The growth of a planetary core by pebble accretion stops at the so-called pebble isolation mass, when the core generates a pressure bump that traps drifting pebbles outside its orbit. The value of the pebble isolation mass is crucial in determining the final planet mass. If the isolation mass is very low, gas accretion is protracted and the planet remains at a few Earth masses with a mainly solid composition. For higher values of the pebble isolation mass, the planet might be able to accrete gas from the protoplanetary disc and grow into a gas giant. Previous works have determined a scaling of the pebble isolation mass with cube of the disc aspect ratio. Here, we expand on previous measurements and explore the dependency of the pebble isolation mass on all relevant parameters of the protoplanetary disc. We use 3D hydrodynamical simulations to measure the pebble isolation mass and derive a simple scaling law that captures the dependence on the local disc structure and the turbulent viscosity parameter α. We find that small pebbles, coupled to the gas, with Stokes number τ f < 0.005 can drift through the partial gap at pebble isolation mass. However, as the planetary mass increases, particles must be decreasingly smaller to penetrate the pressure bump. Turbulent diffusion of particles, however, can lead to an increase of the pebble isolation mass by a factor of two, depending on the strength of the background viscosity and on the pebble size. We finally explore the implications of the new scaling law of the pebble isolation mass on the formation of planetary systems by numerically integrating the growth and migration pathways of planets in evolving protoplanetary discs. Compared to models neglecting the dependence of the pebble isolation mass on the α-viscosity, our models including this effect result in higher core masses for giant planets. These higher core masses are more similar to the core masses of the giant planets in the solar system.

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 Jup . We check the influence of three parameters: the inclination i, eccentricity e, and mass M p 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 p 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 −4 deg/year · M disc /(0.01 M ) with the damping of e on a smaller time-scale. While the inclination of low planetary masses (<5 M Jup ) is always damped, large planetary masses with large i can undergo a Kozai-cycle with the disc. These Kozai-cycles are damped through the disc in time. Eccentricity is generally damped, except for very massive planets (M p ∼ 5 M Jup ) where eccentricity can increase for low inclinations. So the dynamics tends to a final state: planets end up in midplane and can then, over time, increase their eccentricity as a result of interactions with the disc. Conclusions. The interactions with the disc lead to damping of i and e after a scattering event of high-mass planets. If i is sufficiently reduced, the eccentricity can be pumped up because of interactions with the disc. If the planet is scattered to high inclination, it can undergo a Kozai-cycle with the disc that makes it hard to predict the exact movement of the planet and its orbital parameters at the dispersal of the disc.

We represent graphically the evolution of the set of resonances of a quasi-integrable dynamical system, the so-called Arnold web, whose structure is crucial for the stability properties of the system. The basis of our representation is the use of an original numerical method, whose definition is directly related to the dynamics of orbits, and the careful choice of a model system. We also show the transition from the Nekhoroshev stability regime to the Chirikov diffusive one, which is a generic nontrivial phenomenon occurring in many physical processes, such as slow chaotic transport in the asteroid belt and beam-beam interaction.