Secular spin-axis dynamics of exoplanets

Context: Seasonal variations and climate stability of a planet are very sensitive to the planet obliquity and its evolution. This is of particular interest for the emergence and sustainability of land-based life, but orbital and rotational parameters of exoplanets are still poorly constrained. Numerical explorations usually realised in this situation are thus in heavy contrast with the uncertain nature of the available data. Aims: We aim to provide an analytical formulation of the long-term spin-axis dynamics of exoplanets, linking it directly to physical and dynamical parameters, but still giving precise quantitative results if the parameters are well known. Together with bounds for the poorly constrained parameters of exoplanets, this analysis is designed to allow a quick and straightforward exploration of the spin-axis dynamics. Methods: The long-term orbital solution is decomposed in quasi-periodic series and the spin-axis Hamiltonian is expanded in powers of eccentricity and inclination. Chaotic zones are measured by the resonance overlap criterion. Bounds for the poorly known parameters of exoplanets are obtained from physical grounds (rotational breakup) and dynamical considerations (equipartition of AMD). Results: This method gives accurate results when the orbital evolution is well known. The chaotic zones for planets of the Solar System can be retrieved in details from simple analytical formulas. For less constrained planetary systems, the maximal extent of the chaotic regions can be computed, requiring only the mass, the semi-major axis and the eccentricity of the planets present in the system. Additionally, some estimated bounds of the precession constant allow to classify which observed exoplanets are necessarily out of major spin-orbit secular resonances (unless the precession rate is affected by the presence of massive satellites).