  We present a model for the equilibrium of solid planetary cores embedded in a
gaseous nebula. From this model we are able to extract an idealized roadmap of
all hydrostatic states of the isothermal protoplanets. The complete
classification of the isothermal protoplanetary equilibria should improve the
understanding of the general problem of giant planet formation, within the
framework of the nucleated instability hypothesis. We approximate the
protoplanet as a spherically symmetric, isothermal, self-gravitating classical
ideal gas envelope in equilibrium, around a rigid body of given mass and
density, with the gaseous envelope required to fill the Hill-sphere. Starting
only with a core of given mass and an envelope gas density at the core surface,
the equilibria are calculated without prescribing the total protoplanetary mass
or nebula density. The static critical core masses of the protoplanets for the
typical orbits of 1, 5.2, and 30 AU, around a parent star of 1 solar mass are
found to be 0.1524, 0.0948, and 0.0335 Earth masses, respectively, for standard
nebula conditions (Kusaka et al. 1970). These values are much lower than
currently admitted ones primarily because our model is isothermal and the
envelope is in thermal equilibrium with the nebula. For a given core, multiple
solutions (at least two) are found to fit into the same nebula. We extend the
concept of the static critical core mass to the local and global critical core
mass. We conclude that the 'global static critical core mass' marks the meeting
point of all four qualitatively different envelope regions.
