Coupled Interior-Atmosphere Evolution Model

• The coupled interior‐atmosphere evolution model is a computational framework that links a planet's deep interior processes with atmospheric dynamics using first‐principles conservation laws.
• It employs iterative coupling between interior structure models and radiative–convective atmospheric modules to derive critical observable parameters such as radius, luminosity, and spectral features.
• The framework applies to both gas giants and rocky planets, informing their formation histories, thermal evolution, and habitability through rigorous sensitivity tests and benchmarking.

A coupled interior-atmosphere evolution model is a computational framework that self-consistently links the physics and chemistry of a planet’s deep interior (structure, thermal evolution, composition, melting, and volcanism) to the parallel evolution of its atmosphere (composition, temperature, climate, escape), with direct exchange of mass and energy across the boundary layer. This paradigm is central to modern exoplanetary and planetary science because observational constraints—such as radius, luminosity, and atmospheric spectra—are interpreted through the lens of these models. Both gas giants and terrestrial planets require distinct but conceptually related coupled frameworks, as exemplified by the latest models such as GASTLI for gas giants and the PACMAN, CHILI, and SERPINT frameworks for rocky worlds (Acuña et al., 2024, Lichtenberg et al., 20 Nov 2025, Krissansen-Totton et al., 2022, Sahu et al., 17 Jan 2025).

1. Physical and Mathematical Foundations

The foundation of a coupled interior-atmosphere evolution model is a set of first-principles conservation equations for both the spherically symmetric interior and the overlying stratified atmosphere, tightly linked at their interface by mass, heat, and chemical fluxes.

Interior Structure and Evolution

• Hydrostatic Equilibrium and Mass Conservation: The planetary interior is described by ordinary differential equations for pressure $P(r)$, temperature $T(r)$, density $\rho(r)$, gravity $g(r)$, and enclosed mass $m(r)$ as functions of radius $r$. For gas/ice giants, these include:

$$\frac{dP}{dr} = -\rho(r)g(r) \qquad \frac{dm}{dr} = 4\pi r^2 \rho(r) \qquad \frac{dg}{dr} = 4\pi G \rho(r) - 2G \frac{m(r)}{r^3}$$

and thermal structure via the adiabatic/Grüneisen gradient (Acuña et al., 2024).

• Thermal Evolution: Time evolution of internal entropy $S$ and planetary luminosity $L$ is computed via energy conservation:

$$\frac{\partial L}{\partial m} = -T\frac{\partial S}{\partial t}$$

For rocky planets, mantle convection is governed by global/boundary-layer or mixing-length theory, with energy equations incorporating radiogenic heating, secular cooling, and (in early epochs) latent heat from solidification (Krissansen-Totton et al., 2022, Ballmer et al., 2021, Lichtenberg et al., 20 Nov 2025).

• Equation of State (EOS): The density and thermodynamics of all major constituents (H/He/water/rock or silicate/iron) are given by either tabulated (e.g., Mazevet et al., SESAME) or ab initio EOS, including mixture laws for multi-component envelopes or mantles (Acuña et al., 2024, Wilkinson et al., 2024).

Atmospheric Structure and Evolution

• Radiative-Convective Equilibrium: The atmospheric module solves either a line-by-line, correlated-$k$, or gray radiative transfer problem, enforcing thermal balance:

$$\frac{dF_\mathrm{net}}{dz} = 0, \qquad F_\mathrm{net} = F_\uparrow - F_\downarrow$$

With convective adjustment applied when the local lapse rate exceeds the adiabatic gradient.

• Vertical Stratification and Cloud Physics: Modern models interpolate in dense (multi-dimensional) grids of pressure-temperature-composition, or solve 1D radiative-convective transport with explicit opacities, to capture the atmospheric temperature, composition, and emergent spectra (Acuña et al., 2024, Wilkinson et al., 2024, Acuña-Aguirre et al., 17 Nov 2025).
• Atmospheric Escape and Weathering: For rocky planets, atmospheric mass budgets include volcanic outgassing, escape (diffusion-limited or energy-limited), and weathering sinks, forming a climate-regulating feedback loop (Lichtenberg et al., 20 Nov 2025, Baumeister et al., 2023, Krissansen-Totton et al., 2021).

2. Coupling Strategies and Numerical Implementation

The interior and atmosphere modules are tightly coupled through boundary conditions, feedbacks, and iterative solution algorithms.

• Boundary Matching: The surface or atmosphere-interior boundary is set at a chosen pressure (commonly 1–1000 bar). At this interface, the atmospheric $T_\mathrm{surf}$ serves as the upper boundary for the interior solution, while interior structure (radius at this $P$ and $T$, and composition) informs the atmosphere (Acuña et al., 2024, Wilkinson et al., 2024).
• Atmospheric Thickness and Observables: The atmospheric thickness $\Delta R_\mathrm{atm}$ above the interior is computed by integrating the hydrostatic equilibrium up to transit pressure (typically $\sim20$ mbar), yielding the full observable planetary radius $$R_\mathrm{pl} = R_\mathrm{inter} + \Delta R_\mathrm{atm}$$ (Acuña et al., 2024).
• Iterative Coupling Algorithm: A typical coupling loop,
  1. Guess $R_\mathrm{guess}$,
  2. Use $g_\mathrm{surf}$ to compute atmospheric structure and extract $T_\mathrm{surf}$,
  3. Solve interior equations with this boundary $T_\mathrm{surf}$, obtain $R_\mathrm{inter}$,
  4. If $$|R_\mathrm{inter} - R_\mathrm{guess}|/\epsilon < \text{tolerance}$$ exit, else iterate,
  5. Once converged, integrate atmospheric thickness to derive $R_\mathrm{pl}$ (Acuña et al., 2024, Acuña-Aguirre et al., 17 Nov 2025).

• Thermal Evolution Integration: Entropy evolution $S(t)$ is integrated using ODE solvers (e.g., scipy.odeint), with initial hot/cold start conditions, inverting $$S(t)\to T_\mathrm{int}(t)\to R_\mathrm{pl}(t), L(t)$$ (Acuña et al., 2024, Wilkinson et al., 2024, Lichtenberg et al., 20 Nov 2025).

3. Applications: Gas Giants vs. Terrestrial Planets

Gas/Ice Giants

• Compositional Inference and Formation Constraints: Coupled models allow robust inference of core mass fraction (CMF), envelope metal mass fraction ($Z_\mathrm{env}$), and atmospheric metallicity $[\mathrm{Fe}/\mathrm{H}]$, given observational constraints (radius, mass, atmospheric spectra). Monte Carlo MCMC retrievals (e.g., emcee, nested sampling) are employed to probe allowed interior-atmosphere parameter spaces (Acuña et al., 2024, Acuña-Aguirre et al., 17 Nov 2025).
• Applications: GASTLI achieves $\lesssim2\%$ accuracy in Solar System gas giants’ radii, and matches the compositional-metallicity trends seen in exoplanet populations (Acuña et al., 2024). HADES provides degenerate-breaking by leveraging radius constraints and JWST spectra, demonstrating the critical role of self-consistent atmosphere-interior modeling for retrievals (Wilkinson et al., 2024).

Rocky/Intermediate Planets

• Mantle Melting, Volcanism, and Climate Feedback: Solidification of the magma ocean, secular interior cooling, mantle convection style (plate tectonics vs. stagnant lid), and melting depths regulate volatile outgassing and thus atmospheric mass and climate. The regime of lid dynamics is decisive for the evolution of habitability, atmospheric composition, and planetary cooling (Ballmer et al., 2021, Baumeister et al., 2023, Lichtenberg et al., 20 Nov 2025).
• Redox-State Coupling: Mantle $f\!O_2$ directly controls volcanic CO2/H2/H2O outgassing, atmospheric greenhouse strength, and stellar-driven water loss. Models that include O2 sinks (crustal oxidation, outgassed reductants) are essential for Venus and Earth analogs (Baumeister et al., 2023, Krissansen-Totton et al., 2021).
• Monte Carlo Ensembles: Recent models (PACMAN, CHILI) use large parameter-space sampling to map out the joint influence of redox, bulk composition, and thermal evolution on observable atmospheric outcomes (Krissansen-Totton et al., 2022, Lichtenberg et al., 20 Nov 2025).

4. Validation, Sensitivity, and Model Intercomparison

Validation and uncertainty quantification are foundational elements for coupled interior-atmosphere models.

• Solar System Benchmarks: Models are validated against the radii and atmospheric constraints of Jupiter, Saturn, Neptune, and planets like HAT-P-26b, finding typical errors of 1–2% for appropriate input parameters, and pinpointing sources of discrepancy (EOS choice, cloud treatment, pressure referencing) (Acuña et al., 2024).
• Intercomparison Protocols: The CHILI protocol defines standardized tests for coupled models across evolutionary and static regimes, providing a scaffold for comparing treatments of convection, partitioning, atmospheric radiative transfer, and outgassing/escape (Lichtenberg et al., 20 Nov 2025). Preliminary findings indicate significant spread owing to: (i) convective efficiency parameterizations, (ii) atmospheric opacity treatment (gray vs. non-gray), and (iii) outgassing/solidification kinetics.
• Sensitivity Analyses: Parametric studies systematically probe EOS choice, transit pressure, C/O ratio, initial entropy, and atmospheric metallicity, quantifying their impacts on recovered bulk properties and spectral predictions. E.g., switching EOS can induce up to 10% changes in radius for Neptune-mass planets, and atmospheric thickness can add >10% to observed radii for water-rich ice planets (Acuña et al., 2024).

5. Observational and Theoretical Implications

Coupled models deliver direct implications for exoplanetary science and planetary evolution:

• Breaking Degeneracies: Coupled retrievals using both mass/radius and panchromatic JWST/HST spectra can disaggregate covariant parameters such as $[\mathrm{Fe}/\mathrm{H}]$, C/O, core mass, and intrinsic luminosity, especially when incorporating age and multi-modal data (transit + emission) (Acuña-Aguirre et al., 17 Nov 2025, Wilkinson et al., 2024).
• Formation Histories: The deduced bulk (core + envelope) metal fractions and atmospheric compositions inform accretion and enrichment scenarios (e.g., pebble accretion, planetesimal ingestion, "hot" vs. "cold" starts), and can test the predictions of core accretion vs. gravitational instability; for instance, high-core-mass hot Jupiters favor core-accretion histories (Wilkinson et al., 2024, Acuña et al., 2024).
• Habitability Constraints: Stagnant-lid vs. mobile-lid coupled models, with explicit volatile cycling and redox feedbacks, bracket the parameter space for sustained surface water, the likelihood and duration of habitable epochs, and bifurcations into runaway greenhouse or snowball states (Baumeister et al., 2023, Ballmer et al., 2021, Lichtenberg et al., 20 Nov 2025).
• Spectral Diagnostics: Coupled models yield both synthetic spectra and evolutionary tracks, predicting how transmission and emission features evolve with time and chemistry—enabling direct links to JWST/ELT/PLATO observations (Acuña et al., 2024, Wilkinson et al., 2024).

6. Best Practices, Challenges, and Future Directions

• Recommended Practices:

1. Explicit definition of surface (or atmosphere-interior) boundary, including referencing to well-defined pressure and temperature levels.
2. Full tabulation of all physical constants, EOS, and mixture rules used for each module, and maintenance of consistency at the coupling interface.
3. Sensitivity testing of all key parameters (EOS, opacity, initial conditions, outgassing efficiency, escape formulation) (Acuña et al., 2024, Lichtenberg et al., 20 Nov 2025).
4. Consistent numerical implementation, with ODE/PDE solvers for time-evolution, interpolation within precomputed multidimensional grids, and clear iteration schemes for convergence.
5. Comparative benchmarking against validation data (Solar System, exoplanet analogs) and participation in standardized intercomparison efforts (e.g., CHILI protocol) (Lichtenberg et al., 20 Nov 2025).

• Ongoing Challenges: Major sources of uncertainty include incomplete knowledge of high-P/T opacities, phase relations, EOS in multi-component systems, efficiency of mixing and volatile partitioning, and the detailed physics of cloud formation and atmospheric escape.
• Future Directions: Integration of improved radiative transfer solvers, ab initio EOS, updated hydrodynamic escape models, and high-dimensional MCMC or emulator-based retrievals is underway (Wilkinson et al., 2024, Acuña-Aguirre et al., 17 Nov 2025). Progressive refinement and cross-validation between codes (e.g., PACMAN, GASTLI, HADES, SERPINT) are establishing robust community standards for coupled interior-atmosphere modeling.

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The coupled interior-atmosphere evolution model is now an essential tool for interpreting exoplanet observations, reconstructing formation and evolutionary histories, and quantifying the conditions for atmospheric chemistry, planetary structure, and habitability (Acuña et al., 2024, Wilkinson et al., 2024, Lichtenberg et al., 20 Nov 2025, Acuña-Aguirre et al., 17 Nov 2025, Baumeister et al., 2023).