Scaling laws for the geometry of an impact-induced magma ocean

Abstract

Growing protoplanets experience a number of impacts during the accretion stage. A large impactor hits the surface of a protoplanet and produces impact-induced melt, where the impactor's iron emulsifies and experiences metal-silicate equilibration with the mantle of the protoplanet while it descends towards the base of the melt. This process repeatedly occurs and determines the chemical compositions of both mantle and core. The partitioning is controlled by parameters such as the equilibration pressure and temperature, which are often assumed to be proportional to the pressure and temperature at the base of the melt. The pressure and temperature depend on both the depth and shape of the impact-induced melt region. A spatially confined melt region, namely a melt pool, can have a larger equilibrium pressure than a radially uniform (global) magma ocean even if their melt volumes are the same. Here, we develop scaling laws for (1) the distribution of impact-induced heat within the mantle and (2) shape of the impact-induced melt based on more than 100 smoothed particle hydrodynamic (SPH) simulations. We use Legendre polynomials to describe these scaling laws and determine their coefficients by linear regression, minimizing the error between our model and SPH simulations. The input parameters are the impact angle θ (0∘,30∘,60∘, and 90∘), total mass MT (1MMars−53MMars, where MMars is the mass of Mars), impact velocity vimp (vesc−2vesc, where vesc is the mutual escape velocity), and impactor-to-total mass ratio γ (0.03−0.5). We find that the equilibrium pressure at the base of a melt pool can be higher (up to ≈80%) than those of radially-uniform global magma ocean models. This could have a significant impact on element partitioning. These melt scaling laws are publicly available on GitHub (https://github.com/mikinakajima/MeltScalingLaw).