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