Atmospheres of protoplanetary cores: critical mass for nucleated instability

We study quasi-static atmospheres of accreting protoplanetary cores for different opacity behaviors and realistic planetesimal accretion rates in various parts of protoplanetary nebula. Atmospheres segregate into those having outer convective zone which smoothly merges with the nebular gas, and those having almost isothermal outer radiative region decoupling atmospheric interior from the nebula. Specific type of atmosphere depends only on the relations between the Bondi radius of the core, photon mean free path in the nebular gas, and the luminosity radius (roughly the size of the sphere which can radiate luminosity of the core at effective temperature equal to the nebular temperature). Cores in the inner parts of protoplanetary disk (within roughly 0.3 AU from the Sun) have large luminosity radii resulting in the atmospheres of the first type, while cores in the giant planet region (beyond several AU) have small luminosity radii and always accumulate massive atmospheres of the second type. Critical core mass for nucleated instability is found to vary as a function of distance from the Sun. It is 5-20 M_Earth at 0.1-1 AU which is too large to permit the formation of ``hot Jupiters'' by nucleated instability near the cores that have grown in situ. In the region of giant planets critical mass is 20-60 M_Earth (for opacity 0.1 cm^2/g) if planetesimal accretion was fast enough for protoplanetary cores to form prior to the nebular gas dissipation. This might indicate that giant planets in the Solar System have gained their atmospheres by nucleated instability only after their cores have accumulated most of the mass in solids during the epoch of oligarchic growth, subsequent to which planetesimal accretion slowed down and cores became supercritical.