PALEOS: Multiphase Equations of State and Mass-Radius Relations for Exoplanet Interiors

This paper introduces PALEOS, an open-source toolkit that unifies equations of state for iron, silicates, and water across 17 phases to generate self-consistent mass-radius relations, demonstrating that thermal effects and phase transitions (such as magma oceans) significantly alter planetary radii and internal dynamics, thereby resolving degeneracies in interpreting exoplanet observations.

The Gist

Imagine trying to guess what's inside a sealed, glowing ball of rock just by weighing it and measuring how big it looks. For decades, astronomers have done this with exoplanets (planets outside our solar system). They measure the planet's mass (how heavy it is) and its radius (how wide it is). From these two numbers, they try to figure out if the planet is a giant ball of iron, a rocky world like Earth, or a water-covered ice ball.

But there's a problem: The same weight and size can hide two completely different worlds.

This paper introduces a new tool called PALEOS (Planetary Assemblage Layers: Equations of State) to solve this mystery. Here is how it works, explained simply:

1. The "Cookbook" Problem

Think of building a model of a planet like following a complex recipe. To know how dense the planet is, you need to know how the ingredients (iron, rock, and water) behave under extreme pressure and heat.

• The Old Way: Scientists used different "cookbooks" for different ingredients. One book told them how iron behaves, another for rock, and another for water. These books didn't always agree with each other, and they mostly assumed the ingredients were cold and frozen, like a block of ice in a freezer.
• The PALEOS Way: The authors created one master cookbook that speaks the same language for all ingredients. It covers everything from the freezing surface of a planet to the super-hot, super-pressurized core. Crucially, it knows that rock and iron can melt into lava, just like butter melts in a hot pan.

2. The "Hot Rock" Surprise

The biggest discovery in this paper is that temperature changes the size of a planet in a way we used to ignore.

• The Analogy: Imagine a metal bridge. On a cold day, it's short. On a scorching hot day, the metal expands, and the bridge gets longer.
• The Planet Reality: When a planet is very hot (like those orbiting close to their stars), the rock inside it expands. This makes the whole planet bigger without adding any extra mass.
• The Confusion: If you see a large, heavy planet, you might think, "Wow, that must be made of light, fluffy rock!" But PALEOS shows you: "No, it's actually made of heavy iron, but it's so hot that the rock has swollen up to look big."

3. The "Two Faces" of a Planet

The authors used PALEOS to look at two specific planets, WASP-47 e and TOI-1807 b. They found something mind-blowing: For the exact same weight and size, there are two totally different possibilities.

• Scenario A (The "Sleeping Giant"): The planet is relatively cool. It is made mostly of light rock with a tiny iron core. It is geologically dead—no volcanoes, no magnetic field, just a solid, frozen rock.
• Scenario B (The "Active Volcano"): The planet is extremely hot. It is actually made of heavy iron, but the heat has melted the rock into a global ocean of lava (a magma ocean) and the iron core is liquid. This planet would have a magnetic field and active volcanoes.

The Catch: If you only look at the weight and size, you cannot tell which scenario is real. They look identical from the outside, but inside, one is a frozen tomb and the other is a boiling cauldron.

4. Why This Matters

For a long time, scientists assumed planets were like cold, hard stones. PALEOS proves that for many planets, especially those close to their stars, the inside is a thermodynamic system where heat, melting, and pressure are all dancing together.

• The "Degeneracy": This is a fancy word for the paper's main point: Mass and Radius are not enough. You need to know the temperature to know what the planet is actually made of.
• The Solution: PALEOS provides a new map that includes temperature as a key ingredient. It helps astronomers realize that a "rocky" planet might actually be a "hot iron" planet, or vice versa.

In Summary

PALEOS is a new, open-source computer tool that acts like a universal translator for planetary physics. It tells us that we can't just guess a planet's interior by weighing it. We have to ask, "How hot is it?" because heat can make a heavy iron planet look like a light rock, or a dead world look like a living, magnetic one. It turns the simple question of "What is this planet made of?" into a much more complex, but accurate, story about heat, melting, and the hidden lives of distant worlds.

Technical Summary

Problem
Modeling the interiors of rocky and water-rich exoplanets is fundamentally a thermodynamic closure problem. Existing equations of state (EoS) for planetary materials are scattered across disciplines, utilize disparate formalisms, and rarely provide the complete set of thermodynamic quantities (density, entropy, heat capacities, thermal expansion, and adiabatic gradient) required for self-consistent evolutionary models. A critical gap exists in the ability to track the full thermodynamic state of planetary layers across the transition from solid to melt. Current mass–radius (M–R) interpretations often rely on isothermal, cold-EoS assumptions (typically at 300 K), which fail to account for thermal expansion and phase transitions (melting) that significantly alter planetary radii. This leads to a fundamental degeneracy: a single observed mass and radius can correspond to radically different geophysical states (e.g., a cold, solid, dynamo-less world versus a hot, molten world with a liquid core and active magnetic field), a distinction that is observationally relevant for interpreting atmospheric retention and composition in the era of JWST and future missions.

Methodology
The authors present paleos (Planetary Assemblage Layers: Equations of State), an open-source Python toolkit designed to consolidate and unify EoS for three primary planetary materials: iron (Fe), magnesium silicate (MgSiO$_3$), and water (H$_2$O).

• Thermodynamic Framework: The code adopts a modular decomposition of total pressure into static-lattice ($P_{cold}$), thermal-phonon ($P_{th}$), electronic excitation ($P_{el}$), and magnetic ($P_{mag}$) contributions. It employs five distinct cold-pressure formalisms (Birch-Murnaghan, Vinet, Holzapfel, Keane, Kunc) and various thermal models (Mie-Grüneisen-Debye, Einstein, RT-press) selected based on material and pressure range.
• Analytical Derivation: For each material, paleos derives density, specific internal energy, entropy, heat capacities ($C_P, C_V$), thermal expansion ($\alpha$), and adiabatic gradient ($\nabla_{ad}$) analytically from the underlying pressure–volume–temperature relations using Maxwell relations, ensuring internal thermodynamic consistency.
• Phase Handling: The framework covers 17 thermodynamic phases (including solid polymorphs, liquids, and ices) and implements automatic phase selection across a domain of 1 bar to 100 TPa and 100 to $10^5$ K. It includes continuous solid-to-melt transitions, allowing for the modeling of magma oceans.
• Calibration and Validation: The authors perform reference-state calibration to ensure entropy jumps across phase boundaries satisfy the Clausius-Clapeyron relation and the second law of thermodynamics. The framework is validated against the Preliminary Reference Earth Model (PREM).
• Lookup Tables: To facilitate efficient integration in interior structure codes, the EoS are compiled into high-resolution lookup tables on regular logarithmic pressure–temperature grids.

Key Contributions

1. Unified EoS Framework: A single, thermodynamically consistent framework for Fe, MgSiO$_3$, and H$_2$O that provides all necessary thermodynamic derivatives for interior structure integration, including phase-aware transitions.
2. Mass–Radius Grid: The computation and public release of a grid of 17,900 mass–radius relations for rocky (Fe + MgSiO$_3$) and water-rich (Earth-like core + H$_2$O envelope) planets. This grid spans masses from 0.1 to 100 $M_\oplus$ and surface potential temperatures ($T_{pot}$) from 300 to 4000 K.
3. Demonstration of Degeneracy: A detailed analysis of the composition–temperature degeneracy applied to two ultrashort-period (USP) super-Earths, WASP-47 e and TOI-1807 b.

Results

• Validation: When applied to a two-layer Earth model (pure Fe core, pure MgSiO$_3$ mantle), paleos recovers Earth's radius to within 0.3% and lower-mantle densities to within 3%. The model correctly reproduces the dual-phase core structure (solid inner core, liquid outer core) and the mantle phase sequence (enstatite $\to$ high-pressure clinoenstatite $\to$ bridgmanite $\to$ post-perovskite) without ad hoc tuning, provided a core-mantle boundary temperature of $\sim$4370 K is imposed.
• Thermal Expansion Effects: The study finds that thermal expansion becomes a first-order effect for planets with $T_{pot} > 1500$ K. For low-mass silicate planets, the radius offset due to thermal expansion alone exceeds 1% above 1500 K and reaches 16% at 4000 K. This signal is comparable in amplitude to the iron–rock–water composition degeneracy and current transit radius measurement uncertainties.
• Phase Transitions: The inclusion of continuous solid-to-melt EoS reveals that once an adiabat crosses the melting curve, the resulting liquid silicate layer is significantly less dense than crystalline phases, causing the mass–radius curve to lift off steeply from cold benchmarks.
• Case Studies (WASP-47 e and TOI-1807 b): The authors demonstrate that for these USP planets, the observed mass and radius are consistent with two distinct, purely rocky interior solutions:
  • Solution A (Cool): A fully solid interior with a smaller iron core fraction (CMF) and no magma ocean.
  • Solution B (Hot): A fully molten silicate mantle (magma ocean) and a liquid iron core (potentially sustaining a dynamo) with a significantly higher CMF.
    These solutions are indistinguishable in mass and radius but represent radically different geophysical states. The degeneracy implies that a hotter interior requires a more massive iron core to reproduce the same radius as a cooler, less dense silicate mantle.

Significance
The paper argues that phase-aware, thermally resolved EoS are essential for translating astronomical observations into exoplanetary geophysics. The classical assumption of isothermal, cold interiors introduces systematic biases in inferred compositions, particularly for hot USP planets where thermal expansion and melting are significant. By making the mantle potential temperature ($T_{pot}$) an explicit parameter, paleos allows researchers to map observed mass–radius pairs to a curve in the (CMF, $T_{pot}$) plane rather than a single point. This framework provides the necessary prior for future interior retrievals that aim to distinguish between geologically dead and active worlds, a distinction critical for understanding atmospheric retention, magnetic field generation, and surface habitability. The authors position paleos as a foundational tool for the next generation of exoplanet characterization missions (e.g., PLATO, Ariel) where interior model systematics must be controlled to match observational precision.