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PALEOS: Multiphase Equations of State and Mass-Radius Relations for Exoplanet Interiors
Pith reviewed 2026-05-07 00:40 UTC · model grok-4.3
The pith
PALEOS supplies a phase-aware, thermally consistent set of equations of state for iron, MgSiO3, and H2O that recovers Earth's radius to 0.3 percent and produces mass-radius curves in which thermal expansion above 1500 K changes radius by more than 1 percent and up to 16 percent at 4000 K.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Continuous solid-to-melt EoS let thermal expansion span the fully-solid to magma-ocean regime: the radius offset exceeds 1% above 1500 K and reaches 16% at 4000 K for low-mass silicate planets, comparable to composition degeneracy and transit-radius uncertainties.
Load-bearing premise
That the consolidated EoS for the seventeen phases remain thermodynamically consistent and sufficiently accurate across the entire pressure-temperature range encountered in 0.1-100 Earth-mass planets, including the high-temperature melt regime where direct experimental constraints are sparse.
read the original abstract
Modeling the interior of a rocky or water-rich exoplanet is a thermodynamic closure problem: every layer's density, temperature gradient, and phase must follow from an equation of state (EoS) that remains self-consistent across the pressure-temperature range from surface to core. Existing EoS span disciplines, use different formalisms, and rarely supply the full thermodynamic quantities needed by evolutionary models of interior phase transitions. We present PALEOS (Planetary Assemblage Layers: Equations of State), an open-source toolkit consolidating EoS for iron, magnesium silicate (MgSiO$_3$), and water (H$_2$O) into a unified, phase-aware, thermally responsive framework spanning 17 phases. PALEOS derives density, energy, entropy, heat capacities, thermal expansion, and the adiabatic gradient analytically via Maxwell relations, and is released as lookup tables on regular P-T grids. We validate it against the Preliminary Reference Earth Model, recovering Earth's radius to 0.3% and lower-mantle densities to 3%, and compute 17,900 mass-radius relations from 0.1 to 100 $M_\oplus$ for rocky (Fe + MgSiO$_3$) and water-rich (Earth-like core + H$_2$O envelope) compositions at 300-4000 K. Continuous solid-to-melt EoS let thermal expansion span the fully-solid to magma-ocean regime: the radius offset exceeds 1% above 1500 K and reaches 16% at 4000 K for low-mass silicate planets, comparable to composition degeneracy and transit-radius uncertainties. We demonstrate this on two ultrashort-period super-Earths, WASP-47 e and TOI-1807 b: each admits two purely rocky solutions indistinguishable in mass and radius but in radically different states, one fully solid with no dynamo, the other hosting a deep magma ocean and a liquid iron core capable of sustaining a magnetic field. Phase-aware, thermally resolved EoS are essential for translating astronomical observations into exoplanetary geophysics.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces PALEOS, an open-source toolkit that consolidates multiphase equations of state for iron, MgSiO3, and H2O across 17 phases into a unified, thermodynamically consistent framework. It derives all required thermodynamic quantities analytically via Maxwell relations, supplies P-T lookup tables, validates the model against PREM (0.3% radius recovery, 3% lower-mantle density), and computes 17,900 mass-radius curves for 0.1–100 M⊕ planets. The central result is that continuous solid-to-melt EoS produce radius offsets >1% above 1500 K and up to 16% at 4000 K for low-mass silicate worlds, comparable to composition and observational uncertainties, with implications for the thermal state and dynamo potential of specific USP super-Earths.
Significance. If the consolidated EoS remain accurate and consistent in the high-temperature melt regime, the work supplies a practical, phase-aware tool that directly links interior thermal structure to observable mass-radius relations and magnetic-field prospects. The analytic Maxwell-relation approach and public lookup tables are concrete strengths that would allow other modelers to propagate phase boundaries and thermal expansion self-consistently.
major comments (1)
- [Abstract] The headline 1–16% radius offsets (abstract) are obtained by integrating the new continuous solid-to-melt EoS over the full P–T path of 0.1–100 M⊕ planets. The only quantitative validation reported is recovery of PREM (0.3% radius, 3% lower-mantle density). No independent checks against melt-density, sound-speed, or shock data at T>2000 K and P>100 GPa are mentioned; any few-percent systematic density error in the melt branches would propagate directly into the quoted radius shifts and the comparison with composition degeneracy.
minor comments (1)
- [Abstract] The abstract states that quantities are “derived analytically via Maxwell relations” yet also that tables are released “on regular P-T grids.” Clarify whether the released tables are direct analytic evaluations or interpolated, and whether thermodynamic consistency (e.g., Maxwell relations) is preserved to machine precision on the grid.
discussion (0)
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