Obliquity of Mercury: Influence of the precession of the pericenter and of tides

Abstract

Mercury is expected to deviate from the classical Cassini state since this state is defined for a uniformly precessing rigid planet. We develop an extended Cassini state model that includes the variations (or nutations) in obliquity and deviation induced by the slow precession of the pericenter. The model also describes the constant shift over time in mean obliquity and deviation associated with the short-periodic tidal deformations of Mercury, characterized by the tidal love number k2 and by the ratio k2/Q of the tidal Love number over the tidal quality factor, respectively. This model is then used to interpret Mercury’s orientation, including the deviation from the classical Cassini state, in terms of parameters of Mercury’s interior.We determine and solve analytically the angular momentum equation, highlighting the respective roles of the pericenter precession and tidal deformations on the spin precession behavior. We also show explicitly that Peale’s equation is sometimes wrongly cited in the literature, resulting in wrong estimates of the polar moment of inertia, and review the importance of many effects that change the determination of the polar moment of inertia from obliquity measurements.From the observed orientation of Stark, Oberst, Preusker, Gwinner, Peale, Margot, Phillips, Zuber and Solomon (2015b), we estimate that ▪ which is ∼ 0.9% smaller than the estimate by Stark et al. (2015b) themselves. That difference is due to our refinements of the Cassini state model (0.1%) and to their wrong use of Peale’s equation (0.8%). The difference is smaller than the actual precision (3−4%) on the polar moment of inertia but may be of the order of precision that can be reached with BepiColombo mission (≤ 0.3%).The parameter k2 cannot be estimated from the spin axis orientation, because of its correlation with the polar moment of inertia, which is much more important in determining the obliquity in our improved model. However, it is necessary to include its effect in the model to avoid a systematic error of 0.3% on the determination of the polar moment of inertia. The parameter k2/Q can be estimated from the spin orientation, since its effect can be easily separated from the effect of the polar moment of inertia on the deviation, as this latter parameter is already well determined by its contribution to the obliquity. Given the actual precision on the spin axis orientation, we place an upper limit of about 0.02 on the ratio k2/Q and of about 350 on Q (assuming k2=0.5) at the 1σ level. In the future, the relative precision on the determination of k2/Q from the spin axis orientation could be as good as 30% with BepiColombo, so that the non-elastic parameter of Mercury could be estimated for the first time.