Planet Formation by Gas-Assisted Accretion of Small Solids

We compute the accretion efficiency of small solids, with radii 1 cm $\le$ Rs $\le$ 10 m, on planets embedded in gaseous disks. Planets have masses 3 $\le$ Mp $\le$ 20 Earth masses (Me) and orbit within 10 AU of a solar-mass star. Disk thermodynamics is modeled via three-dimensional radiation-hydrodynamic calculations that typically resolve the planetary envelopes. Both icy and rocky solids are considered, explicitly modeling their thermodynamic evolution. The maximum efficiencies of 1 $\le$ Rs $\le$ 100 cm particles are generally $\lesssim$ 10%, whereas 10 m solids tend to accrete efficiently or be segregated beyond the planet's orbit. A simplified approach is applied to compute the accretion efficiency of small cores, with masses Mp $\le$ 1 Me and without envelopes, for which efficiencies are approximately proportional to Mp^(2/3). The mass flux of solids, estimated from unperturbed drag-induced drift velocities, provides typical accretion rates dMp/dt $\lesssim$ 1e-5 Mearth/yr. In representative disk models with an initial gas-to-dust mass ratio of 70-100 and total mass of 0.05-0.06 Msun, solids' accretion falls below 1e-6 Mearth/yr after 1-1.5 million years (Myr). The derived accretion rates, as functions of time and planet mass, are applied to formation calculations that compute dust opacity self-consistently with the delivery of solids to the envelope. Assuming dust-to-solid coagulation times of approximately 0.3 Myr and disk lifetimes of approximately 3.5 Myr, heavy-element inventories in the range 3-7 Me require that approximately 90-150 Me of solids cross the planet's orbit. The formation calculations encompass a variety of outcomes, from planets a few times the Earth mass, predominantly composed of heavy elements, to giant planets. The peak luminosities during the epoch of solids' accretion range from $\approx$ 1e-7 to $\approx$ 1e-6 times the solar luminosity.