Secular spin-axis dynamics of exoplanets

Abstract

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).Remembering that S p is the amplitude of a given term in the inclination series (5), this means that the parameter Article number, page 4 of 23 M. Saillenfest et al.: Secular spin-axis dynamics of exoplanets

Obliquity Evolution of Circumstellar Planets in Sun-like Stellar Binaries

“…Our model uses a constant spin period because the timescale for significant increases in spin period due to tidal interactions with the host star is longer than our integration timescale. As a result, we make use of the secular time-dependant Hamiltonian that includes the canonical variable χ (= cos ) and spin longitude ψ in the following equations of motion (Neron de Surgy & Laskar 1997;Saillenfest et al 2019):…”

Analytical solution of the Colombo top problem

“…Up to some rearrangement of terms, they are similar to those of Saillenfest et al (2019) in type II or III. For type IV, when a 2 3 + b 2 3 < 1, the Cardano form provided by Saillenfest et al (2019) is formally correct, but it gives real values only as the sums of two complex conjugates (casus irreducibilis of the resolvent cubic). In the present work we have preferred to use expressions based on the purely real trigonometric form whenever the quartic has a positive discriminant.…”

“…2.3 (we already did it for z 234 ). The alternative is to find the stationary points (x, y, z) of the Hamiltonian and use them to evaluate the energy by the substitution into H (x, y, z) = E. The latter path is more convenient and was recently taken by Saillenfest et al (2019).…”

“…For easier reference to Saillenfest et al (2019), we abandon the shifted variables and solve w(z − b) = 0, which is the same equation as the one they used. However, thanks to considering the Cartesian variables on a sphere, we do not have to warn about any loss of sign, because it does not occur neither in evaluating R(U 1 ,U 2 ), nor in obtaining W (Z).…”

“…Further applications include spin states of exoplanets (e.g. Atobe et al, 2004;Atobe and Ida, 2007;Saillenfest et al, 2019), or artificial satellites and space debris (e.g Efimov et al, 2018).…”