We study the interaction of a protoplanetary disc with a planet on a highly inclined orbit in the linear regime. The evolution of the planet is dominated by dynamical friction for planet masses above several Earth masses. Smaller planets are dominated by aerodynamic drag, especially for very high inclinations and retrograde orbits.
The time-scales associated with migration and inclination damping are calculated. For certain values of the inclination, the inclination damping time-scale is longer than the migration time-scale and the disc lifetime. This result shows that highly inclined planets cannot (re-)align with the protoplanetary disc.
We discuss the dependence of numerical simulations on the gravitational softening parameter. We find only a logarithmic dependence, making global three-dimensional simulations of this process computationally feasible.
A large fraction of hot Jupiters is on highly inclined orbits with respect to the rotation axis of the star. On the other hand, small mass planetary systems discovered by the Kepler mission have low mutual inclinations. This shows that there are two distinct formation mechanisms at work. The process that creates inclined hot Jupiters does not operate on small mass planets because the damping time-scales are so long that these systems would still be inclined today.