Highly inclined and eccentric massive planets: I. Planet-disc interactions

Context. In the solar system, planets have a small inclination with respect to the equatorial plane of the Sun, but there is evidence that in extrasolar systems the inclination can be very high. This spin-orbit misalignment is unexpected, as planets form in a protoplanetary disc supposedly aligned with the stellar spin. It has been proposed that planet-planet interactions can lead to mutual inclinations during migration in the protoplanetary disc. However, the effect of the gas disc on inclined giant planets is still unknown. Aims. In this paper we investigate planet-disc interactions for planets above 1 M . We check the influence of three parameters: the inclination i, eccentricity e, and mass M of the planet. This analysis also aims at providing a general expression of the eccentricity and inclination damping exerted on the planet by the disc. Methods. We perform three-dimensional numerical simulations of protoplanetary discs with embedded high-mass planets on fixed orbits. We use the explicit/implicit hydrodynamical code NIRVANA in 3D with an isothermal equation of state. Results. We provide damping formulae for i and e as a function of i, e, and M that fit the numerical data. For highly inclined massive planets, the gap opening is reduced, and the damping of i occurs on time-scales of the order of 10 deg/year·M /(0.01 M) with the damping of e on a smaller time-scale. While the inclination of low planetary masses (