The radial dependence of pebble accretion rates: A source of diversity in planetary systems I. Analytical formulation

Context. The classical planetesimal accretion scenario for the formation of planets has recently evolved with the idea that pebbles, centimeter-to meter-sized icy grains migrating in protoplanetary disks, can control planetesimal and / or planetary growth. Aims. We investigate how pebble accretion depends on disk properties and affects the formation of planetary systems. Methods. We construct analytical models of pebble accretion onto planetary embryos that consistently account for the mass and orbital evolution of the pebble flow and reflect disk structure. Results. We derive simple formulas for pebble accretion rates in the so-called settling regime for planetary embryos that are more than 100 km in size. For relatively smaller embryos or in outer disk regions, the accretion mode is three-dimensional (3D), meaning that the thickness of the pebble flow must be taken into account, and resulting in an accretion rate that is independent of the embryo mass. For larger embryos or in inner regions, the accretion is in a two-dimensional (2D) mode, i.e., the pebble disk may be considered infinitely thin. We show that the radial dependence of the pebble accretion rate is different (even the sign of the power-law exponent changes) for different disk conditions such as the disk heating source (viscous heating or stellar irradiation), drag law (Stokes or Epstein, and weak or strong coupling), and in the 2D or 3D accretion modes. We also discuss the effect of the sublimation and destruction of icy pebbles inside the snow line. Conclusions. Pebble accretion easily produces a large diversity of planetary systems. In other words, to infer the results of planet formation through pebble accretion correctly, detailed prescriptions of disk evolution and pebble growth, sublimation, destruction and migration are required.