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In the core accretion model for the formation of gas giant planets, runawaygas accretion onto a core is the primary requisite, triggered when the coremass reaches a critical value. The recently revealed wide diversity of theextrasolar giant planets suggests the necessity to further the understanding ofthe conditions resulting in the critical core mass that initiates runawayaccretion. We study the internal structure of protoplanets under hydrostaticand thermal equilibria represented in terms of a polytropic equation of stateto investigate what factors determine and affect the critical core mass. Wefind that the protoplanets, embedded in protoplanetary disks, have the sameconfiguration as red giants, characterized by the envelope of thecentrally-condensed type solution. Applying the theory of stellar structurewith homology invariants, we demonstrate that there are three types of criteriafor the critical core mass depending on the stiffness of polytrope and thenature of outer boundary condition. For the stiff polytropes of index $N \le 3$with the Bondi radius as the outer boundary, the criterion governing thecritical core mass occurs at the surface. For stiff polytropes with the Hillouter boundary and for soft polytropes of $N>3$, this criterion acts at thebottom of gaseous envelope. Further, we elucidate the roles and effects ofcoexistent radiative and convective zones in the envelope of critical coremass. Based on the results, we discuss the relevance of Bondi and Hill surfaceconditions and explore the parameter dependences of critical core mass.

The structure of gas-accreting protoplanets and the condition of the critical core mass