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

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 et al. (2015b), we estimate that C/MR^2=0.3434+/-0.0102, 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%).