Toward Reliable Interpretations of Small Exoplanet Compositions: Comparisons and Considerations of Equations of State and Materials Used in Common Rocky Planet Models

TL;DR

This work addresses the challenge of reliably inferring the interior compositions of small exoplanets from mass and radius, showing that different EOS+mineral suites yield bulk densities that differ by up to about 10% across plausible compositions. Using a common planet-builder, the authors compare ten widely used EOS+mineral suites and demonstrate that such differences propagate to significant variations in inferred Fe/Mg and WMF, affecting planet classifications (e.g., Earth-like vs. super-Mercury vs. water-world). They also show that many planets exhibit suite-dependent interpretations, which can bias demographic conclusions about planet formation. The study recommends best practices—bracketing core light-element content, including upper mantle minerals, incorporating post-perovskite phases, accounting for mantle melting in hot planets, and propagating EOS uncertainties—aimed at producing robust, self-consistent interpretations of small exoplanet interiors and their formation histories.

Abstract

The bulk compositions of small planets ($R_p< 2 \mathrm{R}_\oplus$) are directly linked to their formation histories, making reliable compositional constraints imperative for testing models of planet formation and evolution. Because exoplanet interiors cannot be directly observed, their make-up must be inferred from mass-radius-composition models that link assumed stellar abundances to the direct observables: planetary mass and radius. There are a variety of such models in the literature, each adopting different equations of state (EOS) to describe the materials' properties at depth and varying assumptions about the minerals present within the planets. These EOS+mineral suites provide the foundations for compositional inferences, but they have not yet been systematically compared. In this work, we review several suites, with a detailed description of the basic structure, mineral physics, and materials within standard small planet models. We show that EOS+mineral suites predict planet densities whose differences are comparable to current observational uncertainties, which present a challenge for robustly interpreting and classifying small planets. We apply a powerful small-planet characterization framework, which illustrates that variations among EOS+mineral suites lead to inconsistent conclusions for both individual planets and sample-level demographics. Our results demonstrate the need for more careful considerations of the materials and EOS used in mass-radius-composition models, especially given the current focus on finding and characterizing potentially habitable rocky planets. We conclude with recommendations for best practices so that future interpretations of small planets and their formation are accurate and consistent.

Figures (7)

• Figure 1: Representative sections of (a) 1$\mathrm{M_\oplus}$ and (b) 10$\mathrm{M_\oplus}$ planets as a function of depth and pressure. Both were created using the Perple_X software with the thermodynamic database of Stixrude_2011 (see e.g., Sect. \ref{['subsection:exoplex']}) assuming an Earth-like composition (Fe/Mg = 0.9, Si/Mg = 0.9, Ca/Mg = 0.07, Al/Mg = 0.09, and 8 wt % mantle FeO). Cpx = clinopyroxene, Opx = orthopyroxene, and C2/C = C2/C orthopyroxene. The garnet is majoritic garnet. Perovskite is silicate-perovskite.
• Figure 2: Predicted material densities as a function of pressure for BM3 (blue lines) and Vinet EOS (black lines) for different values of $K_0^{'}$. All densities and pressures are normalized by the zero-pressure density ($\rho_0$) and bulk modulus ($K_0$), respectively. Dashed, solid, and dashed-dotted lines show $K_0^{'} = 4, 5,$ and 6, respectively. The green filled regions show the $P/K_0$ ranges for the core-mantle-boundary (CMB) pressure ($P_\mathrm{CMB, \oplus}$) and central pressure ($P_\mathrm{c, \oplus}$) of Earth for $K_0$ values typical of silicates and Fe. The orange filled regions similarly show $P/K_0$ ranges for the CMB and central pressure of a 5$M_\oplus$ planet ($P_{\mathrm{CMB}, 5M_\oplus}$ and $P_{\mathrm{c}, 5M_\oplus}$, respectively) with Earth-like Fe/Mg.
• Figure 3: Material density as a function of pressure for the isothermal/isentropic (a) core and (b)-(c) mantle EOS used within the selected suites. Panel (a) spans from the central pressure of a 10$M_\oplus$ planet Boujibar_2019unterborn_panero19 to $200$ GPa. Panel (b) shows the lower-mantle minerals from the core-mantle-boundary pressure of a 10$M_\oplus$ planet to 80 GPa. The inset shows the transition of pv$\to$ppv. Panel (c) shows the lower mantle mineral phases from 30-1 GPa. Solid lines show EOS+mineral suites that include upper mantle minerals and assume the planet core is pure Fe with no light elements (or LE, in the key). Similarly, the dashed lines show those that do not include upper mantle minerals and assume a pure Fe core with no light elements. The dashed-dotted lines show suites that include upper mantle minerals and no light elements in the pure Fe core. MAGRATHEA v1.0 is shortened to 'Mag' and ExoPlex to 'EP.'
• Figure 4: Material density ($\rho$) as a function of radial distance ($r$) from the center of (a) 1$M_\oplus$ and (b) 10$M_\oplus$ rocky planets with Earth-like Fe/Mg = 0.9. The various curves correspond to the EOS+mineral suites in Sect. \ref{['sect:selected_EOS_mineral_suites']} applied using the planet builder described in Sect. \ref{['sect:planet_builder']}. Suites that assume a pure-Fe core and no upper mantle mineral phases are shown via dashed lines. Solid lines show those with a pure Fe core and upper mantle mineral phases. Suites that include light elements (or LE, in the key) in the core and upper mantle mineral phases are shown as dashed-dotted lines. The horizontal lines on the left-hand side of each subplot indicate the calculated bulk density of the planets for each suite.
• Figure 5: (a) Bulk planet density ($\rho_p$) for 1$M_\oplus$ and 10$M_\oplus$ rocky planets as a function of bulk Fe/Mg and EOS+mineral suite. Subplots (b) and (c) show the percent differences in the calculated $\rho_p$ ($\Delta\rho_p$) of each suite relative to ExoPlex at 10$M_\oplus$ and 1$M_\oplus$, respectively. Vertical lines show the Fe/Mg ranges of the RZ (black dashed; Sect. \ref{['sect:planet_interpretations_and_inferences']}) and Earth-like Fe/Mg = 0.9 (green; Sect. \ref{['fig:profile_plots']}). Suites that assume a pure-Fe core and no upper mantle mineral phases are shown via dashed lines. Solid lines show those with a pure Fe core and upper mantle mineral phases. Suites that include light elements in the core and upper mantle mineral phases are shown as dashed-dotted lines.