Models of Accreting Gas Giant Protoplanets in Protostellar Disks

We present evolutionary models of gas giant planets forming in protoplanetary disks. We first consider protoplanet models that consist of solid cores surrounded by hydrostatically supported gaseous envelopes that are in contact with the boundaries of their Hill spheres, and accrete gas from the surrounding disk. We neglect planetesimal accretion, and suppose that the luminosity arises from gas accretion alone. This generally occurs on a long time scale which may be comparable to the protostellar disk lifetime. We classify these models as being of type A, and follow their quasi static evolution until the point of rapid gas accretion is reached. We consider a second class of protoplanet models that have not hitherto been considered. These models have a free surface, their energy supply is determined by gravitational contraction, and mass accretion from the protostellar disk that is assumed to pass through a circumplanetary disk. An evolutionary sequence is obtained by specifying the accretion rate that the protostellar disk is able to supply. We refer to these models as being of type B. An important result is that these protoplanet models contract quickly to a radius ∼ 2 × 10cm and are able to accrete gas from the disk at any reasonable rate that may be supplied without any consequent expansion (e.g. a Jupiter mass in ∼ few ×10 years, or more slowly if so constrained by the disk model). We speculate that the early stages of gas giant planet formation proceed along evolutionary paths described by models of type A, but at the onset of rapid gas accretion the protoplanet contracts interior to its Hill sphere, making a transition to an evolutionary path described by models of type B, receiving gas through a circumplanetary disk that forms within its Hill sphere, which is in turn fed by the surrounding protostellar disk. We consider planet models with solid core masses of 5 and 15 M⊕, and consider evolutionary sequences assuming different amounts of dust opacity in the gaseous envelope. The initial protoplanet mass doubling time scale is very approximately inversely proportional to the magnitude of this opacity. Protoplanets with 5 M⊕ cores, and standard dust opacity require ∼ 3× 10 years to grow to a Jupiter mass, longer than reasonable disk life-times. A model with 1 % of standard dust opacity requires ∼ 3× 10 years. Rapid gas accretion in both these cases ensues once the planet mass exceeds ≃ 18 M⊕, with substantial time spent in that mass range. Protoplanets with 15 M⊕ cores grow to a Jupiter mass in ∼ 3 × 10 6 years if standard dust opacity is assumed, and in ∼ 10 years if 1 % of standard dust opacity is adopted. In these cases , the planet spends substantial time with mass between 30 – 40 M⊕ before making the transition to rapid gas accretion. We emphasize that these growth times apply to the gas accretion phase and not to the prior core formation phase. According to the usual theory of protoplanet migration, although there is some dependence on disk parameters, migration in standard model disks is most effective in the mass range where the transition from type A to type B occurs. This is also the transitional regime between type I and type II migration. If a mechanism prevents the type I migration of low mass protoplanets, they could then undergo a rapid inward migration at around the transitional mass regime. Such protoplanets would end up in the inner regions of the disk undergoing type II migration and further accretion potentially becoming sub Jovian close orbiting planets. Noting that more dusty and higher mass cores spend more time at a larger transitional mass that in general favours more rapid migration, such planets are more likely to become close orbiters. We find that the luminosity of the forming protoplanets during the later stages of gas accretion is dominated by the circumplanetary disk and protoplanet-disk boundary layer. For final accretion times for one Jupiter mass in the range 10y, the luminosities are in the range ∼ 10L⊙ and the characteristic temperatures are in the range 1000− 2000K. However, the luminosity may reach ∼ 10L⊙ for shorter time periods at the faster rates of accretion that could be delivered by the protoplanetary disk.