Critical protoplanetary core masses in protoplanetary disks and the formation of short-period giant planets

We study a solid protoplanetary core undergoing radial migration in a protoplanetary disk. We consider cores in the mass range similar to 1-10 M+ embedded in a gaseous protoplanetary disk at different radial locations. We suppose that the core luminosity is generated as a result of planetesimal accretion and calculate the structure of the gaseous envelope assuming hydrostatic and thermal equilibrium. This is a good approximation during the early growth of the core, while its mass is less than the critical value, M-crit above which such static solutions can no longer be obtained and rapid gas accretion begins. The critical value corresponds to the crossover mass above which rapid gas accretion begins in time-dependent calculations. We model the structure and evolution of the protoplanetary nebula as an accretion disk with constant alpha. We present analytic fits for the steady state relation between the disk surface density and the mass accretion rate as a function of radius. We calculate M-crit as a function of radial location, gas accretion rate through the disk, and planetesimal accretion rate onto the core. For a fixed planetesimal accretion rate, M-crit is found to increase inward. On the other hand, it decreases with the planetesimal accretion rate and hence with the core luminosity. We consider the planetesimal accretion rate onto cores migrating inward in a characteristic time of similar to 10(3)-10(5) yr at 1 AU, as indicated by recent theoretical calculations. We find that the accretion rate is expected to be sufficient to prevent the attainment of M-crit during the migration process if the core starts off significantly below it. Only at those small radii at which local conditions are such that dust, and accordingly 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 mass of the core is too small. However, within the context of our disk models, and if it is supposed that some process halts the migration, massive cores can be built up through the merger of additional incoming cores on a timescale shorter than for in situ formation. A rapid gas accretion phase may thus begin without an earlier prolonged phase in which planetesimal accretion occurs at a reduced rate because of feeding zone depletion in the neighborhood of a fixed orbit. Accordingly, we suggest that giant planets may begin to form through the above processes early in the life of the protostellar disk at small radii, on a timescale that may be significantly shorter than that derived for in situ formation.