The motion of two planets around a Sun-like star under the combined effects of mutual interaction and tidal dissipation is investigated. The secular behaviour of the system is analysed using two different approaches. First, we solve the exact equations of motion through the numerical simulation of the system evolution. In addition to the orbital decay and circularization, we show that the final configuration of the system is affected by the shrinking of the inner orbit. Our second approach consists of the analysis of the stationary solutions of mean equations of motion based on a Hamiltonian formalism.
We consider the case of a hot super-Earth planet with a more massive outer companion. As a real example, the CoRoT-7 system is analysed, solving the exact and mean equations of motion. The star–planet tidal interaction produces orbital decay and circularization of the orbit of CoRoT-7b. In addition, the long-term tidal evolution is such that the eccentricity of CoRoT-7c is also circularized and a pair of final circular orbits is obtained. A curve in the space of eccentricities can be constructed through the computation of stationary solutions of mean equations including dissipation. The application to the CoRoT-7 system shows that the stationary curve agrees with the result of numerical simulations of exact equations.
A similar investigation performed in a super-Earth–Jupiter two-planet system shows that the doubly circular state is accelerated when there is a significant orbital migration of the inner planet, in comparison with previous results where migration is neglected.