RELAXATION OSCILLATIONS IN TIDALLY EVOLVING SATELLITES

We study the rotational evolution under tidal torques of axisymmetric natural satellites in inclined, precessing orbits, In the spin-and orb it-averaged equations of motion, we find that a global limit cycle exi sts for parameter values near the stability limit of Cassini state S-1 . The limit cycle involves an alternation between states of near-synch ronous spin at low obliquity, and strongly subsynchronous spin at an o bliquity near 90 degrees. This dynamical feature is characterized as a relaxation oscillation, arising as the system slowly traverses two sa ddle-node bifurcations in a reduced system. This slow time scale is co ntrolled by epsilon, the nondimensional tidal dissipation rate. Unfort unately, a straightforward expansion of the governing equations for sm all epsilon is shown to be insufficient for understanding the underlyi ng structure of the system. Rather, the dynamical equations of motion possess a singular term, multiplied by epsilon, which vanishes in the unperturbed system. We thus provide a demonstration that a dissipative ly perturbed conservative system can behave qualitatively differently from the unperturbed system.