The growth in discoveries of new exoplanets opens doors to the search for habitable worlds beyond our Solar system. Many of these planets have similar sizes to our Earth and can be found in the habitable zone of low-mass stars, like the M-dwarfs and K-dwarfs stars. These stars evolve at a slow pace, providing enough time for the planets to develop the necessary conditions for habitability. However, the habitable zone is closer to the host star, leading to strong interactions between the star and the planet. Indeed, close-in planets undergo strong tidal interactions with the parent star that modify their spins and orbits. In this work, we study the tidal dynamics of Earth-like planets in the habitable zone of these low-mass stars where we focus on two main contributors: the gravitational tidal effects and the atmospheric thermal tidal effects. To obtain the equations of motion, we develop the quadrupole tidal potential solely in a series of Hansen coefficients, which are widely used in celestial mechanics and depend just on the eccentricity. We derive the secular equations of motion and the orbital tidal power in a vectorial formalism, which is frame independent and valid for any rheological and atmospheric model. With the model defined, we apply it to planetary systems whose host stars are from two different class types: the M-dwarfs stars and the K-dwarfs stars. In the M-dwarf stars systems, the planets in the habitable zone undergo strong gravitational tidal interactions, where the final stage for tidal evolution is the synchronization of the rotation and orbital periods, the alignment of the planet spin axis with the normal to the orbit (zero obliquity) and the orbital circularisation (zero eccentricity), which is not ideal for habitability. Yet, considering two different kinds of rheological models, we show that this configuration can be avoided. When tidal dissipation is not too strong, the eccentricity evolves slowly, and the rotation rate can be trapped in spin-orbit resonances that delay the evolution towards the synchronous state. Also, we show that capture in some spin-orbit resonances may also excite the obliquity to high values rather than damp it to zero. In K-dwarfs stars systems, the habitable zone is further away, so the tidal interactions between the star and the planet are weaker than the interactions in the M-dwarfs stars systems. In this case, together with the gravitational tides, we consider that the planets are subject to atmospheric thermal tides. These tides transfer additional energy to the planet that can oppose the dissipation from the gravitational tides. Indeed, even a relatively thin atmosphere can drive the rotation of these planets away from the synchronous state. Assuming the Andrade rheology model for the gravitational tides, we show that there are possible configurations where the final state corresponds to asynchronous rotation even for zero eccentricity. These asynchronous rotations can be expected for planets above a critical semi-major axis. Interestingly, we find that Earth-like planets in the habitable zone of stars with masses ∼ 0.82 solar masses may end up with an equilibrium rotation of 24 h.
Organized by: Catarina Cosme