Atmospheres of Protoplanetary Cores: Critical Mass for Nucleated Instability

We systematically study quasi-static atmospheres of accreting protoplanetary cores for different opacity behaviors and realistic planetesimal accretion rates in various parts of the protoplanetary nebula. We demonstrate that there are two important classes of atmospheres: (1) those having an outer convective zone that smoothly merges with the surrounding nebular gas, and (2) those possessing an almost isothermal outer radiative region that effectively decouples the atmospheric interior from the nebula. The type of atmosphere accumulating around a given core depends on the core mass, nebular parameters, and accretion luminosity of the core. Cores in the inner parts of the protoplanetary disk (within roughly 0.3 AU from the Sun) have large luminosities resulting in atmospheres of the first type, while cores in the giant planet region (beyond several AU) have small accretion luminosities and always accumulate massive atmospheres of the second type. The critical core mass needed for the nucleated instability to commence is found to vary considerably as a function of distance from the Sun. This mass is 5-20 M_⊕ at 0.1-1 AU, which is too large to permit the formation of "hot Jupiters" by nucleated instability around the cores that have grown in situ. In the region of giant planets the critical core mass depends on the gas opacity and planetesimal accretion rate but is insensitive to the nebular temperature or density provided that the opacity in the outer radiative region does not depend on the gas density (e.g., dust opacity). The critical mass in the region of giant planets can be as high as 20-60 M_⊕ (for an opacity of 0.1 cm² g^-1) if planetesimal accretion is fast enough for protoplanetary cores to form prior to the nebular gas dissipation.