The Dynamics and Helium Distribution in Hydrogen-helium Fluid Planets

In the preceding paper (Paper I) we discussed the thermodynamic and microscopic transport properties of hydrogen-helium fluid mixtures. These results are used in the present paper for a semiquantitative analysis of the thermal and compositional history of an evolving hydrogenhelium planet such as Jupiter or Saturn. First, the evolution of a homogeneous planet with no first-order phase transitions or immiscibilities is considered. The temperature gradient is at least adiabatic (since thermal conduction cannot transport a sufficient heat flux) and is also large enough to ensure that the fluid state prevails everywhere. Convection is therefore uninhibited by molecular viscosity, and the fractional superadiabaticity is very small, despite the inhibitory effects of rotation and magnetic field. Adiabatic, evolutionary models are discussed. The times taken for Jupiter and Saturn to reach their observed luminosities are about 4 x 109 and 2 x 109 years, respectively, essentially independent of formation details. The result for Saturn appears to be inconsistent with its actual age, assumed to be "'4.5 x 109 years. Next, the effects of a first-order molecular-metallic hydrogen transition are discussed for a pure hydrogen planet: A well-defined interface between the phases persists, despite the presence of convection. The temperature is continuous at the interface and the entropy is discontinuous, the change in entropy being equal to the latent heat of transition. Consequently, the heat content and derived "age" differ from that determined for a purely adiabatic model (by a factor between 1 and 2, depending on the unknown latent heat). Convection in the presence of a composition gradient is discussed, and the importance of overstable modes and diffusive-convective equilibria established. The convective transport of helium away from a localized helium source is shown to be inefficient because helium diffusivity is much less than heat diffusivity. Evolutions with helium immiscibility (but no first-order molecular-metallic hydrogen transition) are discussed. Helium droplets nucleate from the supersaturated mixture, grow to "' 1 em radius, and fall under the influence of gravity, despite the convection. Most of the energy release from this differentiation is available for radiation, and the decay time for the planet's excess luminosity is increased, typically by about a factor of 5. Finally, more complicated cases are discussed which include both immiscibility and the firstorder character of the molecular-metallic hydrogen transition. The Gibbs phase rule leads to a discontinuity of the helium fraction at the transition, the formation of a helium-rich core, and an energy release comparable to that for immiscibility. This core can grow at the expense of the helium content in either the metallic or molecular region. In some cases, the molecular envelope helium content is actually enhanced by upward convective transport of helium. The various parameters (especially the critical temperature of the molecular-metallic hydrogen transition) are too uncertain for detailed quantitative conclusions. The success of adiabatic, homogeneous evolutionary calculations for Jupiter suggests that helium differentiation has not yet begun for that planet or has begun very recently ( :( 109 years ago), which in turn suggests that the critical temperature for the molecular-metallic hydrogen transition cannot greatly exceed 20,000 K. Helium differentiation in Saturn (and deviations from primordial abundance for helium and minor constituents in the atmosphere) appears to be required to explain the observed excess luminosity. Subject headings: planets: abundances planets: interiors planets: Jupiter