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Turbulence plays a key role in the transport of pebble-sized particles. It also affects the ability of pebbles to be accreted by protoplanets because it stirs pebbles out of the disk midplane. In addition, turbulence suppresses pebble accretion once the relative velocities become too high for the settling mechanism to be viable. Following Paper I, we aim to quantify these effects by calculating the pebble accretion efficiency ε using three-body simulations. To model the effect of turbulence on the pebbles, we derive a stochastic equation of motion (SEOM) applicable to stratified disk configurations. In the strong coupling limit (ignoring particle inertia) the limiting form of this equation agrees with previous works. We conduct a parameter study and calculate ε in 3D, varying pebble and gas turbulence properties and accounting for the planet inclination. We find that strong turbulence suppresses pebble accretion through turbulent diffusion, agreeing closely with previous works. Another reduction of ε occurs when the turbulent rms motions are high and the settling mechanism fails. In terms of efficiency, the outer disk regions are more affected by turbulence than the inner regions. At the location of the H₂O iceline, planets around low-mass stars achieve much higher efficiencies. Including the results from Paper I, we present a framework to obtain ε under general circumstances.

We developed a pebble-driven core accretion model to study the formation and evolution of planets around stars in the stellar mass range of 0.08 M_⊙–1 M_⊙. By Monte Carlo sampling of the initial conditions, the growth and migration of a large number of individual protoplanetary embryos were simulated in a population synthesis manner. We tested two hypotheses for the birth locations of embryos: at the water ice line or log-uniformly distributed over entire protoplanetary disks. Two types of disks with different turbulent viscous parametersα_tof 10⁻³and 10⁻⁴are also investigated to shed light on the role of outward migration of protoplanets. The forming planets are compared with the observed exoplanets in terms of mass, semimajor axis, metallicity, and water content. We find that gas giant planets are likely to form when the characteristic disk sizes are larger, the disk accretion rates are higher, the disks are more metal rich, and/or their stellar hosts are more massive. Our model shows that first, the characteristic mass of super-Earth is set by the pebble isolation mass. Super-Earth masses increase linearly with the mass of its stellar host, which corresponds to one Earth mass around a late M-dwarf star and 20 Earth masses around a solar-mass star. Second, the low-mass planets, up to 20 M_⊕, can form around stars with a wide range of metallicities, while massive gas giant planets are preferred to grow around metal rich stars. Third, super-Earth planets that are mainly composed of silicates, with relatively low water fractions, can form from protoplanetary embryos at the water ice line in weakly turbulent disks where outward migration is suppressed. However, if the embryos are formed over a wide range of radial distances, the super-Earths would end up having a distinctive, bimodal composition in water mass. Altogether, our model succeeds in quantitatively reproducing several important observed properties of exoplanets and correlations with their stellar hosts.

Context.Coagulation theory predicts that micron-sized dust grains grow into pebbles, which drift inward towards the star when they reach sizes of mm−cm. When they cross the orbit of a planet, a fraction of these drifting pebbles will be accreted. In the pebble accretion mechanism, the combined effects of the planet’s gravitational attraction and gas drag greatly increase the accretion rate.Aims.We calculate the pebble accretion efficiencyε_2D– the probability that a pebble is accreted by the planet – in the 2D limit (pebbles reside in the midplane). In particular, we investigate the dependence ofε_2Don the planet eccentricity and its implications for planet formation models.Methods.We conduct N-body simulations to calculate the pebble accretion efficiency in both the local frame and the global frame. With the global method we investigate the pebble accretion efficiency when the planet is on an eccentric orbit.Results.We find that the local and the global methods generally give consistent results. However, the global method becomes more accurate when the planet is more massive than a few Earth masses or when the aerodynamic size (Stokes number) of the pebble is larger than 1. The efficiency increases with the planet’s eccentricity once the relative velocity between the pebble and the planet is determined by the planet’s eccentric velocity. At high eccentricities, however, the relative velocity becomes too high for pebble accretion. The efficiency then drops significantly and the accretion enters the ballistic regime. We present general expressions forε_2D. Applying the obtained formula to the formation of a secondary planet, in resonance with an already-formed giant planet, we find that the embryo grows quickly due to its higher eccentricity.Conclusions.The maximumε_2Dfor a planet on an eccentric orbit is several times higher than for a planet on a circular orbit, but this increase gives the planet an important headstart and boosts its following mass growth. The recipe forε_2Dthat we have obtained is designed to be implemented into N-body codes to simulate the growth and evolution of planetary systems.

We conduct a pebble-driven planet population synthesis study to investigate the formation of planets around very low-mass stars and brown dwarfs in the (sub)stellar mass range between 0.01 M_⊙ and 0.1 M_⊙. Based on the extrapolation of numerical simulations of planetesimal formation by the streaming instability, we obtain the characteristic mass of the planetesimals and the initial mass of the protoplanet (largest body from the planetesimal populations), in either the early self-gravitating phase or the later non-self-gravitating phase of the protoplanetary disk evolution. We find that the initial protoplanets form with masses that increase with host mass and orbital distance, and decrease with age. Around late M-dwarfs of 0.1 M_⊙, these protoplanets can grow up to Earth-mass planets by pebble accretion. However, around brown dwarfs of 0.01 M_⊙, planets do not grow to the masses that are greater than Mars when the initial protoplanets are born early in self-gravitating disks, and their growth stalls at around 0.01 Earth-mass when they are born late in non-self-gravitating disks. Around these low-mass stars and brown dwarfs we find no channel for gas giant planet formation because the solid cores remain too small. When the initial protoplanets form only at the water-ice line, the final planets typically have ≳15% water mass fraction. Alternatively, when the initial protoplanets form log-uniformly distributed over the entire protoplanetary disk, the final planets are either very water rich (water mass fraction ≳15%) or entirely rocky (water mass fraction ≲5%).