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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Revisiting the First, Second and Combined Laws of Thermodynamics

The present study revisits the first, second, and combined laws of thermodynamics by introducing partial internal energy in the first law, entropy production and partial entropy in entropy change, and partial volume in the context of work exchange. These developments yield a more rigorous and internally consistent thermodynamic framework, within which chemical potential is defined as the partial derivative of internal energy with respect to composition at constant entropy and volume, and is related to partial internal energy, partial entropy, and partial volume.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Thermodynamics and its prediction and CALPHAD modeling: Review, state of the art, and perspectives

Thermodynamics is a science concerning the state of a system, whether it is stable, metastable, or unstable, when interacting with its surroundings. The combined law of thermodynamics derived by Gibbs about 150 years ago laid the foundation of thermodynamics. In Gibbs combined law, the entropy production due to internal processes was not included, and the 2nd law was thus practically removed from the Gibbs combined law, so it is only applicable to systems under equilibrium, thus commonly termed as equilibrium or Gibbs thermodynamics. Gibbs further derived the classical statistical thermodynamics in terms of the probability of configurations in a system in the later 1800's and early 1900's. With the quantum mechanics (QM) developed in 1920's, the QM-based statistical thermodynamics was established and connected to classical statistical thermodynamics at the classical limit as shown by Landau in the 1940's. In 1960's the development of density functional theory (DFT) by Kohn and co-workers enabled the QM prediction of properties of the ground state of a system. On the other hand, the entropy production due to internal processes in non-equilibrium systems was studied separately by Onsager in 1930's and Prigogine and co-workers in the 1950's. In 1960's to 1970's the digitization of thermodynamics was developed by Kaufman in the framework of the CALculation of PHAse Diagrams (CALPHAD) modeling of individual phases with internal degrees of freedom. CALPHAD modeling of thermodynamics and atomic transport properties has enabled computational design of complex materials in the last 50 years. Our recently termed zentropy theory integrates DFT and statistical mechanics through the replacement of the internal energy of each individual configuration by its DFT-predicted free energy. The zentropy theory is capable of accurately predicting the free energy of individual phases, transition temperatures and properties of magnetic and ferroelectric materials with free energies of individual configurations solely from DFT-based calculations and without fitting parameters, and is being tested for other phenomena including superconductivity, quantum criticality, and black holes. Those predictions include the singularity at critical points with divergence of physical properties, negative thermal expansion, and the strongly correlated physics. Furthermore, those individual configurations may thus be considered as the genomic building blocks of individual phases in the spirit of the materials genome®. This has the potential to shift the paradigm of CALPHAD modeling from being heavily dependent on experimental inputs to becoming fully predictive with inputs solely from DFT-based calculations and machine learning models built on those calculations and existing experimental data through newly developed and future open-source tools. Furthermore, through the combined law of thermodynamics including the internal entropy production, it is shown that the kinetic coefficient matrix of independent internal processes is diagonal with respect to the conjugate potentials in the combined law, and the cross phenomena that the phenomenological Onsager flux and reciprocal relationships are due to the dependence of the conjugate potential of a molar quantity on nonconjugate molar quantities and other potentials, which can be predicted by the zentropy theory and CALPHAD modeling.

42 ENGINEERING↗
DOE OSTI2025

Revising the theory of cross phenomena

In the present paper, the theory of cross phenomena (TCP) proposed by the present author is revised by starting from the first law of thermodynamics instead of the combined law of thermodynamics in the previous publication because the potentials in the latter are not independent of each other. TCP aims to derive the flux equations with respect to independent driving forces from the fundamental laws of thermodynamics rather than the phenomenological Onsager flux equations, which are flawed as shown by the revised TCP.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Coupled Aerodynamic and Hydrodynamic Hybrid Simulation of Floating Offshore Wind Turbines

The development and innovation of floating offshore wind energy in the U.S. requires detailed high-fidelity observations and measurements of turbine and platform loading due to wind, waves, and currents. However, full-scale and quasi-full-scale experiments require significant financial and temporal investments for construction, experimental testing, and long-term field campaigns. To support the commercial advancement of the offshore wind energy industry, specialized wind tunnel and wave basin experimental facilities are critical to be able to test FOWT designs at small scale under controlled conditions prior to full-scale deployment. Oregon State University (OSU) is internationally known as a leader in water and energy research, development, and testing. The O.H. Hinsdale Wave Research Laboratory (HWRL) and the Wallace Energy Systems and Renewables Facility (WESRF) at OSU have extensive experience building, modeling, monitoring, controlling, and actuating scaled systems. Experiments on wave-structure interaction have been performed at the HWRL since its establishment in 1972. Studies have included the interaction of waves with coastal structures (breakwaters, seawalls, buildings, cylinders, bridges, fixed foundations of offshore wind turbines, etc.) and with floating structures (e.g., wave energy converters, maneuvering of vessels, etc.). Hinsdale is actively used by marine energy technology developers, both for private testing and OSU-collaborative research projects. However, despite the availability of several large-scale facilities for hydrodynamic testing (at OSU and elsewhere in the U.S.), existing experimental laboratories are generally limited in their ability to accurately generate combined wind and wave conditions. The simulation of both wind and waves in experimental testing is complicated due to a number of constraints, including: [i] incompatible similitude laws governing the wind and waves for scaled experiments, [ii] producing accurate wind over a large enough control volume via fans, and [iii] generating wind that reasonably represents the atmospheric boundary layer in existing wave basins/flumes. Hence, physical test data providing insight into the simultaneous wave- and wind-structure response of floating offshore wind components can be difficult to generate. Given the aforementioned challenges in classic hydrodynamic experiments, the motivation of this project is to establish a real-time hybrid simulation (RTHS) approach that can apply aero- and hydro-dynamic loading by augmenting wave-only experimental facilities with virtual aerodynamic forces through numerical models representing the remaining dynamic forces. RTHS is a physical-numerical approach that partitions a prototype system into physical and numerical sub-assemblies that interact with each other through actuators and sensors in real time. In coupling physical and numerical models, the hybrid simulation approach applied herein is ideal for problems with: (1) structures subjected to different scaling laws, such as floating offshore wind turbines subjected to combined aero/hydro-dynamic loading, (2) structures that are too large or complex to be tested entirely in a laboratory setting, such as deep-water mooring applications, and (3) component testing, where the behavior of a portion of the assembly is uncertain but still interacts with other portions of the structure, such as testing the fatigue life of turbine blades. Few U.S. experimental facilities are able to test simultaneous aero- and hydro-dynamic loading and none can accurately produce aero/hydro-dynamic response on scaled FOWT models due to conflicting similitude laws between the wind (commonly Reynolds) and the waves (commonly Froude). To aid in accelerating the development of the U.S. floating offshore industry, there is a significant need to develop a flexible, modular framework that can expand the capacities of existing wave-only laboratories. The project goal is to demonstrate a hydrodynamic real-time hybrid simulation (hydro-RTHS) framework that couples numerical wind and physical waves acting on a FOWT, thus representing simultaneous aero/hydro-dynamic loading. The FOWT is partitioned into a full-scale numerical sub-assembly associated with the aerodynamics and a model-scale physical sub-assembly associated with the hydrodynamics. The numerical-physical partition associated with hydro-RTHS mitigates scaling constraints by supplying different scaling laws to the physical and numerical sub-assemblies. Herein, length, force, and time are scaled and exchanged between the sub-assemblies using Froude scaling to represent the open-channel flow in the physical sub-assembly. Other similitude laws could also be utilized depending on the problem definition. It is envisioned that the ability to model FOWTs under waves and wind, with mitigation of similitude distortions, would result in reduced development costs (currently, FOWT concept development is performed with full-size pro- totypes at enormous expense and risk) and increase the reliability of the FOWT industry (since extreme wave and wind conditions and contingency events can be tested safely in a controlled environment).

16 TIDAL AND WAVE POWER↗
DOE OSTI2025

Preliminary Study on Fine-Grained Power and Energy Measurements on Grace Hopper GH200 with Open-Source Performance Tools

The increasing adoption of tightly integrated, heterogeneous architectures, combined with the slowdown of Moore’s law, has made application power and energy-driven optimizations critical to efficiently use high-performance computing systems. This paper introduces a newly developed open-source toolkit that seamlessly integrates the Linux real-time hardware monitoring program hwmon with the Performance Application Programming Interface and the Score-P performance measurement system, thereby enabling fine-grained power and energy measurements for high-performance computing applications. Our primary target platform is the Wombat test bed, which is a system based on the NVIDIA GH200 superchip. The toolkit can capture transient power peaks with high temporal resolution (50 ms) and, thanks to Score-P integration, can map power metrics to specific code regions, thereby providing actionable information on power-intensive operations and inefficiencies. The toolkit also provides a holistic view of both the power and the energy consumption of the entire GH200 superchip by covering all major components: the Grace CPU, the Hopper GPU, and the I/O subsystem. Experiments that use Locally Self-consistent Multiple Scattering, which is an application for first-principles calculations of materials developed at Oak Ridge National Laboratory, have demonstrated the tool’s ability to identify transient power spikes and uncover opportunities for energy-aware optimizations. Additionally, we introduce a Python-based utility for converting Open Trace Format 2 traces to Parquet format, thus enabling advanced data analysis for numerical integration methods applied to power data for accurate energy profiling.

Hernandez Mendoza, Oscar [ORNL] (ORCID:00000002538↗
DOE OSTI2026

OzDES Reverberation Mapping of Active Galactic Nuclei: Final Data Release, Black-Hole Mass Results, & Scaling Relations

Over the last decade, the Australian Dark Energy (OzDES) collaboration has used Reverberation Mapping to measure the masses of high redshift supermassive black holes. Here we present the final review and analysis of this OzDES reverberation mapping campaign. These observations use 6-7 years of photometric and spectroscopic observations of 735 Active Galactic Nuclei (AGN) in the redshift range 0.13-3.85 and bolometric luminosity range 44.3 - 47.5 erg/s. Both photometry and spectra are observed in visible wavelengths, allowing for the physical scale of the AGN broad line region to be estimated from reverberations of the H{̱e̱ṯa̱}̱, MgII and CIV emission lines. We successfully use reverberation mapping to constrain the masses of 62 super-massive black holes, and combine with existing data to fit a power law to the lag-luminosity relation for the H{̱e̱ṯa̱}̱ and MgII lines with a scatter of ~0.25 dex, the tightest yet identified, fit specifically for consistency with high redshift AGN. We fit a similarly constrained relation for CIV, resolving a tension with the low luminosity literature AGN by accounting for selection effects arising from finite survey length. We also examine the impact of emission line width and luminosity (related to accretion rate) in reducing the scatter of these scaling relationships and find no significant improvement over the lag-only approach for any of the three lines. Using these relations, we further estimate the masses and accretion rates of 246 AGN with single epoch methods. We also use these relations to estimate the relative sizes of the H{̱e̱ṯa̱}̱, MgII and CIV emitting regions. In short, we provide a comprehensive benchmark of high redshift AGN reverberation mapping at the close of this most recent generation of surveys, including light curves, time-delays, and a set of significantly improved radius-luminosity relations for use with high-redshift populations.

McDougall, Hugh [Queensland U.]↗
DOE OSTI2026

Test of the Gravitational Force Law on Cosmological Scales Using the Kinematic Sunyaev-Zeldovich Effect

The mean pairwise velocity of massive halos reflects the gravitational force law on cosmic scales. For this work, we combine cosmic microwave background intensity maps from the Atacama Cosmology Telescope and a galaxy catalog from the Sloan Digital Sky Survey to estimate the mean pairwise velocity using the kinematic Sunyaev-Zeldovich (kSZ) effect. On scales from 30 to 230 megaparsecs, we constrain the gravitational acceleration between pairs of halos at separation 𝑟 to be 𝑔 ∝ 1/𝑟 𝑛 with 𝑛 = 2.1 ± 0.3, which is consistent with Newtonian gravity in an expanding spacetime (i.e., the standard Λ⁢ CDM model). This constraint shows agreement with an inverse quadratic radial dependence over the large distances separating galaxy halos, as expected in standard cosmology. Upcoming surveys have the potential to rule out 𝑛 = 1 at 10⁢𝜎 significance. Our results establish the kSZ effect as a powerful tool for testing gravity on cosmological scales.

alternative gravity theories↗
DOE OSTI2025

Quantifying fission gas adsorption onto natural clinoptilolite in the presence of environmental air and water

Adsorption of noble gas fission products onto naturally occurring minerals is of interest for its potential to retain or retard emissions from nuclear fuel reprocessing operations or underground nuclear explosions. However, experimental studies of trace noble gas adsorption in the presence of air and water have largely focused on synthetic materials, such as activated carbon or metal-organic frameworks. Here, in this study, adsorption of Kr and Xe onto the naturally occurring zeolitic mineral clinoptilolite is studied in the presence of nitrogen and water. By varying the composition of the gas phase and monitoring the change in the combined adsorbate mass, the adsorbed concentration of noble gas is calculated gravimetrically. For dry clinoptilolite, the concentration of adsorbed Kr and Xe is linearly correlated with noble gas pressure and Henry's Law appears satisfactory, despite the presence of nitrogen at atmospheric pressures. However, the presence of water significantly reduces the adsorbed concentration of both Kr and Xe, which is typical in nanoporous sorbents. Here, an empirical bivariate model is presented, combining the Henry's Law adsorption model for a dry adsorbent with the exponential reduction in the presence of water, as reported by Lungu and Underhill in 1999. This model provides a means to estimate the adsorbate concentration at the trace partial pressures and higher water contents relevant to field-scale modeling of fission gas transport through the vadose zone.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2026

Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation law arising from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By minimizing the reproducing kernel Hilbert space norm while penalizing kernel complexity through maximum likelihood estimation, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface flow in fracture networks and arterial blood flow. Finally, the results demonstrate that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical.

Dirichlet-to-Neumann map↗
DOE OSTI2026

Melter Tests to Define LAW Halide Concentrations, Phase 2

The present work represents the second of two phases to address the formation of secondary sulfate phases while processing high waste loading LAW formulations with elevated halide, chromium, and phosphate feed concentrations over a wide range of LAW waste types, including those with high potassium concentrations. The Test Plan for the present work outlines a series of tests designed to assess the formation of secondary phases over a range of sulfur and halide concentrations for two LAW streams and associated glass formulations as well as crucible testing to identify additives that have the potential to suppress the formation of secondary phases for wastes with high chlorine contents. The combined results from both phases of testing and from previous tests with these LAW waste streams will be used as inputs to future work to develop concentration limits for these components in the LAW glass property composition models and to support the modification and updating of the current WTP LAW glass formulation algorithm.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2025

Infinite temperature at zero energy

We construct a family of static, geometrically local Hamiltonians that inherit eigenstate properties of periodically-driven (Floquet) systems. Our construction is a variation of the Feynman-Kitaev clock -- a well-known mapping between quantum circuits and local Hamiltonians -- where the clock register is given periodic boundary conditions. Assuming the eigenstate thermalization hypothesis (ETH) holds for the input circuit, our construction yields Hamiltonians whose eigenstates have properties characteristic of infinite temperature, like volume-law entanglement entropy, across the whole spectrum -- including the ground state. We then construct a family of exactly solvable Floquet quantum circuits whose eigenstates are shown to obey the ETH at infinite temperature. Combining the two constructions yields a new family of local Hamiltonians with provably volume-law-entangled ground states, and the first such construction where the volume law holds for all contiguous subsystems.

FOS: Physical sciences↗
DOE OSTI2026

Evaluating the limitations of Bayesian metabolic control analysis

Bayesian Metabolic Control Analysis (BMCA) is a promising framework for inferring metabolic control coefficients in data-limited scenarios, combining Bayesian inference with linear-logarithmic (lin-log) rate laws. These metabolic control coefficients quantify how changes in enzyme activities affect steady-state fluxes and metabolite concentrations across a metabolic network. However, its predictive accuracy and limitations remain underexplored. This study systematically evaluates BMCA’s ability to infer elasticity values, flux control coefficients (FCC), and concentration control coefficients (CCC) under varying data availability conditions using three synthetic metabolic network models. We demonstrate that BMCA predictions are highly dependent on the inclusion of flux and enzyme concentration data, with the omission of these datasets leading to severe inaccuracies. In our synthetic, enzyme-perturbation datasets, external metabolite concentrations had minimal impact and, in some cases, their exclusion improved predictions; when external-nutrient perturbations were introduced and those concentrations were observed, gains were at most modest. Additionally, we find that posterior estimation with both ADVI and HMC can underestimate large-magnitude elasticities in our synthetic settings, with ADVI showing somewhat higher variance under strong up-regulation; thus, recovering |elasticity| ≳ 1.5 remains challenging regardless of the inference engine. ADVI also fails to accurately infer allosteric interactions, even when regulatory effects are strong. While BMCA maintains reasonable accuracy in partially recovering the rankings of the highest FCC values, its estimates of absolute values remain constrained by prior assumptions and data limitations. Our findings reveal the BMCA algorithm’s strengths and weaknesses, providing guidance on its application in metabolic engineering, and highlighting the need for methodological refinements to enhance its predictive capabilities.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2026

FunDiff: diffusion models over function spaces for physics-informed generative modeling

Recent advances in generative modeling-particularly diffusion models and flow matching-have been widely used for synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. To address this, we introduce FunDiff, an efficient and robust framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generates continuous functions that can be evaluated at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics-informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, demonstrating that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions. We further demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results indicate that our method can generate physically consistent samples with high fidelity to the target distribution, and exhibit robustness to noisy and low-resolution data.

Wang, Sifan [Yale University, New Haven, CT (Unite↗
DOE OSTI2025

Dominant end-tunneling effect in two distinct Luttinger liquids coexisting in one quantum wire

Luttinger liquids occupy a notable place in physics as one of the most understood classes of quantum many-body systems. The experimental mission of measuring its main prediction, power laws in observable quantities, has already produced a body of exponents in different semiconductor and metallic structures. Here, we combine tunneling spectroscopy with density-dependent transport measurements in the same quantum wires over more than two orders of magnitude in temperature to very low electron temperatures down to ∼40 mK. This reveals that, when the second 1D subband becomes populated, the temperature dependence splits into two ranges with different exponents in the power-law dependence of the conductance, both dominated by the finite-size effect of the end-tunneling process. This result demonstrates the importance of measuring the Luttinger parameters as well as the number of modes independently through spectroscopy in addition to the transport exponent in the characterization of Luttinger liquids. This opens a pathway to unambiguous interpretation of the exponents observed in quantum wires.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2026

Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study

Neural scaling laws play a pivotal role in the performance of deep neural networks and have been observed in a wide range of tasks. However, a complete theoretical framework for understanding these scaling laws remains underdeveloped. In this paper, we explore the neural scaling laws for deep operator networks, which involve learning mappings between function spaces, with a focus on the Chen and Chen style architecture. These approaches, which include the popular Deep Operator Network (DeepONet), approximate the output functions using a linear combination of learnable basis functions and coefficients that depend on the input functions. We establish a theoretical framework to quantify the neural scaling laws by analyzing its approximation and generalization errors. We articulate the relationship between the approximation and generalization errors of deep operator networks and key factors such as network model size and training data size. Moreover, we address cases where input functions exhibit low-dimensional structures, allowing us to derive tighter error bounds. These results also hold for deep ReLU networks and other similar structures. Our results offer a partial explanation of the neural scaling laws in operator learning and provide a theoretical foundation for their applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Multi-modal characterization of nitrate reduction nano-catalysts with periodic strain distribution

Strain engineering serves as a pivotal strategy to optimize catalytic activity in electrocatalysis. However, the catalyst sizes under industrial conditions are usually large and even beyond nanometer regime. The critical methodological limitations on strain imaging of such catalysts with both large field of view and high spatial resolution obscure the mechanistic understanding of strain-performance correlations. Here, we present an optimized four-dimensional scanning transmission electron microscopy (4D-STEM) method to acquire strain mapping of both bulk and surface across particles up to 500 nm with 0.6 nm spatial resolution and 0.55% precision. We observe the ripple-like periodic strain coupled with elemental fluctuations inside a perovskite-type hydroxide CuCoSn(OH) 6 and find it correlated to electrocatalytic nitrate reduction (NO 3 – RR) absorption energy to achieve the 92.6% Faradaic efficiency and long-term test over 1000 h at membrane electrode assembly (MEA) for ammonia electrosynthesis. This universal framework design offers a practical method that not only develops an advanced measurement combining multi-modal characterization techniques but also reveals the intrinsic structure-property constitutive law of industry-level catalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Phononic Casimir Effect in Planar Materials

The phononic Casimir effect between planar objects is investigated by deriving a formalism from the quantum partition function of the system following multiscattering approach. This fluctuation-induced coupling is mediated by phonons modeled as an effective elastic medium. We find that excitations with three types of polarizations arise from the resolved boundary conditions, however the coupling is dominated by only one of these degrees of freedom due to exponential suppression effects in the other two. Here, the obtained scaling laws and dependence on materials properties and temperature suggest effective pathways of interaction control. Scenarios of materials combinations are envisioned where the phononic Casimir effect is of similar order as the standard Casimir interaction mediated by electromagnetic fluctuations.

Acoustic phonons↗
DOE OSTI2026

Constraining the Dynamo Layers in Jupiter and Saturn with Observations and Scaling Laws

The dipole-dominated magnetic fields of Jupiter and Saturn provide evidence for active dynamos operating within their deep interiors, yet the depth of the convecting dynamo layers remains poorly constrained. While magnetic field observations, gravity data, and interior models each provide partial insight, they have not been combined into a single, self-consistent picture of the internal structure. Here, we develop a framework that links observed magnetic field strength with intrinsic heat flux and gravity-constrained interior structure using energy-based dynamo scaling laws. By relating the axial magnetic field strength to the convective power, we infer the radial thickness of the dynamo-generating region for both Jupiter and Saturn. The constants of proportionality in the scaling relations are derived using independent constraints from Earth observations, Jupiter observations, and numerical dynamo simulations. Applied to Jupiter, this framework shows how the inferred dynamo layer thickness is coupled to the outer boundary of the dynamo region. Thinner dynamo layers are predicted when the outer boundary shifts to shallower depths, and no solutions are possible when the outer boundary is less than 73% of Jupiter’s radius. These results constrain plausible geometries for future numerical dynamo simulations. Extending the analysis to Saturn, we find a thick, deep-seated dynamo layer with an outer radius at 42% of the radius to be most plausible. An alternative solution with an inner radius of the dynamo region at 60% of the planetary radius, as suggested by ring seismology models, requires a very thin dynamo layer, occupying only 2%–3% of the total radius.

Geosciences↗