Carbonate-silicate cycle predictions of Earth-like planetary climates and testing the habitable zone concept

The study revisits the stellar habitable zone (HZ) concept, traditionally defined by the ability of a CO2-H2O greenhouse to maintain surface liquid water, with inner and outer edges set by water loss and the CO2 maximum greenhouse, respectively. While many planets can lie within the Sun’s HZ, habitability also depends on greenhouse gas abundances regulated by planetary processes. The prevailing hypothesis is that long-term atmospheric CO2 on Earth-like planets is controlled by the carbonate-silicate weathering cycle, providing a climate-stabilizing negative feedback that has kept Earth temperate despite solar brightening. The central research question is how this weathering feedback sets the relationship between incident stellar flux and atmospheric pCO2 for Earth-like planets across the HZ, and whether this predicted relationship can be empirically detected to test the HZ hypothesis.

Classical HZ limits have been estimated by 1D and 3D climate models focusing on H2O and CO2 greenhouse effects, placing Sun-like system limits near 0.97–1.70 AU. Extensions consider additional greenhouse gases (e.g., H2, CH4), rotation, and planetary conditions potentially shifting HZ boundaries. However, CH4-rich atmospheres may be biotically maintained. Prior HZ studies generally assume, but do not explicitly model, carbonate-silicate feedbacks; qualitative expectations suggest pCO2 should increase with orbital distance. Previous work indicated weathering-mediated climate might alter pCO2 predictions in the HZ and that moving an Earth-like planet outward would require higher CO2. More recent studies discuss thermodynamic limits on weathering and the role of seafloor weathering. The present work integrates climate theory with an explicit carbonate-silicate cycle model to quantify both the expected log-linear trend between incident flux and log(pCO2) and the scatter arising from planetary geophysical and geochemical variability.

The authors couple a 1D radiative-convective climate model (VPL 1D model) with a carbonate-silicate weathering model to compute steady-state surface temperature and pCO2 for Earth-like planets across the conservative HZ of a Sun-like star (incident flux S/S0 from ~1.05 to 0.35). Climate model outputs (surface temperature) are represented by a fourth-order polynomial in ln(pCO2) and normalized flux to enable rapid evaluation. The weathering model balances volcanic CO2 outgassing against continental and seafloor weathering with precipitation of carbonates, tracking carbon and alkalinity in a combined atmosphere–ocean–pore space box (simplified from a prior two-box model), validated to within ~3% against the original formulation. Habitability constraints require steady-state surface temperatures between 248 K (to avoid global glaciation) and 355 K (to avoid rapid ocean loss); pCO2 is limited to ≤10 bar. The model includes numerous parameters with broad ranges reflecting Earth’s plausible history (e.g., outgassing flux, temperature and pCO2 dependence of continental weathering, land fraction, biological weathering factor, ocean sediment thickness, seafloor dissolution kinetics, internal heat scaling), sampled uniformly (Table 1). Of 1200 random parameter combinations, 1050 yield habitable steady states forming the modeled planet set. Analytically, combining energy balance theory (linear relation between outgoing longwave radiation and surface temperature) with weathering parameterizations shows Ts decreases with decreasing S, and, for typical ranges of weathering pCO2 and temperature sensitivity, yields an approximately linear relation between S and log(pCO2) for Earth-like planets. To assess detectability of the weathering-imposed 2D distribution in S–pCO2 space, the authors perform 2D Kolmogorov–Smirnov (KS) tests comparing samples drawn from the model planet distribution to a reference distribution with log-uniform pCO2 (10^-4 to 10 bar) and uniformly sampled S (0.35–1.05), under the same habitability temperature bounds. Repeated resampling (10,000 trials) estimates the probability that the weathering distribution could be mistaken for log-uniform, as a function of the number of observed Earth-like planets.

• The coupled model predicts that atmospheric CO2 should increase with orbital distance within the HZ, with an approximately log-linear relationship between incident flux S and log10(pCO2). Surface temperature decreases with decreasing S when weathering feedbacks are included, contrasting with a constant-temperature assumption.
• Across 1050 simulated habitable steady states, a log-linear fit between S and log10(pCO2) yields slope 3.92 ± 0.24 (95% CI) in units of −log10(pCO2[bar]) per (S/S0), with r^2 = 0.49, indicating substantial but structured scatter due to planetary parameter variability.
• A simple slope measurement alone cannot distinguish weathering-regulated pCO2 from a log-uniform pCO2 distribution subject to habitability temperature bounds; a simulated log-uniform case yields a similar slope (3.76 ± 0.465, 95% CI).
• The weathering-regulated 2D distribution in S–pCO2 space is diagnostically different from log-uniform pCO2: it shows an abundance of high-pCO2 planets in the outer HZ, a paucity of low-pCO2 planets at intermediate S (~0.7–0.9 S/S0), and relatively few very high-pCO2 planets overall.
• Using a 2D KS test, at least 83 Earth-like exoplanet observations are required to rule out a log-uniform pCO2 distribution at 95% confidence if the true distribution follows the model prediction.
• Sensitivity analyses indicate that most scatter in pCO2 arises from intrinsic planetary properties: land fraction, biological weathering factor, and internal heat/outgassing (Q). At higher pCO2, the temperature and pCO2 dependence of continental weathering (parameters akin to α and Te) become increasingly important.
• Only the nominal yield of LUVOIR-A (15 m) among proposed HabEx/LUVOIR concepts would likely provide sufficient Earth-like planet detections to confidently discriminate distributions.

The findings integrate climate energy balance and carbonate-silicate weathering feedbacks to revise expectations for HZ climates: rather than constant surface temperature, Earth-like planets should exhibit decreasing temperature and increasing CO2 with orbital distance, approximately linear in S versus log(pCO2). The structured 2D distribution in S–pCO2 offers a test of the HZ hypothesis and the prevalence of Earth-like weathering regulation, provided sufficient sample sizes. The model assumes Earth-like compositions and broad parameter ranges reflective of Earth’s history; this is reasonable given indications of Earth-like rocky exoplanet geochemistry but may not capture all natural variability. Additional processes (e.g., reverse weathering), planets without surface oceans (edge cases), waterworlds without silicate weathering, or CH4-rich atmospheres could complicate observed trends. Detection of surface oceans is important for interpreting any CO2–flux relation, because the weathering thermostat requires liquid water. If future observations fail to find the predicted S–pCO2 relationship, it may imply limited influence of the carbonate-silicate cycle on habitability or the need to revise conventional HZ bounds.

This work predicts and quantifies an approximately log-linear relationship between incident stellar flux and atmospheric CO2 for Earth-like planets operating a carbonate-silicate weathering thermostat, including the expected scatter from geophysical and geochemical diversity. It demonstrates that testing the HZ hypothesis requires measuring the full 2D S–pCO2 distribution, not just a trend slope, and that at least 83 Earth-like exoplanet observations are needed for a confident statistical discrimination from a log-uniform pCO2 distribution. The results guide future mission strategies (e.g., LUVOIR-A) and emphasize the importance of confirming surface oceans. Future research should incorporate observational uncertainties in pCO2 retrievals, consider additional climate–geochemical feedbacks (e.g., reverse weathering), explore more diverse planetary types (e.g., waterworlds), and refine weathering parameterizations with improved Earth and laboratory constraints.

• Assumes Earth-like planets with liquid surface oceans and a functioning carbonate-silicate cycle; results do not apply to dry planets, waterworlds lacking silicate weathering, or planets outside 248–355 K global mean temperature.
• Parameter ranges are broad and based on Earth’s plausible history; true exoplanet variability may be larger or different (e.g., land fraction, outgassing rate, weatherability).
• Simplifies ocean–pore system into a single box; validated to a few percent but still an approximation of more complex dynamics.
• Caps pCO2 at 10 bar and uses a specific humidity framework; outer HZ behavior beyond this regime and cloud effects are simplified by the 1D model and polynomial fit.
• Does not include processes like reverse weathering or additional greenhouse gases (e.g., H2, CH4) that could alter pCO2–S relations.
• Observational inference analysis via 2D KS test may be less reliable for small samples (<~20) and does not include instrument retrieval uncertainties in pCO2.
• Assumes fixed stratospheric H2O and empirical RH profile; albedo and 3D climate dynamics are parameterized via 1D modeling.