On the chaotic rotation of planetary satellites: The Lyapunov spectra and the maximum Lyapunov exponents

The possibility of dynamic chaos in the spin motion of minor natural planetary satellites is studied numerically and analytically. A satellite is modelled as a tri-axial rigid body in a fixed elliptic orbit. The Lyapunov characteristic exponents (LCEs) are used as indicators of the degree of chaos of the motion. For a set of real satellites (i.e. satellites with actual values of inertial and orbital parameters), the full Lyapunov spectra of the chaotic rotation are computed by the HQR-method of von Bremen et al. (1997). A more traditional “shadow trajectory” method for the computation of maximum LCEs is also used. Numerical LCEs obtained in the spatial and planar cases of chaotic rotation are compared to analytical estimates obtained by the separatrix map theory in the model of nonlinear resonance (here: synchronous spin-orbit resonance) as a perturbed nonlinear pendulum (Shevchenko 2000a, 2002). Further evidence is given that the agreement of the numerical data with the separatrix map theory in the planar case is very good. It is shown that the theory developed for the planar case is most probably still applicable in the case of spatial rotation, if the dynamical asymmetry of the satellite is sufficiently small or/and the orbital eccentricity is relatively large (but, for the dynamical model to be valid, not too large). The theoretical implications are discussed, and simple statistical dependences of the components of the LCE spectrum on the parameters of the problem are derived.