A new ab initio equation of state of hcp-Fe and its implication on the interior structure and mass-radius relations of rocky super-Earths

Abstract

More than a third of all exoplanets can be classified as super-Earths based on radius (1´2 R C ) and mass (< 10 M C ).Here we model mass-radius relations based on silicate mantle and iron core equations of state to infer to first order the structure and composition range of rocky super-Earths assuming insignificant gas envelopes. As their core pressures exceed those in the Earth by an order of magnitude, significant extrapolations of equations of state for iron are required. We develop a new equation of state of hexagonal close packed (hcp) iron for super-Earth conditions (SEOS) based on density functional theory results for pressures up to 137 TPa. A comparison of SEOS and extrapolated equations of state for iron from the literature reveals differences in density of up to 4% at 1 TPa and up to 20% at 10 TPa. Such density differences significantly affect mass-radius relations. On mass, the effect is as large as 10% for Earth-like super-Earths (core radius fraction of 0.5) and 20% for Mercury-like super-Earths (core radius fraction of 0.8). We also quantify the effects of other modeling assumptions such as temperature and composition by considering extreme cases. We find that the effect of temperature on mass (< 5%) is smaller than that resulting from the extrapolation of the equations of state of iron and lower mantle temperatures are too low to allow for rock and iron miscibility for R < 1.75 R C . Our end-member cases of core and mantle compositions create a spread in mass-radius curves reaching more than 50% in terms of mass for a given planetary radius, implying that modeling uncertainties dominate over observational uncertainties for many observed super-Earths. We illustrate these uncertainties explicitly for Kepler-36b with well-constrained mass and radius. Assuming a core composition of 0.8ρ Fe (equivalent to 50 mol% S) instead of pure Fe leads to an increase of the core radius fraction from 0.53 to 0.64. Using a mantle composition of Mg 0.5 Fe 0.5 SiO 3 instead of MgSiO 3 leads to a decrease of the core radius fraction to 0.33. Effects of thermal structure and the choice of equation of state for the core material on the core radius of Kepler-36b are small but non-negligible, reaching 2% and 5%, respectively.

Liquid Iron Equation of State to the Terapascal Regime From Ab Initio Simulations

“…Kepler‐36b is one of the super‐Earth planets with the tightest constraints on mass ( ) and radius (1.486 ± 0.035 R Earth ; Carter et al, ) and has therefore been the subject of extensive modeling of its internal structure (e.g., Dorn et al, ; Hakim et al, ; Unterborn et al, ). Here we use a specific core adiabat from Hakim et al (; the model with a radius ratio of R core / R planet = 0.527, a MgSiO 3 mantle, and a pure Fe isentropic core in their Figure 8) to explore differences between our work and that of other thermodynamic models for liquid iron (Dorogokupets et al, ; Ichikawa et al, ; Komabayashi, ) in terms of a ϱ profile in the core of Kepler‐36b. It is worth reiterating here that our model is the only one constrained to P in the TPa range.…”

“…Density of iron from the various thermodynamic models along a hypothetical T profile in the core of Kepler‐36b, proposed by Hakim et al (). We use the model for a pure Fe isentropic core and MgSiO 3 mantle from Figure 8 of Hakim et al () with a core radius ratio of R core / R planet = 0.527.…”

“…Density of iron from the various thermodynamic models along a hypothetical T profile in the core of Kepler‐36b, proposed by Hakim et al (). We use the model for a pure Fe isentropic core and MgSiO 3 mantle from Figure 8 of Hakim et al () with a core radius ratio of R core / R planet = 0.527. Results from our model (black line) are compared to other liquid density profiles (solid lines) using the models of Komabayashi () (K14, green), Ichikawa et al () (I14, blue), and Dorogokupets et al () (D17, purple), and the ϱ profile for solid ε iron used by Hakim et al () (H18, dashed orange line).…”

“…Due to size and mass, P within a super‐Earth with 10 M Earth exceeds that in the Earth by an order of magnitude and temperature ( T ) by a factor of 2 (Wagner et al, ), making the characterization of thermodynamic parameters of planetary materials to the TPa regime a key to understanding physical processes in their interior. Equations of state of different candidate materials in that P ‐ T range are therefore of central importance to model their internal structure based on first‐order observables, like mass‐radius relations (Hakim et al, ; Wagner et al, ). Being the major core‐forming element of terrestrial‐type planets and the most stable element in the stellar fusion process, iron is of special interest to planetary physics and continues to be the focus of many studies.…”

Mass determinations of the three mini-Neptunes transiting TOI-125

“…In order to characterise the internal structure of TOI-125b, TOI-125c and TOI-125d we construct models considering a pure iron core, a silicate mantle, a pure water layer and a H-He atmosphere. The models follow the basic structure model of Dorn et al (2017), with the equation of state (EOS) for the iron core taken from Hakim et al (2018), and the EOS of the silicate-mantle from Connolly (2009). For water we use the quotidian EOS of Vazan et al (2013) for low pressures and the one of Seager et al (2007) for pressures above 44.3 GPa.…”