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Resolving core-envelope degeneracies in giant planets and sub-Neptunes: constraining the equivalence in the presence of dilute gradients

This paper introduces "fuzzycore," an open-source static structural integrator that resolves core-envelope degeneracies in giant planets and sub-Neptunes by mapping how compositional gradients and dilution widths influence planetary radii, thereby enabling consistent forward-modeling to link atmospheric observations from missions like JWST and Ariel to deep interior architectures.

Original authors: Christian Wilkinson, Benjamin Charnay, Stéphane Mazevet, Baptiste Perrier, Anne-Marie Lagrange, Lukas Delay

Published 2026-07-21

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CC BY 4.0

Original authors: Christian Wilkinson, Benjamin Charnay, Stéphane Mazevet, Baptiste Perrier, Anne-Marie Lagrange, Lukas Delay

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Puzzle: Why Planets Are Hard to Measure

Imagine trying to figure out what's inside a sealed, glowing balloon just by looking at how big it is and how heavy it feels. That is essentially the job of planetary scientists studying exoplanets—worlds orbiting stars other than our Sun. For a long time, scientists thought planets were like Russian nesting dolls: a hard, rocky core at the bottom, a layer of water or ice in the middle, and a fluffy gas atmosphere on top, with sharp lines separating each layer. But recent discoveries have turned this simple picture upside down. We now suspect that inside giant planets and smaller "sub-Neptune" worlds, these layers might not be so distinct. Instead, the heavy stuff (like rock and water) might be slowly mixing into the light gas (hydrogen and helium), creating a "fuzzy" core where everything is blended together, like a smoothie rather than a layered cake.

This mixing creates a massive headache for scientists. If you see a planet with a specific size and weight, there are countless ways to arrange that "smoothie" inside to get the same result. You could have a tiny, dense core with a thin, heavy atmosphere, or a huge, fluffy core with a light atmosphere, and they might look exactly the same from the outside. This is called a "degeneracy," and it means that just measuring a planet's size and mass isn't enough to tell us what it's really made of or how it formed. To solve this, we need a way to test millions of different internal recipes to see which ones actually fit the data, a task that is usually too slow and complicated for our current computer models.

The New Tool: A Cosmic Recipe Tester

Enter fuzzycore, a new open-source computer program designed by C. Wilkinson and their team to tackle this exact problem. Think of fuzzycore as a super-fast, specialized chef who can instantly test millions of different planetary "recipes" without having to wait for the planet to actually cook (evolve) over billions of years. While other advanced programs try to simulate the entire life story of a planet—including how it cools down and how heat moves around—fuzzycore skips the long wait. It focuses purely on the "snapshot" of a planet at a single moment in time, asking: "If we arrange the heavy elements in this specific fuzzy way, does the planet end up the right size?"

The researchers built fuzzycore to handle the messy reality of "fuzzy cores." Instead of assuming a sharp boundary between a solid core and a gas atmosphere, the program allows for a gradual transition where heavy elements slowly fade out as you go up into the atmosphere. They tested this new tool against a known model of Jupiter (our own solar system's giant) and found that it works incredibly well. Even when they changed the shape of the "fuzziness"—making it a smooth curve, a step-ladder, or a sharp drop—the final size of the planet barely changed. In fact, the size stayed within 2.3% of the same result, no matter how they shaped the gradient, as long as the total amount of heavy stuff and the metal content at the edges remained the same. This discovery is huge because it means scientists can use simple, smooth mathematical shapes to represent these complex interiors without losing accuracy.

What They Found: The Great Cosmic Trade-Off

Using fuzzycore, the team mapped out the "degeneracy atlas" for sub-Neptune planets, which are a popular type of world found in our galaxy. They discovered a fascinating trade-off: you can swap a heavy, solid water layer for a lighter, gas-heavy atmosphere, but only if you also change how "diluted" the heavy elements are in that gas.

They ran a massive simulation with 2,155 different successful models for a planet with the mass of 8 M⊕ (8 times the mass of Earth). They found that a planet with a radius of 2.75 R⊕ (2.75 times the radius of Earth) could be built in dozens of different ways. You could have a planet with a thick water ocean and a tiny bit of gas, or one with a thin water layer and a thick, metal-rich atmosphere. The "fuzziness" of the core (represented by a width parameter σ) turned out to be a secret variable that is almost impossible to guess just by looking at the planet's size. Whether the heavy elements were packed tightly at the bottom or spread out loosely, the planet could still end up looking exactly the same from the outside.

One of the most surprising findings was about "super-puffs"—planets that are incredibly light and fluffy. The team showed that if you try to make a super-puff by dumping a lot of heavy metals into its atmosphere to create a fuzzy core, the planet actually shrinks. This is because the heavy metals make the gas denser, crushing the atmosphere inward. This suggests that if we see a super-puff that is also very metal-rich in its atmosphere, something else must be happening to keep it puffed up, like intense heat or escaping winds, because the heavy metals alone would make it too small.

Why This Matters

This paper doesn't claim to have solved the mystery of every planet's interior. Instead, it provides a powerful new map. It shows us exactly where the "blind spots" are. The authors demonstrate that while we can use data from telescopes like JWST to guess the metal content of a planet's atmosphere, we still can't tell how deep that metal goes or how "fuzzy" the core is without more information.

The fuzzycore tool is designed to be a partner to other, slower, more complex computer models. It acts as a rapid scanner, quickly identifying all the possible internal structures that fit the data we have today. Once it finds these possibilities, scientists can then use the slower, more detailed models to see which of those possibilities could actually survive over billions of years. By separating the "what fits now" from the "how it evolved," this work helps astronomers stop guessing and start narrowing down the true nature of the strange, fuzzy worlds orbiting distant stars.

Technical Summary: Resolving Core-Envelope Degeneracies in Giant Planets and Sub-Neptunes

Problem Statement
The internal structure of giant planets and sub-Neptunes represents a highly degenerate, multi-dimensional problem in planetary science. The recognition of "fuzzy" or dilute cores—where heavy elements are distributed in gradients rather than sharp boundaries—and massive volatile mantles has complicated the interpretation of observed planetary radii ( $R_p$ ) and masses ( $M_p$ ). Bridging the "radius valley" requires distinguishing between planets with thick H/He atmospheres, supercritical water oceans, or mixed compositions. Current evolutionary models (Henyey-class codes like MESA, APPLE, CEPAM) are computationally expensive, often prohibiting the dense parameter sweeps necessary to map the full space of static structural degeneracies, particularly regarding the width of compositional gradients ( $\sigma$ ). Furthermore, the specific spatial distribution of heavy elements significantly affects the total radius and gravity moments ( $J_n$ ), yet the functional topology of these gradients remains poorly constrained by observation alone.

Methodology
The authors present fuzzycore, an open-source, static, multi-zone hydrostatic structural integrator designed for the rapid forward-model exploration of interior degeneracies. Unlike evolutionary codes, fuzzycore decouples structural profiling from time-dependent energy transport, convective mixing, and phase-equilibrium critical curves.

Key methodological components include:

Multi-Zone Hydrostatic Equilibrium: The framework solves 1D spherically symmetric equations for mass conservation and hydrostatic equilibrium. It dynamically adapts grid resolution to resolve steep density gradients across interfaces.
Equation of State (EOS) Formulation:
- Mixed Core: Uses an additive volume law (Amagat's law) to mix ab initio tables for iron and multi-phase rock (separating MgSiO3 into MgO and SiO2 at high pressures) to ensure monotonic density stratification.
- Fluid Envelope: Models the envelope as a ternary mixture (H, He, heavy elements) using ideal linear mixing of specific volumes. Heavy elements ( $Z$ ) are proxied by a wide-range H2O EOS to capture mean molecular weight ( $\mu$ ) effects and compressibility across the transition from pristine atmosphere to enriched basal layers.
- Condensed Mantle: Supports a distinct, phase-separated water mantle (Ice VII/X or superionic) for volatile-rich sub-Neptunes, coupled thermodynamically to the envelope base.
Compositional Gradients: Instead of discrete boundaries, heavy-element mass fraction $Z(x)$ is modeled via a continuous Gaussian decay function. To handle numerical stability as gradients sharpen ( $\sigma \to 0$ ), the framework employs analytical sub-grid averaging to conserve the heavy-element budget and prevent unphysical density spikes.
Numerical Integration: The envelope is discretized into a "staircase" of fully mixed convective zones. A robust adiabat stepper method using local linear least-squares regressions over a KD-Tree of the 3D phase space ( $P, T, S$ ) integrates thermodynamic states.
Boundary Shooting: A nested boundary-shooting method (Brent's method) solves the boundary value problem, iterating on interface pressures to match mass, pressure, and temperature continuity between the atmosphere, mantle, and core. It supports both mass-targeting and surface-gravity-targeting modes.
Stability Guards: Includes an "isothermal guard" to prevent numerical cooling artifacts at low-pressure phase transitions and adaptive downsampling for ultra-thin envelopes.

Key Contributions and Results

Functional-Form Robustness: Benchmarking against an evolution-derived Jupiter profile (Vazan et al. 2018) demonstrates that when envelope boundary metallicities and integrated heavy-element budgets are matched, the macroscopic planetary radius is robust to the exact functional topology of the gradient. Six distinct parametric families (Gaussian, sigmoid, exponential, bilinear, single-step, and digitized hydrodynamic staircases) yielded a radius spread of only 2.3%. This validates the use of smooth analytical parameterizations (like Gaussian decay) for mapping degeneracies.
Sub-Neptune Degeneracy Atlas: The framework was applied to a fixed $8 M_\oplus$ $8 M_{\oplus}$ planet to generate a "water-world degeneracy atlas" across a $15 \times 15 \times 12$ $15 \times 15 \times 12$ grid of water mass, envelope mass, and dilution width ( $\sigma$ $σ$ ).
- $\mu$ -Compaction: The study quantifies how redistributing heavy elements from a compact core into a dilute envelope increases the mean molecular weight, physically compacting the atmosphere. This "µ-compaction" can suppress thermal expansion, reducing the planetary radius by up to 45% for fixed heavy-element budgets.
- Thermal Blanketing: Metal-rich envelopes act as thermal throttles, increasing the temperature difference between the surface and core by $\sim 5000$ K, though this is a static structural finding, not a cooling history.
- Observational Limits: For a fixed observed $(M_p, R_p)$ , the dilution width $\sigma$ is essentially unconstrained. The atlas reveals a continuous family of static architectures (varying water mass, envelope mass, and $\sigma$ ) that reproduce the same radius, highlighting that $(M_p, R_p)$ alone cannot resolve the interior structure.
Application to Specific Targets:
- GJ 1214 b (Volatile-rich): Modeled as a compact water-rock body with a thin, highly enriched envelope, converging to a radius within 1% of observations.
- Kepler-11 e (Gas-rich/Super-puff): Modeled as a massive core with a heavily polluted, dilute envelope. The model demonstrates that a deeply suspended heavy-element gradient is required to "ballast" the massive envelope down to the observed compact radius, countering the inflation expected from a pristine H/He envelope.
Atmosphere-Interior Coupling: The framework is designed to serve as a backend for atmospheric retrievals (e.g., coupling with HADES). It can link JWST/Ariel atmospheric metallicity priors to deep interior architectures by solving for self-consistent surface gravity and boundary conditions.

Significance and Scope
The paper positions fuzzycore not as a replacement for fully implicit evolutionary codes, but as a necessary complementary tool. By stripping away the computational overhead of time-dependent mixing and energy transport, fuzzycore enables the systematic, high-resolution mapping of static structural degeneracies that are otherwise too expensive to explore with Henyey-class solvers.

The authors emphasize that fuzzycore identifies mechanically permitted hydrostatic configurations. It does not determine whether these configurations are thermodynamically self-consistent with demixing physics or hydrodynamically stable against turbulent erosion over geological timescales; those questions remain the domain of evolutionary modeling and ab initio phase-equilibrium studies. However, the framework provides the essential "degeneracy atlas" required to interpret current and future observational data, demonstrating that the spatial distribution of heavy elements is a critical, yet observationally degenerate, dimension in planetary characterization.

The Cosmic Puzzle: Why Planets Are Hard to Measure

The New Tool: A Cosmic Recipe Tester

What They Found: The Great Cosmic Trade-Off

Why This Matters

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