Context. The ubiquity of short-period super-Earths remains a mystery in planet formation, as these planets are expected to become gas giants via runaway gas accretion within the lifetime of a protoplanetary disc. Super-Earths' cores should form in the late stage of the disc evolution to avoid runaway gas accretion. Aims. The three-dimensional structure of the gas flow around a planet is thought to influence the accretion of both gas and solid materials. In particular, the outflow in the mid-plane region may prevent the accretion of the solid materials and delay the formation of super-Earths' cores. However, it is not yet understood how the nature of the flow field and outflow speed change as a function of the planetary mass. In this study, we investigate the dependence of gas flow around a planet embedded in a protoplanetary disc on the planetary mass. Methods. Assuming an isothermal, inviscid gas disc, we perform three-dimensional hydrodynamical simulations on the spherical polar grid, which has a planet located at its centre. Results. We find that gas enters the Bondi or Hill sphere at high latitudes and exits through the mid-plane region of the disc regardless of the assumed dimensionless planetary mass m = R Bondi /H, where R Bondi and H are the Bondi radius of the planet and disc scale height, respectively. The altitude from where gas predominantly enters the envelope varies with the planetary mass. The outflow speed can be expressed as |uwhere c s is the isothermal sound speed and R Hill is the Hill radius. The outflow around a planet may reduce the accretion of dust and pebbles onto the planet when m √ St, where St is the Stokes number. Conclusions. Our results suggest that the flow around proto-cores of super-Earths may delay their growth and, consequently, help them to avoid runaway gas accretion within the lifetime of the gas disc.

Context. The pebble accretion model has the potential to explain the formation of various types of planets. The main difference between this and the planetesimal accretion model is that pebbles not only experience the gravitational interaction with the growing planet but also a gas drag force from the surrounding protoplanetary disk gas. Aims. A growing planet embedded in a disk induces three-dimensional (3D) gas flow, which may influence pebble accretion. However, so far the conventional pebble accretion model has only been discussed in the unperturbed (sub-)Keplerian shear flow. In this study, we investigate the influence of 3D planet-induced gas flow on pebble accretion. Methods. Assuming a nonisothermal, inviscid gas disk, we perform 3D hydrodynamical simulations on the spherical polar grid, which has a planet located at its center. We then numerically integrate the equation of motion of pebbles in 3D using hydrodynamical simulation data. Results. We find that the trajectories of pebbles in the planet-induced gas flow differ significantly from those in the unperturbed shear flow for a wide range of investigated pebble sizes (St = 10−3–100, where St is the Stokes number). The horseshoe flow and outflow of the gas alter the motion of the pebbles, which leads to a reduction of the width of the accretion window, wacc, and the accretion cross section, Aacc. On the other hand, the changes in trajectories also cause an increase in the relative velocity of pebbles to the planet, which offsets the reduction of wacc and Aacc. As a consequence, in the Stokes regime, the accretion probability of pebbles, Pacc, in the planet-induced gas flow is comparable to that in the unperturbed shear flow except when the Stokes number is small, St ~ 10−3, in 2D accretion, or when the thermal mass of the planet is small, m = 0.03, in 3D accretion. In contrast, in the Epstein regime, Pacc in the planet-induced gas flow becomes smaller than that in the shear flow in the Stokes regime in both 2D and 3D accretion, regardless of assumed St and m. Conclusions. Our results combined with the spacial variety of turbulence strength and pebble size in a disk, suggest that the 3D planet-induced gas flow may be helpful to explain the distribution of exoplanets and the architecture of the Solar System.

Context. Pebble accretion is among the major theories of planet formation. Aerodynamically small particles, called pebbles, are highly affected by the gas flow. A growing planet embedded in a protoplanetary disk induces three-dimensional (3D) gas flow. In our previous work, Paper I, we focused on the shear regime of pebble accretion and investigated the influence of planet-induced gas flow on pebble accretion. In Paper I, we found that pebble accretion is inefficient in the planet-induced gas flow compared to that of the unperturbed flow, particularly when St ≲ 10−3, where St is the Stokes number. Aims. Following on the findings of Paper I, we investigate the influence of planet-induced gas flow on pebble accretion. We did not consider the headwind of the gas in Paper I. Here, we extend our study to the headwind regime of pebble accretion. Methods. Assuming a nonisothermal, inviscid sub-Keplerian gas disk, we performed 3D hydrodynamical simulations on the spherical polar grid hosting a planet with the dimensionless mass, m = RBondi∕H, located at its center, where RBondi and H are the Bondi radius and the disk scale height, respectively. We then numerically integrated the equation of motion for pebbles in 3D using hydrodynamical simulation data. Results. We first divided the planet-induced gas flow into two regimes: flow-shear and flow-headwind. In the flow-shear regime, where the planet-induced gas flow has a vertically rotational symmetric structure, we find that the outcome is identical to what we obtained in Paper I. In the flow-headwind regime, the strong headwind of the gas breaks the symmetric structure of the planet-induced gas flow. In the flow-headwind regime, we find that the trajectories of pebbles with St ≲ 10−3 in the planet-induced gas flow differ significantly from those of the unperturbed flow. The recycling flow, where gas from the disk enters the gravitational sphere at low latitudes and exits at high latitudes, gathers pebbles around the planet. We derive the flow transition mass analytically, mt, flow, which discriminates between the flow-headwind and flow-shear regimes. From the relation between m, mt, flow and mt, peb, where mt, peb is the transition mass of the accretion regime of pebbles, we classify the results obtained in both Paper I and this study into four groups. In particular, only when the Stokes gas drag law is adopted and m < min(mt, peb, mt, flow), where the accretion and flow regime are both in the headwind regime, the accretion probability of pebbles with St ≲ 10−3 is enhanced in the planet-induced gas flow compared to that of the unperturbed flow. Conclusions. Combining our results with the spacial variety of turbulence strength and pebble size in a disk, we conclude that the planet-induced gas flow still allows for pebble accretion in the early stage of planet formation. The suppression of pebble accretion due to the planet-induced gas flow occurs only in the late stage of planet formation, more specifically, in the inner region of the disk. This may be helpful for explaining the distribution of exoplanets and the architecture of the Solar System, both of which have small inner and large outer planets.