The habitable zone (HZ) concept, first alluded to by Newton, describes the orbital region around a star where a planet could maintain liquid surface water. Modern definitions of the HZ rely on climate models considering factors like greenhouse gas concentrations and planetary albedo. While current estimations place the HZ for a sun-like star between 0.97 and 1.70 AU, this is based on a simplified model using only H2O and CO2 as greenhouse gases, neglecting others like H2 or CH4. More complex models suggest the HZ might extend further, depending on planetary conditions. However, residing within the HZ doesn't guarantee habitability; sufficient greenhouse warming is crucial. Mars, for example, lies within the Sun's HZ but lacks sufficient greenhouse warming due to insufficient CO2. The carbonate-silicate weathering cycle is hypothesized to regulate CO2 levels on Earth-like planets, maintaining a clement climate despite the Sun's brightening over geologic time. This cycle involves atmospheric CO2 dissolving in water, weathering silicates, and ultimately forming marine carbonate minerals. The process is reversed through outgassing. This feedback loop is posited to stabilize planetary climates over millions of years. Previous climate calculations assumed a functioning carbonate-silicate cycle but didn't explicitly model it. This study addresses the need to quantify the relationship between atmospheric CO2 and orbital distance to enable testing of the HZ hypothesis using future telescopic observations from missions such as HabEx and LUVOIR. The authors aim to determine the number of exoplanet observations required to confirm the relationship and validate the HZ hypothesis.

The concept of a habitable zone has evolved significantly since Newton's initial observations. Early work by Kasting and others established foundational models for HZ limits, mainly focusing on CO2 and H2O as the primary greenhouse gases. Subsequent research incorporated more complex factors like planetary rotation, atmospheric composition (including CH4), and the impact of other greenhouse gases. The role of the carbonate-silicate cycle in regulating long-term climate stability has been a significant focus. Walker et al. proposed the weathering thermostat hypothesis, explaining Earth's climate stability despite the Sun's changing luminosity. Studies by Berner, Sleep and Zahnle explored the interplay between the carbon cycle and the evolution of Earth's climate. Krissansen-Totton et al. developed sophisticated geological carbon cycle models, enhancing our understanding of this complex interplay. However, much uncertainty remains about the exact relationship between incident stellar flux, atmospheric CO2, and long-term habitability. This study builds upon this foundation by incorporating the carbonate-silicate cycle explicitly into a model to predict the distribution of habitable planets in the HZ.

The study employs a coupled climate and carbonate-silicate weathering model to simulate the climates of Earth-like planets within the HZ. The model incorporates various planetary properties (detailed in Table 1), including CO2 outgassing fluxes, weathering coefficients, land area, ocean sediment thickness, and biological weathering contributions. The model calculates steady-state pCO2 and surface temperature for each planet. Planets with globally averaged surface temperatures below 248 K (completely frozen) or above 355 K (rapid water loss) are considered uninhabitable. The authors randomly generated 1050 habitable, stable climates using uniform distributions of the model parameters, eliminating cases that fell outside the habitability limits. They also revisited habitable zone climate theory, showing that the assumption of a constant, temperate surface temperature maintained by the carbonate-silicate cycle is not fully accurate. The model suggests a nearly linear relationship between incident flux (S) and log(pCO2), arising from energy balance, water vapor feedback, and the temperature and pCO2-dependent nature of the continental weathering flux (Fw). A key element of their approach is using a 1D radiative-convective climate model (VPL model) to generate surface temperatures for various pCO2 and incident flux combinations, fitted with a fourth-order polynomial for efficiency. The model accounts for the atmosphere's composition (CO2 and H2O) and includes a simplified version of the original two-box model by Krissansen-Totton et al, merging the ocean-atmosphere and pore space into a single entity for improved convergence. The internal heat of a planet is parameterized using its age, impacting the outgassing rate. Finally, the authors employ a 2D Kolmogorov-Smirnov (KS) test to assess the probability of observing a log-uniform distribution of pCO2 in the HZ, contrasting it with the distribution predicted by their model. This allowed them to estimate the number of Earth-like exoplanets needed to differentiate between a carbonate-silicate-regulated pCO2 distribution and a random one.

The coupled climate and weathering model predicts that atmospheric CO2 abundances should broadly increase with orbital distance in the HZ, though with significant scatter (Fig. 1). This scatter is attributed to variations in the model parameters, representing uncertainties in the carbonate-silicate cycle. The authors demonstrate that the relationship between incident flux (S) and log(pCO2) is approximately log-linear, contrasting with models assuming a constant surface temperature. This log-linear relationship emerges from energy balance considerations and the temperature-dependent nature of weathering. The slope of the best fit line to the simulated planets shows that about half the variance in log(pCO2) is explained by changes in incident flux (R^2 = 0.49, slope = 3.92 ± 0.24). However, a simple log-linear trend alone is insufficient to validate the HZ hypothesis, as a similar trend could arise from a log-uniform distribution of pCO2. Therefore, the authors propose testing the two-dimensional S-pCO2 distribution. They show that at least 83 (2σ) Earth-like exoplanet observations are needed to confidently distinguish between a carbonate-silicate-regulated distribution and a random log-uniform distribution (Fig. 3). Analysis of proposed space telescope capabilities (HabEx and LUVOIR) indicates that only LUVOIR-A (15 m diameter) would likely provide sufficient observations to achieve this. Furthermore, the spread in pCO2 at different orbital distances is predominantly driven by variations in intrinsic planetary properties (CO2 outgassing rate and continental properties) rather than uncertainties in the weathering model parameters. Figure 5 shows how different parameters affect the spread of steady-state pCO2 values. Figure 6 further shows that even with fixed continental properties, the temperature and pCO2 dependencies of weathering will still generate significant spread in the pCO2 values particularly in the outer HZ.

The model's assumption that Earth's carbon cycle parameter variations are representative of habitable exoplanets is a reasonable first approximation but acknowledges limitations. The validity of this assumption likely depends on the specific parameter. Future improvements in understanding Earth's weathering processes could reduce these uncertainties. Other weathering feedbacks, such as reverse weathering, weren't included in this model due to limited constraints, but their inclusion may affect future predictions. The study notes that near the HZ boundaries, where surface liquid water may not exist, the carbonate-silicate cycle's dominance may cease. The model limitations highlight that the predicted relationship between incident flux and pCO2 may not perfectly represent all habitable exoplanets. Planets vastly different from Earth (e.g., water worlds, methane-rich worlds) could behave differently. Despite uncertainties, future missions should search for the predicted S-pCO2 relationship, including possible sharp transitions at the HZ inner edge. The absence of this relationship would necessitate a revision of the conventional HZ concept.

This study provides a quantitative framework for testing the conventional HZ concept using observations of exoplanet atmospheres. The authors' coupled climate and weathering model predicts a specific two-dimensional relationship between incident stellar flux and atmospheric CO2, which would serve as evidence supporting the HZ hypothesis and the prevalence of the carbonate-silicate cycle. The required number of observations highlights the need for large-scale exoplanet surveys with advanced instrumentation. Future research should focus on refining the model by incorporating other weathering feedbacks and expanding its applicability to a wider range of planetary types.

The model relies on the assumption that variations in Earth's carbon cycle parameters through time are representative of habitable exoplanets. While a reasonable first approximation, this may not hold true for all exoplanets. The model simplifies the carbonate-silicate cycle and neglects other potential feedback mechanisms, such as reverse weathering. The habitability limits of 248 K and 355 K are approximations, and the actual limits might vary depending on other planetary conditions. The model does not account for the observational uncertainties associated with exoplanet characterization, which could affect the ability to detect the predicted relationships. Furthermore, the study focuses only on Earth-like planets; the relationships might be different for planets with significantly different compositions or internal structures.