On the tidal evolution of Hot Jupiters on inclined orbits

Tidal friction is thought to be important in determining the long-term spin-orbit evolution of short-period extrasolar planetary systems. Using a simple model of the orbit-averaged effects of tidal friction, we study the evolution of close-in planets on inclined orbits, due to tides. We analyse the effects of the inclusion of stellar magnetic braking by performing a phase-plane analysis of a simplified system of equations, including the braking torque. The inclusion of magnetic braking is found to be important, and its neglect can result in a very different system history. We then present the results of numerical integrations of the tidal evolution equations, where we find that it is essential to consider coupled evolution of the orbital and rotational elements, including dissipation in both the star and planet, to accurately model the evolution. The main result of our integrations is that for typical Hot Jupiters, tidal friction aligns the stellar spin with the orbit on a similar time as it causes the orbit to decay. This tells us that if a planet is observed to be aligned, then it probably formed coplanar. This reinforces the importance of Rossiter–McLaughlin effect observations in determining the degree of spin-orbit alignment in transiting systems. We apply these results to the only observed system with a spin-orbit misalignment, XO-3, and constrain the efficiency of tidal dissipation (i.e. the modified tidal quality factors Q′) in both the star and the planet in this system. Using a model in which inertial waves are excited by tidal forcing in the outer convective envelope and dissipated by turbulent viscosity, we calculate Q′ for a range of F-star models, and find it to vary considerably within this class of stars. This means that using a single Q′, and assuming that it applies to all stars, is probably incorrect. In addition, we propose an explanation for the survival of two of the planets on the tightest orbits, WASP-12 b and OGLE-TR-56 b, in terms of weak dissipation in the star, as a result of their internal structures and slow rotation periods.

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