Celest Mech Dyn Astr DOI 10.1007/s10569-017-9787-3 ORIGINAL ARTICLE Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury Christoph Lhotka 1 Received: 30 May 2017 / Revised: 16 August 2017 / Accepted: 22 August 2017 © Springer Science+Business Media B.V. 2017 Abstract We investigate the stable Cassini state 1 in the p:q spin–orbit resonant problem. Our study includes the effect of the gravitational potential up to degree and order 4 and p:q spin–orbit resonances with p, q ≤ 8 and p ≥ q . We derive new formulae that link the gravitational field coefficients with its secular orbital elements and its rotational parameters. The formulae can be used to predict the orientation of the spin axis and necessary angular momentum at exact resonance. We also develop a simple pendulum model to approximate the dynamics close to resonance and make use of it to predict the libration periods and widths of the oscillatory regime of motions in phase space. Our analytical results are based on averaging theory that we also confirm by means of numerical simulations of the exact dynamical equations. Our results are applied to a possible rotational history of Mercury. Keywords Cassini state · Spin–orbit resonances · Gravity field · Mercury 1 Introduction Stable Cassini states correspond to stable equilibria of the orientation of the spin axis of a rotating body with respect to its orbit normal that is parametrized by an angle ε called obliquity. Two or four equilibrium orientations may exist, but with different stability indices. The phenomenon has first been observed in case of the Moon (Cassini 1693) and stated in terms of the second and third Cassini laws. The first Cassini law, which is independent of the other two, corresponds to a resonance between the rotation frequency and the mean motion that is called a spin–orbit resonance (for a complete description of Cassini states with or without resonances, see Correia 2015). The Moon and the majority of planetary moons in the Solar system are situated in a 1:1 spin–orbit resonance. Other spin–orbit resonances may exist. Mercury is known to be trapped in a 3:2 spin–orbit resonance (Pettengill and Dyce B Christoph Lhotka christoph.lhotka@oeaw.ac.at 1 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse, 6, 8042 Graz, Austria 123