Abstract. In this paper we apply a new technique alternative to the numerically computed Lyapunov Characteristic Number (LCN) for studying the dynamical behaviour of planetary systems in the framework of the gravitational N -body problem. The method invented by P. Cincotta and C. Simó is called the Mean Exponential Growth of Nearby Orbits (MEGNO). It provides an efficient way for investigation of the fine structure of the phase space and its regular and chaotic components in any conservative Hamiltonian system. In this work we use it to study the dynamical behaviour of the multidimensional planetary systems. We investigate the recently discovered υ And planetary system, which consists of a star of 1.3 M and three Jupiter-size planets. The two outermost planets have eccentric orbits. This system appears to be one of the best candidates for dynamical studies. The mutual gravitational interaction between the two outermost planets is strong. Moreover the system can survive on a stellar evolutionary time scale as it is claimed by some authors (e.g., Rivera & Lissauer 2000b). As the main methodological result of this work, we confirm important properties of the MEGNO criterion such as its fast convergence, and short motion times (of the order of 10 4 times the longest orbital period) required to distinguish between regular and chaotic behaviors. Using the MEGNO technique we found that the presence of the innermost planet may cause the whole system to become chaotic with the Lyapunov time scale of the order of 10 3 -10 4 yr only. Chaos does not induce in this case visible irregular changes of the orbital elements, and therefore its presence can be overlooked by studying variations of the elements. We confirm explicitly the strong and sensitive dependence of the dynamical behaviour on the companion masses.
Abstract. The 3:2 spin-orbit resonance between the rotational and orbital motions of Mercury (the periods are P φ = 56.646 and P λ = 87.969 days respectively) results from a functional dependance of the tidal friction adding to a non-zero eccentricity and a permanent asymmetry in the equatorial plane of the planet. The upcoming space missions, MESSENGER and BepiColombo with onboard instrumentation capable of measuring the rotational parameters stimulate the objective to reach an accurate theory of the rotational motion of Mercury. For obtaining the real motion of Mercury, we have used our BJV model of solar system integration including the coupled spin-orbit motion of the Moon. This model, expanded in a relativistic framework, had been previously built in accordance with the requirements of the Lunar Laser Ranging observational accuracy. We have extended the BJV model by generalizing the spin-orbit couplings to the terrestrial planets (Mercury, Venus, Earth, and Mars). The updated model is called SONYR (acronym of Spin-Orbit N-BodY Relativistic model). As a consequence, the SONYR model gives an accurate simultaneous integration of the spin-orbit motion of Mercury. It permits one to analyze the different families of rotational librations and identify their causes such as planetary interactions or the parameters involved in the dynamical figure of the planet. The spin-orbit motion of Mercury is characterized by two proper frequencies (namely Φ = 15.847 yrs and Ψ = 1066 yrs) and its 3:2 resonance presents a second synchronism which can be understood as a spin-orbit secular resonance (Π = 278 898 yrs). A new determination of the mean obliquity is proposed in the paper. By using the SONYR model, we find a mean obliquity of 1.6 arcmin. This value is consistent with the Cassini state of Mercury. Besides, we identify in the Hermean librations the impact of the uncertainty of the greatest principal moment of inertia (C/MR 2 ) on the obliquity and on the libration in longitude (2.3 milliarcsec and 0.45 arcsec respectively for an increase of 1% on the C/MR 2 value). These determinations prove to be suitable for providing constraints on the internal structure of Mercury.
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