On the equilibrium rotation of Hot Jupiters in eccentric and excited orbits

Hot-Jupiters are a common sub-class of exoplanets, which are enough close to the star to undergo tidal dissipation. The continuous action of tides modify the rotation of the planets until an equilibrium situation is reached. It is often assumed that synchronous motion is the most probable outcome of tidal evolution, since synchronous rotation is observed for the majority of the satellites in the Solar System. This is true for circular orbits, but when the orbits are eccentric, tidal effects are stronger when the planets are closer to the star, and therefore, the rotation rate tends to equalize the orbital speed rate at the pericenter (which is faster than synchronous rotation). An additional complication arises if the eccentricity is not constant and undergoes periodic perturbations from an external companion. Here we obtain an expression for the equilibrium rotation of Hot-Jupiters undergoing tidal dissipation and planetary perturbations. We show that for these planets, the equilibrium rotation rate is faster than for non-perturbed eccentric orbits.