Gas flow around a planet embedded in a protoplanetary disc

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. The cores of super-Earths should form in the late stage of 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 midplane region may prevent the accretion of solid materials and delay the formation of the super-Earth 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 midplane region of the disc regardless of the assumed dimensionless planetary mass m = RBondi∕H, where RBondi 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 planetary mass. The outflow speed can be expressed as $|u_{\textrm{out}}|\;{=}\;\sqrt{3/2}mc_{\textrm{s}}$ (RBondi ≤ RHill) or $|u_{\textrm{out}}|\;{=}\;\sqrt{3/2}(m/3)^{1/3} c_{\textrm{s}}$ (RBondi ≥ RHill), where cs is the isothermal sound speed and RHill is the Hill radius. The outflow around a planet may reduce the accretion of dust and pebbles onto the planet when $m\;{\gtrsim}\;\sqrt{\textit{St}}$, where S t 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