## Introduction

Newton first alluded to the concept of a stellar habitable zone (HZ) in his 1687 Principia1 by noting that Earth’s liquid water would vaporize or freeze at the orbits of Mercury and Saturn, respectively2. Later, Whewell noted that “the Earth’s orbit is in the temperate zone of the Solar System”3. Since then, the definition of the stellar HZ has been refined, reaching its modern incarnation based on climate models4,5.

Current HZ calculations6 find that around a Sun-like star, an Earth-like planet could remain habitable between 0.97 and 1.70 AU. The inner edge of the HZ is set by loss of surface water and the outer edge is set by the maximum greenhouse of a CO2 atmosphere where extensive CO2 condensation and increased Rayleigh scattering prevent any further greenhouse warming from CO2 (refs. 6,7). This definition of the HZ only considers H2O and CO2 as greenhouse gases, so Earth-like planets warmed by other greenhouse gases (e.g., H2 or CH4) could remain habitable at bigger orbital distances5,8,9. However, CH4-rich atmospheres in the HZ may not be possible without life to generate substantial CH4 (refs. 10,11). In addition, more complex climate models have shown the HZ might extend to smaller orbital distances, perhaps interior to Venus’ orbit, with appropriate planetary conditions12,13,14,15.

Residing within the HZ does not guarantee habitable surface conditions. Crucially, greenhouse gas abundances (and planetary albedo) must conspire to produce clement surface conditions. For example, by most estimates, Mars resides within the Sun’s HZ but is not habitable because there is insufficient greenhouse warming from CO2, in part because of the lack of volcanic outgassing of CO2. Thus, considering the planetary processes that control atmospheric CO2 abundances on Earth-like planets in the HZ is necessary to constrain planetary habitability.

The prevailing hypothesis is that CO2 levels are controlled by a weathering thermostat16. This can explain why Earth has maintained a clement surface throughout its history despite the  ~30% brightening of the Sun over the past  ~4.5 Gyr17,18,19,20,21. The changing luminosity of the Sun with time is similar to moving a planet through the HZ, and so the same CO2 weathering process responsible for maintaining habitability on the Earth through time, the carbonate–silicate weathering cycle, may similarly stabilize planetary climates within the HZ.

In the carbonate–silicate cycle, atmospheric CO2 dissolves in water and weathers silicates on both the continents and seafloor, which releases cations and anions16,22,23,24,25,26,27. On the continents, the weathering products, including dissolved SiO2, HCO$${\,}_{3}^{-}$$, and Ca++, wash into the oceans where the HCO$${\,}_{3}^{-}$$ combine with cations like Ca++ to create CaCO3, which precipitates out of solution. The net process converts atmospheric CO2 into marine carbonate minerals (i.e., CaCO3). Also, seafloor weathering occurs when seawater releases Ca++ ions from the seafloor basalt and CaCO3 precipitates in pores and veins. Subsequently, the carbonates within sediments and altered seafloor can be subducted.

Carbon returns to the atmosphere from outgassing. If CO2 outgassing increases above the steady-state outgassing rate, a planet’s surface temperature rises. This leads to increased rainfall and continental weathering as well as potentially warmer deep-sea temperatures and more seafloor weathering21,24,28. Increased weathering draws down atmospheric CO2 and stabilizes the climate over   ~106-year timescales on habitable, Earth-like planets29.

One- and three-dimensional (1D, 3D) climate calculations of HZ limits4,6,14 assume that a carbonate–silicate weathering cycle is functioning but do not explicitly include it. The assumed presence of the carbonate–silicate cycle would predict that atmospheric CO2 of Earth-like planets increases with orbital distance in the HZ4,6,29. In particular, future telescopic observations, e.g., NASA’s Habitable Exoplanet Imaging Mission (HabEx)30 and Large Ultraviolet Optical Infrared Surveyor (LUVOIR)31, could search for the CO2 trend to test the HZ hypothesis32,33,34. Previous studies29,35 have suggested the carbonate–silicate weathering cycle could alter predictions of pCO2 in the HZ, but it is important to know the exact relationship we are looking for. Also, while an increase of pCO2 with orbital distance in the HZ may be true if all Earth-like exoplanets have the exact same properties as the modern Earth, the trend becomes less certain if HZ planetary characteristics deviate from those of the modern Earth. There could be considerable variability in atmospheric CO2 throughout the HZ, perhaps even enough to obscure a monotonic trend with orbital distance.

Here, we show that uncertain physicochemical and geophysical parameters in the carbonate–silicate weathering cycle26 cause scatter in pCO2 with orbital distance. We then demonstrate that future telescopes must observe at least 83 (2σ) HZ planetary atmospheres to confidently detect our predicted relationship between atmospheric CO2 and orbital distance, and confirm the HZ hypothesis.

## Results

### Stable pCO2 abundances from our numerical model

We use a coupled climate and carbonate–silicate weathering model (see Methods, subsection “Numerical carbonate–silicate cycle modeling”) to explore pCO2 on Earth-like planets in the HZ. The model considers numerous planetary properties, listed in Table 1, and their effect on the carbonate–silicate weathering cycle to calculate a planet’s steady-state pCO2 and surface temperature. If the globally averaged, steady-state surface temperature is below 248 K, we assume the planet is completely frozen and uninhabitable at the surface, as shown by 3D climate models36. Similarly, we assume planets are uninhabitable beyond 355 K, above which surface water would be rapidly lost to space37 (see Methods, subsection “Numerical carbonate–silicate cycle modeling” for additional details on these assumed temperature constraints).

We randomly generated 1050 habitable, stable, Earth-like exoplanet climates using uniform distributions of the model parameters in Table 1. A total of 1200 random, initial parameter combinations were considered but we eliminated those that resulted in planets that froze completely or were too hot to retain their surface oceans. As colored dots, Fig. 1 shows habitable, steady-state solutions.

Our model predicts that atmospheric CO2 abundances should broadly increase and narrow in their spread with orbital distance in the HZ (Fig. 1), consistent with other models of CO2 in the HZ29,38. As justified next in section “Habitable zone climate theory revisited”, the scatter is about a nominal linear trend between incident flux, S, and log(pCO2), which is different from a non-linear trend in models that assume a constant surface temperature in the HZ from negative feedbacks32,34 but do not actually model the carbonate–silicate feedbacks. If future missions are to test the HZ concept by searching for a trend between incident flux, S, and pCO2 (refs. 32,33,34), they could search for the fundamental S–pCO2 relationship shown in Fig. 1.

Below, we show that a log-linear relationship between pCO2 and S may be the default in the HZ if Earth-like carbonate–silicate weathering is ubiquitous on habitable planets. In fact, the trend is elucidated by combining climate theory with carbonate–silicate cycle theory in what follows.

### Habitable zone climate theory revisited

A conventional assumption is that the carbonate–silicate weathering cycle will approximately maintain a stable, temperate surface temperature for an Earth-like planet moved about in the HZ6,7,39 or even a constant temperature32,34. Thus, if we moved the modern Earth outward in the HZ, the smaller incident flux would initially cause the planet to cool. The cooler temperature would lower the CO2 weathering rate causing CO2 to accumulate in the atmosphere until the temperature returned to its nominal value of 289 K. Figure 2 shows this scenario with the dotted blue 289 K contour, which gives the pCO2 value required to maintain a 289-K surface temperature for the modern Earth as it moves about the HZ. The line was calculated from a radiative-convective climate model described in the Methods below, subsection “Habitable zone 1D climate model” (see Eq. (8)).

The constant, 289 K surface temperature contour in Fig. 2 is a non-linear relationship between incident flux, S, and log(pCO2) but it does not consider the temperature and pCO2 feedbacks inherent to the carbonate–silicate weathering cycle. We demonstrate that if these feedbacks are taken into account, surface temperature declines with orbital distance, as mentioned in previous work29, and the relationship between S and log(pCO2) is actually approximately linear for Earth-like planets in the HZ.

If Bond albedo is fixed, the surface temperature, Ts, for an Earth-like planet in steady state varies approximately linearly with incident flux, S5,40,41. This linear relationship between Ts and S arises from energy balance and from water vapor feedback and can be expressed as

$${F}_{{\rm{SOL}}}={F}_{{\rm{OLR}}}=\left(\frac{1-{A}_{{\rm{B}}}}{4}\right)S=a+b{T}_{{\rm{s}}},$$
(1)

where FSOL is the incoming solar radiation flux, FOLR the outgoing long-wavelength radiation flux, AB the Bond albedo, and a and b are empirical constants. From satellite measurements of the modern Earth and radiative calculations, for Ts in K, the empirical constants in Eq. (1) are approximately a = −370 W m−2 and b = 2.2 W m−2 K−1 (ref. 41).

Solving for Ts in Eq. (1), the surface temperature is given by

$${T}_{{\rm{s}}}=\left(\frac{1-{A}_{{\rm{B}}}}{4b}\right)S-\frac{a}{b}.$$
(2)

Under the conventional assumption that the HZ is regulated by a CO2–H2O greenhouse effect where H2O concentrations respond to changes in pCO2, the temperature offset in Eq. (2),  −a/b, is a function of pCO2. Thus, surface temperature, as a function of S and pCO2, is given by

$${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)=\left(\frac{1-{A}_{{\rm{B}}}}{4b}\right)S+f\left({{\rm{pCO}}}_{2}\right),$$
(3)

where $$f\left({{\rm{pCO}}}_{2}\right)$$ is a function that depends on pCO2. For the modern Earth at 1 AU, f(pCO2) = −a/b. For pCO2 ≤ 0.1 bar, the CO2 greenhouse effect is logarithmic in pCO2, i.e., $$f({{\rm{pCO}}}_{2})\propto \mathrm{ln}\,({{\rm{pCO}}}_{2})$$42,43. Above   ~0.1 bar, weaker CO2 absorption features become important and f(pCO2) deviates from $$\propto \mathrm{ln}\,({{\rm{pCO}}}_{2})$$43,44.

As pCO2 increases for an Earth-like planet moved outward in the HZ, the surface temperature will follow Eq. (3) while the rate at which CO2 is removed from the atmosphere will adjust according to the carbonate–silicate weathering feedback. Quantitatively, the pCO2- and Ts-dependent flux of CO2 removal due to the continental weathering flux, Fw (in mol CO2 per unit time) is described by

$${F}_{{\rm{w}}}=\rho {\left(\frac{{{\rm{pCO}}}_{2}}{{{\rm{pCO}}}_{2}^{{\rm{mod}}}}\right)}^{\alpha }\exp \left(\frac{{T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)-{T}_{{\rm{s}}}^{{\rm{mod}}}}{{T}_{{\rm{e}}}}\right),$$
(4)

where ρ is a constant determined by the continental weathering properties of the modern Earth, α a dimensionless constant between 0.1 and 0.5 and regulates the pCO2 dependence of continental silicate weathering, Te a constant between 10 K and 40 K and represents the e-folding temperature dependence of continental weathering. The range for Te has been empirically constrained for the surface temperatures relevant to habitable, Earth-like planets from lab measurements and Phanerozoic geologic constraints16,26. Finally, pCO$${\,}_{2}^{{\rm{mod}}}=288\times 1{0}^{-6}$$ bar and $${T}_{{\rm{s}}}^{{\rm{mod}}}=289$$ K are the modern Earth’s preindustrial pCO2 and surface temperature, respectively21.

The range for α on the Earth was determined empirically from geologic constraints over the past 100 Myr26. We assume that this derived range for α applies to the Earth through time21,45,46 and the Earth-like exoplanets modeled here that have a carbonate–silicate cycle. However, better proxy data for the ancient Earth or observing the carbonate–silicate cycle on habitable exoplanets32,34 may be necessary to understand if the assumed range for α applies more generally to habitable planets.

In Eq. (4), we assume seafloor weathering is negligible, which is a reasonable approximation for the modern Earth21, and illustrative for our purposes of deriving a simple, analytic relationship between S and pCO2. Here, we seek to predict the behavior of the modern Earth in the HZ to gain intuitive understanding, whereas in our numerical model we consider a broad range of properties for Earth-like planets on which seafloor weathering may be important.

The modern Earth, and all Earth-like planets considered in this work, are assumed to be in steady state, in which the flux of CO2 from volcanic outgassing is equal to the rate of CO2 removal from weathering, Fw. If we assume a HZ planet with CO2 outgassing the same as the modern Earth’s, Fw remains constant despite changes in S and pCO2. Setting $${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)={T}_{{\rm{s}}}^{{\rm{mod}}}$$ and $${{\rm{pCO}}}_{2}={{\rm{pCO}}}_{2}^{{\rm{mod}}}$$ for the modern Earth, from Eq. (4), we see that Fw = ρ and

$$1={\left(\frac{{{\rm{pCO}}}_{2}}{{{\rm{pCO}}}_{2}^{{\rm{mod}}}}\right)}^{\alpha }\exp \left(\frac{{T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)-{T}_{{\rm{s}}}^{{\rm{mod}}}}{{T}_{{\rm{e}}}}\right).$$
(5)

Equation (5) must hold for a modern Earth within the HZ. If it did not, Fw would not balance CO2 outgassing, which would result in either complete CO2 removal, or CO2 accumulation without bound.

Solving for $${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)$$ in Eq. (5), we find

$${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)={T}_{{\rm{s}}}^{{\rm{mod}}}-\alpha {T}_{{\rm{e}}}\mathrm{ln}\,\left(\frac{{{\rm{pCO}}}_{2}}{{{\rm{pCO}}}_{2}^{{\rm{mod}}}}\right).$$
(6)

Equating Eq. (6) to Eq. (3) and rearranging gives

$$\left(\frac{1-{A}_{{\rm{B}}}}{4b}\right)S={T}_{{\rm{s}}}^{{\rm{mod}}}-\left[\alpha {T}_{{\rm{e}}}{\mathrm{ln}}\,\left(\frac{{{\rm{pCO}}}_{2}}{{{\rm{pCO}}}_{2}^{{\rm{mod}}}}\right)+f\left({{\rm{pCO}}}_{2}\right)\right].$$
(7)

If $$f({{\rm{pCO}}}_{2})\propto \mathrm{ln}\,({{\rm{pCO}}}_{2})$$, which is the case for pCO2 ≤ 0.1 bar43,44, then $$S\propto -\mathrm{ln}\,({{\rm{pCO}}}_{2})$$. However, even if f(pCO2) deviates from log-linearity with pCO2, S will become increasingly log-linear with pCO2 as αTe increases. In Eq. (7), increasing αTe will cause the $$\mathrm{ln}\,({{\rm{pCO}}}_{2})$$ term to dominate the relationship between S and pCO2. Intuitively, increasing αTe decreases the temperature dependence of continental weathering relative to its pCO2 dependence. Note that bigger Te reduces the temperature dependence of continental weathering while bigger α increases the pCO2 dependence of continental weathering (Eq. (4)).

In addition to predicting a linear relationship between $${\mathrm{log}}\left({\mathrm{pCO}}_{2}\right)$$ and S, the carbonate–silicate cycle implies that moving an Earth-like planet outward in the HZ will cause $${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)$$ to decrease. For increasing orbital distance, pCO2 must increase for $${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)$$ to increase in the HZ. From Eq. (5), pCO2 will be greater than pCO$${\,}_{2}^{{\rm{mod}}}$$ in such cases so $${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)$$ must be less than $${T}_{{\rm{s}}}^{{\rm{mod}}}$$. This decrease in $${T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)$$ as S decreases is shown in Fig. 2. Physically, the power law dependence of weathering on pCO2 can balance volcanic outgassing at lower surface temperatures in the outer HZ.

Figure 2 shows the approximately log-linear relationship between pCO2 and S for the modern Earth moved outward in the HZ. The gray lines and colored circles in Fig. 2 show the expected pCO2 value for the given incident flux S, calculated from Eq. (6). For each S value in Fig. 2, Eq. (6) was solved for pCO2 by using Eq. (8), the polynomial fit for surface temperature based on a 1D climate model (described in the Methods, subsection “Habitable zone 1D climate model”), assuming values of αTe.

The value of αTe affects the slope of the relationship between S and pCO2 due to the carbonate–silicate weathering cycle, shown in Fig. 2. From above, the ranges for α and Te are 0.1 ≤ α ≤ 0.5 and 10 ≤ Te ≤ 40 (ref. 21), so 1 ≤ αTe ≤ 20. If we consider uniform distributions of α and Te, then roughly 95% of αTe values will be greater than 2.3. If α = 0.23 and Te = 10 K then αTe =  2.3, which is used for the Strong T-dep. curve in Fig. 2. The mean of each parameter, α = 0.3 and Te = 25 K gives αTe = 7.5, which corresponds to the Moderate T-dep. curve in Fig. 2. For αTe ≤ 2.3 the colored points and gray curves become increasingly similar to the constant 289 K surface temperature contour in Fig. 2. However, for uniform distributions of α and Te,   ~95% of αTe values are greater than 2.3, so an approximately log-linear relationship between S and $$\mathrm{log}\,({{\rm{pCO}}}_{2})$$ is the default expectation for Earth-like planets in the HZ.

### Observational uncertainty for pCO2 in the HZ

In the log-linear fit shown as the solid red line in Fig. 1, which is the expected relationship between pCO2 and S that we have derived above, the r2-value is 0.49. Thus, about half the variance in log(pCO2) is described by changes in incident flux. The slope is 3.92 ± 0.24 (95%) with units −$${\mathrm{log}\,}_{10}$$(pCO2 [bar])/(S/S), so our model predicts a trend of increasing atmospheric CO2 with orbital distance, which future missions might detect32,33,34. However, there is sufficient spread in our simulated planets that confirming the HZ hypothesis from such a trend may be difficult.

This difficulty is readily seen if we consider a log-uniform distribution for pCO2 on Earth-like planets in the HZ. If we randomly generate 1050 such planets, where 10−4 ≤ pCO2 ≤ 10 bar is sampled log-uniformly, 0.35 ≤ S ≤ 1.05 is sampled uniformly, and impose the same constraints on surface temperature for habitability as in Fig. 1, then the log-linear line of best fit through the uniform planet data has a slope of 3.76 ± 0.465 (95%) with units −$${\mathrm{log}\,}_{10}$$(pCO2 [bar])/(S/S). Thus, measuring just the log-linear trend between pCO2 and S in the HZ is unlikely to test the HZ hypothesis as this measurement cannot confidently detect the presence of the carbonate–silicate weathering cycle—it is indistinguishable from that of randomly distributed pCO2 between the surface temperature limits for habitability.

The inability to differentiate between the log-linear trends for weathering-mediated and random pCO2 vs S in the HZ is due to the assumed surface temperature constraints we impose in our model (between 248 and 355 K for planets in the HZ, see Methods, subsection “Numerical carbonate–silicate cycle modeling”). Such temperature constraints are necessary as the carbonate–silicate weathering cycle can only operate when water, as liquid, is present at the planetary surface. Even without the carbonate–silicate weathering cycle, a minimum surface temperature for habitable planets, which must exist, will result in increasing pCO2 with orbital distance, as shown by the constant temperature contours in Fig. 1.

To test the HZ hypothesis, we propose searching for the two-dimensional (2D) distribution of planets in the S-pCO2 phase space that arises from the carbonate–silicate weathering cycle. This S-log(pCO2) relationship is shown by the point density in Fig. 1, where the distribution of habitable, stable planets is not log-uniformly distributed over pCO2. Rather, around the best-fit line, there is an abundance of planets in the outer HZ at high pCO2, a dearth of low pCO2 planets between   ~0.9 and   ~0.7 S/S, and few high-pCO2 planets throughout the HZ compared to the log-uniform pCO2 case. These differences are expected features of the carbonate–silicate weathering cycle due to the temperature- and pCO2-dependent nature of the weathering feedback. Recall from section “Habitable zone climate theory revisited”  that, as S decreases, the lowered temperature will reduce weathering causing pCO2 to increase. This results in the lack of low-pCO2 planets in the middle of the HZ and the high abundance of habitable planets in the outer HZ (purple shaded region in Fig. 1). Similarly, for large pCO2, the temperature is warmer and pCO2 higher than that of modern Earth so the carbonate–silicate weathering cycle acts to lower pCO2, which reduces the number of high-pCO2 planets throughout the HZ relative to the outer HZ.

To detect the prevalence of the carbonate–silicate weathering cycle and test the validity of the HZ concept, future observations should measure the 2D S-pCO2 distribution of habitable, Earth-like exoplanets. This measured distribution can be compared to the distribution of habitable planets we predict in Fig. 1 to determine if Earth-like planets in the HZ are consistent with the S-pCO2 predictions of the carbonate–silicate weathering cycle.

A test of the 2D phase space of S and pCO2 in the HZ is shown in Fig. 3, which was produced using a 2D Kolmogorov–Smirnov (KS) test. The 2D KS test compares the statistical similarity of a sample distribution to a reference distribution47,48,49. For Fig. 3, the reference distribution was comprised of 500 randomly generated planets from the log-uniform distribution for pCO2 described above (10−4 ≤ pCO2 ≤ 10 bar, 0.35 ≤ S ≤ 1.05S/S, and surface temperature between 248 and 355 K). The sample distribution was generated by randomly selecting a number of planets from Fig. 1 equal to the number of observed exoplanets. For a given number of observed exoplanets in Fig. 3, the horizontal axis, we ran the KS test 10,000 times then calculated the mean and standard deviation from those runs, shown by the gray contour and shaded region. This resampling is necessary as the 2D KS test is a nonparametric approximation that two data sets come from the same underlying population48. We note that below  ~20 observed planets and for probabilities above   ~0.1, the 2D KS test used here can be unreliable49. These limitations do not invalidate the analysis shown in Fig. 3, as we want to know, with 95% confidence, that a log-uniform pCO2 distribution can be ruled out if real exoplanets follow the distribution shown in Fig. 1, which corresponds to the gray line and shaded contour dipping below the 0.05 probability value, shown by the horizontal black line, at 83 observations in Fig. 3.

Thus, confidently detecting the carbonate–silicate weathering cycle will require many exoplanet observations, as shown in Fig. 3. Proposed NASA telescopes, HabEx and LUVOIR, are expected to observe between 3 and 115 Earth-like exoplanets30,31 (see Table 2). The ranges for each mission concept are shown by the colored circles with error bars in Fig. 3. Only the nominal capability of LUVOIR-A, the variant of the proposed LUVOIR space telescope with a primary mirror diameter of 15 m, would provide sufficient Earth-like exoplanet detections to confidently discriminate between a log-uniform pCO2 distribution in the HZ and a pCO2 distribution regulated by the carbonate–silicate weathering cycle. A caveat is that this calculation does not consider the instrument uncertainty in derived pCO2 measurements for each telescope or that other processes not considered in our model may alter pCO2 in the HZ, as discussed below.

## Discussion

Our model assumes that the full variation and uncertainty in Earth’s carbon cycle parameters through time (Table 1) are representative of habitable Earth-like exoplanets generally. This assumption is a reasonable first-order approximation as the bulk composition and geochemistry of rocky exoplanets appear similar to Earth’s50. However, the validity of this assumption likely depends on the parameter in question. For example, it is probably reasonable to expect habitable exoplanets to have a wide range of land fractions and outgassing fluxes, but it is unclear whether there is as much natural variability in the temperature dependence of silicate weathering. An improved mechanistic understanding of weathering on Earth51,52 might reduce these uncertainties.

Other weathering feedbacks have been proposed to operate on the Earth through time, such as reverse weathering53. In reverse weathering, cations and dissolved silica released from silicate weathering are sequestered into clay minerals rather than carbonates so that CO2 remains in the atmosphere, warming the climate and reducing ocean pH. Reverse weathering is thought to be strongly pH dependent and as ocean pH decreases, reverse weathering turns off, acting as a climate stabilization mechanism similar to the carbonate–silicate cycle. The importance of reverse weathering is so poorly constrained through Earth’s history45 that it does not make sense to consider it in our model. However, with future constraints from geology and lab measurements, reverse weathering might alter the stable CO2 abundances of our modeled atmospheres shown in Fig. 1.

At both the inner and outer edges of the HZ, our model assumes that abundant liquid water exists at the planetary surface because, without a liquid surface ocean, the carbonate–silicate weathering cycle ceases and CO2 cannot be sequestered after outgassing. Beyond these temperature bounds, other processes must regulate pCO2. This is a caveat to consider in future observations. As we see from Fig. 1, Mars has low atmospheric CO2 and low incident flux. Frozen exoplanets similar to Mars, populating the white area under the 248 K contour in Fig. 1, could exist in exoplanet surveys. Similarly, planets devoid of surface water, such as Venus, might exist at high pCO2 within the HZ. If future observations detect such planets without confirming the existence of a liquid surface or surface temperature, it could introduce additional uncertainties in any relation between orbital distance and atmospheric CO2. Detecting a surface ocean, one of the most important surface features to confirm when searching for biosignatures and habitability54,55,56, is also important to interpret trends of CO2 in the HZ.

Because we only consider variations on an Earth-like planet, our model predictions may underestimate the inherent variability in habitable exoplanetary conditions. Planets very different from the modern Earth, such as waterworlds without a carbonate–silicate weathering cycle57 or CH4-rich worlds58,59, could introduce additional uncertainty in an observed relationship between S and pCO2 in the HZ. Despite such uncertainties, future missions should measure the relationship between S and pCO2 in the HZ, or possibly a sharp transition in pCO2 at the inner edge of the HZ due to loss of surface water and subsequent shutoff of surface weathering38,60. A more complex model than presented here is necessary to predict such a jump in pCO2 at the inner edge of the HZ. However, if the carbonate–silicate weathering cycle is indeed ubiquitous, as is typically assumed in HZ calculations, then the relationship between incident flux and pCO2 may follow the S-pCO2 relationship predicted in Fig. 1. If no such relationship is observed, then the carbonate–silicate weathering cycle may have limited influence on planetary habitability and the limits of the conventional HZ could need revision. Alternatively, the HZ hypothesis could be incorrect and the long-term climate of HZ planets could be set by phenomena beyond those considered here.

A previous version of this work was published as part of a Ph.D. thesis61.

## Methods

### Habitable zone 1D climate model

We use the Virtual Planetary Laboratory (VPL) 1D radiative-convective climate model5,62 to generate surface temperatures for an Earth-like planet at various pCO2 and incident fluxes. We consider incident fluxes between 1.05S and 0.35S, the HZ limits for a Sun-like star6, and atmospheric CO2 partial pressures between 10−6 and 10 bar. We assume the atmosphere is comprised of CO2 and H2O. If the CO2 partial pressure is below 1 bar, we set the initial atmospheric pressure to 1 bar and add N2 to the atmosphere such that the total surface pressure is 1 bar. We fix the stratospheric water vapor concentration to the modern Earth value and follow the Manabe–Wetherald relative humidity distribution in the troposphere with empirical constraints based on the modern Earth63.

We fit the surface temperature output, Ts in K, from the climate model with a fourth-order polynomial in $$\mathrm{ln}\,({{\rm{pCO}}}_{2})$$ and normalized stellar flux, as follows:

$$\begin{array}{l}{T}_{{\rm{s}}}\left(S,{{\rm{pCO}}}_{2}\right)=4.809-222.0X-68.44{X}^{2}-6.737{X}^{3}-0.206{X}^{4}\\ +1414XY+446.4{X}^{2}Y+44.41{X}^{3}Y+1.364{X}^{4}Y\\ -2964X{Y}^{2}-978.4{X}^{2}{Y}^{2}-98.86{X}^{3}{Y}^{2}-3.059{X}^{4}{Y}^{2}\\ +2655X{Y}^{3}+907.5{X}^{2}{Y}^{3}+92.87{X}^{3}{Y}^{3}+2.892{X}^{4}{Y}^{3}\\ -868.4X{Y}^{4}-304.6{X}^{2}{Y}^{4}-31.48{X}^{3}{Y}^{4}-0.985{X}^{4}{Y}^{4}\\ +1045Y-1496{Y}^{2}+1064{Y}^{3}-281.1{Y}^{4}.\end{array}$$
(8)

Here, CO2 partial pressure pCO2 is in bar, $$X=\mathrm{ln}\,({{\rm{pCO}}}_{2})$$, and Y = S/S is the incident flux, S, normalized to the solar constant, S. Figure 4 shows the agreement between the 1D climate model and the polynomial fit used in this work.

### Numerical carbonate–silicate cycle modeling

To calculate the steady-state pCO2 in the atmospheres of Earth-like planets in the HZ, we use a weathering model that describes pCO2 on the Earth through time21,26. We summarize the model below and highlight how the model in this work differs from previous implementations21,26. These previous implementations provide a comprehensive explanation and justification of the model parameterizations, and empirical and theoretical basis. The model, as a Python script, is available in the Supplementary Data and contains a complete description of the model equations and parameters (see the file weathering_model.py).

The weathering model balances the flux of outgassed CO2 against the loss of carbon due to continental and seafloor weathering, which result in precipitation of carbonates in the ocean and seafloor pore space. Quantitatively, for time t, this is described by time-dependent equations where we normalize to the mass of the ocean, Mo (nominally, an Earth ocean, 1.35 × 1021 kg):

$$\frac{{\mathrm{d}}C}{{\mathrm{d}}t} =\frac{{F}_{{\rm{out}}}\,+\,{F}_{{\rm{carb}}}\,-\,{P}_{{\rm{o}}}\,-\,{P}_{{\rm{p}}}}{{M}_{{\rm{o}}}}\\ \frac{{\mathrm{d}}A}{{\mathrm{d}}t} =2\times \frac{{F}_{{\rm{sil}}}\,+\,{F}_{{\rm{carb}}}\,+\,{F}_{{\rm{diss}}}\,-\,{P}_{{\rm{o}}}\,-\,{P}_{{\rm{p}}}}{{M}_{{\rm{o}}}}.$$
(9)

Here, C is the non-organic carbon content of the atmosphere–ocean system in mol C kg−1, and A is the carbonate alkalinity in mol equivalents (mol eq). Carbonate alkalinity (henceforth alkalinity) is the charge-weighted sum of the mol liter−1 concentration of bicarbonate and carbonate anions, [HCO$${\,}_{3}^{-}$$] + 2[CO$${\,}_{3}^{2-}$$]. Fout is the global CO2 outgassing flux, Fcarb and Fsil are the continental carbonate and silicate weathering fluxes, Fdiss is the rate of seafloor basalt dissolution, and Pp and Po are the pore and ocean precipitation fluxes. The fluxes on the right-hand side of Eq. (9) (Fout, Fcarb, Po, Pp, Fsil, Fcarb, and Fdiss) are given in mol C yr−1 for dC/dt and in mol eq yr−1 for dA/dt.

The alkalinity that enters the ocean from weathering will balance a +2 charge cation (e.g., Ca++), which is why a factor of 2 enters in the definition of dA/dt in Eq. (9). Hence, geochemists often think of alkalinity in terms of the balance of cations produced in weathering, principally Ca++. This reasoning arises because the weighted sum of carbonate and bicarbonate concentrations must balance the charge of conservative cations minus conservative anions $$\left({\mathrm{i}}.{\mathrm{e}}., \, 2[{\mathrm{Ca}}{^{++}}]\,+\,2[{\mathrm{Mg}}{^{++}}]\,+\,{\mathrm{Na}}^{+}+\ldots{\hbox{-}}[{\mathrm{Cl}}^{-}]{\hbox{-}}{\ldots}{\!}\right)$$, ignoring minor contributions from weak acid anions and water dissociation. Weathering releases cations and carbon speciation adjusts to ensure charge balance, so that the cation release is effectively equivalent to carbonate alkalinity.

To improve the rate of model convergence and range of model inputs over which Eq. (9) converges, we do not consider the seafloor pore space and atmosphere–ocean as separate systems. This differs from previous versions of the model21, which considered the atmosphere–ocean and pore space independently. Rather, we approximate the atmosphere–ocean and pore space as a single entity in Eq. (9). This simplification does not appreciably change the model output for atmospheric CO2 because we run the model to steady state in all cases, where the atmosphere–ocean and pore space reach approximate equilibrium. In the next section, we present additional details on our model implementation and discuss the agreement between our no-pore model and the original, two-box model21.

A second modification is the range of incident stellar fluxes over which the model can be run. Previously, the model described here was used to study the Earth through time21 and thus only considered solar fluxes between S (the modern solar constant) and early Earth’s 0.7S (S = 1360 W m−2). We extend that range to include the entire conservative HZ of a Sun-like star, roughly 1.05S to 0.35S6. We use Eq. (8), the fourth-order polynomial fit to a 1D climate model, to calculate surface temperatures throughout the HZ. The Bond albedo of the planet is calculated dynamically by the climate model and thus included implicitly in our polynomial fit.

With the coupled climate and weathering model, we generate steady-state, Earth-like planets by randomly sampling plausible initial model inputs. The ranges for each parameter we consider are representative of the Earth through time21 and shown in Table 1. These ranges represent very broad uncertainties of the carbonate–silicate cycle on the Earth through time and so are appropriate for Earth-like planets. We conservatively assume a uniform distribution for each parameter range shown in Table 1.

We parameterize the internal heat of an Earth-like planet conservatively using the planet’s age, ranging 0–10 Gyr, which is the approximate habitable lifetime of an Earth-like planet around a Sun-like star64. The equation for planetary heat relative to the modern Earth, Q, is given by

$$Q={\left(1-\frac{4.5-\tau }{4.5}\right)}^{-{n}_{{\rm{out}}}},$$
(10)

where τ is the age of the planet in Gyr, and nout is the scaling exponent for internal heat, with a range given in Table 1.

The parameter ranges shown in Table 1 represent the uncertainty of the carbonate–silicate weathering cycle on the Earth through time21. Implicit in our assumed parameter ranges is that continental land fraction, fland, and biological weathering fraction, fbio, have increased from 0 when the Earth formed to 1 on the modern Earth. Similarly, the relative internal heat, Q, is assumed to be large when the Earth is young and unity for the modern Earth. Therefore, on the modern Earth, where fland = 1, fbio = 1, and Q = 1, the weathering rate is maximized and outgassing rate is relatively small (see Methods, subsection “Validity of carbon cycle parameterizations to exoplanets” for a discussion on the importance of these three parameters in our model). This is seen in Fig. 1, where the modern Earth appears near the lower bound for predicted pCO2 in the HZ. If the continents on an exoplanet were more easily weathered or outgassing much lower than on the modern Earth, such exoplanets could have pCO2 values well below the modern Earth value shown in Fig. 1. We do not consider such exoplanets in this model, so the results presented here are only applicable to planets similar to the Earth through time.

Our model assumes that each simulated planet is habitable, i.e., it has a stable, liquid surface ocean, a necessity for the carbonate–silicate cycle to operate. For a mean surface temperature below 248 K, Earth-like planets are likely completely frozen36, which we use as a lower temperature bound in the model. While 248 K is below the freezing point of water, it is a global mean surface temperature and 3D models show that the range 248–273 K for this parameter does not preclude the existence of a liquid ocean belt near the equator. At the other temperature extreme, a hot, Earth-like planet can rapidly lose its surface oceans due to high atmospheric water vapor concentrations that are photolyzed and subsequently lost to space. This upper temperature bound on habitability occurs at   ~355 K37. Above 355 K, Earth-like planets are unlikely to remain habitable for more than   ~1 Gyr37 and cannot operate a carbonate–silicate cycle over geologic timescales. We use these two temperature bounds, 248 K and 355 K, as the limits for habitability in our model. Any modeled planet with a final surface temperature outside these limits is uninhabitable and removed from our results.

We limit HZ planets to those with pCO2 below 10 bar. For most Earth-like planets in the HZ, 10 bar of CO2 results in planets with surface temperatures well above 355 K, which are not habitable on long time scales. If we impose a fixed stratospheric water vapor concentration in the 1D climate model and modify the tropospheric water vapor concentration based on empirical data from the modern Earth, we enable the 1D climate model to accurately model habitable, Earth-like planets through much of the HZ. But in the outer HZ, with more than  ~10 bar of CO2, this assumption overestimates atmospheric water vapor concentrations and leads to artificially warm planets, so we reject such cases. Above   ~10 bar of CO2 in the outer HZ, assuming a saturated troposphere for water vapor, increasing atmospheric CO2 may not lead to additional warming6. Rather, the surface cools in such scenarios because additional CO2 leads to increased Rayleigh scattering and no additional warming. Because Earth-like planets in the outer HZ would be frozen and uninhabitable even with CO2 partial pressures above  ~10 bar, we impose a 10-bar limit for CO2 in the outer HZ. This limit agrees with previous CO2 limitations in coupled climate and weathering models29.

### Combined ocean and pore space model justification

The carbon cycle model used in this work was previously derived as a two-box model21, where the atmosphere–ocean and the seafloor pore space were separated. In this work, we combine the ocean–atmosphere and the pore space into a single unit. This modification can be implemented in the original model21 by assuming that the pH of the pore space is the same as the pH of the ocean, and assuming that the alkalinity and carbon content of the ocean and pore space are the same. The dissolution and precipitation fluxes can then be calculated without treating the ocean–atmosphere and the pore space as different systems. This modification allows the model to converge quicker over a wider range of parameter combinations.

To validate our combined model, we ran the modern Earth through both the original, two-box model21 and our modified model at ten different incident fluxes between S and 0.7S. The average error in predicted CO2 values between our model and the two-box model was 2.8%, with a minimum error of 2.3%, and a maximum error of 3.6%. Given the large uncertainties in model inputs (Table 1), the few percent error introduced by our simplified model is unimportant.

For each parameter combination in our simplified model, we start with the modern Earth then impose a step change for each model parameter. We then run the simulation for 10 Gyr or until the system reaches steady state. We consider the model to have reached steady state when extrapolation of the rate of change of pCO2 for 1 Gyr changes pCO2 by <1%. Typically, the model converges within a few Myr to a few tens of Myr. Rarely (2 of the 1200 planets simulated in this work), parameter combinations will not reach steady state after 10 Gyr. Simulations with combinations of exceptionally high outgassing rates and low CO2 weathering rates can enter a regime were atmospheric CO2 builds without bound, never converging. Such model results are beyond the range of validity of our model.

### Validity of carbon cycle parameterizations to exoplanets

The parameterization of weathering in our model has been empirically validated for the modern Earth16,21,26. The exponential temperature dependence of continental weathering is a reasonable approximation that agrees with field and lab measurements16 and can reproduce the climate results of more complex models21,26. Similarly, the power-law parameterization for the pCO2 dependence of continental weathering agrees with data from the modern Earth26 and can even be approximately derived from equilibrium chemistry arguments for an Earth-like exoplanet38. The bulk geochemistry of rocky exoplanets may be similar to Earth’s50, so we expect our weathering parameterization to reasonably approximate Earth-like planets in the HZ. However, uncertainties in how the carbonate–silicate weathering cycle regulates climate on Earth persist21, so the predicted variations in pCO2 in our model may not capture the true variability of pCO2 in the HZ. Below, we show that our broad parameterization of the carbonate–silicate weathering cycle may encompass the plausible range of pCO2 for the Earth through time, but improved understanding the carbonate–silicate weathering cycle may be necessary to know if such variations are indeed representative of the Earth through time and applicable to Earth-like planets generally.

The rate of weathering depends strongly on the intrinsic features of a planet, such as the CO2 outgassing rate and the properties of its continents. Changes in continental uplift rate, lithology, and configuration are parameterized in our model through the fland and fbio terms. The parameters fland and fbio linearly scale the weathering flux and could analogously be considered a continental weatherability scaling factor. For the ranges of fland and fbio considered in our model (see Table 1), changes in the continental weatherability alone can generate pCO2 values spanning  ~4 orders of magnitude. This broad parameterization likely encompasses pCO2 perturbations due to continental weatherability changes caused by large volcanic eruptions or changes in continental configuration. Indeed, the largest, constrained change in pCO2 due to such events on Earth may be closer to  ~1 order of magnitude, coeval with the eruption of the Siberian Traps65.

The importance of continental weatherability (fland and fbio) on pCO2, relative to other parameters, is shown in Fig. 5. Figure 5 was generated by sampling uniform distributions for each model parameter shown in Table 1 across its listed range. When one parameter was varied, all other parameters were held constant at their modern Earth value, which we define as: $${F}_{{\rm{mod}}}^{{\rm{out}}}=6$$ Tmol C yr−1, n = 1.75, x = 1, Te = 25 K, α = 0.3, ξ = 0.3, fland = 1, Sthick = 1, $${F}_{{\rm{carb}}}^{{\rm{mod}}}=10$$ Tmol C yr−1, fbio = 1, $${a}_{{\rm{grad}}}=1.075$$, γ = 0.2, β = 0.1, m = 1.5, Ebas = 90 kJ mol−1, and Q = 1. Note that we incorporate nout and τ from Table 1 into Q, the internal heat (see Eq. (10)), which is the parameter of interest. We show two different values for S in Fig. 5, S = 1.0S in the left panel and S = 0.5S in the right panel. For both values of S, Fig. 5 shows that variations in fland and fbio alone can alter pCO2 by orders of magnitude.

The internal heat of the planet, Q, plays a similarly important role in setting pCO2. The rate of CO2 outgassing is determined by Q and our broad parameterization of Q allows pCO2 to vary by orders of magnitude throughout the HZ, as shown in Fig. 5.

The rate of CO2 outgassing and continental weatherability drive the majority of the spread in pCO2 shown in Fig. 1. This is readily seen in Fig. 6, which shows the results of 300 random parameter combinations from uniform distributions of the parameters in Table 1 except for Q, fland, and fbio, which were all fixed to 1. Of the 300 parameter combinations, 235 remained above 248 K and are shown in Fig. 6. Comparing Fig. 6 to Fig. 1, it is readily apparent that the broad uncertainty in pCO2 from our results is due to variations in intrinsic planetary properties (Q, fland, and fbio) rather than uncertainties in the tuning parameters of our carbon cycle parameterization.

The outgassing rate and continental properties of habitable exoplanets remain unknown. Thus, our broad parameterization of those terms, which align with possible conditions on Earth throughout its history, are a reasonable approximation. If an Earth-like, carbonate–silicate weathering cycle is common on habitable planets, then these parameters may largely determine pCO2 on such planets and generate a range for pCO2 at a given orbital distance similar to that shown in Fig. 1.