Formation of Giant Planets: Dependences on Core Accretion Rate and Grain Opacity

We have investigated the formation of gaseous envelopes of giant planets with wide ranges of parameters through quasi-static evolutionary simulations. In the nucleated instability model, rapid gas accretion is triggered when the solid core mass exceeds a critical mass. The gas accretion should be regulated essentially by core accretion rate and grain opacity in the outermost envelope. The conventional critical core mass ~5-20 M⊕ (M⊕: Earth's mass), however, is based on some nominal values of these quantities. The discovery of extrasolar giant planets requires investigation of the gas accretion processes under various circumstances. Furthermore, the current planetary accretion theory points out that the cores of Jupiter and Saturn would have been isolated from planetesimals and the core accretion would have almost stopped in their later stage of formation before their masses reached the conventional critical core mass. Through numerical simulations of quasi-static evolution of the gaseous envelope, we have investigated the characteristic growth times of the envelope mass for wide ranges of core accretion rate and grain opacity. We also studied the case where core accretion stops before onset of rapid gas accretion. Our main results are (1) the growth time of the envelope mass τg depends strongly on the core mass, moderately on the grain opacity, and weakly on the past core accretion process, and (2) τg is expressed approximately as τg ~ 108(Mcore /M⊕)-2.5(κ gr/1 cm 2 g-1) yr, where Mcore is the core mass and κgr is the grain opacity. Our results combined with the recent planetary accretion theory suggest surface density of solid materials twice as massive as that of the minimum-mass solar nebula model and the longer lifetime of the nebula than the 108 yr needed to form Jupiter and Saturn; otherwise migration of protoplanets may have to be considered. Our extensive parametric study not only confirms the difficulty in the formation of the giant planets quantitatively and rigorously, it also gives essential information in considering the problem of the formation, which is quite useful in applications.