The effects of disc warping on the inclination of planetary orbits

The interaction between a planet located in the inner region of a disc and the warped outer region is studied. We consider the stage of evolution after the planet has cleared-out a gap, so that the planetary orbit evolves only under the gravitational potential from the disc. We develop a secular analysis and compute the evolution of the orbital elements by solving Lagrange's equations valid to second order in the eccentricity. We also perform numerical simulations with the full disc potential. In general, the interaction between the disc and the planet leads to the precession of the orbit. The orbital plane therefore becomes tilted relative to the disc's inner parts, with no change in the eccentricity. When the inclination approaches 90°, there is an instability and the eccentricity increases. In this case, both the inclination and the eccentricity develop large variations, with the orbit becoming retrograde. As the eccentricity reaches high values, we would expect tidal capture on a short orbit of the planet by the star to occur. This instability happens when the disc is severely warped, or if there is a significant amount of mass in a ring inclined by at least 45° relative to the initial orbital plane. The inclination of the orbit does not depend on the semimajor axis nor on the planet's mass. However, for a significant inclination to be generated on a time-scale of at most a few Myr, the planet should be beyond the snow line. The process described here would therefore produce two distinct populations of inclined planets: one with objects beyond the snow line with at most moderate eccentricities, and another with objects on short circularized orbits.