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The dynamical evolution of close-in binary systems formed by a super-Earth and its host star. Case of the Kepler-21 system

The aim of this work is to develop a formalism for the study of the secular evolution of a binary system which includes interaction due to the tides that each body imparts on the other. We also consider the influence of the $J_2$-related secular terms on the orbital evolution and the torque, caused by the triaxiality, on the rotational evolution, both of which are associated only to one of the bodies. We apply these set of equations to the study of the orbital and rotational evolution of a binary system composed of a rocky planet and its host star in order to characterize the dynamical evolution at work, particularly near spin-orbit resonances. We used the equations of motion that give the time evolution of the orbital elements and the spin rates of each body to study the time evolution of the Kepler-21 system as an example of how the formalism that we have developed can be applied. We obtained a set of equations of motion without singularities for vanishing eccentricities and inclinations. This set gives, on one hand, the time evolution of the orbital elements due to the tidal potentials generated by both members of the system as well as the triaxiality of one of them. On the other hand, it gives the time evolution of the stellar spin rate due to the corresponding tidal torque and of the planet's rotation angle due to both the tidal and triaxiality-induced torques. We found that for the parameters and the initial conditions explored here, the tidally and triaxiality-induced modifications of the tidal modes can be more significative than expected and that the time of tidal synchronization strongly depends on the values of the rheological parameters.