The efficient delivery of highly-siderophile elements to the core creates a mass accretion catastrophe for the Earth

Abstract

The excess abundance of highly siderophile elements (HSEs), as inferred for the terrestrial planets and the Moon, is thought to record a `late veneer' of impacts after the giant impact phase of planet formation. Estimates for total mass accretion during this period typically assume all HSEs delivered remain entrained in the mantle. Here, we present an analytical discussion of the fate of liquid metal diapirs in both a magma pond and a solid mantle, and show that metals from impactors larger than approximately 1 km will sink to Earth's core, leaving no HSE signature in the mantle. However, by considering a collisional size distribution, we show that to deliver sufficient mass in small impactors to account for Earth's HSEs, there will be an implausibly large mass delivered by larger bodies, the metallic fraction of which lost to Earth's core. There is therefore a contradiction between observed concentrations of HSEs, the geodynamics of metal entrainment, and estimates of total mass accretion during the late veneer. To resolve this paradox, and avoid such a mass accretion catastrophe, our results suggest that large impactors must contribute to observed HSE signatures. For these HSEs to be entrained in the mantle, either some mechanism(s) must efficiently disrupt impactor core material into $\leq0.01$ mm fragments, or alternatively Earth accreted a significant mass fraction of oxidised (carbonaceous chondrite-like) material during the late veneer. Estimates of total mass accretion accordingly remain unconstrained, given uncertainty in both the efficiency of impactor core fragmentation, and the chemical composition of the late veneer.

Figures (10)

• Figure 1: Collisions between leftover planetesimals drives the redistribution of mass between bodies of different sizes, generating a so-called collisional size-frequency distribution (SFD) dominated, in number, by the smallest bodies. The Earth will therefore, unavoidably, accrete planetesimals of a wide range of sizes during the late veneer. For the canonical collisional SFD Dohnanyi1969, total mass is concentrated in the largest planetesimals (plotted above). The geodynamical processes responsible for the delivery of HSEs to the mantle, are controlled, primarily, by the size of the impactor, and therefore have the capacity to dramatically bias estimates of total mass accretion during the late veneer. We identify three possible regimes of HSE delivery, with illustrative schematic diagrams inset above. (a) Small impactors at low velocity will generate little melt, and are expected to fragment into millimetric pieces (see §\ref{['sec:post_impact_HSE_distribution']}). The ability of these impactors to affect mantle geochemistry will depend on whether the tectonic regime of the planet enables them to be recycled into the mantle. (b) Small impactors at high velocity will generate significant melt, from both the target and impactor Melosh1989. We expect metal diapirs to quickly enter the solid mantle (§\ref{['sec:metal_entrainment_magma_pond']}), bringing with them the impactor's HSEs. The ability of these impactors to affect mantle geochemistry will depend on whether the frictional force resisting diapir descent is larger than the negative buoyancy. (c) Large, differentiated impactors will generate large volumes of melt. Unless impactor core material can be fragmented into very small droplets, large diapirs will enter the solid mantle, and quickly sink to Earth's core (§\ref{['sec:metal_entrainment_mantle']}).
• Figure 2: The maximum diameter of metal drops that can be entrained by turbulent convection in a fully liquid magma pond, $d_{\text{entr.}}$ (corresponding to the condition $\theta_S=\theta_c$; equation \ref{['eq:crit_radius_turb']}), as a function of magma pond depth for three plausible values of the heat flux $F$. The friction speed $u$ in equation (\ref{['eq:crit_radius_turb']}) is computed from equations \ref{['eq:friction']}-\ref{['eq:ConvectiveSpeed_Turbulent']}. Note that $\theta_S=\theta_c$ is a necessary, not sufficient condition for entrainment, and so metal drops smaller than this maximum diameter will not all remain suspended in equilibrium (see equation \ref{['eq:entrained_fraction']}, figure \ref{['fig:suspended_metal_MO_Earth']}).
• Figure 3: Volume fraction of suspended metal in a turbulent magma ocean on an Earth-sized planet as a function of the metal drop diameter, for three plausible values of the heat flux $F$. The green shaded band locates the suspended volume fraction needed so that the HSE concentration in the magma pond matches that of the present-day Earth's mantle. We use equations \ref{['eq:entrained_fraction']}-\ref{['eq:Stokes']} assuming $g=9.8\,{\rm m\,s}^{-2}$, $\rho_m=9000\,{\rm kg\,m}^{-3}$, $\rho_s=4500\,{\rm kg\,m}^{-3}$, $\mu_s=0.05$ Pa s and $H = 2000\,$km. (a) Lower-end estimates assume UserColor UserColor $\epsilon=0.2\%$ . (b) Upper-end estimates with $\epsilon=0.9\%$.
• Figure 4: The maximum diameter of metal diapirs that can be entrained by mantle convection, $d_{\text{entr.}}$ (equation \ref{['eq:critical_diapir_diameter']}), as a function of mantle viscosity. The dashed green line corresponds to the lower estimate, using $c=0.05$, $\Delta T=500\,$K, $\theta_c=0.4$, while the dotted green line shows the higher estimate, with $c=0.02$, $\Delta T=2500\,$K, $\theta_c=0.2$. The shaded region indicates typical values expected for a partially solid mantle on the early-Earth.
• Figure 5: Total mass accretion to the Earth is calculated, assuming all observed HSEs ($M_{\rm HSE, \oplus}$) are delivered by impactors smaller than $D_{\rm crit}$, from a collisional size distribution <$n(D)dD \propto D^{-7/2}dD$;>Dohnanyi1969 with maximum impactor diameter $D_{\rm max}$. (a) We vary $D_{\rm max}$, while keeping $D_{\rm crit}=1\,$km fixed, and find total mass accretion from larger bodies quickly becomes unrealistically large. Estimates for the main asteroid belt, and streaming instability are included for reference, in blue and yellow respectively, which deliver even more mass in $D>D_{\rm crit}$ impactors. (b) Total mass accretion is calculated as a function of $D_{\rm crit}$, which is largely determined by the Earth's early-atmosphere (see \ref{['sec:appendix_melting']}). Total mass accretion remains implausibly large in the presence of large ($\sim\,$500 km) planetesimals, as are found in present-day asteroid belt, and predicted by the streaming instability.