Spin evolution of Venus-like planets subjected to gravitational and thermal tides
Authors: Revol A. , Bolmont E. , Tobie G. , Dumoulin C. , Musseau Y. , Mathis S. , Strugarek A. , Brun A.S. 
Published: April 2023

Context. The arrival of powerful instruments will provide valuable data for the characterization of rocky exoplanets. Rocky planets are mostly found in close-in orbits. They are therefore usually close to the circular-coplanar orbital state and are thus considered to be in a tidally locked synchronous spin state. For planets with larger orbits, however, exoplanets should still have nonzero eccentricities and/or obliquities, and realistic models of tides for rocky planets can allow for higher spin states than the synchronization state in the presence of eccentricities or obliquities.
Aims. This work explores the secular evolution of a star–planet system under tidal interactions, both gravitational and thermal, induced by the quadrupolar component of the gravitational potential and the irradiation of the planetary surface, respectively. We show the possible spin–orbit evolution and resonances for eccentric orbits and explore the possibility of spin-orbit resonances raised by the obliquity of the planet. Then, we focus on the additional effect of a thick atmosphere on the possible resulting spin equilibrium states and explore the effect of the evolution of the stellar luminosity.
Methods. We implemented the general secular evolution equations of tidal interactions in the secular code called ESPEM. In particular, we focus here on the tides raised by a star on a rocky planet and consider the effect of the presence of an atmosphere, neglecting the contribution of the stellar tide. The solid part of the tides was modeled with an anelastic rheology (Andrade model), while the atmospheric tides were modeled with an analytical formulation that was fit using a global climate model simulation. We focused on a Sun-Venus-like system in terms of stellar parameters, orbital configuration and planet size and mass. The Sun-Venus system is a good laboratory for studying and comparing the possible effect of atmospheric tides, and thus to explore the possible spin state of potential Venus-like exoplanets.
Results. The formalism of Kaula associated with an Andrade rheology allows spin orbit resonances on pure rocky worlds. Similarly to the high-order spin–orbit resonances induced by eccentricity, the spin obliquity allows the excitation of high-frequency Fourier modes that allow some spin-orbit resonances to be stable. If the planet has a dense atmosphere, like that of Venus, another mechanism, the thermal tides, can counterbalance the effect of the gravitational tides. We found that thermal tides change the evolution of the spin of the planet, including the capture in spin–orbit resonances. If the spin inclination is high enough, thermal tides can drive the spin toward an anti-synchronization state, that is, a the 1:1 spin–orbit resonance with an obliquity of 180 degrees.
Conclusions. Through our improvement of the gravitational and thermal tidal models, we can determine the dynamical state of exo-planets better, especially if they hold a thick atmosphere. In particular, the contribution of the atmospheric tides allows us to reproduce the spin state of Venus at a constant stellar luminosity. Our simulations have shown that the secular evolution of the spin and obliquity can lead to a retrograde spin of the Venus-like planet if the system starts from a high spin obliquity, in agreement with previous studies. The perturbing effect of a third body is still needed to determine the current state of Venus starting from a low initial obliquity. When the luminosity evolution of the Sun is taken into account, the picture changes. We find that the planet never reaches equilibrium: the timescale of the rotation evolution is longer than the luminosity variation timescale, which suggests that Venus may never reach a spin equilibrium state, but may still evolve.