Evolution of inclined planets in three-dimensional radiative discs

Institut für Astronomie & Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
e-mail: bertram.bitsch@uni-tuebingen.de

Received: 22 November 2010
Accepted: 4 April 2011

Abstract

Context. While planets in the solar system only have a low inclination with respect to the ecliptic there is mounting evidence that in extrasolar systems the inclination can be very high, at least for close-in planets. One process to alter the inclination of a planet is through planet-disc interactions. Recent simulations considering radiative transport have shown that the evolution of migration and eccentricity can strongly depend on the thermodynamic state of the disc. So far, this process has only been studied for a few selected planet masses using isothermal discs.

Aims. We extend previous studies to investigate the planet-disc interactions of fixed and moving planets on inclined and eccentric orbits. We also analyse the effect of the disc’s thermodynamic properties on the orbital evolution of embedded planets in detail.

Methods. The protoplanetary disc is modelled as a viscous gas where the internally produced dissipation is transported by radiation. To solve the equations we use an explicit three-dimensional (3D) hydrodynamical code NIRVANA that includes full tensor viscosity, as well as implicit radiation transport in the flux-limited diffusion approximation. To speed up the simulations we apply the FARGO-algorithm in a 3D context.

Results. For locally isothermal discs, we confirm previous results and find inclination damping and inward migration for planetary cores. For low inclinations (i ≲ 2H/r), the damping is exponential, while it follows di/dt ∝ i^-2 for larger i. For radiative discs, the planetary migration is very limited, as long as their inclination exceeds a certain threshold. If the inclination is damped below this threshold, planetary cores with a mass up to  ≈33   M_Earth start to migrate outwards, while larger cores migrate inwards right from the start. The inclination is damped for all analysed planet masses.

Conclusions. In a viscous disc an initial inclination of embedded planets will be damped for all planet masses. This damping occurs on timescales that are shorter than the migration time. If the inclination lies beneath a certain threshold, the outward migration in radiative discs is not handicapped. However, only planets with a mass up to  ≈33   M_Earth are prone to this outward migration. Outward migration is strongest for circular and non-inclined orbits.