Core Formation Timescale, Silicate-Metal Equilibration, and W Diffusivity

The extent to which material accreted to the proto-Earth and segregated to form the core was chemically and isotopically equilibrated with the silicate mantle is an outstanding problem in planetary science. This is particularly important when attempting to assign a meaningful age for planetary accretion and core formation based on Hf-W isotope systematics. The Earth and other terrestrial planets likely formed by accretion of previously differentiated planetesimals. For the planetesimals themselves the most important energy source for metal-silicate differentiation is the combined radioactive heating due to decay of 26Al (half-life 0.7 Ma) and 60Fe (half-life 1.5 Ma). It is expected that the fractionation of Hf and W during planetesimal core formation will lead to a divergence in the W isotopic compositions of the core and silicate portions of these bodies. This expectation is supported by the enormously radiogenic 182W signatures reported for basaltic eucrites. The observation that the W isotopic compositions of the silicate portions of Earth, Moon and Mars are similar and markedly less radiogenic than eucrites suggests that during planet accretion the pre-differentiated metallic core material containing low 182W must have equilibrated extensively with the more radiogenic (high 182W) silicate material to subdue the ingrowth of 182W in the silicate mantle of the planets. The standard theory of planet formation predicts that after runaway and oligarchic growth, the late stage of planet formation is characterized by impact and merging of Mars-sized objects. This is a tremendously energetic process estimated to raise the temperature of the proto-Earth to about 7000K (a temperature equivalent to a mass spectrometer's plasma source, which indiscriminately ionizes all incoming elements). After the giant impacts, the proto-Earth had a luminosity and surface temperature close to a low mass star for a brief period of time. Stevenson (1990) argued that emulsification caused by large-scale Rayleigh-Taylor instabilities following giant impacts breaks up the metallic core of the impactor into centimeter-sized droplets within minutes. We have constructed a simple model to study the kinetics of isotope equilibration between metal-silicate during the "rain-fall" of metal droplets descending through the terrestrial magma ocean. This model highlights the importance of the kinetics of ion mobility of W for assessing quantitatively the degree of metal-silicate equilibration during core formation. We have determined for the first time that the W self-diffusion coefficient in basaltic liquid is 4.98E-7 cm2/s at 3GPa, 1500 C. We assume this is a minimum value in the magma ocean scenario, and the equilibration is rate-limited by diffusion in the silicate liquid. Applying this value and taking a reasonable estimate of viscosity for silicate liquids from the literature, we show that the degree of equilibration asymptotically approaches 100% within the timescale of metal-silicate segregation when the metallic droplets are <20 cm in diameter.