Fig. 2: The relationship between incident flux and atmospheric CO₂ for Earth-like planets regulated by a carbonate–silicate weathering cycle.

The horizontal axis shows incident flux, S, normalized to the solar constant (S_⊕) and the corresponding orbital distance in Astronomical Units (AU) above the plot. The vertical axis shows the atmospheric CO₂ partial pressure (pCO₂) in bar. The dotted blue curve labeled 289 K shows the pCO₂ value required to maintain a 289 K surface temperature for the given incident flux, S. The conventional assumption of CO₂ in the HZ stipulates that pCO₂ will adjust to maintain a temperate or even constant surface temperature. Under this assumption, moving the modern Earth (labeled black square) outward in the HZ would have the planet approximately follow the dotted blue 289 K contour. The colored points and gray curves show the modern Earth moving outward in the HZ with a functioning carbonate–silicate weathering cycle, calculated from Eq. (6). We consider two temperature and pCO₂ dependencies for continental weathering in this plot. The strong temperature dependence contour (labeled Strong T-dep.), uses a temperature and pCO₂-dependent weathering factor of αT_e = 2.3, which implies a strong temperature feedback on continental weathering compared to the pCO₂ feedback (see Eq. (7)). The moderate temperature dependence contour (labeled Moderate T-dep.), uses a temperature and pCO₂-dependent weathering factor of αT_e = 7.5. These two values for αT_e result in two different paths the Earth can take as it moves outward in the HZ. The planet color shows the mean surface temperature. Log-linear fits to the colored points of the Strong T-dep. and Moderate T-dep. contours have r² values of 0.959 and 0.999, respectively. Thus, even for a strong temperature dependence of continental weathering, our coupled climate and weathering model predicts an approximately log-linear relationship between incident flux and pCO₂ on Earth-like planets in the HZ.