Giant planet formation: a first classification of isothermal protoplanetary equilibria

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. In this way, a variety of hydrostatic core-envelope equilibria has been obtained. Two types of envelope equilibria can be distinguished: uniform equilibrium, were the density of the envelope gas drops approximately an order of magnitude as the radial distance increases to the outer boundary, and compact equilibrium, having a small but very dense gas layer wrapped around the core and very low, exponentially decreasing gas density further out. The effect of the envelope mass on the planetary gravitational potential further discriminates the models into the self-gravitating and the non-self gravitating ones. 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 (M⊙) are found to be 0.1524, 0.0948, and 0.0335 Earth masses (M⊕), 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. Our solutions show a wide range of possible envelopes. For a given core, multiple solutions (at least two) are found to fit into the same nebula. Some of those solutions posses equal envelope mass. This variety is a consequence of the envelope's self-gravity. We extend the concept of the static critical core mass to the local and global critical core mass. Above the global critical mass, only compact solutions exist. We conclude that the 'global static critical core mass' marks the meeting point of all four qualitatively different envelope regions.