Effects of core formation on the Hf-W isotopic composition of the Earth and dating of the Moon-forming impact

Earth's core formation set the initial compositions of the core and mantle. Various aspects of core formation, such as the degree of metal-silicate equilibration, oxygen fugacity, and depth of equilibration, have significant consequences for the resulting compositions, yet are poorly constrained. The Hf-W isotopic system can provide unique constraints on these aspects relative to other geochemical or geophysical methods. Here we model the Hf-W isotopic evolution of the Earth, improving over previous studies by combining a large number of N-body simulations of planetary accretion with a core formation model that includes self-consistent evolution of oxygen fugacity and a partition coefficient of tungsten that evolves with changing pressure, temperature, composition, and oxygen fugacity. The effective average fraction of equilibrating metal is constrained to be k>0.2 for a range of equilibrating silicate masses (for canonical accretion scenarios), and is likely <0.55 if the Moon formed later than 65 Ma. These values of k typically correspond to an effective equilibration depth of ∼0.5-0.7× the evolving core-mantle boundary pressure as the planet grows. The average mass of equilibrating silicate was likely at least 3× the impactor's silicate mass. Equilibration temperature, initial fO2, initial differentiation time, semimajor axis, and planetary mass (above ∼0.9 M⊕) have no systematic effect on the 182W anomaly, or on fHf/W (except for fO2), when applying the constraint that the model must reproduce Earth's mantle W abundance. There are strong tradeoffs between the effects of k, equilibrating silicate mass, depth of equilibration, and timing of core formation, so the terrestrial Hf-W isotopic system should be interpreted with caution when used as a chronometer of Earth's core formation. Because of these strong tradeoffs, the Earth's tungsten anomaly can be reproduced for Moon-forming impact timescales spanning at least 10-175 Ma. Early Moon formation ages require a higher degree of metal-silicate equilibration to produce Earth's 182W anomaly.