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

Kaustubh Hakim*, Attilio Rivoldini, Tim Van Hoolst, Stefaan Cottenier, Jan Jaeken, Thomas Chust, Gerd Steinle-Neumann

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


More than a third of all exoplanets can be classified as super-Earths based on radius (1–2 R) and mass (<10 M). 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. 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 Mg0.5Fe0.5SiO3 instead of MgSiO3 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.

Original languageEnglish
Pages (from-to)61-78
Number of pages18
Publication statusPublished - 1 Oct 2018


We thank Diana Valencia and an anonymous reviewer for their insightful comments in improving this manuscript. This research has been supported by the Planetary and Exoplanetary Science Network (PEPSci), funded by the Netherlands Organization for Scientific Research ( NWO ) Project no. 648.001.005 , led by Carsten Dominik and Wim van Westrenen. TVH and AR acknowledge the financial support from the Belgian PRODEX program (grant no. 4000120791) managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office and from the BRAIN-be program of the Belgian Federal Science Policy Office. SC acknowledges financial support from OCAS NV by an OCAS-endowed chair at Ghent University. Work by GSN is supported by Deutsche Forschungsgemeinschaft (DFG, German Science Foundation) through Research Unit 2440 (Matter Under Planetary Interior Conditions, STE1105/13-1 ). Appendix A

FundersFunder number
Planetary and Exoplanetary Science Network
European Space Agency
Deutsche ForschungsgemeinschaftSTE1105/13-1
Belgian Federal Science Policy Office
Nederlandse Organisatie voor Wetenschappelijk Onderzoek4000120791, 648.001.005
Universiteit Gent


    • Equation of state of iron
    • Exoplanets
    • Interior structure
    • Mass-radius relations
    • Super-Earths


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