Investigation of the Mimas-Tethys system wassuccessfully conducted by means of a Hamiltonian chaos con-trol method (Khan & Shahzad 2008). Regard-ing rotation of an oblate moon, the full set of modified Eulerequations was investigated numerically with various methods inthe context of chaos control (Tsui & Jones 2000) for a satellitewith thrusters. ![]() In general, a chaos control scheme already demon-strated its usefulness in astrodynamical applications, for ex-ample in maneuvering the ISEE-3 / ICE-3 satellite to reach theGiacobini-Zinner comet in 1985 (Shinbrot et al. Finally, regarding the influence of a secondarybody on the rotation of an oblate moon, Tarnopolski (2017)showed, using the correlation dimension and bifurcation dia-grams, that the introduction of an additional satellite can changethe rotation from regular to chaotic.Well-defined methods to reduce chaos in physical systemshave been known for a long time now (e.g. The Hamiltonian formalism has also beenemployed in the research of secondary resonances (Gkolias et al.2016), and has taken non-conservative forces into account (Gko-lias et al. An interesting pos-sibility that Enceladus might be locked in a 1:3 librational-orbitalsecondary resonance was investigated using the model of pla-nar rotation within the Hamiltonian formalism (Wisdom 2004).The dynamical stability was examined for a number of knownsatellites by Melnikov & Shevchenko (2010) in particular, thesynchronous spin-orbit resonance in case of Hyperion was con-firmed to be unstable. The Lyapunov spectra were exhaustively examined for severalsatellites (Shevchenko 2002 Shevchenko & Kouprianov 2002 Kouprianov & Shevchenko 2003, 2005), and the Lyapunov timesspanned 1.5–7 orbital periods for Hyperion. The stability of spin-orbit res-onances, with application to the solar system satellites, was in-ferred based on a series expansion of the equation of planar ro-tation (Celletti & Chierchia 1998, 2000 see also Celletti 1990). (1996) examined a number of theoreticalmodels, including the gravitational, magnetic, and tidal momentsand analyzed the rotation in the gravitational field of two centers.The structure of the respective phase spaces was investigatedwith Poincaré surfaces of section. It was also shown that the Voyager 2 images of Hyper-ion indicate that the motion was predictable at the time of thepassage (Thomas et al. (1995) performed numerical simulationswith the full set of Euler equations to model the long-term dy-namical evolution, and found that the chaotic tumbling of Hype-rion leads to transitions between temporarily regular and chaoticrotation with a period of hundreds of days up to thousands ofyears. These findings agree withthe recent results of Tarnopolski (2015), who showed that to ex-tract a maximal Lyapunov exponent from ground-based photo-metric observations of Hyperion’s light curve, at least one year ofwell-sampled data is required, but a three-year data set would bedesirable. (1994) employed the method ofclose returns to a sparse data set of dynamical states of Hype-rion simulated with Euler equations, and found that a time seriesspanning about 2.6 years of data is su ffi cient to infer the tempo-rary rotational state (chaotic / regular). Since then, the analyses of chaotic rotationof an oblate celestial body has become very common, regardingHyperion, per se, as well as other solar system satellites.For instance, Boyd et al. (1984) predicted that Hyper-ion rotates chaotically, analyzed the phase space of a model ofplanar rotation, showed that it has an unstable attitude, and com-puted the Lyapunov time to be about 10 times the orbital period(i.e. 1982) obtained high-quality imagesof Hyperion (Bond 1848 Lassel 1848), the highly asphericalmoon of Saturn, it became clear that it remained in an exotic ro-tational state (Klavetter 1989a,b Black et al. celestial mechanics – chaos – methods: numericalĪfter Voyager 2 (Smith et al. ![]() This allows not only forsignificantly diminished di ff usion of the trajectory in the phase space, but turns the purely chaotic motion into strictly periodic motion. The Hamiltonian formalism wasutilized to employ a control method for suppressing chaos.Īn additive control term, which is an order of magnitude smaller than the potential, is constructed. This paper investigates the chaotic rotation of an oblate satellite in the context of chaos control.Ī model of planar oscillations, described with the Beletskii equation, was investigated. Tarnopolski Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Polande-mail: Rotation of an oblate satellite: Chaos control
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