Metal-silicate mixing upon Giant Impacts into magma oceans

Introduction: Planetary formation models suggest that Earth experienced multiple high-energy impacts. They can produce substantial melt in the proto-Earth’s silicate mantle, possibly forming a global magma ocean. Mixing of the impactor’s metallic core into the molten Earth's silicate mantle controls the chemical equilibration between metal and silicates, which defines the respective compositions of Earth's core and mantle. Previous studies explore mixing upon large impacts either with numerical modelling or with analog laboratory experiments. Numerical simulations are efficient in reproducing the shock physics of hypervelocity impacts. However, their spatial resolution is limited and does not allow for reproducing the turbulent features that are responsible for metal-silicate mixing in a magma ocean. On the other hand, liquid impact experiments that do produce small-scale mixing and turbulence are subsonic: they neglect compressibility effects.  Here, we investigate the degree of mixing upon impact by coupling various results from fluid impact experiments [1] and numerical modelling extending the crater depth from experiments to supersonic conditions [2].

Methods: The laboratory experiments used to explore mixing and extend to the supersonic regimes consist in a volume of fluid impacting into a tank of water. The denser impactor fluid is dyed, enabling for the optical estimate of the volume of the evolving sinking plume right after the impact. The volume of the plume gives, once corrected from the volume of the impactor material, an analogue of the volume of entrained silicates upon a given collision. We have extended laboratory results on the crater growth from subsonic to supersonic conditions using the grid-based Eulerian shock physics code iSALE [3,4,5,6] to simulate fluid impacts. The scaling-law that is produced [2] to extend laboratory experiments results to supersonic, hypervelocity conditions, is used here to be further applied to the mixing. The experiments from [1] suggest that the impact between an impactor of density ρ_i and a lighter target of density ρ_t  involves two main stages: the opening of the crater at early times and the fall of central jet that had been formed from the collapse of the crater. That latter event controls the release of the impactor material into the target. At later times, buoyancy forces become important, which controls the sinking of the impactor material into a turbulent thermal descending into the target. The competition between total buoyancy and the momentum of the collapsing jet controls the dynamics of the descending thermal [1], hence the mixing between impactor material and surrounding target silicates. It has been showed that the jet height scales as the maximum crater depth  and that the maximum jet volume scales as the maximum crater volume [7]. We use these scalings, coupled to those regarding mixing from [1] and those on the extension of maximum crater depth from experiments to supersonic conditions from [2] to extend the mixing results from laboratory experiments to hypervelocity regimes accounting for both the effects of the Froude number (measure of the importance of the impactor kinetic energy to its gravitational energy at impact) and the Mach number (impact velocity to sound speed ratio).

Results: Figure 1 shows the mass of silicates that is entrained in the thermal prior to its descent to the magma. It provides a direct measure of the so-called mixing between metal and silicates during a large impact into a magma ocean. We find that the Mach number decreases the estimates of metal-silicate mixing upon impacts derived from laboratory experiments. The extent of its effect however depends on the other impact parameters such as the impactor size. The larger the impactor compared to the target size, the larger the effect of the Mach number. For M>3, the mixing estimates can be underestimated by more than a factor 3 if not accounting for the shock.

Figure 1. Mass of silicates mixed with metal by the impact stage, prior to the descent into the magma, Δ, as a function of the impactor to target radius, R/R_t , varying the Mach number, M, for two cases: a) impact velocity is escape velocity, U = Ue  and b) impact velocity is twice larger than the escape velocity, U = 2 Ue. Circle: impactor of 100 km in radius onto an Earth-sized target. Diamond: canonical Moon-forming scenario with a Mars-sized impactor and U = Ue [8]. Square: Moon-forming scenario with an impactor mass 20 times smaller than the target onto a fast-spinning
Earth at U = 2 Ue [9].

Discussion: We have studied the statistics on how often in typical classical scenarios of accretion such collisions would occur for Earth analogs through its growth history. We use collision files from N-body simulations in the Grand Tack scenario [10] and estimate that, depending on the sound speed of the impacted material, 24% to 74% of the total amount of collisions endured by Earth analogs may occur at M>3. If considering only giant impacts, as these are those when the effect of the Mach number is the most significant, this drops to 4% to 28%. The effect of the Mach number on the mixing, however not extreme, may need to be accounted for when following the metal silicates reequilibration through an entire stage of planetary formation.  

Acknowledgments: We gratefully acknowledge the developers of iSALE-2D, including Gareth Collins, Kai Wünnemann, Dirk Elbeshausen, Tom Davison, Boris Ivanov and Jay Melosh. This work was funded by the Deutsche Forschungsgemeinschaft (SFB-TRR170, subproject C2 and C4).

References: [1] Landeau M. et al. (2021) Earth Plan. Sci. Lett. 564, 116888. [2] Allibert L. et al. (2023) JGR: Planets 128 (8), e2023JE007823. [3] Collins. G. S. et al. (2004) Meteoritics & Planetary Science 39:217-231. [4] Wünnemann K. et al. (2006) Icarus 180:514-527. [5] Elbeshausen, D. et al. (2009). Icarus, 204(2), 716-731. [6] Elbeshausen et al. (2011). Proceedings of 11th Hypervelocity Impact Symposium (HVIS), Fraunhofer Verlag. [7] Ghabache É. et al. (2014) . Fluid Mech. 761, 206–219. [8] Canup R. M. (2004) Icarus 168 (2), 433–456. [9] Ćuk, M. and Stewart S. (2012) Science 338 (6110), 1047–1052. [10] Jacobson S. A. and Morbidelli A. (2014) Phil. Trans. Roy. Soc. A 372, 20130174.