Hot Hydrogen Climates near the inner edge of the Habitable Zone

TL;DR

This paper demonstrates that hydrogen-rich atmospheres can cause significant warming near the inner edge of the habitable zone, potentially leading to hot early planetary conditions, through thermodynamic effects and water vapor amplification.

Contribution

It reveals that H2 can induce strong warming near the inner habitable zone edge via thermodynamics and water vapor effects, expanding understanding of early planetary climates.

Findings

1 bar of H2 raises temperatures above 340 K

50 bar of H2 raises temperatures above 450 K

H2 enables 'Souffle' climates with enhanced warming

Abstract

Young terrestrial planets can capture or outgas hydrogen-rich atmospheres with tens to hundreds of bars of H2, which persist for 100 Myrs or longer. Although the earliest habitable conditions on Earth and terrestrial exoplanets could thus arise while the atmosphere is still dominated by H2, the climatic effects of H2 remain poorly understood. Previous work showed that H2 induces strong greenhouse warming at the outer edge of the habitable zone. Here we use a 1D radiative-convective model to show that H2 also leads to strong warming near the inner edge of the habitable zone. Unlike H2's greenhouse warming at the outer edge, however, its effect near the inner edge is driven by thermodynamics: H2's large thermal scale height allows the atmosphere to store more water vapor than either a pure-H2O atmosphere or an atmosphere with a heavy background gas, such as N2 or CO2, thereby amplifying the greenhouse effect of H2O. Using idealized grey calculations, we then present a general argument for how different background gases affect the inner edge of the habitable zone. H2 stands out for its ability to induce novel "Souffle" climates, which further support its warming effect. Our results show that if the earliest conditions on a planet near the inner edge of the habitable zone were H2-rich, they were likely also hot: 1 bar of H2 is sufficient to raise surface temperatures above 340 K, and 50 bar of H2 are sufficient to raise surface temperatures above 450 K.

Tables

Table 1. Table 1: An atmosphere’s dry background pressure determines whether an atmosphere can enter a dilute runaway, or transitions directly into the steam runaway. For the dilute runaway we assume a diatomic background gas, R d / c p subscript 𝑅 𝑑 subscript 𝑐 𝑝 R_{d}/c_{p} =2/7, for the steam runaway we use N 2 as the background gas.

	Dilute Runaway	Steam Runaway
Condensing gas	$p_{s,dry}$ greater than (bar)	$p_{s,dry}$ less than (bar)
H2O	2.0	0.004
CH4	13	0.05
NH3	16	0.03
CO2	420	4.8

Equations

$$\frac{1}{\mathrm{OLR}}\frac{d\mathrm{OLR}}{d{T}_s}+\frac{1}{(1-\alpha )}\frac{d\alpha }{d{T}_s}>0.$$

$$\tau$$

$$=$$

$$H_v$$

$$\frac{1}{{\rho }_v}\frac{d{\rho }_v}{dz}$$

$$=$$

$$\approx$$

$$H_v$$

$$H_T^{dilute}$$

$$H_T^{steam}$$

$$=$$

$$e^{\ast }$$

$$\tau$$

$$=$$

$$\tau$$

$$\mathrm{OLR}$$

$${\mathrm{OLR}}_{\infty }$$

$$=$$

$${\mathrm{OLR}}_{\infty }$$

$${\tau }_0$$

$$\frac{{\tau }_{\mathrm{dilute}}}{{\tau }_{\mathrm{steam}}}$$

$$=$$

$${\frac{dT}{dp}|}_{dilute}$$

$$H_T$$

$$=$$

$${\frac{dp}{dz}|}_{steam}$$

$$\approx$$

$$H_T$$

$${\tau }_{LW}(T_1)$$

$$p_{s,dry}$$

$$R_b$$

Full text

planetary atmospheres – astrobiology – exoplanets

1 Introduction

Hydrogen plays an important role in the early stages of terrestrial planet evolution. Planets that form inside a nebular disk are likely to gravitationally capture a primordial H2 atmosphere (Rafikov, 2006; Pierrehumbert & Gaidos, 2011). Even after the disk disperses, planets can outgas H2 during the solidification of the planet’s magma ocean (Elkins-Tanton & Seager, 2008; Schaefer & Fegley, 2010), or can form H2 through the oxidation of iron-rich impactors during the giant impact phase (Genda et al., 2017). The amount of H2 generated varies with formation mechanism but can be substantial, ranging from several to more than 1000 bars (Pierrehumbert & Gaidos, 2011; Elkins-Tanton & Seager, 2008; Genda et al., 2017).

H2 eventually dissipates because its low molecular weight allows it to escape to space. In the meantime, however, it can shape a planet’s earliest surface conditions. To order of magnitude, we estimate that it takes $\sim$100 Myr for a young Earth-like planet around a Sun-like star to lose 100 bars of H2 to space (see Appendix A). This timescale becomes longer if H2 is produced or outgassed later, due to the rapid decrease over time in the XUV output of young Sun-like stars. Our estimate agrees with the results of Genda et al. (2017), who argued that early Earth could have had a $\sim$70 bar H2 atmosphere which gradually escaped over 200 Myr, produced by the oxidation of fragments from a giant impact.

These timescales are longer than the $1-10$ Myr needed to solidify a magma ocean and condense a steam atmosphere (Hamano et al., 2013; Lebrun et al., 2013). They are also long enough to shape prebiotic chemistry and, potentially, for an early biosphere to develop (Lazcano & Miller, 1994; Bell et al., 2015; Betts et al., 2018). Some terrestrial planets, including ancient Earth, could therefore undergo an early habitable phase with liquid water oceans underneath a H2 atmosphere. What would such a climate be like?

The habitable zone has conventionally only been studied for an atmosphere dominated by high mean-molecular-weight (MMW) background gases, in particular N2 and CO2 (e.g. Kasting et al., 1993; Kopparapu et al., 2013). There are relatively few studies that have considered the habitability of H2-rich planets, and these studies either focused on thick H2 atmospheres at the outer edge of the habitable zone or atmospheres with relatively small amounts of H2. In both cases, hydrogen can have a significant effect on climate. Thick H2 envelopes, with tens of bars of H2, generate a strong greenhouse effect via H2-H2 collision-induced absorption that can allow planets to remain habitable far beyond the outer edge of the conventional N2-CO2 habitable zone (Stevenson, 1999; Pierrehumbert & Gaidos, 2011; Wordsworth, 2012). Similarly, even small amounts of H2, on the order of 1-30% molar fraction, can induce greenhouse warming via H2-N2 or H2-CO2 collision-induced absorption. H2 could thus also help resolve the Faint Young Sun problem on early Earth and early Mars, and could expand the outer edge of the habitable zone (Wordsworth & Pierrehumbert, 2013; Ramirez et al., 2014; Batalha et al., 2015; Ramirez & Kaltenegger, 2017).

Less attention has been paid to how a transient H2 atmosphere, with 1 or more bars of H2, would affect a planet’s climate closer to the inner edge of the habitable zone, which we consider in this work. Unlike at the outer edge, where H2’s radiative effect dominates, the interaction between water vapor and the H2 background becomes increasingly important near the inner edge, which we define as roughly the region where the incident stellar flux, or stellar constant, exceeds 1000 W m*-2* ($\sim$70% of the flux received by Earth today). We explore these interactions with a 1D radiative-convective model that represents a planet orbiting a Sun-like star. To relate our results to conventional N2-CO2 habitable zone calculations, we consider both high MMW and H2-rich atmospheres (the model is described in Section 2). Our results fall into two categories. First, we show that H2 has a strong warming effect near the inner edge. This effect is independent of H2’s greenhouse effect and instead arises from hydrogen’s thermodynamic properties (Section 3). Second, we present a general consideration for how a planet’s background gases affect its climate, with a particular focus on the approach to the runaway greenhouse. We show that a habitable planet has distinct climate feedbacks with H2 versus high-MMW backgrounds, and that these feedbacks give rise to novel climate states in H2-rich atmospheres which further support H2’s warming effect (Section 4). Finally, we discuss our findings and its implications for the origin of life in the early Solar System and around other stars (Section 5).

2 Methods

We the Python line-by-line RADiation model for planetary atmosphereS (PyRADS), which is a 1D model representing an atmosphere in radiative-convective equilibrium. We have validated PyRADS against other line-by-line calculations in the longwave (Koll & Cronin, 2018), as well as the shortwave (Appendix B). The model code is open source and freely available on github111github.com/ddbkoll/PyRADS-shortwave.

The atmosphere is composed of a dry background gas (e.g., N2 or H2) plus condensing water vapor. The amount of atmospheric water vapor increases rapidly with temperature following the Clausius-Clapeyron relation, so all atmospheres become steam-dominated at high temperatures. For example, the saturation vapor pressure of water is $\sim 6\times 10^{-3}$ bar at the triple point (273 K), whereas it is one bar at the boiling point (373 K). A planet with 1 bar of background N2 at a surface temperature of 273 K therefore has an atmosphere that is largely made of N2, whereas at 373 K the total surface pressure rises to 2 bar and the air near the ground contains as many H2O molecules as N2 molecules.

To compute convective temperature profiles we use the general moist adiabat, which is valid for both a dry atmosphere and a steam-dominated atmosphere (Ding & Pierrehumbert, 2016). All thermodynamic constants are taken from Pierrehumbert (2010). We choose a variable vertical resolution to ensure that we resolve the upper atmosphere in the runaway limit, yet also have good resolution near the surface at intermediate surface temperatures.

For the radiative transfer we perform two separate calculations in the longwave and shortwave. Our longwave calculations cover the spectrum up to 5,000 cm*-1* (2 $\mu$m) with a spectral resolution of 0.01 cm*-1*. In the shortwave we cover the spectrum between 1,000 and 110,000 cm*-1* (0.09-10$\mu$m) with a spectral resolution of 0.01 cm*-1* up to 30,000 cm*-1* and a resolution of 10 cm*-1* at higher wavenumbers (see below). We assume a Sun-like host star, with a solar spectrum taken from the VPL database (Segura et al., 2003), and a surface albedo of 0.12. To solve the shortwave radiative transfer we use DISORT (Stamnes et al., 1988), implemented via PyDISORT222github.com/chanGimeno/pyDISORT. We run DISORT at line-by-line resolution using 4 angular streams. Sensitivity tests indicate that our results are essentially identical if we use more angular streams.

To compute opacities we use a combination of H2O lines from HITRAN2016 (Gordon et al., 2017) and the more comprehensive ExoMol BT2 line list (Barber et al., 2006). At the temperatures we are considering, the choice of line list only has a minor impact on longwave fluxes because HITRAN2016 already contains most strong absorption lines in the infrared. In contrast, shortwave fluxes are sensitive to the choice of line list because HITRAN2016 does not cover the UV even though the near and middle UV contains a moderate fraction of the incoming stellar flux ($\sim 8\%$ for a Sun-like star). As a compromise between computational tractability and complete spectral coverage we adopt HITRAN2016 lines below 15,000 cm*-1* and BT2 lines between 15,000 cm*-1* and 30,000 cm*-1*. Similarly, for computational tractability in our CO2 calculations we use the HITRAN2016 line list. We use a Lorenz line shape throughout instead of a Voigt line shape. This approximation means we do not resolve line cores, but the resulting impact on OLR is small (Koll & Cronin, 2018).

H2O line broadening could behave differently in a N2 versus a H2 atmosphere. To explore this possibility we compared line widths for H2O-air broadening (Voronin et al., 2010) with line widths for H2O-H2 broadening (Barton et al., 2017). Despite our expectation that collisional line width should scale with intermolecular velocity (Pierrehumbert, 2010), and hence with the inverse square root of the broadening gas’s molecular mass, H2O line widths are roughly the same for H2 and N2 broadening. In both cases the widths of the strongest lines are about 0.1 cm*-1* atm*-1*. Although this invariance might be particular to H2 and N2, because line widths for H2O-He collisions are noticeably smaller (Barton et al., 2017), we use H2O-air broadening parameters for both N2 and H2.

In addition to line opacities we include several other opacity sources. We compute H2O-H2O and H2O-foreign collision-induced absorption (CIA) using the semi-empirical MTCKD continuum model (Mlawer et al., 2012). For H2-H2 CIA we use data from HITRAN. We are not aware of any H2O-H2 CIA data, so we use H2O-N2 CIA in both our N2 and our H2 calculations. Our use of H2O-N2 data to mimic H2O-H2 CIA could induce errors but is reasonable given the lack of laboratory measurements: N2 and H2 are both homonuclear diatomics, so to first order one might expect them to similarly interact with the H2O molecule and its electric field moments. For CO2-CO2 and CO2-foreign CIA we use data from Pierrehumbert (2010). For Rayleigh scattering we use the N2 and H2O scattering cross-sections from Goldblatt et al. (2013), the H2 scattering cross-sections from Dalgarno & Williams (1962), and the CO2 scattering cross-sections from Pierrehumbert (2010). We parameterize absorption in the UV by setting the absorption cross-section of H2O at frequencies higher than 50,000 cm*-1* (0.2 $\mu$m) equal to a representative value based on the MPI-Mainz Atlas, $\kappa_{UV}\approx 10^{-18}$ cm2 molec*-1* $=3\times 10^{3}$ m2 kg*-1*.

Consistent with our use of a 1D model we do not include effects that are determined by 3D dynamics, such as clouds or changes in relative humidity. Instead we assume clear-sky profiles and a relative humidity of unity. We also assume an all-adiabat atmosphere in which convection reaches to the top of the atmosphere, and ignore the stratosphere. We discuss the impact of these modeling assumptions in Section 5.

For our analytical results in Section 4 we note that the definition of optical thickness often includes a implicit average cosine of the zenith angle, $\overline{\cos\theta}$, which accounts for the angular distribution of longwave radiation (Pierrehumbert, 2010). Our numerical calculations use $\overline{\cos\theta}=3/5$. To reproduce our analytical values, optical thickness $\tau$ along a vertical path should be divided by $\overline{\cos\theta}$.

Below we will present our results in terms of the planet’s surface temperature $T_{s}$ as a function of incoming stellar flux (Figure 1). To present our results in this way we first run our model across a large grid of surface temperatures. The model outputs albedo $\alpha$ and OLR, which then allows us to infer the stellar flux $L_{*}$ that would be required to maintain a given surface temperature via planetary energy balance, $L_{*}=4\mathrm{OLR}(T_{s})/(1-\alpha(T_{s}))$. We refer to a climate state as stable if warming increases the planet’s OLR more than it increases the planet’s absorbed shortwave flux:

[TABLE]

3 H2 warming near the inner edge.

3.1 Spectral calculations

It is not obvious that background gases like H2 or N2 should have a strong impact near the inner edge of the habitable zone. Earth’s greenhouse effect is already dominated by H2O, and H2O becomes even more important at higher temperatures. We therefore consider the inner edge for four atmospheric scenarios. First, we consider a pure H2O atmosphere as the simplest model of the runaway greenhouse (Pierrehumbert, 2010; Goldblatt, 2015). Second, we consider an Earth-like N2-H2O atmosphere with 1 bar of N2, which is typical of conventional habitable zone calculations (Kasting et al., 1993; Kopparapu et al., 2013). Third, we consider the extreme case of a CO2-H2O atmosphere with 1 bar of CO2. Usually CO2 concentrations are assumed to be negligibly low near the inner edge of the habitable zone due to the silicate-weathering thermostat (Walker et al., 1981). Here we include CO2 as a limiting case of a high MMW background gas with a strong greenhouse effect to draw out the distinction between high MMW gases and H2. Fourth, we consider the new limit of a H2-H2O atmosphere with 1 bar of H2.

Figure 1a shows that in all four scenarios albedo first decreases and then rises again with warming. This behavior is due to the increase of atmospheric water vapor with warming. Below about 250 K all atmospheres contain little water vapor, and are dominated by atmospheric scattering and surface absorption. At these temperatures H2 creates a significantly higher albedo than both N2 and CO2, because H2 has the highest scattering efficiency among common gases on a per-mass basis (a kg of H2 contains more molecules than any other gas)333Note that our calculations compare different atmospheric compositions at fixed surface pressure, which is equivalent to holding the total atmospheric mass fixed.. Above 250 K shortwave absorption increases in the near-IR, where H2O is a strong absorber, so albedo drops. Above 350 K the atmospheric mass starts to increase in all scenarios, which increases Rayleigh scattering at short wavelengths where H2O is a relatively poor absorber, so albedo rises again. Finally, above 400 K, all four scenarios become dominated by water vapor so the albedo converges.

Figure 1b shows that OLR in all scenarios approaches a single limiting value at high temperatures. The limiting value at high temperatures is called the runaway greenhouse or Simpson-Nakajima limit (Goldblatt et al., 2013). The runaway arises from the rapid increase of water vapor with warming: once water vapor becomes optically thick at all frequencies, longwave emission decouples from the surface and OLR is determined by the temperature of the upper atmosphere, which is constant once the atmosphere becomes dominated by water vapor (Nakajima et al., 1992; Pierrehumbert, 2010; Goldblatt et al., 2013).

The behavior of OLR leading up to the runaway, however, depends strongly on the assumed background gas. The simplest case is a pure H2O atmosphere, in which OLR approaches the runaway greenhouse monotonically. Adding 1 bar of N2 does not change OLR much relative to a pure H2O atmosphere, because N2 has no strong greenhouse effect of its own, but it introduces an important qualitative difference: N2 causes OLR to first overshoot the runaway limit before it slowly falls back (Nakajima et al., 1992). Adding 1 bar of CO2 reduces OLR at low temperatures, due to CO2’s greenhouse effect, but at high temperatures CO2 again causes OLR to overshoot the runaway limit. Previous work argued that the overshoot occurs because adding a dry background gas steepens the atmospheric lapse rate relative to that of a steam atmosphere (in the sense of increasing $dT/dp$; Nakajima et al., 1992; Pierrehumbert, 2010). This means the upper atmosphere is colder for a N2-H2O or CO2-H2O atmosphere than for pure H2O, which reduces the amount of water vapor in the upper atmosphere and allows the atmosphere to emit more radiation. Importantly, this explanation seems insensitive to the composition of the background gas, so one might expect that H2 also creates an OLR overshoot.

Surprisingly, we find that OLR changes very differently with warming in a H2-rich atmosphere (Fig. 1b). At cold temperatures H2 reduces OLR due to its CIA greenhouse effect. The change in OLR is much larger for 1 bar of H2 than for 1 bar of CO2, so on a per-mass basis H2 is a stronger greenhouse gas than CO2. Above $250$ K OLR shoots up, however, then abruptly flattens out at $320$ K, before slowly approaching the runaway limit. Crucially, whereas N2 and CO2 cause OLR to overshoot, H2 causes OLR to undershoot the runaway greenhouse limit. This undershoot strongly affects the climates of H2-rich atmospheres near the inner edge of the habitable zone.

Figure 1c combines our albedo and OLR calculations, and shows how different background gases influence a planet’s climate stability. We find that N2-rich and CO2-rich atmospheres abruptly cease to be stable above surface temperatures of 310 K and 380 K. There are no nearby stable states, so a N2-rich or CO2-rich planet transitions from a moderately warm climate straight into the runaway greenhouse and would only re-equilibrate again once the surface temperature exceeds about $1,600$ K (Goldblatt et al., 2013). The abrupt onset of the runaway greenhouse in N2-rich and CO2-rich atmospheres is largely due to the overshoot in the OLR-$T_{s}$ curve (Fig. 1b): once peak OLR is reached, the planet cannot emit more longwave radiation with warming, so the only way for the climate to remain stable is by reflecting more sunlight with further warming. Albedo does increase slightly above $350$ K, due to increased Rayleigh scattering as the atmosphere thickens, but the albedo increase is small and largely cancelled by the decrease in OLR with further warming at these high temperatures444Our N2 calculations do show a small island of stability corresponding to surface temperatures of $390-410$ K and stellar fluxes of about $1356-1358$ W m*-2*. It is highly unlikely that these states would actually be stable: our code still underestimates shortwave absorption in the near UV, and thus produces higher-than-realistic albedos, which would further destabilize warm climates. Moreover, the incident stellar flux would have to be extremely fine-tuned to maintain this state..

In contrast to high MMW atmospheres, pure H2O and H2-rich atmospheres enter the runaway gradually which allows them to sustain unusually warm climates near the inner edge. First, the inner edge of the habitable zone moves slightly outward in pure H2O and H2-rich atmospheres when compared to high MMW atmospheres due to a the lack of an OLR overshoot. A high MMW background thus allows a planet to remain habitable at moderately higher stellar fluxes than is possible with a pure H2O or a H2-rich atmosphere. Focusing on those climates that are inside the habitable zone, however, we find that pure H2O and H2-rich atmospheres can sustain much higher surface temperatures than high MMW atmospheres. For example, an atmosphere with 1 bar of N2 is essentially limited to surface temperatures below 310 K because higher surface temperatures are unstable and lead to further warming. In comparison, an atmosphere with 1 bar of H2 does not lose its climate stability as it approaches the inner edge, and can sustain surface temperatures of $340$ K or more without entering the runaway. This gradual approach to the runaway arises from the OLR undershoot in H2 (Fig. 1).

To explore how the warming effect of H2 depends on surface pressure, we performed additional simulations in which we increased the surface pressure of H2 up to 50 bars. Figure 2 shows that 50 bars of H2 are sufficient to warm the surface to 450 K or more near the inner edge of the habitable zone. For reference, the critical point of H2O occurs at about 647 K. As long as the surface temperature remains below the critical point, a planet can still maintain an ocean and thus remains habitable according to the classical definition of the habitable zone (Kasting et al., 1993). Surface temperatures of 450 K or more, sustained over a timescale of 100 Myrs, would strongly shape a planet’s earliest surface environment and the development of prebiotic chemistry. Moreover, although these temperatures might seem high, the reported upper limit for extant life on Earth is around 400 K (Takai et al., 2008), so these conditions could still be compatible with a thermophile biosphere.

One might think that the strong H2 warming near the inner edge is caused by H2’s radiative properties. That is not the case. Figure 3 shows that H2 still leads to strong warming at surface temperatures above 320 K when we disable H2-H2 CIA in our OLR calculations. The inset in Figure 3 shows a measure of the atmosphere’s broadband longwave optical thickness, which we define as $\tau_{LW}=\ln[(\sigma T_{s}^{4})^{-1}\times\int B_{\lambda}(T_{s})e^{-\tau_{\lambda}}d\lambda]$, where $B_{\lambda}$ is the Planck function and $\tau_{\lambda}$ is the atmospheric optical thickness at a given wavelength $\lambda$. The inset shows that the optical thickness of a H2-rich atmosphere is due to H2’s CIA opacity below 320 K, but becomes dominated by H2O opacity at high temperatures. At low temperatures the climate effect of H2 is therefore due to its role as a greenhouse gas, but at high temperatures its impact must be due to other physics.

3.2 An intuitive explanation for why MMW matters

Why does H2 lead to strong warming near the inner edge, even without any greenhouse effect of its own, whereas a heavy background gas like N2 does not? We show next that background gases influence OLR through their influence on an atmosphere’s temperature structure, which we quantify in terms of the thermal scale height. In turn, H2’s ability to inflate the thermal scale height is unique among common atmospheric gases.

First, the optical thickness of an atmospheric column is directly proportional to the scale height of water vapor. To show this we write the atmosphere’s optical thickness555Although we will assume grey radiation in the next section, the derivation in this section is general and also holds monochromatically, i.e., at a single wavelength. as

[TABLE]

where $z$ is height, $\kappa_{v}$ is the absorption cross-section, and $\rho_{v}$ is the density of water vapor (in general, the subscript v will denote quantities related to water vapor). We assume $\kappa_{v}$ is independent of height and $\rho_{v}$ decays exponentially following the water vapor scale height $H_{v}$, $\rho_{v}=\rho_{v,0}e^{-z/H_{v}}$. In general $H_{v}$ can also vary with height, but due to the exponentially rapid falloff of $\rho_{v}$ we treat $H_{v}$ as constant and equal to its value at the surface. We additionally define the absorption length $l_{0}\equiv 1/(\kappa_{v}\rho_{v,0})$, which is the characteristic distance over which a longwave photon is absorbed by the atmosphere. The atmospheric optical thickness is then simply

[TABLE]

The absorption length $l_{0}$ does not depend on background gases, so N2 or H2 affect OLR only via changes in the scale height of water vapor $H_{v}$.

Next, background gases affect $H_{v}$ only via changes in the temperature scale height $H_{T}$. To show this we start with the general definition for the scale height of water vapor,

[TABLE]

We evaluate $H_{v}$ using the ideal gas law for water vapor, $\rho_{v}=e^{*}/(R_{v}T)$ where $e^{*}$ is the saturation vapor pressure, and the Clausius-Clapeyron relation, $d\ln e^{*}/d\ln T=L_{v}/(R_{v}T)$ where $L_{v}$ is the latent heat of vaporization,

[TABLE]

Here we used $L_{v}/(R_{v}T)\gg 1$, which holds for water vapor and a wide range of other condensible gases. We recognize $(-1/T\times dT/dz)^{-1}$ as the temperature scale height $H_{T}$, which sets the vertical length scale of temperature decrease, so

[TABLE]

Again, $R_{v}$, $T$ and $L_{v}$ are independent of any background gases, so N2 or H2 affect optical thickness only via changes in $H_{T}$.

Next, we constrain the value of $H_{T}$. At cold temperatures water vapor is highly dilute and has essentially no impact on the vertical structure of the atmosphere. In this limit the lapse rate follows the standard dry adiabat, $dT/dz=-g/c_{p}$, and

[TABLE]

Here $c_{p}$ is the specific heat capacity of the background gas, e.g., of N2.

At high temperatures the atmosphere becomes steam-dominated, and total pressure becomes equal to the saturation vapor pressure of water. In this limit the temperature scale height is equal to

[TABLE]

where we used the hydrostatic relation in the third step.

If we evaluate $H_{T}$ for H2 versus N2, we find that different background gases lead to dramatically different temperature scale heights. With an Earth-like gravity and at a temperature of 300 K, $H_{T}^{dilute}\approx 30$ km in N2, whereas $H_{T}^{dilute}\approx 430$ km in H2. The difference arises from the low molecular weight of H2 relative to that of N2. Even though both are diatomic molecules with similar degrees of freedom, one kilogram of H2 contains 14 times more molecules than one kilogram of N2, which means the specific heat capacity $c_{p}$ of H2 is roughly 14 times larger than that of N2.

Now, as both N2 and H2 atmospheres warm up, they eventually end up being dominated by water vapor. The temperature scale height of a steam atmosphere is $H_{T}^{steam}\approx 250$ km. This means $H_{T}$ grows with warming if an atmosphere starts out N2-rich, but $H_{T}$ shrinks with warming if the atmosphere starts out H2-rich.

Because $H_{T}$ controls $H_{v}$, and therefore the atmosphere’s optical thickness $\tau$, we see that warming leads to a compositional climate feedback which depends on the molecular weight of the background gas. An N2-rich atmosphere starts out dense and compact, with a small thermal scale height $H_{T}$. Warming moves the atmospheric composition closer to the steam limit, so $H_{T}$ increases. This implies $H_{v}$ also increases with warming, so moving from a N2-rich to a H2O-rich atmosphere means the atmosphere moistens even faster than a pure H2O atmosphere would. The moistening increases $\tau$, tends to reduce OLR, and thus acts to amplify warming.

In contrast, a H2-rich atmosphere starts out with a large scale height $H_{T}$. Warming moves the atmospheric composition closer to the steam limit, so $H_{T}$ and $H_{v}$ eventually have to decrease. Consequently, the shift from H2 to H2O means the atmosphere moistens less quickly than a pure H2O or a N2-rich atmosphere does – a stabilizing climate feedback.

The existence of this feedback, and its sensitivity to the MMW of the background gas, provide a first clue as to why H2-rich atmospheres become much warmer than N2-rich, or even CO2-rich, atmospheres (Fig. 1). Moreover, the sign of this feedback appears to be stabilizing only for H2. Even if we consider Helium as the next-heavier background gas after H2, such as a remnant atmosphere clinging to the core of a evaporated mini-Neptune (Hu et al., 2015), we find $H_{T}^{dilute}\approx 160$ km, which is smaller than $H_{T}^{steam}\approx 250$ km. The compositional feedback therefore requires the molecular weight of the background gas to be small enough so that the sensible heat of the background gas, $c_{p}T$, becomes larger than the latent heat of steam, $L_{v}$. H2 appears to be unique in this regard, which means H2-rich habitable atmospheres are distinct in terms of generating a stabilizing feedback as they transition from a cold, dry climate to a hot, steam-dominated climate.

4 New climate states in H2-rich atmospheres

Different background gases have a strong influence on a planet’s climate, so can we understand their effect more generally? In this section we first present grey radiative calculations which qualitatively recover the onset of the runaway greenhouse in our spectral calculations, and which therefore allow us to understand in detail how different background gases affect the inner edge of the habitable zone (Section 4.1). Second, we present analytical results which explain why OLR overshoots the runaway limit with a high MMW gas like N2, but undershoots the runaway limit with H2. In doing so we present a new climate state which we call the dilute runaway, and elucidate its underlying physics (Sections 4.2 and 4.4). Third, we discuss how H2’s thermodynamics give rise to a non-monotonic climate feedback which we call the Soufflé effect (Section 4.3). Finally, the calculations in this section assume idealized grey radiation, which allows us to understand the thermodynamic effect of different background gases, but do not capture all aspects of our spectral calculations. We discuss the remaining differences between grey and spectral calculations at the end of this section (Section 4.5).

4.1 Numerical calculations

For our grey radiative calculations we use the same grey longwave opacity for water vapor as in Nakajima et al. (1992), $\kappa_{v}=0.01$ m2 kg*-1*. The background gas is assumed to be completely transparent to longwave radiation, so its only influence on OLR is via its influence on the atmosphere’s temperature structure. The colored lines in Figure 4 show the response of OLR to warming for three different atmospheric compositions: a pure H2O atmosphere, a N2-rich atmosphere, and a H2-rich atmosphere. We vary the surface pressure of the background gas between 0.1 and 10 bar for H2 and 0.1 and 100 bar for N2.

Figure 4 recovers the main qualitative features of our spectral calculations. First, the OLR of a grey pure H2O atmosphere closely resembles our spectral calculations for pure H2O (Fig. 1). At temperatures below about 250 K there is little water vapor in the atmosphere, the atmosphere is largely transparent to longwave radiation, and most emission originates from the surface, so $\mathrm{OLR}\approx\sigma T_{s}^{4}$. With further warming the atmosphere rapidly becomes optically thick and monotonically approaches a limiting value of about $290$ W m*-2*. This limiting OLR has previously been called the Simpson-Nakajima limit (Goldblatt et al., 2013), for reasons we explain below we will call it here the “steam limit”. Second, adding N2 as a background gas leads to an overshoot and allows OLR to exceed the steam limit, as in our spectral calculations. Adding H2 has a similar but opposite effect, in which OLR undershoots the steam limit.

Figure 4 also shows several new features which we did not find in our spectral calculations. Although some of these features are sensitive to the details of our radiative assumptions (see Section 4.5), we focus on them here because they allow for important insight into the thermodynamic effect of different background gases. In particular, at moderate surface temperatures and high surface pressures, OLR hits an asymptotic limit that is distinct from the steam limit. This new OLR limit is higher than the steam limit for N2 and lower for H2. The temperature range over which OLR remains close to this limit depends on the amount of background gas. For example, atmospheres with less than 1 bar of background gas do not reach this limit whereas an atmosphere with 100 bar of N2 remains at this limit for more than 100 K. We note that this asymptotic limit is distinct from the Komabayashi-Ingersoll limit (Komabayasi, 1967; Ingersoll, 1969), which requires a stratosphere, whereas our calculations do not include a stratosphere (we discuss this point in Section 5).

The asymptotic OLR limit at high surface pressures was first discussed by Nakajima et al. (1992). Imagine a moderately warm planet with large amounts of dry background gas, so that its atmosphere contains some H2O but water vapor does not yet affect the temperature structure and the lapse rate is still dry-adiabatic. As the planet warms beyond a threshold (e.g., $\sim$ 300 K in our spectral calculations; Fig. 1), water vapor becomes optically thick and the planet’s OLR decouples from the surface temperature. In this case the atmosphere’s emission temperature remains constant with warming – the planet is in a runaway state. The invariance to warming is only disrupted once water vapor reaches a sufficient abundance to affect the lapse rate, and thus modify the temperature at the emission level. Because this radiation limit resembles the runaway greenhouse, but requires enough background gas so that H2O is dilute compared to the background gas, we call this OLR limit the “dilute runaway”. The dilute runaway stands in contrast to the traditional Simpson-Nakajima runaway in which the atmosphere is dominated by water vapor, and which we call the “steam runaway”.

Nakajima et al. (1992) discussed the physics that can lead to the dilute runaway, but did not discuss its physical relevance in detail. Below we show that the dilute runaway, and how it depends on atmospheric MMW, is the underlying reason why H2-rich atmospheres approach the steam runaway gradually, whereas N2-rich atmospheres are unstable at high temperatures and enter the steam runaway abruptly.

4.2 The dilute runaway

As an atmosphere warms its OLR first tends towards the dilute limit, and then towards the steam limit (Fig. 4). If one could derive a simple expression for OLR in both limits, one could therefore predict whether a background gas causes an atmosphere to over- or undershoot the steam limit. In this section we do so, by deriving a closed-form analytical expression for the OLR of a runaway greenhouse atmosphere which was previously only solved implicitly (Nakajima et al., 1992; Pierrehumbert, 2010).

First, we derive an approximate form of the Clausius-Clapeyron relation. The Clausius-Clapeyron relation is often written as $d\ln e^{*}/d\ln T=L_{v}/(R_{v}T)$, which can be integrated to solve for $e^{*}(T)$. This form of $e^{*}(T)$ does not lead to closed-form expressions for optical thickness and OLR, however, so we instead treat the right-hand side as constant, $d\ln e^{*}/d\ln T\approx L_{v}/(R_{v}T_{0})\equiv\gamma$. In this case $e^{*}$ becomes a simple power law,

[TABLE]

Here $\gamma$ controls how rapidly saturation vapor pressure increases with temperature, and $T_{0}$ is a reference temperature which we will pick to be close to the temperature range of interest.

Our approximation allows us to express the relation between optical thickness and temperature as a power law, which turns out to be highly useful. We combine Equation 4 with the ideal gas law for water vapor, $\rho_{v}=e^{*}/(R_{v}T)$. We find

[TABLE]

This expression depends on atmospheric composition and background gases only through the temperature scale height $H_{T}$. As we showed before, $H_{T}^{dilute}=c_{p}T/g$ and $H_{T}^{steam}=L_{v}/g$. If we further approximate the temperature dependency of $H_{T}^{dilute}$ as $H_{T}^{dilute}\approx c_{p}T_{0}/g$, we can express $\tau$ as single power law in both limits,

[TABLE]

Here $\tau_{0}=(c_{p}T_{0}/L_{v})\times\kappa_{v}e_{0}^{*}/g$ in the dilute limit while $\tau_{0}=\kappa_{v}e_{0}^{*}/g$ in the steam limit.

We are now able to find a closed solution for the runaway greenhouse radiation limit. For a grey gas, OLR is equal to

[TABLE]

As the atmosphere becomes optically thick the surface contribution from the first term can be neglected and the integral’s upper limit replaced with infinity,

[TABLE]

Here $\Gamma$ is the gamma function, and $\tau_{0}$ is defined as

[TABLE]

Our solution in Equation 25 has a similar form to Pierrehumbert (2010)’s solution for the OLR of a dry atmosphere, but instead holds for a moist runaway atmosphere.

To evaluate our solution we only need to choose a reference temperature $T_{0}$ where our approximation of Clausius-Clapeyron will be most accurate. For the runaway greenhouse $T_{0}$ should be close to the surface temperature at which the atmosphere becomes optically thick. A convenient choice for all three atmospheric scenarios is the temperature at which the optical thickness of a pure-H2O atmosphere equals unity.

Figure 4 compares our analytical solution against our numerical calculations. We find that the analytical solution closely match the numerical results, so our expression successfully captures the physics of both the dilute and the steam runaway.

Our solution now allows us to understand why OLR overshoots in a N2-rich atmosphere but undershoots in a H2-rich atmosphere (Fig. 4). The reference temperature $T_{0}$ is independent of the background gas, so the greater value of $\mathrm{OLR}_{\infty}$ for N2 relative to H2 depends entirely on $\tau_{0}$. In turn, background gases only affect $\tau$ through the temperature scale height $H_{T}$. The ratio of optical thicknesses in the dilute versus the steam limit is

[TABLE]

which expresses our result from Section 3.2 in terms of a single non-dimensional parameter. Again, for H2O condensing in N2 at around 300 K this ratio is less than unity, $c_{p}T_{0}/L_{v}\sim 0.1$, whereas in H2 this ratio is $c_{p}T_{0}/L_{v}\sim 1.7$. Comparing the two values, the ratio is closer to unity for H2 than for N2, which explains why the OLR undershoot in H2 is small whereas the OLR overshoot in N2 is large (Fig. 4). Moreover, as we showed in Section 3.2, H2’s impact as a background gas is essentially unique. Even if we consider condensible gases other than H2O, we are not able to find a likely combination for which $c_{p}T_{0}/L_{v}>1$ without H2 as the background gas666For CO2 condensing in a CH4-rich atmosphere, $c_{p}T_{0}/L_{v}\sim 1.1$ due to CO2’s small latent heat combined with CH4’s relatively small MMW. However, such an atmosphere would likely not be chemically stable unless it was constantly replenished through a disequilibrium process.. Conversely, even if we consider gases other than H2O that could condense in a H2-rich atmosphere (e.g., an early Titan with a H2 atmosphere and a CH4 surface ocean, or a super-Mars with a H2 atmosphere and large CO2 glaciers), we find that $c_{p}T_{0}/L_{v}>1$ for all these condensible substances.

In conclusion, H2 is the only common background gas for which the dilute runaway limit lies below the steam limit, which explains why H2-rich atmospheres approach the runaway greenhouse gradually while all high MMW atmospheres enter the runaway greenhouse abruptly. Next, we show why the evolution of OLR is not monotonic with warming in a H2 atmosphere.

4.3 Non-monotonic OLR: the Soufflé Effect in H2

In the previous section we derived asymptotic expressions for the dilute and steam limits. In this section we go one step further and consider first-order effects that appear when an atmosphere is slightly warmer than the cold dilute limit, or slightly colder than the hot steam limit. We find two thermodynamic feedbacks that shape the evolution of OLR with warming, and that explain the non-monotonicity of OLR in H2 (Fig. 4).

At relatively low temperatures, we find that the latent heat release of water vapor dominates over water vapor’s other thermodynamic effects. This means the addition of water vapor will always first warm and moisten an atmosphere, regardless of the dry background gas. As proof we consider the general moist adiabat (Pierrehumbert, 2010; Ding & Pierrehumbert, 2016) in the limit of a nearly dry atmosphere. We perform a series expansion, assuming the molar mixing ratio is small, $e^{*}/p_{d}\ll 1$, and additionally use $L_{v}/(R_{v}T)\gg 1$ (which holds for a wide range of condensibles) while $R/c_{p}$ is always of order unity. We find

[TABLE]

By combining this result with the hydrostatic relation for dry air we can solve for the thermal scale height in the near-dilute limit,

[TABLE]

Here the factor $R_{d}/c_{p}\times(L/(R_{v}T))^{2}\times e^{*}/p$ captures the first-order thermodynamic effect of water vapor. Because $R_{d}/c_{p}\times(L/(R_{v}T))^{2}\times e^{*}/p$ is always positive, it acts to increase $H_{T}$ and thus $\tau$. Intuitively, adding trace amounts of water vapor to a dry parcel releases latent heat, which decreases the lapse rate and allows the atmospheric column to hold more water vapor. This provides a moistening feedback which increases $H_{v}$ and $\tau$, and thus decreases OLR. Importantly, the moistening feedback is largely independent of background gas because the background gas only appears via the parameter $R_{d}/c_{p}$ which is approximately constant (e.g., $R_{d}/c_{p}=2/5$ for He, whereas $R_{d}/c_{p}\approx 2/7$ for N2 or H2). This feedback explains why, after an atmosphere hits the dilute runaway, further warming then leads to a OLR decrease in both N2- and H2-rich atmospheres (Figure 4).

At high temperatures the compositional feedback we already described in Section 3.2 dominates. To quantify it we consider the effect of a background gas on the mean molecular weight of a steam atmosphere, which shows up via the gas constant used in the hydrostatic relation, and perform a series expansion in the limit $p_{d}/e^{*}\ll 1$. We find

[TABLE]

We solve for the thermal scale height by combining this result with the lapse rate of a steam atmosphere and find

[TABLE]

The sign of $(R_{d}-R_{v})/R_{d}\times p_{d}/e^{*}$ depends on the MMW of the background gas, so a background gas can either increase or decrease $H_{T}$ and $\tau$ near the steam limit. For N2 in H2O, $R_{d}<R_{v}$, so N2 has a net drying effect. However, as a steam atmosphere warms, this drying effect weakens and $\tau$ will increase even faster than it would in a pure H2O atmosphere - a positive feedback, just as in Section 3.2. In contrast, H2 has a net moistening effect because $R_{v}<R_{d}$. This effect weakens with warming, which slows the increase of $\tau$ relative to that in a pure H2O atmosphere, and thus provides a negative feedback.

For a N2-dominated atmosphere, both the latent heating feedback in Equation 31 and the drying feedback in Equation 34 have the same sign with warming and act to decrease OLR. That is why, after an initial overshoot, OLR decreases monotonically towards the steam limit (Figure 4). Conversely, for a H2-dominated atmosphere the latent heating feedback first decreases OLR before the moistening feedback eventually sets in and causes OLR to rise again.

Based on what a H2-rich atmosphere looks like as it warms up, we call its temperature-dependent succession of feedbacks the “Soufflé effect”. Figure 5 shows how the Soufflé effect plays out in our calculations. As the atmosphere warms up, the pressure scale height increases. At cold temperatures this increase is linear with temperature, and arises from the atmosphere’s thermal expansion. Starting at 280 K the H2-rich atmosphere begins to puff up a lot faster. This effect is due to the latent heat release of water vapor, which moistens the atmosphere and reduces OLR (Fig. 4). However, just as too much steam causes a Soufflé to collapse, too much steam becomes fatal for a H2-rich atmosphere: once heavy condensing H2O begins to displace light dry H2, at around 340 K, the atmospheric scale height collapses. This collapse slows down the further increase of atmospheric water and increases OLR (compare Fig. 4 and Fig. 5b).

4.4 How much dry background gas is required?

We have shown that different background gases can have a major influence on a planet’s climate, but how much background gas is required for this influence to become apparent? We address this question by considering whether an atmosphere can transition straight from a cold climate into the steam runaway, or whether it enters a dilute runaway phase first.

To enter either runaway state the atmosphere first needs to become optically thick so that OLR decouples from the planet’s surface emission, which means

[TABLE]

Here $T_{1}$ is the surface temperature necessary so that the overlying atmospheric column contains enough water vapor to be optically thick. For example, with a grey absorption cross-section of $\kappa_{v}=0.01$ m2 kg*-1*, $T_{1}\sim 310$ K in a thick N2 background and $T_{1}\sim 270$ K in a thick H2 background. In general $T_{1}$ depends on the detailed radiative properties of the condensing gas. However, it turns out that several condensible greenhouse gases start to become optically thick roughly around surface temperatures corresponding to their triple point (see SI in Koll & Cronin, 2018). In the following we therefore approximate $T_{1}$ as the triple point temperature of each condensible gas777By equating $T_{1}$ with the triple point, our criterion is overly stringent for H2O: H2O’s triple point is 273 K, whereas the H2O runaway occurs at around 310 K in a saturated N2-rich atmosphere (Koll & Cronin, 2018)..

To sustain the dilute runaway an atmosphere needs to have sufficient background gas, otherwise it only experiences a transient overshoot/undershoot (Fig. 4). We quantify the required amount of background gas by considering the first-order correction term to the dry adiabat in Equation 29, which is equal to $R_{d}/c_{p}\times(L_{v}/(R_{v}T))^{2}\times e^{*}/p$. For the atmosphere to be dilute this term has to be much less than unity, which leads to the following threshold on the dry (background) surface pressure,

[TABLE]

Conversely, if the background pressure is very low the atmosphere transitions from a cold climate straight into the steam runaway. Equation 33 shows that the first-order correction term near the steam limit is equal to $(R_{d}-R_{v})/R_{d}\times p_{d}/e^{*}$, which leads to the following threshold,

[TABLE]

We evaluate the thresholds from Equations 36 and 37 in Table 1, replacing inequalities with a factor of three.

Table 1 shows that even small amounts of dry background gas are sufficient to modify the behavior of a steam atmosphere, while moderately large amounts of dry background gas can sustain a dilute runaway. A few millibar of dry background gas, or about the surface pressure of Mars, already suffice to modify the H2O steam runaway. This value increases to several bar of dry gas for CO2, while methane and ammonia are intermediate between the two. It is thus relatively easy for a minor background gas to modify the OLR of a condensible-rich atmosphere. Of course, the exact climatic impact of the background gas (i.e., whether it creates an OLR overshoot or undershoot) depends on its MMW. The dilute runaway is easiest to achieve for H2O, with around two bar of dry background gas, and most difficult to achieve for CO2, with several hundred bar of dry background gas (Table 1). The difference between condensible gases is primarily driven by the triple point vapor pressures, with a small vapor pressure for H2O and a large vapor pressure for CO2.

To put these calculations into context, present-day Earth has 1 bar of N2-O2 background while Titan has about 1.5 bar of N2. In both cases there is enough dry background gas to significantly affect the atmosphere’s thermal structure and OLR, but both fall short of the dilute runaway threshold for H2O and CH4. In contrast, Venus’ atmosphere currently has about 3 bar of N2 background pressure. If this N2 entered the atmosphere early in the planet’s history, it could have had a significant influence on the onset of the runaway greenhouse on Venus.

4.5 Explaining the difference between grey and spectral calculations

The comparison between Figures 1 and 4 shows that H2 atmospheres undershoot the steam limit as our grey model predicts. However, unlike our grey calculations, our spectrally-resolved calculations do not enter a dilute runaway and, for H2-rich atmospheres, do not show a non-monotonic OLR with warming.

The difference between our grey and our spectral calculations arises from the opacity of the background gas. Whereas our grey calculations assume that any background H2 is transparent in the infrared, H2 actually has a strong CIA greenhouse effect which is dominant at cold surface temperatures. In a H2-rich atmosphere the impact of H2O is thus first felt via the impact of H2O condensation on the lapse rate (Section 4.3), which influences the temperature of the emission level. It is only at relatively high surface temperatures that H2O starts to be the dominant opacity source at the emission level so that OLR approaches the steam runaway.

To illustrate how H2’s greenhouse effect affects our grey calculations, we repeat them with a dry background gas that is opaque in the longwave. This is a general proxy for cases in which the atmosphere’s background greenhouse effect is non-zero but remains fixed with surface temperature (e.g., atmospheres dominated by H2 or CO2, but also Earth-like atmospheres with a mixed N2-CO2 background). We use the following opacities, $\kappa_{v}=10^{-2}$ m2 kg*-1* and $\kappa_{dry}=10^{-3}$ m2 kg*-1*. With 1 bar of dry background gas, this means the atmosphere has an optical thickness of about $10$ even without any water vapor.

Figure 6 shows why a background greenhouse gas is sufficient to obscure the dilute runaway and any non-monotonic OLR evolution with warming. At cold temperatures the atmosphere is already optically thick, and OLR is substantially lower than the surface emission $\sigma T_{s}^{4}$. Instead of entering the dilute runaway at around $270$ K, OLR stays low until shooting up at around $300$ K. The rapid increase occurs because the reduction of the lapse rate with surface warming (see Section 4.3) causes the atmosphere’s emission temperature to increase faster than the surface temperature. With sufficient warming H2O dominates the opacity near the emission level, at which point calculations with an opaque background become identical to calculations with a transparent background (Fig. 6). Further warming then results in the familiar asymptote of OLR towards the steam limit.

Figure 6 also shows how additional greenhouse gases affect the initiation of the runaway greenhouse in a high MMW atmosphere, such as adding CO2 to a N2-rich atmosphere. With a transparent N2-rich background the OLR overshoot is large and the OLR “bump” occurs at a low temperature of about 300 K, whereas with an optically thick N2-rich background the amplitude of the overshoot shrinks and the OLR bump moves to about 370 K. Additional greenhouse gases such as CO2 can therefore shift the initiation of the runaway to higher temperatures in a N2-rich atmosphere, but they do not alter our result that high MMW atmospheres experience an OLR overshoot whereas H2-rich atmospheres experience an OLR undershoot.

In conclusion, H2’s CIA greenhouse explains why the dilute runaway is not directly apparent in our spectral calculations for H2-rich atmospheres. Nevertheless, the dilute runaway is still a useful theoretical limit because it provides a simple way of understanding why different background gases lead to an OLR overshoot or undershoot, and thus explains why N2-rich atmospheres abruptly jump from temperate climates into the steam runaway, whereas H2-rich atmospheres approach the steam runaway smoothly (Figure 1). Similarly, although H2’s CIA greenhouse prevents OLR from becoming non-monotonic, the stabilizing climate feedback induced by the Soufflé effect is the underlying reason why H2 atmospheres can remain stable at extremely elevated surface temperatures. Finally, Soufflé dynamics also shape the remote appearance of H2 atmospheres, which would be important for interpreting potential transit observations of H2-rich exoplanets.

5 Discussion

We have shown that H2 leads to a number of novel thermodynamic feedbacks that allow terrestrial planets with H2-rich atmospheres near the inner edge of the habitable zone to sustain hot surface climates. Our work relates to previous studies on the origin of life and the habitability of H2 climates in multiple ways:

Previous work on the origin of life. Our work has interesting consequences for early Earth. Several bars of H2 are sufficient to create global-mean surface temperatures above 60∘ C, either via a direct H2-H2 greenhouse or through H2’s amplification of the H2O greenhouse (Fig. 1). Such a hot start for life is compatible with the view that the last common ancestor of life on Earth was a thermophile (Giulio, 2003), and has the additional benefit that hot surface conditions could have dramatically accelerated prebiotic chemical reactions (Stockbridge et al., 2010). By the same token life would also experience a hot start on habitable evaporated cores, i.e., mini-Neptunes which lost their hydrogen envelopes and turned into super-Earths (Luger et al., 2015). Theoretical predictions and exoplanet observations suggest that this formation pathway is common (Lopez et al., 2012; Fulton & Petigura, 2018), so many habitable-zone planets would have sustained thick hydrogen envelopes early on. As a mini-Neptune loses its hydrogen and surface conditions drop below the critical point of H2O ($\sim$220 bar, 647 K), the planet would be able to sustain a liquid ocean. As long as any remnant H2 persists, however, the planet’s surface would remain hot (Fig. 1). Life on H2-rich exoplanets would thus arise under hyperthermal and reduced conditions, but might subsequently struggle to adapt to the planet’s gradual cooling and oxidation as the atmospheric hydrogen was lost to space (see Wordsworth, 2012; Abbot, 2015).

Previous work on H2 climates. Our results complement previous work on the impact of H2 in colder climates, where its strong CIA greenhouse effect is dominant. Several previous studies considered H2-rich atmospheres but assumed a fixed dry adiabatic temperature structure in their calculations (Pierrehumbert & Gaidos, 2011; Wordsworth, 2012), which precluded them from identifying the unusual climate feedbacks and Soufflé dynamics of H2 atmospheres we discuss here. Other studies included full moist thermodynamics but limited themselves to atmospheres in which H2 did not dominate the mass of the assumed background gas (Wordsworth & Pierrehumbert, 2013; Ramirez et al., 2014; Batalha et al., 2015; Ramirez & Kaltenegger, 2017), so these studies could similarly not identify the feedbacks that occur once H2 becomes the dominant background gas.

Finally, our results could be affected by additional physics as follows:

Lifetime of H2 atmospheres. Our observation that young H2-rich atmospheres can be sustained long enough to influence the origin of life is sensitive to a number of factors. One important factor is the planet’s host star. We considered planets around Sun-like stars, but planets around M dwarfs would lose hydrogen much more rapidly due to the elevated XUV emission of young M dwarfs (Penz et al., 2008). Nevertheless, even in this case some rocky planets around M dwarfs might outgas sufficient hydrogen during the solidification of the magma ocean to set the stage for an early H2-rich climate (Elkins-Tanton & Seager, 2008). Another important factor is the formation time of the planet’s hydrogen envelope. Since H2 is constantly escaping to space, and since the host star’s XUV flux diminishes rapidly over time, any delay in outgassing would substantially lengthen the lifetime of a H2 atmosphere. For example, this might be possible if hydrogen was produced by a very late giant impact, several hundred million years after the star’s formation (e.g., Quintana et al., 2016).

Mixed backgrounds. Although we have focused on pure background compositions, it is straightforward to extend our results to atmospheres with mixed compositions such as a primordial H2-He or a H2-N2 atmosphere. Equation 28 shows that in this case the mass-weighted mean heat capacity, $\bar{c}_{p}$, compared to the ratio $L_{v}/T$ determines whether the background gas leads to an OLR overshoot or undershoot. Evaluating this criterion for a wide range of mixed backgrounds, we find that the OLR undershoot remains robust as long as H2 dominates the background’s overall mass (i.e., as long as H2’s mass mixing ratio exceeds 50-55%). In the scenario of early Earth with 1 bar of N2, the addition of slightly more than 1 bar of H2 would thus be sufficient to create an atmosphere with climate feedbacks that are driven by H2 thermodynamics.

Stratospheres. Our calculations assume an all-adiabat atmosphere and neglect the radiative impact of a stratosphere. To test our results we repeated our grey radiative calculations with a stratosphere. We find that our results are largely robust, even though the addition of a stratosphere lowers OLR in the dilute runaway limit while barely affecting OLR in the steam runaway limit. The underlying reasons for how a stratosphere affects our grey calculations are not easily summarized, and will be explained in a forthcoming paper (Koll & Cronin, in prep). Here we simply note that there is an intimate connection between our results for the dilute runaway and previous work on the Komabayashi-Ingersoll (KI) limit, which is a runaway limit analogous to the steam runaway but which considers only the stratosphere’s radiative balance (Komabayasi, 1967; Ingersoll, 1969).

Additional effects of H2. More work is needed to explore additional effects which we were not able to address in this study. Previous work on the runaway greenhouse in N2-rich atmospheres underlined that sub-saturation and clouds can have a large impact on the runaway greenhouse threshold (Pierrehumbert, 1995; Leconte et al., 2013; Yang et al., 2013). Similarly, moist convection could develop strong spatial and time variability in H2-dominated atmospheres (Li & Ingersoll, 2015). These dynamics cannot be fully represented in 1D equilibrium models. Our study is therefore only a first look at hot habitable climates in hydrogen-dominated atmospheres, and should be followed up with 3D modeling.

We thank Edwin Kite and Sara Seager for detailed feedback and helpful discussions, and an anonymous reviewer for valuable comments. This article uses the ApJ template888https://github.com/pcubillos/ApJtemplate. D.D.B.K. was supported by a James S. McDonnell Foundation postdoctoral fellowship. T.W.C. was supported by NSF grant AGS-1623218.

Appendix A Escape Timescale of H2

We first investigate the possibility of direct atmospheric boil-off (Owen & Wu, 2016). In this scenario the planet’s atmosphere is so hot and extended that its photosphere (i.e., the altitude at which the atmosphere absorbs the bulk stellar flux) becomes comparable to the planet’s Bondi radius. The atmosphere then undergoes hydrodynamic escape that is powered by the broadband absorbed stellar flux, which allows the planet to shed large amounts of gas over extremely short timescales. To evaluate this possibility we first compute the planet’s Bondi radius,

[TABLE]

where $M_{p}$ is the planet’s mass, $c_{s}=\sqrt{\tilde{\gamma}kT/m}$ is the speed of sound in H2, and $\tilde{\gamma}=1.4$ for a diatomic gas. We compare the Bondi radius to the planet’s photospheric radius. We assume the photosphere is located at $0.1$ bar (representative of H2’s emission level in the infrared; this pressure will be larger at shorter wavelengths so our choice is conservative), and estimate the photospheric radius as

[TABLE]

where $p_{s}$ is the surface pressure and $H_{p}=kT/(mg)$ is the atmospheric scale height. We find that Earth-sized planets are stable against atmospheric boil-off. For example, even with $T=1600$ K and $p_{s}=1000$ bar, representative of a planet in the late magma ocean stage with a thick H2 envelope, $R_{photo}/R_{b}\sim 0.3$.

This leaves hydrodynamic escape powered by the star’s extreme ultraviolet (XUV) flux as the main mechanism capable of eroding an early hydrogen-rich atmosphere (Pierrehumbert & Gaidos, 2011; Wordsworth, 2012). The escape flux for energy-limited hydrodynamic escape is given by

[TABLE]

where $\epsilon\sim 0.3$ is an efficiency factor, $M_{p}$ and $R_{p}$ are the planetary mass and radius, and $F_{XUV}$ is the stellar XUV flux. Our formulation assumes that the main escaping species is atomic hydrogen, i.e., H2 is photodissociated in the upper atmosphere.

To model the stellar XUV we use the Sun’s current XUV flux at Earth’s orbit in the range $\lambda\leq 0.92$ $\mu$m, which is equal to $F_{0}=3.9\times 10^{-3}$ W m*-2* (Ribas et al., 2005). This excludes Ly$\alpha$ radiation, which does not contribute to the escape flux in the absence of high-altitude absorbers (Pierrehumbert & Gaidos, 2011). To bracket the time evolution of the stellar XUV flux we use two different power-laws derived from Solar analogs (Ribas et al., 2005; Penz et al., 2008),

[TABLE]

and

[TABLE]

We integrate the escape flux, assuming planet formation ends 100 Myr after the formation of the host star. Figure 7 shows how much H2 is lost as a function of time, and how these amounts compare to different mechanisms for forming an early H2-rich atmosphere. We find that thick H2-envelopes with 70-100 bars of H2 can be retained over timescales of 60-120 Myr.

Appendix B Validation of Shortwave Radiative Transfer

Figure 8 shows that our code compares favorably against the line-by-line calculations of Goldblatt et al. (2013). To focus on the validity of our radiative transfer we use the same atmospheric temperature and humidity profiles as in Goldblatt et al, and use PyRADS to compute top-of-atmosphere planetary albedos. For reference, other radiative transfer codes produce typical differences of about 1-3% in the top-of-atmosphere planetary albedo for identical atmospheric profiles (Yang et al., 2016). We find that the albedo computed with PyRADS agrees with the results from Goldblatt et al to better than 1%, even though our albedo values are systematically higher.

The remaining small differences in albedo are likely due to different modeling assumptions and the use of different opacity sources. We use the ExoMol BT2 line list, which has a more complete coverage than the line list used by Goldblatt et al, so our computed albedos should be lower in the near-IR and visible. At the same time we only parameterize H2O absorption in the UV (see Section 2), so we could be underestimating atmospheric absorption at short wavelengths. In addition, we have found that changes in the assumed stellar source function can modify our results by up to several percent, which would be large enough to explain the remaining offset in Figure 8.