On the evolution of eccentric and inclined protoplanets embedded in protoplanetary disks

Young planets embedded in their protoplanetary disk interact gravitationally with it leading to energy and angular momentum exchange. This interaction determines the evolution of the planet through changes to the orbital parameters. We investigate changes in the orbital elements of a 20 Earth--mass planet due to the torques from the disk. We focus on the non-linear evolution of initially non-vanishing eccentricity $e$ and/or inclination $i$. We treat the disk as a two- or three-dimensional viscous fluid and perform hydrodynamical simulations with an embedded planet. We find rapid exponential decay of the planet orbital eccentricity and inclination for small initial values of $e$ and $i$, in agreement with linear theory. For larger values of $e > 0.1$ the decay time increases and the decay rate scales as $\dot{e} \propto e^{-2}$, consistent with existing theoretical models. For large inclinations ($i$ > 6 deg) the inclination decay rate shows an identical scaling $di/dt \propto i^{-2}$. We find an interesting dependence of the migration on the eccentricity. In a disk with aspect ratio $H/r=0.05$ the migration rate is enhanced for small non-zero eccentricities ($e < 0.1$), while for larger values we see a significant reduction by a factor of $\sim 4$. We find no indication for a reversal of the migration for large $e$, although the torque experienced by the planet becomes positive when $e \simeq 0.3$. This inward migration is caused by the persisting energy loss of the planet. For non gap forming planets, eccentricity and inclination damping occurs on a time scale that is very much shorter than the migration time scale. The results of non linear hydrodynamic simulations are in very good agreement with linear theory for small $e$ and $i$.