TOWARD THE MINIMUM INNER EDGE DISTANCE OF THE HABITABLE ZONE

The T–P profile and volume mixing ratios are initialized with a convective zone that follows the moist adiabat from the surface to the tropopause, and an isothermal stratosphere. The atmosphere is described by the surface conditions (pressure, temperature, and albedo), relative humidity, CO₂ mixing ratio, background gas (typically N₂), and planetary surface gravity. Unless otherwise stated, we use 60 layers per pressure decade. The top-of-atmosphere pressure is fixed at 10⁻⁴ bars because the atmosphere is optically thin in all spectral bands above this level. The surface pressure is varied between 0.1 and 100 bars.

First, we establish the mixing ratios of CO₂ and the background gases on a dry pressure grid (without water vapor). We assume that these gases are well mixed in the atmosphere and their mixing ratios are constant as a function of pressure. This assumption is correct if the mixing timescale in the atmosphere is much shorter than the timescale on which the sources and sinks of the atmospheric constituents operate.

Next, the moist adiabat is built from the surface upward given the dry surface pressure (P_surf, dry), temperature (T_surf), and relative humidity (Φ) following Pierrehumbert (2010):

$$\begin{equation} \frac{d \ln T}{d \ln P_{\mathrm{dry}}} = \frac{R_{{\rm bg}}}{c_{p,\,{\rm bg}}} \frac{1 + \frac{L}{R_{{\rm bg}} T} f_{{\rm H_2O}}}{1 + \left[ \frac{c_{p,\,{\rm H_2O}}}{c_{p,\,{\rm bg}}} + \left(\frac{L}{R_{{\rm H_2O}} T} - 1 \right) \frac{L}{c_{p,\,{\rm bg}} T} \right]f_{{\rm H_2O}}}, \end{equation} \tag{ 1 }$$

where R_bg and $R_{{\rm H_2O}}$ are the specific gas constant of the background gas and water vapor, respectively; c_p, bg and $c_{p,\,{\rm H_2O}}$ are the specific heat capacity of the background gas and water vapor, respectively; L is the latent heat of water vapor; and $f_{{\rm H_2O}}$ is the water vapor mass mixing ratio:

$$\begin{equation} f_{{\rm H_2O}} = \frac{m_{{\rm H_2O}}}{m_{{\rm bg}}} \frac{ \Phi P_{\mathrm{sat}}(T)}{ P_{\mathrm{dry}}}, \end{equation} \tag{ 2 }$$

where m_bg and $m_{{\rm H_2O}}$ are the molecular masses of the background gas and water vapor, respectively, and P_sat(T) is the saturation vapor pressure at temperature T calculated according to the Smithsonian Meteorological Tables. The moist adiabat is followed until the temperature drops below the prescribed initial stratosphere temperature. The stratospheric water vapor mixing ratio is constant and is determined by the tropopause temperature and relative humidity. Finally, the total atmospheric pressure is calculated as the sum of the water vapor partial pressure and the dry pressure.

We develop a specific numerical approach to calculate the T–P profiles: we fix the surface temperature and determine the semi-major axis where the atmosphere is in radiative equilibrium. Our approach is untypical in a sense that commonly the exoplanet's semi-major axis is fixed and the corresponding surface temperature is calculated (see, e.g., the EXO-P model of Kaltenegger & Sasselov 2010). But, as we want to determine the equilibrium planet–star separation for given surface temperature and pressure, our method is better suited for this study.

The equilibrium T–P profile is determined by iterating the following steps.

• 1.  
  We calculate the fluxes propagating through each atmospheric layer in several wavelength bands; we treat the thermal emission of the exoplanet's surface/atmosphere and the stellar irradiation separately and evaluate the stellar irradiation fluxes at an arbitrary distance.
• 2.  
  We determine the semi-major axis where the top-of-atmosphere fluxes are in equilibrium using the Bond albedo of the exoplanet and the outgoing long wave radiation.
• 3.  
  The stellar irradiation fluxes are rescaled such that the incoming stellar flux corresponds to the value received at the new semi-major axis.
• 4.  
  The stratosphere temperature profile is updated based on the net fluxes propagating through the layers.
• 5.  
  The tropopause height is updated if necessary.

The iteration stops when the T–P profile reaches equilibrium. These steps are described in the following.

2.2.1. Radiative Transfer

2.2.2. Climate Model

The goal of the climate code is to iterate the temperature and water vapor profiles to a radiative-convective equilibrium based on the fluxes propagating through the layer interfaces. This is achieved in three stages. First, we determine the planet–star separation where the top-of-atmosphere incoming and outgoing fluxes are in equilibrium and rescale the stellar fluxes using the new distance. Then we update the stratospheric temperature profile, and finally the tropopause height is determined.

The distance where the top-of-atmosphere fluxes are balanced is calculated as

$$\begin{equation} {\rm OTE} = \frac{1}{4} (1 - \alpha) F_{{\rm in}}(a), \end{equation} \tag{ 3 }$$

where OTE is the top-of-atmosphere outgoing thermal emission, α is the Bond albedo of the planet, and F_in is the top-of-atmosphere incoming stellar irradiation. As the T–P and mixing ratio profiles are assumed constant within one iteration, the OTE is independent of distance a. The albedo (the ratio of top-of-atmosphere outgoing and incoming stellar fluxes) is also distance-invariant, and it is evaluated at an arbitrary distance, typically where the incoming stellar flux equals the solar constant. Therefore, the unknown in Equation (3) is F_in and the distance a where F_in is received at the top of the atmosphere. Once the equilibrium incoming stellar flux is determined, the downwelling and upwelling stellar fluxes are scaled.

We update the stratospheric temperature in the next step. We adopt a numerically efficient and simple approach to update the T–P profile: the temperature of a layer is increased by dT, if the net flux propagating through the layer (dF_i—the difference between incoming and outgoing fluxes) is positive, and the temperature decreases otherwise. The temperature step (dT) is 0.5 K in our simulations. Our method differs from traditional approaches that calculate the derivative dT_i/dt (change of temperature over time) in each atmospheric layer i:

$$\begin{equation} \frac{dT_i}{dt} = \frac{g}{c_{pi}} \frac{dF_i}{dP_i}, \end{equation} \tag{ 4 }$$

where g is the gravitational constant, c_pi is the total heat capacity of the layer, dF_i is the net flux propagating through the layer, and dP_i is the pressure difference at the top and bottom of the layer. If the net flux is negative, the layer cools. The time step (dt) is adaptively determined by setting an upper limit on dT_i (dT_max). We find that although the traditional approach properly captures how the temperature profile evolves in time, it converges slowly if the tropopause pressure level is orders of magnitude larger than the top-of-atmosphere pressure level. Furthermore, it is unimportant how the temperature profile evolves in time from an arbitrary initial state, as the goal is to determine an equilibrium profile.

Finally, the tropopause height and the stratospheric water mixing ratio are updated in the climate code. The lapse rate between the tropopause layer and the layer above is calculated. The tropopause height is increased by one layer if the lapse rate is steeper than the moist adiabat. The tropopause height is decreased by one layer if the lapse rate is larger than 0 K km⁻¹ (i.e., the layer above the tropopause is warmer). As the tropopause temperature changes, so do the tropopause and stratospheric water mixing ratios.

The steps outlined in the previous two sections, namely, radiative transfer, determining the equilibrium distance, stratosphere temperature, tropopause, and water mixing ratio updates, are iteratively performed until the temperature profile relaxes to equilibrium. Our convergence criteria measure the temperature change of each layer for n iterations. Convergence is reached, if the temperature change is less than 2 dT in all of the layers during n successive iterations, where n is typically 10. Model validation and tests are described in the Appendix.

The temperature-pressure profiles described in Section 3 have some general characteristics: a convective zone close to the surface, a cold tropopause, and inversion in the stratosphere. We assume that regions close to the surface are convective. The stratospheric inversion occurs because water vapor absorbs the incoming stellar radiation in the near-infrared. CO₂ does not have absorption bands at such short wavelengths, but it emits in the infrared; thus, CO₂ acts as a stratospheric coolant (Clough & Iacono 1995).

We provide a qualitative description on how the various atmospheric parameters influence the outgoing thermal flux, the albedo of the planet, and thus the inner edge distance in this section. In general, increasing the OTE and/or the albedo of the planet moves the inner edge closer to the star because more incoming stellar flux is necessary to achieve radiative-convective equilibrium. The most important effects of each parameter are described in detail in Section 3, while other parameters are fixed during the simulations.

Stellar type. We perform simulations using various stellar types ranging from M6V to G0V (or stellar masses ranging from 0.09 to 1.1 M_☉, respectively). We use the "BT_Settl" model spectra grid of Allard et al. (2003, 2012 to make comparison easier with Kopparapu et al. 2013). The stellar effective temperatures range between 2700 K and 6000 K with a step of 300 K. We choose model spectra with solar metallicity (Asplund et al. 2009) and assume a log g of 4.5. The stellar mass and luminosity are assigned to the stellar spectra from the stellar evolution models of Baraffe et al. (1998, 2002) assuming an age of 5 × 10⁹ yr.

The spectral energy distribution of the star has several effects on planetary climate and habitability. The first-order effect of stellar type is related to the luminosity of main-sequence stars. An exoplanet has to be much closer to an M dwarf than to an F star to receive the same total flux at the top of its atmosphere. However, the spectral energy distribution of stars has higher order effects on the inner edge distance. These effects become apparent, if the top-of-atmosphere fluxes are compared at the inner edge around various stars (Kopparapu et al. 2013). As an example, the spectral energy distributions of low-mass stars peak at longer wavelengths where Rayleigh scattering is less efficient, and thus the planetary albedo is reduced. Furthermore, a larger fraction of the incoming stellar radiation is directly absorbed by the atmosphere around low-mass stars because greenhouse gases such as H₂O and CO₂ absorb in the near-IR. To a smaller extent, the UV flux of the host star (especially the extreme-UV) influences the composition of the stratosphere, which in turn can slightly influence the surface climate (see Figure 4(a) of Rugheimer et al. 2013).

Surface gravity. The effects of surface gravity are twofold. If the surface pressure is fixed, the cumulative optical depth of the atmosphere is somewhat reduced at all wavelengths for a larger surface gravity because the atmospheric scale height and thus the column density of greenhouse gases are reduced. On one hand, this increases the OTE of the exoplanet and shifts the inner edge closer to the star. On the other hand, more stellar flux reaches the surface because less stellar flux is absorbed by the atmosphere, and the column density of Rayleigh scattering background gas is reduced. If the surface albedo is low, the Bond albedo of the planet is reduced and the inner edge is pushed farther away from the star. If the surface is reflective, the inner edge is pushed closer to the star. Therefore, the albedo effect of surface gravity depends on the relative contributions of two effects to the Bond albedo: Rayleigh scattering and surface reflection. The mass and radius of an exoplanet also influence how much atmosphere the planet can retain (see, e.g., Lammer et al. 2009) and what the volatile content of the mantle is that can potentially be outgassed.

We use values between g = 5 m s⁻² for a planet somewhat larger than Mars (10 m s⁻² for an Earth-like planet) and 25 m s⁻², which is a typical surface gravity of a 10 M_⊕ super-Earth.

Surface pressure. The effect of surface pressure on the inner edge distance is twofold. Larger surface pressure increases the albedo of the planet due to Rayleigh scattering. However, an increasing surface pressure also raises the column density of the greenhouse gases and thus the cumulative optical depth of the atmosphere for a fixed mixing ratio. Therefore, the effect of surface pressure on the inner edge distance cannot be readily estimated; thus, we treat it as a free parameter. We consider planets with 0.1, 1, 10, and 100 bars of surface pressure.

CO₂ mixing ratio. The lower the CO₂ mixing ratio is, the closer the inner edge of the HZ is to the star. CO₂ in combination with water vapor is a very effective greenhouse gas because these gases have non-overlapping absorption bands in the infrared. If CO₂ were removed from the atmosphere of Earth, the global average temperature would be below the freezing point of water (Pierrehumbert 2010). Therefore, it is crucial to calculate how the mixing ratio of CO₂ influences the inner edge distance.

The CO₂ mixing ratio is a free parameter in our study owing to the uncertainties in the outgassing and deposition rates. The mixing ratio of CO₂ is varied between 10⁻⁵ and 10⁻¹. We do not consider higher CO₂ mixing ratios because such planets are expected to be unhabitable close to the host star due to the greenhouse effect of CO₂. Thus, an assumption in our model is that some process (e.g., the carbon–silicate cycle) keeps the atmospheric CO₂ at bay. If this assumption is not met, the atmosphere becomes CO₂-dominated like the atmosphere of Venus.

Relative humidity. The relative humidity is a key parameter in our study because it has a strong influence on the inner edge distance. Unfortunately, it is exceedingly difficult to model the relative humidity profile of an exoplanet because it is determined by complex 3D processes such as atmospheric circulation, cloud formation, and precipitation. The water cycle is also influenced by the spatial distribution of liquid water on the planetary surface. We do not attempt to model these effects. Therefore, the relative humidity is a free parameter in our study, and it is changed between 0.01% and 70%. The nominal value is 1%, which we determine by estimating the dominant form of precipitation (see Section 2.4).

Surface temperature. The inner edge of dry planets is set by the boiling point of water because dry planets do not enter the moist runaway greenhouse stage (even at these high surface temperatures; see Section 3). The boiling point, and thus the surface temperature at the inner edge, is determined by the surface pressure and the relative humidity:

$$\begin{equation} T_{\mathrm{surf}} = \max (T:P_{\mathrm{surf}} + \Phi P_{\mathrm{sat}}(T) > P_{\mathrm{sat}}(T)\,), \end{equation} \tag{ 5 }$$

where P_surf is the dry surface pressure and ΦP_sat(T) is the partial pressure of water on the surface. The inner edge surface temperature of Earth could be 410 K for a dry surface pressure of 1 bar, and the surface relative humidity is 70%. If Earth had 1% relative humidity and all other properties kept constant, the inner edge surface temperature would be lower, 373 K because the partial pressure of water is reduced.

The maximum surface temperature is also limited by the chemical stability of complex organic molecules. If only Equation (5) is used to calculate the inner edge surface temperature, its allowed range is between 273 K and 647 K given by the triple and critical points of water, respectively. Can life exist at temperatures as high as 647 K? On Earth, organisms grow at temperatures up to 395 K (Kashefi & Lovley 2003; Takai et al. 2008). However, the DNA and amino acids become chemically unstable generally at temperatures above 500 K (Lang 1986). Therefore, we limit the surface temperature to a maximum of 500 K. We note that this is a globally averaged surface temperature because we use a 1D climate model. The surface temperature on the globe is expected to be scattered around the mean value as shown by global circulation models (see, e.g., Leconte et al. 2013). Therefore, the surface temperature must remain well below 500 K at a substantial part of the planet (see Section 2.4).

These inner edge surface temperatures differ from previous estimates. Most previous studies assumed that the troposphere is saturated with water vapor, the tropopause and stratosphere temperatures are set to 200 K, and Earth's water reservoir is present on the surface (Kasting et al. 1993; Kopparapu et al. 2013). Under these conditions, the onset of the moist greenhouse stage is one way to define the inner edge of the HZ on an Earth-like planet that occurs at surface temperatures of 340 K (below the boiling point of water). The moist greenhouse stage starts when the water loss timescale becomes comparable to the age of Earth. We show in Section 3 that the water loss timescale is typically long on dry planets even when the surface temperature is close to the boiling point, because the stratosphere is dry.

Surface albedo. We assume that our exoplanets are hot and dry; thus, most of the surface at low zenith angles is possibly covered by deserts or barren rocky surfaces. The average surface albedo of Earth is 0.15 (Chapter 5 of Pierrehumbert 2010, based on ERBE measurements), and the main contributors are water, land, vegetation, and ice/snow surfaces. As a reference, the albedo of liquid water is 0.05 (Clark et al. 2007), and the albedo of the Sahara is 0.4 (Tetzlaff 1983). We consider surface albedos between 0 and 0.8. The high value is a hypothetical (maybe unrealistic) upper limit because typical rock-forming minerals have a low albedo (Clark et al. 2007). A realistic surface albedo might be 0.2, and we choose this as our nominal value (see Table 1). We find that although higher surface albedos of low surface pressure atmospheres have an impact on the inner edge distance, the surface reflection becomes less important at high surface pressures because most of the incoming stellar flux is scattered back to space by Rayleigh scattering, or absorbed directly by the atmosphere. At 100 bars, typically less than 1% of the stellar flux reaches the surface. If the surface pressure is low, the surface albedo influences the tropopause height because the convective zone is driven by the absorption of stellar light at the surface.

Table 1. Overview of Parameter Values in the Nominal Simulation

Parameter	Value
Surface gravity	gsurf = 25 m s−2
Surface pressure	Psurf = 1 bar
CO2 mixing ratio	$X_{{\rm CO_2}}$ = 10−4
Relative humidity	Φ = 1%
Surface temperature	Tsurf = 370 Ka
Surface albedo	asurf = 0.2
Background gas	N2

Notes. These values correspond to a 10 Earth mass super-Earth with a nitrogen-dominated atmosphere and low relative humidity. ^aThe surface temperature is set by the boiling point of water and is a function of relative humidity and surface pressure (see Equation (5)).

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Nominal parameter values. The parameter values of our nominal simulation are shown in Table 1. The most important effect of these parameters is shown in Section 3. We typically change one parameter at a time, while the other parameters are held fixed at the value given in Table 1. In comparison, Abe et al. (2011) assume a surface gravity of 10 m s⁻², a surface pressure of 1 bar, an Earth-like N₂/O₂-dominated atmosphere with a present Earth CO₂ mixing ratio, a minimum of 50% relative humidity, and a surface albedo of 0.3.

It has been hypothesized that dry planets could trap their water reservoir permanently on cold areas like the poles and the night side (Menou 2013; Leconte et al. 2013). Although such a scenario can provide habitable regions on the planet (ice sheets flowing toward the day side and melting there), the habitable area is rather limited. It needs to be studied whether future observatories (such as the James Webb Space Telescope; Clampin 2010) will be able to pick up biosignature gases from planets that are only partially habitable.

One way to enlarge the habitable region is to consider scenarios where precipitation occurs predominantly in the form of rain. The water reservoir is still limited to the night side or poles. However, most of the surface waters are in liquid form; thus, a larger fraction of the planet's surface is habitable, and the water cycle is not broken.

We estimate that if the atmospheric relative humidity is 1%, rain is the dominant form of precipitation for a variety of surface pressures and temperatures. We use the dew point temperature to estimate the dominant form of precipitation. The dew point temperature (T_d) is given by the globally averaged temperature of the air above the surface and the relative humidity. As a point of reference, if the globally averaged surface temperature is 370 K and the relative humidity is 1%, the parcel has to cool down to T_d = 280 K to reach saturation. The dew point temperature for the same relative humidity value is 337 K, if the surface temperature is 500 K. Generally, dew point temperatures above 273 K ensure that liquid water precipitation is present on the planet. If the dew point temperature is below 273 K, the dominant form of precipitation could be snow.

Cloud formation by vertical mixing is not considered because precipitation from cumulus clouds might not reach the surface (called virga). If condensation occurred, e.g., on the upwelling part of the Hadley circulation around the equator, it is uncertain whether rain drops reach the surface. It is likely that the drops evaporate as they fall through gradually warmer and drier regions of the troposphere.

Large temperature differences are necessary to initiate condensation and precipitation. Therefore, the goal of this section is to study the properties of atmospheric circulation that result in a high temperature difference and see whether the circulation pattern is plausible. We distinguish two limiting cases: slowly rotating planets, and fast rotators like Earth. The distinction is necessary because different circulation regimes dominate the heat transfer depending on the rotation period.

2.4.1. Slow Rotators

The equator-to-pole temperature difference is typically small on planets with long periods (several tens to hundreds of days) because the Hadley cell extends all the way to the poles. As Hadley circulation transports heat very efficiently, the equator-to-pole temperature difference is expected to be small (Held & Hou 1980; Showman et al. 2011). However, the day-night temperature difference can be significant, if the atmosphere efficiently cools on the night side. In the most extreme case, the atmosphere could collapse if the temperature falls below the condensation point of nitrogen on the night side (see, e.g., Joshi et al. 1997; Heng & Kopparla 2012).

The speed of the zonal surface winds and the radiative cooling timescale determine the day-night temperature difference. Using the dew point temperature (T_d) and the radiative cooling timescale of the atmosphere, the zonal advection timescale is estimated (Showman & Guillot 2002, Equation (11)):

$$\begin{equation} \tau _{\mathrm{zonal}} = - \tau _{\mathrm{rad}} \log \left(1 - \frac{T_\mathrm{surf} - T_d}{T_\mathrm{surf}} \right), \end{equation} \tag{ 6 }$$

where τ_rad is the radiative timescale of the atmosphere:

$$\begin{equation} \tau _{\mathrm{rad}} = \frac{\Delta P}{g_{\mathrm{surf}}} \frac{c_p}{4 \sigma T_\mathrm{surf}^3}, \end{equation} \tag{ 7 }$$

where ΔP is the pressure difference at the top and bottom of the layer above the surface, c_p is the heat capacity of the layer, and σ is the Boltzmann constant. The zonal wind speed necessary to reach T_d is

$$\begin{equation} u_{\mathrm{zonal}} = R_p / \tau _{\mathrm{zonal}}, \end{equation} \tag{ 8 }$$

where R_p is the planet radius.

We accept exoplanet scenarios for which the estimated zonal wind necessary to preserve the required temperature difference is large. The reason is that large temperature differences drive strong winds, but strong winds reduce temperature differences. If the estimated u_zonal is large, it is expected that the actual zonal wind speed is smaller than u_zonal because the friction between the surface and the atmosphere slows the winds. Therefore, temperature gradients sufficiently large to initiate precipitation can be sustained over the globe. We use the mean surface wind speed of Earth as a reference value (7 m s⁻¹; Capps & Zender 2008) above which we consider the circulation properties to be plausible.

2.4.2. Fast Rotators

The day–night temperature difference on fast rotators is small, but the equator-to-pole temperature difference can be significant. The Hadley cell cannot extend all the way to the poles on fast rotators, such as Earth (Held & Hou 1980). Instead, heat is transported from the maximum latitude of the Hadley cell to the poles by baroclinic instability (Stone 1978). In energy balance models, baroclinic instability is approximated as a large-scale diffusion process (see, e.g., Held 1999; Spiegel et al. 2008). We also take advantage of this formalism and calculate the diffusion coefficient necessary to create a typical temperature difference of T_surf − T_d. The actual equator-to-pole temperature difference is possibly larger, as T_surf is a global average temperature, not the temperature at the substellar point.

The diffusion equation of the energy balance model is (Showman et al. 2011)

$$\begin{equation} c_p \frac{\partial T(\phi)}{\partial t} = \nabla (c_p D \Delta T) + S(\alpha ,\phi) - {\rm OTE}, \end{equation} \tag{ 9 }$$

where T, D, S, ϕ, and α are the temperature, diffusion coefficient, absorbed stellar flux, latitude, and the albedo, respectively. An analytical solution exists for Equation (9), if the albedo, D, and c_p are constants, if the OTE is fitted as A + BT, and if the stellar flux is parameterized as a constant plus a term proportional to the Legendre polynomial P₂(cos (ϕ)) (Held 1999). The diffusion coefficient expressed from the analytical solution is

$$\begin{equation} D = \frac{\left(\frac{T_\mathrm{surf}}{T_\mathrm{surf} - T_d}-1\right)B R_p^2}{6 c_p}. \end{equation} \tag{ 10 }$$

The parameter B is determined by calculating radiative-convective equilibrium profiles with surface temperatures 1 K above and below the nominal T_surf and fitting the OTE. Our reference value for D is the diffusion coefficient that reproduces the zonally averaged equator-to-pole temperature profile of Earth (10⁶ m² s⁻¹; Suarez & Held 1979). Following a similar argument as in the previous section, if an atmospheric scenario requires D < 10⁶ m² s⁻¹ to preserve the necessary temperature difference, we reject the scenario.

The water reservoir of the planet is gradually depleted because the stellar UV radiation dissociates the stratospheric water vapor, and hydrogen subsequently escapes to space. The goal of this section is to estimate the loss timescale of the exoplanet's liquid reservoir assuming diffusion-limited escape, and we follow the description of Kasting et al. (1993). If the water loss time is too short, the planet could lose its water reservoir before life develops. On the other hand, if the timescale is on the order of several billion years, it is safe to assume that the surface waters are stable and long-lasting.

The most important reason to calculate radiative-convective T–P profiles is to reliably estimate the stratospheric water mixing ratio and through that the diffusion-limited escape of H₂O. The tropopause pressure, temperature, and relative humidity set the stratospheric water mixing ratio. It is expected that both the tropopause pressure and temperature vary with atmospheric and stellar parameters. Often it is assumed that the stratosphere has a constant 200 K temperature (Kasting et al. 1993; Kopparapu et al. 2013). However, the water loss timescale might be inaccurate as the tropopause properties are not self-consistently calculated. We note that a constant 200 K stratosphere temperature does not significantly affect the OTE and thus the radiative equilibrium distance of the planet.

The water loss timescale is calculated in the following way. We estimate that our hot desert worlds have a 100 times smaller liquid water reservoir than Earth (Shiklomanov 1993); thus, the total mass of the liquid water reservoir is 1.4 × 10²² g, which amounts to $N_{\mathrm{H_2O}} = 4.6\times 10^{44}$ H₂O molecules. The water loss timescale is calculated as

$$\begin{equation} \tau _{w} = \frac{N_{\mathrm{H_2O}}}{A_\mathrm{surf} n^{\mathrm{top}}_{\mathrm{H_2O}} V_{\mathrm{diff}}}, \end{equation} \tag{ 11 }$$

where A_surf is the surface area of the exoplanet, $n^{\mathrm{top}}_{\mathrm{H_2O}}$ is the number density of H₂O molecules at the top of the atmosphere, and V_diff is the diffusion velocity calculated according to Hu et al. (2012b, see their Equations (3) and (20)). If the top-of-atmosphere temperature is 200 K and the water vapor mixing ratio is 3 × 10⁻⁵ on a planet with 10 m s⁻² surface gravity and one Earth radius, the water loss timescale is 5.2 × 10⁹ yr, in close agreement with the calculations of Kasting et al. (1993) and Kopparapu et al. (2013).

The inner edge of the HZ for dry planets with a relative humidity of 1% can be as close as 0.59 AU around a solar-like star. The relative humidity of the atmosphere is one of the most important factors controlling how close a planet can be to a star and still maintain surface liquid water. We consider tropospheric relative humidity values between 0.01% and 70%. If the relative humidity is 1% or larger, the dominant form of precipitation is expected to be rain (as shown in Section 2.4). If the relative humidity is lower than 1%, the necessary temperature difference (from day to night side or from equator to pole) to initiate condensation would be too high and precipitation would occur in the form of snow. The inner edge of the HZ for planets with various relative humidity levels is shown in Figure 1. In the following we discuss the top-of-atmosphere radiative fluxes: the OTE and the albedo of the planet.

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The inner edge distance is close to the host star mainly because the OTE is high if the atmosphere is dry. In other words, the emitted thermal radiation escapes easily if the greenhouse gas concentrations are low. The OTE does not depend on the stellar type (see Figure 2(a)); it is mainly influenced by the pressure-temperature profile and the greenhouse gas concentrations (shown in Figure 3 for the nominal atmosphere of Table 1).

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The second factor influencing the inner edge distance is the albedo of the planet. The planetary albedo is strongly influenced by the host star's spectral energy distribution and the relative humidity (Marley et al. 1999). Low-mass stars radiate at long wavelengths where absorption by greenhouse gases such as water and CO₂ is present. Therefore, the incoming stellar light is taken up by the atmosphere before it reaches the surface. The larger the relative humidity, the more stellar light is absorbed (see Figure 2(b)). On the other hand, massive stars radiate at short wavelengths where Rayleigh scattering dominates. Therefore, the planetary albedo around massive stars can be larger than the surface albedo.

The atmospheric CO₂ mixing ratio influences the tropopause temperature, which itself affects the water loss timescale. We discuss the effects of CO₂ on habitability from two perspectives: in this work, changes in the CO₂ level slightly modify the planet's radiative equilibrium distance (aka the inner edge distance) because the surface temperature is fixed. The other perspective is to fix the planet's semi-major axis and investigate the influence of CO₂ on the climate. In the latter case, the CO₂ mixing ratio influences the surface temperature. We shortly discuss the second perspective at the end of this section.

The inner edge distance of planets with CO₂-rich atmospheres is large. The already known reason is that CO₂ is a potent greenhouse gas, and thus the OTE decreases with higher CO₂ levels. For example, the OTE is 800 W m⁻² and 600 W m⁻² for $X_{\mathrm{CO_2}} = 10^{-5}$ and 10⁻¹, respectively. However, the albedo remains largely unaffected. As a result, the inner edge distance is 0.54 AU and 0.64 AU for CO₂ mixing ratios of 10⁻⁵ and 10⁻¹ around a solar-like star, respectively.

The tropopause and stratosphere of exoplanets at the inner edge with low levels of atmospheric CO₂ are warm (see Figure 4). This effect is explained by the variation of the inner edge distance with the CO₂ mixing ratio and by the radiative properties of CO₂. Close-in planets are strongly illuminated; therefore, the tropopause and the stratosphere are warm. For example, the tropopause temperature is 240 K and 180 K for CO₂ levels of 10⁻⁵ and 10⁻¹, respectively. CO₂ also acts as a coolant in the stratosphere because it emits radiation that is lost to space, and in opposition to water vapor, it does not absorb in optical wavelengths (Clough & Iacono 1995). The more CO₂ the stratosphere has, the cooler it will be.

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Close-in planets with low levels of CO₂ might quickly lose their surface waters and become unhabitable. The water loss timescale as a function of CO₂ mixing ratio is shown in Figure 5. For a fixed relative humidity, the tropopause temperature determines the water-loss timescale (see Section 2.1). As discussed in the previous paragraph, the troposphere is cold if the atmosphere is CO₂-rich at the inner edge. Therefore, the atmospheric CO₂ level has an indirect influence on the water loss timescale, because it regulates the tropopause temperature. This result has been independently verified and studied in detail by Wordsworth & Pierrehumbert (2013).

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We note that for a fixed semi-major axis, the CO₂ mixing ratio has a similar (but somewhat smaller) effect on the water loss timescale. Radiative-convective models of Earth's atmosphere show that doubling the atmospheric CO₂ warms the surface but cools the tropopause (Schlesinger & Mitchell 1987). The result is that the stratospheric water mixing ratio reduces, and the water loss timescale is prolonged.

The impact of surface pressure on the inner edge distance is complex.

Low surface pressures (∼0.1 bar) are a problem because the water loss timescale is short and complex life might not have enough time to evolve (see Figure 6). The water loss timescale of a hot desert world is only 0.1 billion years if the surface pressure is 0.1 bar; this is because the troposphere is confined to low altitudes, and the tropopause temperature only slightly differs from the surface temperature (see Figure 7(a)). Therefore, the stratosphere is water-rich and the water loss timescale is short. At large surface pressures, the troposphere is extended, the tropopause is cold (see Figure 7(b)), and the water loss timescale is always in excess of 10 billion years (see Figure 6).

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High surface pressures (100 bars) on slow rotators are problematic for habitability. The liquid water cycle cannot plausibly operate because the day–night side temperature difference is too low to initiate condensation. The rotation period of the exoplanet is an important factor influencing the water cycle because it prescribes the dominant circulation mode that redistributes heat. On slow rotators, heat from the day side is transported by zonal flows to the night side (see Section 2.4.1). If the surface pressure is 100 bars, the radiative cooling timescale of the atmosphere is long compared to the advection timescale. That is to say, an atmospheric parcel travels across the night side of the planet before it can cool down to the dew point temperature. To give a more qualitative argument, we calculate the surface zonal wind speed necessary to initiate condensation on the night side (Equation (8)). The zonal wind speed is on the order of 1000 m s⁻¹, 100 m s⁻¹, 10 m s⁻¹, and 1 m s⁻¹ for P_surf = 0.1, 1, 10, and 100 bars, respectively. The inverse correlation between surface pressure and zonal wind speed is due to the long radiative cooling timescale at large pressures. The necessary zonal wind speed on exoplanets with 100 bars of surface pressure is too low; therefore, such atmospheres cannot plausibly cool down to the dew point temperature on the night side, and water condensation does not occur. On fast rotators like Earth, baroclinic instability transports heat from the equatorial regions to the poles (see Section 2.4.2). We find that the baroclinic diffusion coefficient required to initiate condensation around the poles (Equation (10)) is on the order of 10¹⁰–10¹¹ m² s⁻¹ for all surface pressures considered. These values are 4–5 orders of magnitude larger than the baroclinic diffusion coefficient on Earth. Therefore, large equator-to-pole temperature differences and liquid precipitation are plausible on fast rotators at all surface pressures. The baroclinic diffusion coefficient is large because the OTE changes rapidly as a function of surface temperature in dry and hot atmospheres (parameter B in Equation (10)).

A habitable planet with the smallest semi-major axis should have a surface pressure around 1 bar for stars less massive than 1.1 M_☉ (Figure 8). The inner edge distance is a non-linear function of the surface pressure and the host star properties, because both the OTE and the albedo show complex features (Figure 9). The OTE is maximized if the surface pressure is 1 bar. However, water boils at low temperatures, if the pressure is reduced. Therefore, the surface temperature and thus the OTE are small at low pressures. Although the boiling point and the surface temperature are high at large pressures, the optically thick atmosphere reduces the OTE at large pressures (see Figure 9(a)). The albedo of low-pressure exoplanets is very close to the surface albedo. On the other hand, exoplanets with large surface pressures (100 bars) simultaneously have the lowest albedo around low-mass stars due to greenhouse gas absorption, and they also have the highest albedo around massive stars due to Rayleigh scattering (Figure 9(b)).

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An exoplanet's surface pressure is actually difficult to ascertain whether it be a priori from theory or from observations. The surface pressure of exoplanets is influenced by processes such as accretion from the protoplanetary disk, atmospheric escape, atmosphere accretion and/or erosion during impacts, and outgassing. In fact, the most challenging problem in proving that an exoplanet is habitable will be to observationally constrain its surface pressure. Even if the atmosphere is cloud-free, the atmosphere could be so optically thick that the surface remains hidden and unconstrained.

The effect of surface gravity on the inner edge distance is small compared to other factors discussed previously (see Figure 10). Surface gravity influences the pressure scale height of the atmosphere and thus the column density of gases for a fixed surface pressure. The atmosphere of a Mars-sized small planet is expanded, and thus the column density of all gases is raised. This has implications for the OTE as well as the planetary albedo. Increasing the greenhouse gas column density reduces the OTE from 800 W m⁻² to 700 W m⁻² for planets with surface gravities of 25 m s⁻² and 5 m s⁻², respectively. The albedo of a Mars-sized small planet shows the largest deviation from the surface albedo (see Figure 11). The albedo around low-mass stars is the lowest for small planets, because the greenhouse gas column density is large and the incoming stellar light is absorbed. At the same time, the albedo around massive stars is also the largest for small planets because Rayleigh scattering is more efficient due to the large column density of N₂.

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Although the differences are small, the inner edge is slightly closer to the host star for super-Earth planets (see Figure 8(b)). The distance around a solar-like star is 0.57 AU, 0.56 AU, and 0.55 AU for surface gravities of 5, 10, and 25 m s⁻². As the inner edge distance changes rather moderately with planet mass and size, it is a good approximation to use one common inner edge limit for all super-Earth and Earth-like planets.

If the surface pressure is moderate (P_surf ⩽ 10 bars), the atmosphere is optically thin in optical wavelengths. Therefore, a significant fraction of the stellar radiation reaches the surface, and the surface albedo has a strong influence on the inner edge distance (Figure 12). If the surface albedo is 0.8 (a hypothetical, possibly unrealistically high value), the inner edge of the HZ is at 0.38 AU around a solar-like star (within the orbit of Mercury). Such a high surface albedo pushes the inner edge to the extremes, and it remains to be seem whether we find any planet with a highly reflective surface. In the following we adopt two inner edge limits: the nominal atmosphere with properties outlined in Table 1), and one with a surface albedo of 0.8.

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We approximate our inner edge distances using a simple power function in the form

$$\begin{equation} d_{{\rm in}} = d_{{\rm in}\odot }L^s, \end{equation} \tag{ 12 }$$

where d_in is the inner edge distance in AU, L is the stellar luminosity in units of solar luminosity (L_☉), d_in☉ is the inner edge distance at L = 1 L_☉, and s is the slope of the power function. We find that d_in☉ = 0.59 AU and s = 0.495 for our nominal scenario with parameters shown in Table 1. If the surface albedo is 0.8 in the nominal case, d_in☉ = 0.38 AU and s = 0.474. The goodness-of-fit estimates on these parameters (1σ) are in the third decimal point.

We show currently confirmed/validated and candidate super-Earth exoplanets in relation to our fundamental inner edge boundary and to the HZ limits of Kopparapu et al. (2013). We use the Open Exoplanet Catalogue of Rein (2012) as the source of stellar and exoplanetary data for confirmed/validated exoplanets. The NASA Exoplanet Archive is used to access the Kepler exoplanet candidates of the Q1–Q12 data sets (Burke et al. 2013). The stellar and planetary properties of low-mass stars are updated according to Dressing & Charbonneau (2013). We consider exoplanet (candidates) that have a mass (or msin i) less than 10 M_⊕ and/or a radius less than 2.5 R_⊕.

There are 10 radial velocity super-Earths, eight validated transiting exoplanets, and one exoplanet discovered by microlensing that could potentially be habitable given suitable atmospheric conditions. The confirmed/validated exoplanets are shown in Figure 13(a). The potentially habitable radial velocity planets with increasing host star mass are Gliese 581c (Udry et al. 2007), Gliese 667Cc (Anglada-Escudé et al. 2012), Gliese 163c (Bonfils et al. 2013), HD 85512b (Pepe et al. 2011), HD 20794c and d (Pepe et al. 2011), and HD 40307f and g (Tuomi et al. 2013). The transiting potentially habitable exoplanets with increasing stellar mass are Kepler 61b (Ballard et al. 2013), Kepler 52c (Steffen et al. 2013), Kepler 55b and c (Steffen et al. 2013), Kepler 62e and f (Borucki et al. 2013), Kepler 69c (Barclay et al. 2013), and Kepler 22b (Borucki et al. 2012). The microlensing exoplanet OGLE-2005-BLG-390Lb (Beaulieu et al. 2006) could also be habitable, if its atmosphere is hydrogen-rich (Pierrehumbert & Gaidos 2011).

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We identify on the order of 100 potentially habitable Kepler candidates that have a radius below 2.5 R_⊕. Although the Kepler mission is not designed to study nearby low-mass stars, it did discover a large population of planetary candidates around low-mass stars (Dressing & Charbonneau 2013). We anticipate that the Transiting Exoplanet Survey Satellite (Ricker et al. 2010) will discover a large number of super-Earth and Earth-sized planets around nearby low-mass stars. Due to the proximity of the HZ to the host star, a large fraction of those planets could potentially be habitable.

We find that the occurrence rate of terrestrial planets (with radius between 0.5 and 2 R_⊕) in the HZ for M dwarfs is around 0.7–0.9. In comparison, it is 0.5–0.6 based on the Earth-analog HZ limits of Kopparapu et al. (2013; Kopparapu 2013). The occurrence rate is $0.70^{+0.15}_{-0.09}$ for the nominal scenario with a surface albedo of 0.2, and $0.88^{+0.09}_{-0.05}$ for the high surface albedo case. We apply the method of Dressing & Charbonneau (2013) to estimate the occurrence rate and assume that the outer edge of the HZ is at infinity. One caveat of the estimate is that the occurrence rate estimate is not statistically robust yet because the number of planets detected at large separations is small. KOI 2418.01 contributes 0.1 planets/star to the total occurrence rate, and KOI 1686.01 contributes 0.24 planets/star. This means that the true occurrence rate within the HZ might be significantly lower if either KOI 2418.01 or KOI 1686.01 is a false positive.

Where should we look for hot desert worlds? Is it possible to estimate their occurrence rate? What factors play a significant role in forming desert worlds? To answer these questions, we rely on N-body simulations of interacting planetary embryos and planetesimals (e.g., Morbidelli et al. 2000; Chambers 2001; Raymond et al. 2004, 2006, 2007). Such models simulate how planetary embryos accrete water-rich km-sized planetesimals coming past the snow line.

N-body simulations show that water delivery onto rocky planets is a stochastic process and the water reservoir of exoplanets might be difficult to constrain from models. One general trend that emerges from these simulations is that the water reservoir of the planet correlates with the surface density of icy solids around the snow line. However, as water delivery occurs mostly during the planet formation stage, the surface density of solids might be impossible to observationally constrain for old systems.

Water-rich planets close to the host star could naturally end up with small water reservoirs after the formation process (Hamano et al. 2013). It is expected that newly formed rocky planets are in a molten state because they form via giant impacts. If the planet is close to the host star, the surface is unable to cool and solidify efficiently due to the greenhouse effect of the water-vapor-dominated atmosphere. As a result, the planet can experience a prolonged runaway greenhouse stage and lose most of its water even if the planet started out with a larger water reservoir than Earth. This process provides a robust way to form dry planets close to the host star.

Surface water can be removed by the hydration of a planet's crust and potentially upper mantle. On Earth, most hydration occurs along mid-ocean ridges, caused by plate tectonics (see, e.g., Kasting & Holm 1992 and references therein). However, during subduction water is again released through volcanism—possibly stabilizing the amount of surface water on plate tectonics planets. Hence, as a first estimate, plate tectonics planets seem to be capable of keeping small surface oceans.

Planets lacking plate tectonics (stagnant lid planets) are expected to have lowered crustal and mantle hydration rates due to the lack of mid-ocean ridges, but it is not clear how effective volcanism returns the absorbed water to the atmosphere. We could assume that rock hydration occurs primarily by, e.g., serpentinization (or by the formation of clays). Both are low-temperature processes and generally do not occur above 500 K–700 K (for serpentinization, Oze & Sharma 2007) and demand large permeabilities and pore spaces for water to penetrate. Hence, any mineral hydration strongly decays with depth due to increasing interior temperatures and decreasing permeabilities.

Taking Earth as an example (although Earth has plate tectonics), the current average geotherm of ∼500 K–700 K is reached at maximally ∼10–15 km depth. This is assuming a 290 K surface temperature and a 25 K km⁻¹ continental thermal gradient (e.g., Turcotte & Schubert 2002). However, the permeability limits water penetration down to maximally ∼5 km (Vance et al. 2007). Assuming the unrealistic case that 5 km on Earth are being completely serpentinized (molar weight of forsterite ∼0.14 kg mol⁻¹, water 0.018 kg mol⁻¹, average forsterite density of ∼3300 kg/m⁻³, and water-to-forsterite molar reaction ratio of 1.5; e.g., Oze & Sharma 2007) would indeed allow complete absorption of Earth's ocean after ∼5 billion years. However, the assumptions above are highly unrealistic as they assume that the whole crust is serpentinized and that it consists of pure forsterite. As we know from Earth, serpentinization is a highly local and irregular phenomenon, and hence the maximally serpentinized crustal volume is expected to be by orders of magnitude smaller than our worst-case estimate (i.e., Nazarova 1994).

Moreover, our estimation neglects the return of water by volcanic outgassing and the fact that hot desert planets (especially the more massive ones) are different from Earth: due to larger surface temperatures (and especially for more massive planets) the interior is hotter and hence the maximally hydrated lithospheric volume is smaller than estimated for Earth.

As a first approximation, small oceans seem to be feasible over geologic timescales on plate tectonics, but also on stagnant lid planets, though clearly more work is needed to understand the geophysical conditions of hot desert worlds, especially if they do not facilitate plate tectonics.

A conclusive evidence of habitability would be to detect liquid water on the surface of a rocky exoplanet. Transmission spectroscopy can only infer habitability because it probes the atmosphere and not the surface. In other words, it will be a necessary but not sufficient proof of habitability to identify water vapor in the atmosphere of a rocky exoplanet using transmission spectroscopy.

We show the effective height as a function of wavelength (Kaltenegger & Traub 2009) for three types of atmospheres in Figure 14: a hot desert world with 10 m s⁻² surface gravity (see Table 1 for other parameters); the effective height of a CO₂-dominated Venus-like atmosphere with a water mixing ratio of 3 × 10⁻⁵; and the effective height of an Earth-like atmosphere with N₂/CO₂/H₂O (other gases are removed for easier comparison). The spectra are computed at high resolution (λ/δλ = 10⁵) and binned down to a resolution of 1000, which is comparable to the medium resolution of NIRSPEC on the James Webb Space Telescope (Bagnasco et al. 2007). The illustrated transmission signals are noise-free, but a typical signal-to-noise ratio is expected to be ∼10 for water features at 2 μm for systems out to 12 pc, assuming a total observing time of 200 hr (Deming et al. 2009).

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The transmission signals of an Earth-like and hot desert exoplanets are similar, because the stratospheres of these atmosphere types are similar and transmission spectroscopy mainly probes the stratosphere. Water vapor is abundant in the troposphere close to the surface, if a surface reservoir is present. However, transmission spectroscopy typically constrains the stratospheric compositional information only, because the surface region is expected to be optically thick at all wavelengths, and thus the transmission signal is, for the most part, unaffected by the troposphere. This is a problem, because surface properties such as surface pressure and water reservoir remain unconstrained with observations. This point is illustrated in Figure 14 because the water features of a hot desert world and an Earth-like planet are similar, although the surface relative humidity differs by almost two orders of magnitude. Furthermore, the effective height is above 5 km for both cases, and thus the surface regions are unconstrained. The largest difference in the transmission signal is at wavelengths shorter than 1 μm because this part of the spectrum is influenced by the pressure scale height of the atmosphere. The pressure scale height is

$$\begin{equation} H_p = \frac{kT}{Mg_{\mathrm{surf}}}, \end{equation} \tag{ 13 }$$

where k is the Boltzmann constant, T is the temperature of the atmosphere, and M is the mean molecular mass. The mean molecular mass is similar in the two cases, and the surface gravities are identical. However, the temperatures on a hot desert world are higher than on Earth. The average surface temperature of the hot desert world is 370 K in contrast to the 290 K surface temperature of Earth. The amplitude of the molecular features differs at all wavelengths. This difference is also explained by the large pressure scale height on hot desert worlds.

The transmission signal of a Venus-like atmosphere is significantly different from the expected signal of a hot desert exoplanet. Therefore, it might be possible to identify habitable planets with confidence. If the stratospheric mixing ratios of CO₂ and water are retrievable from the transmission spectrum, it is possible to distinguish a CO₂-dominated atmosphere from an N₂-dominated atmosphere (Benneke & Seager 2012). Variations in the effective height are modest in a CO₂-dominated atmosphere for two reasons. First, the CO₂ absorption lines are wide and the atmosphere absorbs significantly even in between band center regions. Second, the scale height of a CO₂-dominated atmosphere is small as the molecular weight of CO₂ is larger than the molecular weight of N₂. However, if the CO₂ and water mixing ratios are small, the variations in effective height are large (see Figure 14). Measuring the relative effective heights of band centers and window regions is a way to distinguish CO₂-dominated atmospheres from Earth-like and hot desert world atmospheres.

We emphasize the previously known result that the transmission signal of low surface gravity planets is easier to detect than the signal of high surface gravity planets because of a larger scale height (and thus larger variations in the transmission signal) for the former. We illustrate the influence of surface gravity on the effective height for hot desert worlds in Figure 15.

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4.4.1. Clouds

4.4.2. 1D versus 3D

4.4.3. Relative Humidity Profile