# Relaxation Oscillations in Tidally Evolving Satellites

Authors: Quinn, Dane1; Gladman, Brett2; Nicholson, Phil2; Rand, Richard3

Source: Celestial Mechanics and Dynamical Astronomy, Volume 67, Number 2, February 1997 , pp. 111-130(20)

Publisher: Springer

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Abstract:

We study the rotational evolution under tidal torques of axisymmetric natural satellites in inclined, precessing orbits. In the spin- and orbit-averaged equations of motion, we find that a global limit cycle exists for parameter values near the stability limit of Cassini state $$\mathcal{S}_1$$ . The limit cycle involves an alternation between states of near-synchronous spin at low obliquity, and strongly subsynchronous spin at an obliquity near 90°. This dynamical feature is characterized as a relaxation oscillation, arising as the system slowly traverses two saddle-node bifurcations in a reduced system. This slow time scale is controlled by ε, the nondimensional tidal dissipation rate. Unfortunately, a straightforward expansion of the governing equations for small ε is shown to be insufficient for understanding the underlying structure of the system. Rather, the dynamical equations of motion possess a singular term, multiplied by ε, which vanishes in the unperturbed system. We thus provide a demonstration that a dissipatively perturbed conservative system can behave qualitatively differently from the unperturbed system.

Document Type: Research Article

Affiliations: 1: Department of Mechanical Engineering, the University of Akron, Akron, OH, 44325-3903, U.S.A., Email: quinn@quinn.mecc.uakron.edu 2: Department of Astronomy, Cornell University, Ithaca, NY, 14853, U.S.A., 3: Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14853, U.S.A.,

Publication date: February 1997

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