period variations of several orbital elements. However, some of these variations that can now be neglected may once have had longer periods, comparable to the 'pendulum' period, if the system was formerly in deep resonance (pairs of periods even closer to the ratio 2:1 than they are now). In that case, the dynamical system cannot be reduced to fewer than nine dimensions. The nine-dimensional system is linearized here in order to study small variations about equilibrium. When tidal effects are included, the resulting evolution is substantially the same as was indicated by the pendulum approach, except that evolution out of deep resonance is found to be somewhat slower than suggested by extrapolation of the pendulum results. This slower rate helps support the hypothesis that the system may have evolved from deep resonance.