Earth–Sun biotic and abiotic cases

Figures 2, 3 and 5 present the base model for the Earth–Sun system comparing them with geochemical estimates. Figure 4 presents the land W _l and seafloor W _sf weatherings for the cases of sensitive W _l and insensitive W _l to land area and biotic and abiotic base cases. The base case for geophysical variables is presented in the next section.

Fig. 2. Evolution of Earth's surface temperature given by the model for the Earth/Sun case. The green cross (Blake et al., Reference Blake, Chang and Lepland2010) is a geochemical and sedimentary estimate for ocean water temperature (26–35$^\circ$C) in the Archaean. The aliened numbered arrows pointing down mark evidence of glacial deposits in the past and the maximum temperature ($\sim \!25^\circ$C) for such to appear, see the text for more details. The red arrow (Hren et al., Reference Hren, Tice and Chamberlain2009) marks maximum ocean temperature ($\sim \!40^\circ$C) based on geochemical and sedimentary evidence. The pre-industrial mean temperature of 287 K is marked as a black dot outlined in white.

Fig. 3. Evolution of Earth's CO₂ partial pressure given by the model for the Earth/Sun case compared with geochemical estimates. Pre-industrial mean pCO₂ of 280 ppm is marked as a black dot outlined in white.

Fig. 4. Weathering rates, W _i, normalized to total present biotic weathering, ${\rm W}^{0}_{total}$, in the Earth/Sun case. Weathering is separated in the cases of sensitive W _l (full lines) and insensitive W _l (dashed lines) to land area, and seafloor weathering, W _sf. Green lines, W _biotic, are the same weatherings but with a land biosphere.

Fig. 5. Evolution of Earth's ocean (surface) water reservoir and the effective ocean depth in our Earth/Sun model. Bars and cross indicate uncertainties of estimates of past Earth's ocean volume based on geochemical evidences (Hren et al., Reference Hren, Tice and Chamberlain2009; Pope et al., Reference Pope, Bird and Rosing2012) and model (Dong et al., Reference Dong, Fischer, Stixrude and Lithgow-Bertelloni2021). The range of Hren et al. (Reference Hren, Tice and Chamberlain2009) is the ocean mass required to satisfy their hydrogen isotope values according to Pope et al. (Reference Pope, Bird and Rosing2012). Full thick lines are for the base initial (and present) surface/mantle partition of 1.0 oceanic mass, M_oc, on the surface and 2.0 oceanic masses in the mantle. The dotted lines are for variations in one more (or less) oceanic mass in the mantle recalibrating to reach 1.0 M_oc today. Present values are marked as black dots outlined in white.

For surface temperatures (Fig. 2), the biotic and abiotic cases in the past held temperatures always above freezing and below ~300 K ($\sim \!25^\circ$C), the inferred maximum temperatures from the evidence of glacial deposits in ancient Earth (numbered arrows pointing down). Those were estimated following Krissansen-Totton et al. (Reference Krissansen-Totton, Arney and Catling2018b) supposing that the existence of glacial deposits anywhere on past Earth would bound conservatively average temperatures below 25$^\circ$C in the same period because Antarctic glaciations started around 35 Mya when global mean temperatures fell below 20$^\circ$C.

CO₂ partial pressures (Fig. 3) were much higher in the past, above 0.1 bar 4.5 Gyr ago in all the cases, biotic and abiotic, and for sensitive and insensitive W _l.

If W _l is sensitive, the small land area in the first 2 Gyr of our planet would decrease W _l and promote higher levels of CO₂ than in the insensitive case, which we see in Fig. 4. Consequently, W _sf is much more important in the past, by a combination of higher spreading rates and deeper melt generation depth, as also reported in Charnay et al. (Reference Charnay, Le Hir, Fluteau, Forget and Catling2017) and Krissansen-Totton et al. (Reference Krissansen-Totton, Arney and Catling2018b), surpassing W _l as the main weathering mechanism. Because W _sf has low sensitivity to surface temperatures, this also leads to a greater CO₂ partial pressure and greater surface temperatures until CO₂ partial pressure becomes greater enough for W _sf to compensate.

When W _l is insensitive, W _l and W _sf have more comparable values in the past, decreasing CO₂ partial pressure and surface temperatures to the point of frigid temperatures (<280 K) in the first billion years.

Going back to Figs. 2 and 3. In the biotic case, the enhancement of W _l via the B _f factor leads to the fall of CO₂ levels for 1.5 billion years when biological productive is high, decreasing temperatures more than ~2 K in modern Earth when compared to the abiotic case. So the biota as modelled has a cooling effect on the climate. The difference could be higher if we had used a larger enhancement factor, but this can increase pCO₂ in the past beyond the geochemical estimates. A solution interlinked with a limitation in the model is the use of CH₄. Biogenic methane was much more abundant in the past Earth, when oxygen levels were lower, and could have risen temperatures considerably. This could have contributed to increased weathering and reduced CO₂ levels even with a larger biotic enhancement factor.

Vascular plants can enhance weathering by concentrating CO₂ in the soil to levels higher than in the atmosphere (Walker et al., Reference Walker, Hays and Kasting1981; Schwartzman and Volk, Reference Schwartzman and Volk1989; Kump et al., Reference Kump, Brantley and Arthur2000). If this can be done almost regardless of the atmospheric CO₂ levels, W _l becomes independent of pCO₂ (making b ≈ 0 in equation (15)). In this case, more CO₂ could be accumulated in the atmosphere before an increase in weathering (driven by high temperatures), and planets with vascular plants could be more climate stable and be habitable further away from its star (Menou, Reference Menou2015). A possible example of life exerting a feedback on its environment for climate stability.

The thickness of sediments covering the seafloor is very heterogeneous on Earth, with most of the seafloor being covered by a layer of a few dozens to hundreds of meters, but reaching many kilometres close to continental margins (Straume et al., Reference Straume, Gaina, Medvedev, Hochmuth, Gohl, Whittaker, Abdul Fattah, Doornenbal and Hopper2019). Seafloor weathering, W _sf, depends more strongly on seafloor temperature, T _sf, than on pCO₂ (see equation (16)), and T _sf depends importantly on thickness of sediments, s _thick, covering the seafloor. To increase s _thick is to isolate more of the seafloor from the cold deep water, increasing T _sf and seafloor weathering. However, this also isolates the seafloor from seawater and CO₂, which decreases seafloor weathering. These competing behaviours, and the fact that seafloor weathering occurs mostly on young oceanic crust, which is not heavily covered in sediments and is hotter because of more heat flux, make it difficult to predict the combined behaviour of seafloor weathering for different sediment thicknesses or which of the variables is the limiting factor in every situation. But is conceivable that CO₂ supply could be a limiting factor if sediment thicknesses is too thick and this is not modelled here in detail.

In the model, changing s _thick from the base 700 m to zero, increase surface temperatures during the beginning of the Archaean (t = 0.5 Gyr) by 10 K. If s _thick is set to 350 m, the increase is about 5 K. If s _thick is increased to 1050 m, surface temperatures drop by 5 K, and if s _thick is increased to 1400 m, surface temperatures drop by 20 K. The effect is more intense in the past, when there is a lot of CO₂ in the atmosphere, continents are small (W _sf dominates over W _l) and the Sun was dimmer. Today, the difference between the five scenarios is barely noticeable. This large drop in surface temperatures for thicker sediment covers may not be realistic thought, as CO₂ would be more heavenly limited on the seafloor and W _sf would be less efficient.

A derivable general consequence could be that planets with thinner sediment covers on the seafloor probably have a less intense seafloor weathering (if deep oceans are cold), higher pCO₂ and surface temperatures. Since part of the sediments has a biological origin, life could also have some power over seafloor weathering. And, as part of the sediments come from runoff from continents, aquaplanets with very small continents could have low deposition of sediments on the seafloor, weaker seafloor weatherings and higher surface temperatures. Because the seafloor weathering is a complex process that depends on many parameters and the strength of weathering, in general, is poorly known (and we do not model the CO₂ limitation for weathering when increasing s _thick), the magnitude and dependency of these results can be smaller than we derived here and should be seen as just exploratory.

As the planet gets old and spreading rates fall, pCO₂ also falls in the atmosphere, the C3 limit will be reached in about 190 Myr in the biotic case, and the C4 limit will be reached in about 810 Myr. The reduced CO₂ levels may impact or hinder the existence of some photosynthetic life forms and an oxygen-rich atmosphere, also affecting especially eukaryotes. This result is in line with other estimates of future CO₂ levels (Lovelock and Whitfield, Reference Lovelock and Whitfield1982; Caldeira and Kasting, Reference Caldeira and Kasting1992; Franck et al., Reference Franck, Kossacki and Bounama1999, Reference Franck, Block, Von Bloh, Bounama, Schellnhuber and Svirezhev2000b; Lenton and von Bloh, Reference Lenton and von Bloh2001), but it is a little shorter from the 1.08 ± 0.14 Gyr life span of Earth's oxygenated atmosphere by Ozaki and Reinhard (Reference Ozaki and Reinhard2021). Following up, temperatures will surpass 100$^\circ$C at 117% of the present solar flux, in about 1.66 Gyr (at an age of 6.23 Gyr), when Earth will then enter a runaway greenhouse and loses its water. Other works in the literature reached similar results (Caldeira and Kasting, Reference Caldeira and Kasting1992; Kasting et al., Reference Kasting, Whitmire and Reynolds1993; Franck et al., Reference Franck, Kossacki and Bounama1999) but some encountered values differing even more than a billion years from our result (Schwartzman, Reference Schwartzman1999; Abe et al., Reference Abe, Abe-Ouchi, Sleep and Zahnle2011; Kopparapu et al., Reference Kopparapu, Ramirez, Kasting, Eymet, Robinson, Mahadevan, Terrien, Domagal-Goldman, Meadows and Deshpande2013; O'Malley-James et al., Reference O'Malley-James, Greaves, Raven and Cockell2013). (see Mello and Friaça, (Reference Mello and Friaça2020) for a comparison)

The maximum stable stellar flux Earth can receive before its climate become unstable and it enters a moist or runaway greenhouse is uncertain. For a fully water-saturated atmosphere in 1D models, the limit is around 102% (moist greenhouse) to 106% (runaway greenhouse) (Kopparapu et al., Reference Kopparapu, Ramirez, Kasting, Eymet, Robinson, Mahadevan, Terrien, Domagal-Goldman, Meadows and Deshpande2013), very close to the present flux. It can be extended up to 121% in GCMs (Leconte et al., Reference Leconte, Forget, Charnay, Wordsworth and Pottier2013; Wolf and Toon, Reference Wolf and Toon2015), and can reach 135% for aquaplanets, and 180% for land planets in the GCMs of Abe et al. (Reference Abe, Abe-Ouchi, Sleep and Zahnle2011) and Kodama et al. (Reference Kodama, Nitta, Genda, Takao, O'ishi, Abe-Ouchi and Abe2018). Our result of 117% on an aquaplanet is on the conservative side.

In Fig. 5, for the base surface/mantle partition of 1 : 2 oceanic masses, oceans reach a peak of 1.6 oceanic masses at around age 1.4 Gyr, from changes in gaseous exchanges with the mantle, but only later, at 2.3 Gyr, that the oceans reach the maximum effective ocean depth of d _m = 5 km. This is because ocean depth in the model depends on the quantity of water on the surface (oceanic masses) and the area and depth of ocean basins. Initially, effective ocean depth increases because water is being degassed more than regassed by the mantle, and a little later, effective ocean depth also increases because continents are growing and the continental growth makes continents occupy more planetary surface, and the continental freeboard constraint in the model force the deepening of ocean basins in the process. In the future, oceans will decrease in volume and effective depth as degassing decreases and continental growth stops (in our model) but not to the point of affecting habitability by much. If continents continue to grow, effective depth may not decrease so much as shown here.

The mass of water in the mantle today is poorly constrained, with estimates varying more than one order of magnitude (Cowan and Abbot, Reference Cowan and Abbot2014). The present reservoir of water in the mantle can be lower than the maximum water storage capacity of the mantle, as there is exchange of water between the mantle and the surface. Dong et al. (Reference Dong, Fischer, Stixrude and Lithgow-Bertelloni2021) showed that the mantle water storage capacity decreases with mantle potential temperature. Because mantle potential temperature is estimated to be higher in the past (as also in our model), if the mantle today holds more water than its capacity in the Archaean, the past surface should have accumulated the excess water. For two ocean masses in the mantle today (what we used in our model), Dong et al. (Reference Dong, Fischer, Stixrude and Lithgow-Bertelloni2021) estimate that the surface should hold $2.27^{ + 0.20}_{-0.96}$ oceanic masses in the Archaean, compatible with our result in Fig. 5.

For more massive planets or with a higher abundance of radioactive isotopes, so with higher internal mantle temperatures, this may be important, because it indicates that the oceans would be deeper and land area rarer. But as mantle volumes (and so storage capacity) would also be bigger in massive planets, it could depend more on how water-rich the planet is.

Keeping the same calibration, increasing the water reservoir in the mantle from 2.0 to 3.0 oceanic masses increases degassing in the past and we end up today with oceans 20% more massive, and decreasing it from 2.0 to 1.0 oceanic mass decreases degassing in the past and we end up today with oceans 25% less massive. If we recalibrate to obtain a surface reservoir of 1.0 oceanic mass today, we obtain the dotted lines in Fig. 5. Considering all three estimates for Archaean oceans volume (Hren et al., Reference Hren, Tice and Chamberlain2009; Pope et al., Reference Pope, Bird and Rosing2012; Dong et al., Reference Dong, Fischer, Stixrude and Lithgow-Bertelloni2021), the range from 1.0 to 2.0 oceanic masses in the mantle today seems to be the most compatible. But we are not very confident in this result because our mantle's viscosity does not contain any dependency on water content and this could partially influence mantle flux and degassing.

In general, for our purposes, our base model for the Earth–Sun system seems to fit reasonably well with the scant data available about the past of our planet. Nonetheless, the first half billion years in our planet's history, the Hadean, were marked by violent events, from the impact and formation of the Moon, the ocean of magma, to the initial and the late heavy bombardment. These events are not modelled here and they may have influenced considerably the initial habitability of our planet, with impacts heating the crust temporarily (Zahnle et al., Reference Zahnle, Arndt, Cockell, Halliday, Nisbet, Selsis and Sleep2007) or cooling the atmosphere by weathering of impact ejecta (Kadoya et al., Reference Kadoya, Krissansen-Totton and Catling2020). So our results for the first billion years should be seen with less confidence. If the initial history of our planet can be generalized for the formation of other terrestrial planets, this concern should be kept in mind for the other planetary masses modelled in the following sections.

Varying planetary mass

Having a base model (green lines in the following figures) we can compare results for different planetary masses, keeping the Sun as the parent star. Varying planetary mass (Fig. 6) has a considerable effect on CO₂ partial pressure and a smaller, but more important, effect on surface temperature in both cases of W _l sensitive and W _l insensitive to land area. The 0.1 M$_{\oplus }$ planet is the less habitable one, presenting the lowest initial temperatures, spending substantial time, ~3.7 Gyr, below the 273 K limit at 1.0 au, also experiencing the fastest drop in pCO₂, from a maximum of less than 0.1 bar, reaching the C3 and C4 limits in just a billion years. One-Earth-mass planet and more massive ones start with pCO₂ just short of 0.3 bar, with small differences between them, and are the more capable ones of maintaining pCO₂ above the C3 and C4 limits for long periods.

Fig. 6. Habitability results for 0.1, 0.25, 0.5, 1.0, 2.0 and 4.0 M$_{\oplus }$ planets at 1 au from the Sun. Full lines are for sensitive W _l, and dashed lines are for insensitive W _l. Oceans’ results presented no difference for sensitive or insensitive W _l. Oceanic masses were normalized to the planet mass for a compact visualization, so the 0.1 M$_{\oplus }$ planet has an initial surface ocean of 1.4 × 10²⁰ kg, the 1.0 M$_{\oplus }$ has 1.4 × 10²¹ kg and the 4.0 M$_{\oplus }$ has 5.6 × 10²¹ kg, but all start at 1.0.

As before, W _l sensitivity to land area permits higher levels of CO₂ when continents are small. Insensitiveness to land area generally amplifies the differences in temperature and pCO₂ levels between masses. With sensitive W _l, the initial temperature for all masses, besides the 0.1 M$_{\oplus }$, is a warm ~285 K, and fairly warm temperatures are maintained for Earth and above one-Earth-mass for 5 Gyr. In the case of insensitive W _l, temperatures are fairly colder below one-Earth-mass.

In all cases, no matter the weathering sensitivity, after ~5.0 Gyr, all curves of mean surface temperature converge. Because of increasing solar luminosity and low pCO₂, this is when the temperature behaviour is driven very little by CO₂ (decaying fast) and increasingly more by H₂O, taking the atmosphere, in a feedback loop, directing the planet into a runaway greenhouse.

Planetary mass can also influence weathering in other ways apart from degassing rates. A more massive planet is also bigger, has a large radius and has a bigger area. We do not know if this would be translated necessarily to larger continents in all real cases, but it is what we considered in our model. So, considering sensitive W _l, a more massive planet has more reaction area to consume its atmospheric CO₂ via land weathering. The less massive planets have comparatively less land area for land weathering to act and would be able to retain higher levels of atmospheric CO₂ than expected for its thermal evolution alone.

This compensates in part for lower internal temperature and degassing rates in low-mass planets. But a complication not included here is that the higher gravity of more massive planets would limit the range of its relief, having shallow seas and short mountains, while a less massive planet could produce deep seas and tall mountains more easily. This difference in terrain verticality may contribute to increasing the area of weathering in less massive planets and decreasing it in more massive planets.

Overall, in our model, internal evolution plays a bigger role in the long run and more massive planets are able to maintain higher levels of CO₂ than a lower mass planet, because of higher rates of degassing.

Degassing behaviour is driven by the melt generation volume S _r × d _m. Higher seafloor spreading rates S _r produce higher heat fluxes, and higher mantle temperatures produce higher melt generation depths d _m. More massive planets have the highest seafloor spreading rates (Fig. 7(e)) and mantle potential temperatures (Fig. 7(b)), but also have a higher acceleration of gravity, which decreases the melt generation depths (Fig. 7(i)). The combined effect is still a higher melt generation volume for more massive planets but the advantage is small. The biggest difference is in longevity. Because d _m goes to zero for potential temperatures below 1100$^\circ$C, degassing in less massive planets will stop much sooner than for more massive planets.

Fig. 7. Geodynamic results for the 0.1, 0.25, 0.5, 1.0, 2.0 and 4.0 M$_{\oplus }$ planets at 1 au from the Sun. Mantle results presented no difference for sensitive or insensitive W _l and changes in surface oceanic masses because the geodynamics of the mantle is, in our model, insensitive to them.

Water degassing happens more intensively for lower planetary masses during the first 2 billion years (Fig. 6(c)) and effective ocean depths keep increasing for less massive planets than Earth. Our parameterization for water exchange is an approximate equilibrium between water being degassed (going out of the mantle) and regassing (going into the mantle). Besides other factors, degassing depends on how much water there is in the mantle (mass of water by mantle volume) and the efficiency with which water can be expelled out of the mantle, which depends on the inverse of pressure over degassing regions (more pressure, less degassing), this pressure comes from the water column above it, so a higher effective ocean depth creates higher pressure. Besides other factors, regassing depends on the pressure on regassing regions (more pressure, more regassing).

A low-mass planet is also a small planet. Because volume goes with r ³, its volume decreases proportionally faster than its mass, and the density of water per volume in the mantle increases when compared with the Earth model (considering the same water content). A less massive planet also has lower gravity and the pressure along (de)regassing regions becomes smaller too. The melt generation depth increases in less massive planets and degassing can occur from deep in the mantle. This leads to an imbalance favouring degassing. This goes on until the density of water in the mantle drops substantially and, with more water on the surface and deeper oceans, the pressure increases. On higher planetary masses, the process is the opposite.

The behaviour in ocean depth is a reflex of the above, with low-mass planets being able to create deep ocean basins and deep oceans (Fig. 6(d)). Continents occupy space on the planetary surface and decrease space for ocean basins, which, under our assumption of continental freeboard, must deepen to accommodate the same volume of water as before. Even then, the ocean's effective depths remained in the habitable range for all planetary masses.

Regarding the mantle (Fig. 7), the differences between masses are clearer. Most masses have an initial heating period, causing an initial increase in the mantle potential temperature, and this period is longer as massive as the planet is. The two lower masses are since the beginning net losing heat, being the first to tend asymptotically to zero in the rate of change in mantle temperature (Fig. 7(c)). The lowest-mass planet also reaches much earlier the same levels of spreading rate and mantle viscosity as the basic model reaches only now (4.57 Gyr). This slows down gaseous exchange between the mantle and the surface, possibly hindering feedback mechanisms for habitability, like the geologic carbon cycle. On higher-mass planets, lower viscosity facilitates convection and heat transport.

Overall, the lower-mass planets cool down faster than the more massive ones, which slows down gaseous exchange earlier, maintaining geologic activity for less time. Concerning habitability, the difference is more striking below Earth's mass than above it. The 0.5 M$_{\oplus }$ planet may mark the threshold for maintaining enduring habitability and less massive planets could have difficulty maintaining habitability by geophysical means for many billions of years. Considering the search for habitable exoplanets, this could be partially bypassed in small planets focusing on young planets (meaning also young parent stars), when mantle temperature, heat flux, CO₂ partial pressure and gaseous exchange can all still be high.

The hotter interiors of massive planets may present some complicated perturbations to habitability. A hotter mantle decreases the temperature gradient for convective transport of energy from the core (Nimmo et al., Reference Nimmo, Primack, Faber, Ramirez-Ruiz and Safarzadeh2020), one of the three requirements for the existence of a strong magnetic field by a dynamo effect, so reducing dynamo activity and potentially the strength of the planetary magnetic field. A colder mantle may present the reverse effect. As both – high geological activity for many billion years with plate tectonics and an intense dynamo effect for a planetary magnetic field – are desirable to habitability, intermediate masses may present better results.

CHZ in time

We can run the model at different distances (different stellar fluxes) to the Sun to examine not just at which times the planet is habitable but also at what distances, delimiting the CHZ. First, we do this just for the base abiotic model. In Fig. 8, the left panels (a, c, e) are the boundaries of the CHZ for a 1 M$_{\oplus }$ planet discriminated between the regions with (i) habitable temperatures (0–100$^\circ$C) and CO₂ partial pressure above the C4 limit (dark green); and regions with (ii) habitable temperatures but CO₂ partial pressure below the C4 limit (light green). The white dashed lines are the case for insensitive W _l. The right panels (b, d, f) show the width of the CHZ for habitable temperatures. In light blue is the width of the continuous CHZ, the size of the region continually habitable since the beginning, not allowing for ‘cold start’ planets (planets that start outside the outer edge of the CHZ but can come inside of it as the CHZ can move outwards with time). Full lines are for sensitive W _l and dashed lines are for the insensitive case. The abiotic Earth model is shown in the first row (a, b) of panels, and two special cases are shown in the rows below.

Fig. 8. Boundaries of the CHZ over time for a 1 M$_{\oplus }$ planet in the Solar System. The left panels are for the CHZ with habitable temperatures. Dark greens are habitable regions with pCO₂ above the C4 limit, light greens are habitable regions with pCO₂ below the C4 limit. The white dashed lines mark the CHZ outer edge in the case of W _l insensitive to land area. The right panels are for the width of the CHZ for habitable temperatures. Full lines are for sensitive W _l, and dashed lines are for the insensitive case. Light blue lines are for the continuous CHZ. The first row (a, b) is for the complete model, the second row (c, d) is for fixing the mantle potential temperature at $T_{\rm m} = 1331^\circ$C (the present temperature given by our model), and the bottom row (e, f) is for fixing Sun's luminosity at the present value.

Solar flux has a big effect on determining terrestrial surface temperatures, but atmospheric CO₂ and H₂O do the fine balancing. In our model, the weathering equations drive the surface part of the geological carbon cycle. At the outer edge, where the solar flux is low, temperatures are low and there is less evaporation, so less rain and weathering, driving less CO₂ out of the atmosphere and facilitating the maintenance of a higher CO₂ partial pressure, which increases temperatures until equilibrium is reached. At the inner edge, where the solar flux is high, temperatures are high, which evaporates more water, which increases raining, weathering and chemical reactions, driving CO₂ out of the atmosphere and carbon to the oceans. With less CO₂ in the atmosphere, the temperatures drop until equilibrium is reached. Inside the CHZ you have intermediate cases between the two, but this only has a relevant effect when there is already enough CO₂ in the atmosphere to be driven out or enough geological activity to drive it in. Too close to the Sun and the CO₂ partial pressure may be too low to be relevant for temperature control.

In our model, we suppose sources (degassing through the spreading rate areas on the ocean floor) and sinks (weathering) of CO₂ to maintain equilibria, besides also a large surface liquid water ocean. In the absence of any of the three, the model is not applicable. Probably this is the case considering the recent state of Venus and Mars, where, in the absence of important carbon sinks, even small sources of CO₂ may lead to its accumulation in the atmosphere. As we are more interested in the habitable regime, when all three are acting, this might not be a cause for much concern.

By the first row (a, b) of panels of Fig. 8, Earth's orbit was always inside the CHZ, as surface temperatures always remained above 273 K, even in the case of insensitive W _l, which produces lower levels of CO₂ in the past. The CHZ width, with present width of ~0.16 au, was much wider in the past, reaching ~0.3 au for the insensitive case, and ~0.55 au for the sensitive case. The continuous CHZ only decreases in size and no region can be permanently habitable for more than 7.8 Gyr. As Earth is expected to have a total life span of approximately 6.2 Gyr, the best location to maximize habitability longevity would be in a higher orbit, at around 1.08 au in the case of sensitive W _l.

The region with enough CO₂ for C4 plants also gets thinner with time, until vanishing completely at around 6.5 Gyr. This is a combination of increasing solar luminosity and decreasing geological activity. Also, compared with Fig. 7(i), 6.5 Gyr is when the melt generation depth goes to zero for a planet of one-Earth-mass. We broke this behaviour into two components in the two bottom rows of panels to analyse the role of geodynamics alone in habitability.

In the second row (c, d), we fixed the potential temperature of the mantle at $T_{\rm m} = 1331^\circ$C, the temperature given by our model now, at t = 4.57 Gyr, so effectively ‘freezing’ all the geodynamics to the present state. Notice that the habitable region below the C4 limit (light green) has no change in width in time and that the displacement away from the Sun is a result of increasing solar luminosity, and that the subtle increasing width of the CHZ is more easily seen (panel d).

In this case, Earth spent most of its first 2–3 billion years uninhabitable, beyond the CHZ outer edge and below 273 K. This indicates that present geodynamics would struggle to sustain an atmosphere sufficiently rich in CO₂ to maintain temperatures above freezing during the Hadean and early Archaean.

On the bottom row (e, f), we fixed solar luminosity to the present value. In this case, the changes in time are all due to changing geodynamics. The inner edge of the CHZ is unchanged in time because the runaway greenhouse is dependent mostly on the behaviour of water and the value of solar flux. The outer edge is considerably extended in the first billions of years in comparison to the base model (panel a) because solar luminosity is the present one. The (mostly) receding outer edge with time is an effect of the insufficient build-up of CO₂ in the atmosphere. The outer edge is farther away when the planet is young and has a high internal temperature, surface heat flux, seafloor spreading rate and more intense gaseous exchanges between mantle and surface, in general, more geodynamic vigour, which is what permits it to pump more CO₂ into the atmosphere. As the planet gets older, it gets colder internally, also having a lower quantity of radioactive isotopes, the heat flux and the spreading rate decrease, and the gaseous exchanges weaken, making it harder and harder to maintain high levels of CO₂ partial pressure in the atmosphere. This impacts the location of the outer edge, but also in the location of the C4 inner limit because shallow oceans (small d _m) decrease degassing, making it easier to drive CO₂ out the atmosphere via weathering at increasingly lower thresholds of temperatures. At the right side of the outer edge in the diagrams, the temperature is below 273 K, but pCO₂ is above the C4 limit (0.01 mbar), at least initially, while the melt generational volume is not zero.

In general, solar evolution's contribution to the location of the CHZ is mostly due to increasing solar luminosity, moving the CHZ away, and widening it a bit. But geodynamics plays a bigger role in shaping the width of the CHZ, which is much larger on the younger planet, decreasing in size with time.

CHZ and planetary mass

As shown in Figs. 6 and 7, varying planetary mass has a considerable impact on the geodynamics of the planet and on its habitability in time and space. In Fig. 9, we can see this effect in the boundaries of the CHZ.

Fig. 9. CHZ over time for planets of different masses in the Solar System. For every mass, the left panels (a, c, e, g, i and k) are for the CHZ with habitable temperatures. Dark greens are habitable regions with pCO₂ above the C4 limit, light greens are habitable regions with pCO₂ below the C4 limit. The white lines and the semi-transparent green regions of the two higher mass planets mark the CHZ outer edge in the case of insensitive W _l. The right panels (b, d, f, h, j and l) are for the width of the CHZ for habitable temperatures. Full lines are for sensitive W _l, and dashed lines are for the insensitive W _l. Light blue lines are for the continuous CHZ.

The inner edge of the CHZ is the same for all six planetary masses considered, but the outer edge is further away as more massive the planet is. This is a consequence of higher internal temperature, surface heat flux and seafloor spreading rate, permitting an equilibrium with higher levels of CO₂ in the atmosphere. This has been pointed out before (Kite et al., Reference Kite, Manga and Gaidos2009; Abbot, Reference Abbot2016), and Kopparapu et al. (Reference Kopparapu, Ramirez, SchottelKotte, Kasting, Domagal-Goldman and Eymet2014) have derived that massive terrestrial planets could have wider habitable zones than less massive ones. Nonetheless, in Kopparapu et al. (Reference Kopparapu, Ramirez, SchottelKotte, Kasting, Domagal-Goldman and Eymet2014), the CHZ would expand via the inner edge, and the outer edge would vary very little because of a contra-balance of more CO₂ also increasing planetary albedo. A possible explanation for the disagreement may be that our albedo does change in function of pCO₂ levels in the atmosphere, but CO₂ is much more limited by degassing and geodynamics in our model in lower-mass planets, making for a shorter CHZ far from the limit presented in Kopparapu et al. (Reference Kopparapu, Ramirez, SchottelKotte, Kasting, Domagal-Goldman and Eymet2014) with higher levels of CO₂, also we did not scale background atmospheric pressure via planetary mass. A more embracing model, involving both features, could shed some light on this.

Focusing on the end behaviour, all masses except the 4.0 M$_{\oplus }$ one end up in a stage with the same CHZ shape and with the same width. What happens is that right after the fall of pCO₂ below the C4 limit, d _m goes to zero (compare the superior boundary of the dark green regions in Fig. 9 with where the curves go to zero in Fig. 7(i)) and the geological carbon cycle is broken, because only the regassing part, via weathering, stays functional.

Sparse volcanism could still pump some CO₂ into the atmosphere, so the drop would not be as severe and sharp as depicted here, also a frozen planet could accumulate CO₂ over long periods, when weathering is low, or if it lacks abundant liquid water for weathering. But the geologic carbon cycle as it is generally modelled would be greatly reduced in effectiveness as a climate thermostat, the geosphere would lose influence over habitability and the CHZ would be almost uniquely dictated by solar evolution. This would be the last stage in planetary habitability, coming earlier as less massive the planet is, and after 11 Gyr only the 4.0 M$_{\oplus }$ has not reached it.

Because the CHZ is changing in width and position with time, some planets could enter and exit the CHZ even twice in the simulated time. For example, a planet close to the outer edge would start inside the CHZ, exit through the outer edge as seafloor spreading rates and mantle temperatures drop, but then it would re-enter the CHZ by the outer edge as the Sun evolves and becomes more luminous, finally exiting the CHZ again for the last time by the inner edge. This initial habitable lobe away from the star is an effect of an initial hotter and active mantle and of reduced W _l by smaller continents, but also from initial warming of the mantle in planetary masses above 0.25 M$_{\oplus }$ (see Fig. 7(b)), as the behaviour of mantle temperature is reflected on the convective flux and seafloor spreading rate.

For the two more massive planets, the CHZ after 2 Gyr is wider for the case of insensitive W _l, not for the sensitive W _l. This occurs because land area is larger on those planets and more weathering can occur, restricting how much CO₂ can accumulate in the atmosphere. If there is no sensitivity to land area, the W _l is less intense (equal to the one-Earth-mass planet, because of our parameterization) and more CO₂ accumulates, because a higher mass planet will be more geologically active. The inverse happens on low-mass planets, as they have smaller continents, so the sensitive case should produce the larger CHZ later in their evolution. But this is not observed, there is no difference between the two cases. This is because the less massive planets cool too fast and there is no opportunity to see this difference.

In general, the effects of planetary mass and geodynamics on the CHZ seem very relevant, with more massive planets having a much wider and long-lived CHZ than less massive ones and the continuous CHZ more than doubling from the 0.1 M$_{\oplus }$ to the 4.0 M$_{\oplus }$ planet. The outer edge also seems very dependent on planetary mass and age. A further predicament not studied here, for less massive planets, as they cool faster, is that they should also have a shorter-lived strong magnetic field, with a higher chance of suffering from atmospheric erosion via stellar winds after some billion years, maybe in a timescale even shorter than the habitability timescale derived here for them.

Because Venus is similar to Earth in size, mass, bulk composition and proximal location in the Solar System, if Venus were indeed habitable in the past, the specific process by which it became uninhabitable would be of great interest in planetary habitability, the future of our planet, and in characterizing Earth-like exoplanets, showing similar planets following different evolutionary paths (Kane et al., Reference Kane, Kopparapu and Domagal-Goldman2014) or, in another possibility, being separated by just time.

It is generally assumed that a runaway greenhouse initiated by high solar flux is responsible for Venus’ present hellish state. This could have been primordial, and Venus was never habitable, or it happened later, and the planet had an initial habitable period. On the positive side, some 3D models (Way et al., Reference Way, Del Genio, Kiang, Sohl, Grinspoon, Aleinov, Kelley and Clune2016; Way and Del Genio, Reference Way and Del Genio2020) have been shown to be able to maintain clement temperatures in young Venus via a combination of water cloud feedbacks and slow rotation effects. And isotopic abundances of hydrogen and deuterium (Donahue, Reference Donahue1999; Kulikov et al., Reference Kulikov, Lammer, Lichtenegger, Terada, Ribas, Kolb, Langmayr, Lundin, Guinan and Barabash2006) in Venus's atmosphere indicate considerable water reservoirs in the past. On the negative side, the water may have contributed greatly to heating the planet and may not have been able to condense in any oceans (Turbet et al., Reference Turbet, Bolmont, Chaverot, Ehrenreich, Leconte and Marcq2021), making the planet uninhabitable from the start.

In a different direction, Way and Del Genio (Reference Way and Del Genio2020) speculated that the hellish present Venus’ state could be due to simultaneous large igneous provinces (LIPs) suddenly liberating large quantities of CO₂ and driving the planet into an extremely moist or runaway greenhouse. LIPs are episodic in nature, with their record dating back to billions of years, but their frequency is uncertain due to incomplete data (Ernst et al., Reference Ernst, Bond, Zhang, Buchan, Grasby, Youbi, El Bilali, Bekker and Doucet2021). They are associated with global environmental and biological changes on Earth, including mass extinctions. Simultaneous LIPs could be frequent enough on a billion-year scale to be a concern for habitability in terrestrial planets in general (Way et al., Reference Way, Ernst and Scargle2021, Reference Way, Ernst and Scargle2022).

In the case of LIPs happening randomly, there would be a constant threat of coincident events wrecking planetary habitability. Mantle plumes and plate tectonics are popular hypotheses for the origin of LIPs, and, as those are sensible to planetary internal temperature, which changes with age, LIPs could change in frequency with planetary age. Their frequency could also depend on planetary mass and radioactive isotope content, as those can change planetary internal temperatures. In this case, some ages and types of planets could be less or more susceptible to such events and face more or fewer risks. In any case, the geosphere could have a larger impact on habitability than previously considered.

In our 1.0 M$_{\oplus }$ planet, pCO₂ is low on a planet closer to its star as Venus at 0.72 au, but modern Venus does not present plate tectonics and water weathering (Lewis and Grinspoon, Reference Lewis and Grinspoon1990; Nimmo and McKenzie, Reference Nimmo and McKenzie1998), so there is no mechanism, or very low capacity, to take CO₂ out of the atmosphere, and it accumulates over time.

Our CHZ does not reach Venus orbit during the ZAMS but goes more than two-thirds of the way from Earth to Venus orbit. The effective albedo in our model includes surface and some cloud albedo, being rather low (for a global albedo, and rather high for a surface albedo). A higher albedo from cloud covering could help maintain temperatures down and bridge the remaining distance gap.

In the case of Mars, it is too small to maintain geodynamic vigour for long periods, but the absence of sinks for CO₂ permits its accumulation in the atmosphere from small sources. The lack of a strong magnetic field and its small gravity permits more easily than Earth the loss of heavier molecules, like O₂, CO₂ and N₂, contributing to thinning its whole atmosphere.

The conditions could have been better in the past, with geomorphological evidence (Craddock and Howard, Reference Craddock and Howard2002; Baker, Reference Baker2006; Di Achille and Hynek, Reference Di Achille and Hynek2010) indicating that considerable reservoirs of water modified its surface and were lost, and chemical evidence of the D/H ratio indicating a vast hydrogen escape (Mahaffy et al., Reference Mahaffy, Webster, Stern, Brunner, Atreya, Conrad, Domagal-Goldman, Eigenbrode, Flesch and Christensen2015). A cocktail of different greenhouse gases and other phenomena could have been able to warm the early Martian atmosphere (Franck et al., Reference Franck, Bounama and von Bloh2004; Pierrehumbert and Gaidos, Reference Pierrehumbert and Gaidos2011; Wordsworth and Pierrehumbert, Reference Wordsworth and Pierrehumbert2013; Ramirez et al., Reference Ramirez, Kopparapu, Zugger, Robinson, Freedman and Kasting2014; Ramirez and Kaltenegger, Reference Ramirez and Kaltenegger2017). Especially, a dense atmosphere of H₂ and CO, not of just H₂O and CO₂, could have warmed early Mars before the loss of the H₂ and of water (Pahlevan et al., Reference Pahlevan, Schaefer, Elkins-Tanton, Desch and Buseck2022).

Even then, this could not have been enough, and liquid running water could have been only episodic, with Mars being still fairly cold (mean annual temperatures below 240 K), so out of the classical CHZ in a stable way, with only episodic surface liquid water early in its history, later transitioning to the now know dry and more oxidized conditions (Wordsworth et al., Reference Wordsworth, Forget, Millour, Head, Madeleine and Charnay2013, Reference Wordsworth, Knoll, Hurowitz, Baum, Ehlmann, Head and Steakley2021; Wordsworth, Reference Wordsworth2016).

In our model, the 0.1 M$_{\oplus }$ planet never reaches even 0.1 bar of CO₂ at any point of its evolution at 1.52 au, being severely limited by its geodynamics, and has a too narrow CHZ and too low temperatures (~200 K) during its first billion years to be easily viable. At so low temperatures, the oceans could freeze and the weathering drop even lower, permitting an increased accumulation of CO₂ in the atmosphere. But this could be episodic or cyclic, and the geologic carbon cycle would still be broken and the planet out of the classical CHZ.

As it is, our model fails to account for the possible past habitability of Venus or Mars, probably because a simple 0D model of effective planetary annual temperatures may not be the best model to probe climatic subtleties, episodic periods of habitability or cloud effects. Also, other greenhouse gases besides CO₂ and H₂O could have contributed to warm planets close to the edges of the classical CHZ. However, our results of uninhabitability of past Venus and Mars are in line with the results of Turbet et al. (Reference Turbet, Bolmont, Chaverot, Ehrenreich, Leconte and Marcq2021) for a never habitable Venus and of Wordsworth (Reference Wordsworth2016) and Wordsworth et al. (Reference Wordsworth, Knoll, Hurowitz, Baum, Ehlmann, Head and Steakley2021) for only episodic current water in past Mars.

An important general result ended being that the derived boundaries for the CHZ point to a highly dynamic behaviour in time and space, and planetary habitability would depend both on stellar properties and, especially, planetary geodynamics.

CHZ in the present Solar System

Since past habitability in the Solar System may be uncertain besides Earth, the present boundaries for the CHZ may be more easily studied, and comparing the different boundaries presented in the literature may give us some information about progress or convergence in the models.

Figure 10 compares most of the works presenting the CHZ for the present Solar System. It is frequent the investigation of habitability for more than just Earth or a planet very like Earth in base parameters (mass, radius, water content, atmospheric composition, atmospheric pressure, etc.), extending the investigation for a wider volume in the space of planetary and atmospheric parameters. We represented this via colour. The green bars are the CHZ for the most Earth-like run in that specific work (so the CHZ would be at least as wide as the green bars), orange and blue bars are extensions of the CHZ when exploring a different combination of planetary parameters, modelling planets possibly rather different than Earth. Some works do estimate the extreme minimum or maximum boundaries for the CHZ, so studying extreme conditions. These cases are indicated by thin bars and green arrows.

Fig. 10. Comparison of the CHZ for the present Solar System to previous works. The green bars are for the Earth case or Earth-like cases, planets very close to Earth in the space of parameters (same mass, radius, atmospheric pressure, atmospheric composition, water content, etc.). Orange and blue bars are extensions of the classical CHZ considering other planetary and atmospheric parameters. Dark orange or dark blue bars with arrows indicate the minimum and maximum boundaries in extreme cases, respectively.

The models, selected parameters and methodology vary considerably from one work to another and we recommend checking them individually for more in-depth details. Some of these differences could invalidate comparison (as we are not comparing the same thing), but they all present an attempt to derive the boundaries of planetary habitability, a habitable zone, so they seem comparable in our judgement. Also, if rather different models and approaches converge to the same or close results and conclusions, their results could be considered more robust.

Fortunately, in all works, Earth's orbit is inside all the CHZ for Earth or Earth-like planets, generally closer to the inner edge.

In Fig. 10, our CHZ is one of the narrowest considering only the result for a 1.0 M$_{\oplus }$ planet. The inner edge at 0.92 au is not so different from other works; the difference is larger in relation to our 1.09 au outer edge. In our model, geodynamics seems to heavily limit maximum CO₂ levels in the atmosphere and would be hard to achieve in normal circumstances many bars of CO₂ in the atmosphere even far away from the Sun, limiting the width of the CHZ more strongly than in other works. Increasing planetary mass increases internal temperature, heat flux and seafloor spreading rate, maintaining more intense geological activity for more time, which is, in the case of the 4.0 M$_{\oplus }$ planet, what extends the present CHZ to 1.24 au (for sensitive W _l), and 1.36 au (for insensitive W _l).

The CHZ of Hart (Reference Hart1978) is the narrowest one, barely containing Earth's orbit. His model is very unstable, easily leading to runaway greenhouses or glaciations. The problem seems to be that it lacks a negative feedback for climate stability (in general, weathering, as in our model), so the conditions should need to be very right to allow for habitability in Earth's orbit. He uses a continuously circumstellar habitable zone (CCHZ), not a CHZ, making for a more strong case of habitability which will produce narrow habitable spaces. But his model is unstable enough that the difference to the CHZ might be small. It is because of that that we included it in the comparison.

The models seem to vary considerably in the size of the CHZ and do not converge over time to the same boundaries. However, an interesting result is that the inner edge of the CHZ is better defined than the outer edge. Counting only the Earth-like CHZ boundaries, the difference in the position of the inner edge is 0.17 au, but 0.70 au for the outer edge. This difference is toned down considering the cases for other planetary parameters but it remains visible. Even if setting working boundaries remains important, it is also important to remember that the CHZ might not have absolute boundaries and may depend on a lot of different parameters, so the boundaries may more properly be fuzzy and planetary habitability varies more continuously with distance.

The trend for small variation in the location of the inner edge is not so much a surprise. The inner edge is classically defined by the runaway greenhouse or wet greenhouse and loss of water. So the fate of the planet is heavily influenced by the behaviour of water in its atmosphere and the changing solar luminosity, which will approximately be the same in the models. But only if the planet is an aquaplanet. For much dryer planets, the inner edge can be pushed closer to the Sun (Abe et al., Reference Abe, Abe-Ouchi, Sleep and Zahnle2011; Zsom et al., Reference Zsom, Seager, De Wit and Stamenković2013; Kodama et al., Reference Kodama, Nitta, Genda, Takao, O'ishi, Abe-Ouchi and Abe2018). The same with an increase in cloud cover, and so in albedo (Selsis et al., Reference Selsis, Kasting, Levrard, Paillet, Ribas and Delfosse2007), which could considerably extend the CHZ inwards.

Earth's orbit is also much closer to the inner edge than the outer edge in most works. This implies that, by most models, Earth's habitability is closer to its end in astronomical timescales.

In the classical CHZ, the outer edge depends on the effectiveness of CO₂ in warming up the atmosphere and maintaining habitable temperatures. How much CO₂ could accumulate in the atmosphere and its effectiveness as a greenhouse gas, considering its sensitiveness with geologic activity, albedo and the formation of CO₂ clouds, is something easily variable from model to model. Because methane, H₂ and other gases can contribute to the greenhouse effect, the outer edge is poorly constricted, potentially being extendible today beyond the orbit of Mars (Kasting et al., Reference Kasting, Whitmire and Reynolds1993) even for Earth-like planets if the greenhouse effect of CO₂ cloud cover or very high pCO₂ (>1 bar) are considered (Selsis et al., Reference Selsis, Kasting, Levrard, Paillet, Ribas and Delfosse2007; Kopparapu et al., Reference Kopparapu, Ramirez, Kasting, Eymet, Robinson, Mahadevan, Terrien, Domagal-Goldman, Meadows and Deshpande2013; Ramirez and Kaltenegger, Reference Ramirez and Kaltenegger2017). Very high pCO₂ in habitable planets may be less likely because of geodynamics limitations as we have discussed here, but a cocktail of different greenhouse gases seems more likely at least when the planet is young.

As some of the CHZ admit Mars’ orbit, they pose the question of why Mars is not habitable today. The reason, as mentioned before, is probably its mass. The CHZ is for a planet with one-Earth-mass (green) and the outer extensions (blue) for the CHZ are more generally for planets with one Earth mass or above it or that can retain a dense atmospheric envelope. The use of the widest CHZ in other systems works as a first approximation but may pose problems if the planets on them are too light, old or closer to the uncertain CHZ's outer edge.

CHZ for F, G, K and M stars

By changing stellar luminosity and albedo response to different stellar spectra we can simulate different stars and estimate a more general CHZ. Figure 11 shows the inner and outer boundaries of the CHZ for the six planetary masses considered. The left panel (a) is the CHZ at the ZAMS. There is only one inner edge for all masses (black line) and different outer edges, one for every mass. For easier visualization, we show just the case of sensitive W _l, but the insensitive case would be analogous, with just shorter outer edges. The right panel (b) is the continuous circumstellar habitable zone for 5 Gyr (region continually inside the CHZ for 5 Gyr). We did not consider ‘cold start’ planets for the continuous CHZ.

Fig. 11. Inner (black line) and outer (coloured lines) boundaries of the CHZ: (a) boundaries for the ZAMS and (b) boundaries for the 5 Gyr continuous CHZ. Internal Solar System planets are marked as black dots with a white outline. The two thick dashed grey lines in (b) are the inner and outer boundaries of the ZAMS CHZ of (a) repeated for comparison. Above the horizontal dashed grey line at 1.3 M$_{\odot }$, stars have a main sequence lifetime inferior to 5 Gyr as given by equation (21).

We can see the general behaviour of the CHZ for various stars at the ZAMS. The CHZ is narrow and more close to the less massive and less luminous stars, being wider and more distant to more massive and more luminous stars. The difference in location of the outer edge is greater for less massive planets than for more massive ones, so the CHZ at the ZAMS is about the same size for planets of masses above 0.5 M$_{\oplus }$.

In the continuous CHZ, we see that the CHZ can contract in both directions: from the inner edge moving outwards, as the stars evolve, and from the outer edge moving inwards, as the planets get old and are less capable of maintaining high levels of CO₂ in their atmospheres. The narrowest continuous CHZ is for the 0.1 M$_{\oplus }$ planet, already non-existent for solar-mass stars and close to disappearing completely in lower-mass stars. The widest region for a 5 Gyr continuous CHZ seems to be for the more massive planets around stars of 0.9–1.2 M$_{\odot }$.

Comparing the two, we can derive evolutionary trends. Below 0.8 solar mass, stellar evolution is slow in the 5 Gyr interval, so luminosity does not increase by much and the inner edge moves little or barely. However, for the planets, the 5 Gyr interval is significant, and the outer edge contracts as the planets get old and cool down. Around low-mass star, this contraction of the outer edge is visible even for the planets more massive than Earth although less than for the less massive ones.

Above 1.1 solar mass, stellar evolution is fast in the 5 Gyr interval, so the inner edge moves by a lot, but the outer edges for the 2.0 and 4.0 M$_{\oplus }$ are curiously in the same position as in the ZAMS. More massive planets would be able to maintain a wider CHZ, but, as we saw, even their outer edge should already have receded at least a bit at this point. Here, stellar evolution is so fast that it compensates and surpasses the effects of planetary evolution. Because we are showing the continuous CHZ and no ‘cold start’ planets are allowed, the initial outer edge stays fixed. If we were allowing ‘cold start’, showing the ‘instantaneous’ CHZ at 5 Gyr, the outer edges for the more massive stars would be much outwards than at the ZAMS, even when considering the planets’ thermal decay. As low-mass planets have the narrowest starting CHZ, they are the most affected by the fast stellar evolution, and for any planet below one-Earth-mass, there is no 5 Gyr continuous CHZ left above 1.2 M$_{\odot }$. Stellar mass of 1.3 M$_{\odot }$ may mark the limit for the 5 Gyr continuous CHZ, as it is almost non-existent, and above 1.3 M$_{\odot }$, stars have a main sequence lifetime inferior to 5 Gyr as given by equation (21).

This behaviour, where the CHZ for planets around low-mass stars is limited more by geodynamics and the CHZ for planets around high-mass stars is limited more by stellar evolution, was mentioned before in Cuntz et al. (Reference Cuntz, von Bloh, Schröder, Bounama and Franck2012).

Figure 12 shows similar cases, but we compare the CHZ for different models in the literature. We had a smaller amount of models to use here than in Fig. 10 because not all models are presented for many different stellar masses, the continuous CHZ is not presented, or the CHZ is presented only for stellar effective temperatures or fluxes and the conversion to stellar mass and age would not be certain in every case. Because of this, the models presented in the ZAMS case (a) are generally not the same as the ones presented in the continuous CHZ (b). Also, the 5 Gyr continuous CHZ of Kasting et al. (Reference Kasting, Whitmire and Reynolds1993) presented here is a mean of the values for a continuous CHZ of 4 and 6 Gyr as presented in their paper. The presented continuous CHZ of Hart (Reference Hart1979) is for just 4.5 Gyr. As his continuous CHZ is already very narrow, a half-billion-year-older continuous CHZ would be even narrower and more pessimistic than shown. For our continuous CHZ, for easier visualization, we only show the case for 4.0 M$_{\oplus }$, the widest CHZ and CCHZ, but we included the variation of sensitive W _l (full black lines) and insensitive W _l (dashed black lines).

Fig. 12. Comparison of the boundaries of the CHZ as given by different models in the literature: (a) boundaries for the ZAMS and (b) boundaries for the 5 Gyr continuous CHZ. The 5 Gyr continuous CHZ of Kasting et al. (Reference Kasting, Whitmire and Reynolds1993) is a mean of a continuous CHZ of 4 and 6 Gyr as presented in their paper. The presented continuous CHZ of Hart (Reference Hart1979) is for just 4.5 Gyr. In both panels, we show our results for only the 4.0 M$_{\oplus }$ to facilitate visualization and better distinguish between lines. Internal Solar System planets are marked as black dots with a white outline.

Looking at the ZAMS case, and comparing it with Fig. 10, the inner edges are still closer together than the outer edges, but the difference is much less significant. The inner edges are very close to Venus’ orbit, if not including Venus in some models. The outer edges are also close to Mars’ orbit but this may include conditions improbable for a planet small as Mars, as in our case it is because we are using the CHZ for a high mass terrestrial planet, a 0.1 M$_{\oplus }$ has a much narrower CHZ even at the ZAMS. Others’ outer edges seem to go further out than ours.

At the 5 Gyr continuous CHZ, the inner edges are still mostly coincident and very close to Earth orbit except for Kasting et al. (Reference Kasting, Whitmire and Reynolds1993), which moves outwards the less above ~0.8 M$_{\odot }$, while the outer edges are much more spread out. As said before, this implies that, by most models, Earth's habitability is closer to its end in astronomical timescales. The narrowest CHZ is the one of Hart (Reference Hart1979), barely containing Earth and non-existent for most stellar masses. The motives are the same as mentioned before for Hart (Reference Hart1978).

Our continuous CHZ has a much closer outer edge, making it the second narrowest here presented, in contrast to other continuous CHZ. Our modelled planets struggle to build dense CO₂ atmospheres and our CHZ contracts with time because of an advancing inner edge (by stellar evolution) and also contracts because of a receding outer edge (via effects of geodynamics). The difference between sensitive and insensitive W _l is more striking here because the insensitive case has a non-existent 5 Gyr continuous CHZ above 1.15 M$_{\odot }$. This is because insensitive W _l makes surface temperature lower initially, as W _l is not weakened by initial small continents, making the CHZ outer edge initially closer to the star.

Width of the CHZ

The widest the CHZ in a stellar system is, the more likely that at least one planet will be inside of it and that it can be habitable. The same for the continuous CHZ with the advantage of possibly long-lived habitability. We saw that the width of the CHZ depends not only on the stellar luminosity, but also on planetary mass, geodynamics and age. So, a more precise or useful way to access the width of the CHZ would be along these dimensions.

In Fig. 13, we compare the width of the CHZ for different planetary and stellar masses, and ages, condensing everything we saw before except for the location of the CHZ. We show both the cases for sensitive W _l (full lines) and insensitive W _l (dashed lines). As before, we see that the low-mass stars have the most long-lived CHZ, going all the 13 Gyr of the simulation's runs, but also the narrowest ones. More massive stars have the widest CHZ, but also the shortest lifespans.

Fig. 13. Width of the CHZ for the six planetary masses and 12 stellar masses considered. Full lines are for sensitive W _l, and dashed lines are for insensitive W _l. Numbers on subtitles and colours correspond to solar masses.

The difference between the cases of sensitive W _l and insensitive W _l is less prominent on low-mass stars and low-mass planets. The insensitive W _l case starts with a narrow CHZ but converges with the sensitive case as the planets get cold internally. Recalling Fig. 9, the moment where CO₂ levels are too low and the outer edge starts to be dictated primarily by the star is clearly visible, at 1 Gyr for the 0.1 M$_{\oplus }$ planets, 2.5 Gyr for the 0.25 M$_{\oplus }$ planets, then 4 Gyr for the 0.5 M$_{\oplus }$ planets and so on to beyond 13 Gyr for the 4.0 M$_{\oplus }$ planets.

For above-solar-mass stars, the difference between the cases of sensitive W _l and insensitive W _l is larger, but more a result of increased stellar flux. On the two higher mass planets, the behaviour is more complex, the width of the CHZ is initially the narrowest for the insensitive W _l case, but later the insensitive W _l case shows the widest CHZ. As seen before, this is a result of calibrating W _l on the smaller continents of the present Earth. In general, we showed that the sensitivity of W _l to land area can have a considerable effect on the outer edge of the CHZ. The two cases studied here are just extremes of the range most commonly presented in the literature for this sensitivity and perhaps a more realistic modelling would present an intermediate sensitivity, equivalent to raising f _land in equation (15) to some power lower than 1.0, but stronger dependencies may not be impossible. If such a poorly studied parameter can have an important impact on the CHZ boundaries, it should be given more attention, as well as the parameterization of the weathering equations used in the models.

Concerning long-lived habitability, a better metric than CHZ is the continuous CHZ presented in Fig. 14. Here, the continuous CHZ is not of a fixed age (5 Gyr as in the anterior cases) and we can follow its width in time. The low-mass stars also present the longest-lived continuous CHZ, but they are even more narrow than the CHZ and do not reach the age of 13 Gyr for the two less massive planets. High-mass stars present the widest continuous CHZ initially, but they shrink fast. Sensitive W _l clearly produces the widest continuous CHZ, initially almost double the width of the insensitive W _l case. This is even the case for planets above Earth mass, because the insensitive W _l case starts with a narrower CHZ, and so the continuous CHZ cannot become wider than that, even if the insensitive case maintains a wider CHZ later as seen in Fig. 13.

Fig. 14. Width of the continuous CHZ for the six planetary masses and 12 stellar masses considered. To facilitate visualization, we separate the two cases of W _l sensitive to land area (top a–f panels, full lines) and W _l insensitive to land area (bottom g–l panels, dotted lines). Numbers on subtitles and colours correspond to solar masses.

Because planets lose geophysical vigour as they age, the CHZ gets narrower with time, and then the continuous CHZ may even become non-existent for systems of a certain age, making old planetary systems weak targets for astrobiology study in the timescale of the age of our Galaxy or in the timescale stars spend in the main sequence. In other words: most habitable planets may have a shorter lifespan than their stars and stellar lifetime should not be equated with planetary habitability lifespan, especially in the case of continuous habitability for many billions of years.

These results may also have consequences for the quantity and age of habitable planets in our galaxy. As reported by some models, three-quarters of Earth-like planets in the Universe may be 1.8 ± 0.9 Gyr older than Earth (Lineweaver, Reference Lineweaver2001), and 75% of the stars in the Galactic habitable zone of our galaxy may be older than the Sun (Lineweaver et al., Reference Lineweaver, Fenner and Gibson2004). All these older planets would have much more time than Earth to pass through abiogenesis, for life to evolve, and for high intelligence and a high technological civilization to appear. However, considering what we discussed in this paper, a substantial fraction of planets previously considered habitable could be geologically dead or with compromised habitability if aged 5 Gyr or more. In this manner, time could also be seen as a Goldilocks’ parameter: too little of it, and still not enough steps have been taken for life to appear and evolve, but too much of it, and the planet may become rigid and uninhabitable, greatly diminishing any chance of any evolutionary steps to take place. This time window would be shorter for low-mass planets and longer for more massive planets, but other planetary and stellar parameters could also influence the size, location and lifespan of the CHZ.

Life, when it appears on a planet, can have some stabilizing effect on its climate, potentially prolonging its habitability, and a technologically advanced species can enrol in geoengineering, terraforming or just leave its system behind, colonizing the Galaxy in a short time frame when compared with the Galaxy's age. But, considering planetary geodynamics more seriously, most current habitable exoplanets may be younger than Earth, not older, and so still had considerably less time than Earth to produce life, evolve highly intelligent species or technologically advanced civilizations.

We did not study different radioactive budgets in this work, but as pointed out in Gonzalez et al. (Reference Gonzalez, Brownlee and Ward2001), a higher stellar birth rate in the past of our Galaxy could have increased interstellar radioactive elements content, making some of the initial generations of habitable planets (old planets now) somewhat richer in radioactive elements than more recent generations. This could have lengthened their habitability timespan by increasing internal temperatures, heat fluxes and geologic vigour. But the extent to which this affects long planetary habitability still needs to be properly studied in conjunction with all discussed in this work.

Also, nothing was told about the distribution of planets by mass (or radius) in our Galaxy. Similar to the power law of initial mass function of stars, in which the formation of low-mass stars is much more frequent than that of high-mass stars, it could be naively expected that small rocky planets could be more common than more massive rocky planets. In this case, most terrestrial planets would be small and could present just a narrow and short-lived CHZ. Nonetheless, observational data, even if restricted by observational biases and incompleteness, suggest a more complex behaviour for the distribution over the entire radius range and no big increase in frequency (and maybe even a decrease in frequency) is observed for small planets in the range considered here (0.1–4.0 M$_{\oplus }$ or 0.49–1.44 R$_{\oplus }$) (Kunimoto and Matthews, Reference Kunimoto and Matthews2020; Dai et al., Reference Dai, Liu, An and Zhou2021). Future exoplanet demographic surveys may be better able to indicate a more reliable distribution.

All these limitations point to different strategies in the search for habitable planets.

The widest CHZ would be found in young systems (<2 Gyr) with planets more massive than Earth. And this also includes planets around stars more massive than the Sun, even though they evolve fast, they still did not have time to increase their luminosity by much and still have a wider continuous CHZ. A potential biosphere around these planets would also be young.

Around low-mass stars (<1.0 M$_{\odot }$) lie the highest chances to find any old planet (>5 Gyr) still habitable or an old biosphere. These stars have the narrowest CHZ but are the most numerous in the Galaxy, and focus on planets above 0.5 M$_{\oplus }$, and especially above one-Earth-mass, could increase the odds considerably.

Finally, a compromise for a long continuous CHZ lifespan and a wide CHZ could be found in systems with planets with mass above 0.5 M$_{\oplus }$ and around stars of approximately solar mass.