Theory of the Mercury's spin-orbit motion and analysis of its main librations

Abstract

The 3:2 spin-orbit resonance between the rotational and orbital motions of Mercury (the periods are Pø = 56.646 and Pλ = 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 (∏ = 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/MR2) 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/MR2 value). These determinations prove to be suitable for providing constraints on the internal structure of Mercury.