Critical Protoplanetary Core Masses in Protoplanetary Disks and the Formation of Short-Period Giant Planets

Abstract

We study a solid protoplanetary core of 1-10 earth masses migrating through a disk. We suppose the core luminosity is generated as a result of planetesimal accretion and calculate the structure of the gaseous envelope assuming equilibrium. This is a good approximation when the core mass is less than the critical value, M_{crit}, above which rapid gas accretion begins. We model the structure of the protoplanetary nebula as an accretion disk with constant α. We present analytic fits for the steady state relation between disk surface density and mass accretion rate as a function of radius r. We calculate M_{crit} as a function of r, gas accretion rate through the disk, and planetesimal accretion rate onto the core \dot{M}. For a fixed \dot{M}, M_{crit} increases inwards, and it decreases with \dot{M}. We find that \dot{M} onto cores migrating inwards in a time 10^3-10^5 yr at 1 AU is sufficient to prevent the attainment of M_{crit} during the migration process. Only at small radii where planetesimals no longer exist can M_{crit} be attained. At small radii, the runaway gas accretion phase may become longer than the disk lifetime if the core mass is too small. However, massive cores can be built-up through the merger of additional incoming cores on a timescale shorter than for in situ formation. Therefore, feeding zone depletion in the neighborhood of a fixed orbit may be avoided. Accordingly, we suggest that giant planets may begin to form early in the life of the protostellar disk at small radii, on a timescale that may be significantly shorter than for in situ formation. (abridged)