Hot Hydrogen Climates Near the Inner Edge of the Habitable Zone

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, which 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 1(a) 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 (1 kg of H₂ contains more molecules than any other gas).³ 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 1(b) 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 (Figure 1(b)). 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 1(c) 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 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 1600 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 (Figure 1(b)): 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 temperatures.⁴

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 (Figure 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 bar. Figure 2 shows that 50 bar 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 Myr, 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.

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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 ${\tau }_{{LW}}=\mathrm{ln}[{(\sigma {T}_{s}^{4})}^{-1}\times \int {B}_{\lambda }({T}_{s}){e}^{-{\tau }_{\lambda }}d\lambda ]$, where B_λ is the Planck function and τ_λ is the atmospheric optical thickness at a given wavelength λ. 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.

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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 thickness⁵ as

$$\begin{eqnarray}&&\tau ={\int }_{0}^{\infty }{\kappa }_{v}{\rho }_{v}{dz},\end{eqnarray} \tag{ 2 }$$

where z is height, κ_v is the absorption cross-section, and ρ_v is the density of water vapor (in general, the subscript v will denote quantities related to water vapor). We assume κ_v is independent of height and ρ_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 ρ_v we treat H_v as constant and equal to its value at the surface. We additionally define the absorption length l₀ ≡ 1/(κ_vρ_v,0), which is the characteristic distance over which a longwave photon is absorbed by the atmosphere. The atmospheric optical thickness is then simply

$$\begin{eqnarray}&&\tau ={\kappa }_{v}{\rho }_{v,0}{\int }_{0}^{\infty }{e}^{-z/{H}_{v}}{dz},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&=\ \displaystyle \frac{{H}_{v}}{{l}_{0}}.\end{eqnarray} \tag{ 4 }$$

The absorption length l₀ 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:

$$\begin{eqnarray}&&{H}_{v}={\left(\displaystyle \frac{-1}{{\rho }_{v}}\displaystyle \frac{d{\rho }_{v}}{{dz}}\right)}^{-1}.\end{eqnarray} \tag{ 5 }$$

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, $d\mathrm{ln}{e}^{* }/d\mathrm{ln}T={L}_{v}/({R}_{v}T)$ where L_v is the latent heat of vaporization:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\rho }_{v}}\displaystyle \frac{d{\rho }_{v}}{{dz}}=\displaystyle \frac{1}{{e}^{* }}\displaystyle \frac{{{de}}^{* }}{{dz}}-\displaystyle \frac{1}{T}\displaystyle \frac{{dT}}{{dz}},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&=\ \left(\displaystyle \frac{1}{{e}^{* }}\displaystyle \frac{{{de}}^{* }}{{dT}}-\displaystyle \frac{1}{T}\right)\displaystyle \frac{{dT}}{{dz}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&=\ \left(\displaystyle \frac{{L}_{v}}{{R}_{v}T}-1\right)\displaystyle \frac{1}{T}\displaystyle \frac{{dT}}{{dz}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\approx \ \displaystyle \frac{{L}_{v}}{{R}_{v}T}\displaystyle \frac{1}{T}\displaystyle \frac{{dT}}{{dz}}.\end{eqnarray} \tag{ 9 }$$

Here we used L_v/(R_vT) ≫ 1, which holds for water vapor and a wide range of other condensible gases. We recognize (−1/T × dT/dz)⁻¹ as the temperature scale height H_T, which sets the vertical length scale of temperature decrease, so

$$\begin{eqnarray}&&{H}_{v}=\displaystyle \frac{{R}_{v}T}{{L}_{v}}{H}_{T}.\end{eqnarray} \tag{ 10 }$$

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 low 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

$$\begin{eqnarray}&&{H}_{T}^{\mathrm{dilute}}=\displaystyle \frac{{c}_{p}T}{g}.\end{eqnarray} \tag{ 11 }$$

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

$$\begin{eqnarray}&&{H}_{T}^{\mathrm{steam}}=-T\displaystyle \frac{{dz}}{{dT}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&=\ -T\displaystyle \frac{{dz}}{{{de}}^{* }}\displaystyle \frac{{{de}}^{* }}{{dT}},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&=\ T\displaystyle \frac{1}{{\rho }_{v}g}\displaystyle \frac{{L}_{v}{e}^{* }}{{R}_{v}{T}^{2}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&=\ \displaystyle \frac{{L}_{v}}{g},\end{eqnarray} \tag{ 15 }$$

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, ${H}_{T}^{\mathrm{dilute}}\approx 30\,\mathrm{km}$ in N₂, whereas ${H}_{T}^{\mathrm{dilute}}\approx 430\,\mathrm{km}$ 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, 1 kg of H₂ contains 14 times more molecules than 1 kg 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 ${H}_{T}^{\mathrm{steam}}\,\approx 250\,\mathrm{km}$. This means H_T grows with warming if an atmosphere starts out N₂-rich, but shrinks with warming if the atmosphere starts out H₂-rich.

Because H_T controls H_v, and therefore the atmosphere's optical thickness τ, 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 τ, 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 (Figure 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 ${H}_{T}^{\mathrm{dilute}}\approx 160\,\mathrm{km}$, which is smaller than ${H}_{T}^{\mathrm{steam}}\approx 250\,\mathrm{km}$. 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_pT, 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.

For our gray radiative calculations we use the same gray longwave opacity for water vapor as in Nakajima et al. (1992), κ_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.

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Figure 4 recovers the main qualitative features of our spectral calculations. First, the OLR of a gray pure H₂O atmosphere closely resembles our spectral calculations for pure H₂O (Figure 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., ∼300 K in our spectral calculations; Figure 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.

As an atmosphere warms its OLR first tends toward the dilute limit, and then toward the steam limit (Figure 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. This is often written as $d\mathrm{ln}{e}^{* }/d\mathrm{ln}T={L}_{v}/({R}_{v}T)$, 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, $d\mathrm{ln}{e}^{* }/d\mathrm{ln}T\approx {L}_{v}/({R}_{v}{T}_{0})\equiv \gamma$. In this case e* becomes a simple power law

$$\begin{eqnarray}&&e* \ =\ {e}_{0}^{* }{\left(\displaystyle \frac{T}{{T}_{0}}\right)}^{\gamma }.\end{eqnarray} \tag{ 16 }$$

Here γ controls how rapidly saturation vapor pressure increases with temperature, and T₀ is a reference temperature which we will choose 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, ρ_v = e*/(R_vT). We find

$$\begin{eqnarray}&&\tau ={\kappa }_{v}{H}_{v}{\rho }_{v},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&=\ {\kappa }_{v}{H}_{T}\displaystyle \frac{{e}^{* }}{{L}_{v}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&=\ {\kappa }_{v}{H}_{T}\displaystyle \frac{{e}_{0}^{* }}{{L}_{v}}\times {\left(\displaystyle \frac{T}{{T}_{0}}\right)}^{\gamma }.\end{eqnarray} \tag{ 19 }$$

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

$$\begin{eqnarray}&&\tau ={\tau }_{0}{\left(\displaystyle \frac{T}{{T}_{0}}\right)}^{\gamma }.\end{eqnarray} \tag{ 20 }$$

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 gray gas, OLR is equal to

$$\begin{eqnarray}&&\mathrm{OLR}=\sigma {T}_{s}^{4}{e}^{-\tau }+{\int }_{0}^{\tau }\sigma T{\left(\tau ^{\prime} \right)}^{4}{e}^{-\tau ^{\prime} }d\tau ^{\prime} .\end{eqnarray} \tag{ 21 }$$

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

$$\begin{eqnarray}&&{\mathrm{OLR}}_{\infty }\approx {\int }_{0}^{\infty }\sigma T{\left(\tau \right)}^{4}{e}^{-\tau }d\tau ,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&=\ \sigma {T}_{0}^{4}{\int }_{0}^{\infty }{\left(\displaystyle \frac{\tau }{{\tau }_{0}}\right)}^{4/\gamma }{e}^{-\tau }d\tau ,\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&=\ \sigma {T}_{0}^{4}{\tau }_{0}^{-4/\gamma }{\int }_{0}^{\infty }{\tau }^{4/\gamma }{e}^{-\tau }d\tau ,\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\mathrm{OLR}}_{\infty }=\sigma {T}_{0}^{4}\times \displaystyle \frac{{\rm{\Gamma }}\left(1+4/\gamma \right)}{{\tau }_{0}^{4/\gamma }}.\end{eqnarray} \tag{ 25 }$$

Here Γ is the gamma function, and τ₀ is defined as

$$\begin{eqnarray}&&{\tau }_{0}=\left\{\begin{array}{l}\displaystyle \frac{{\kappa }_{v}{e}^{* }({T}_{0})}{g},\,\,\mathrm{steam}\,\mathrm{limit}\\ \displaystyle \frac{{\kappa }_{v}{e}^{* }({T}_{0})}{g}\displaystyle \frac{{c}_{p}{T}_{0}}{{L}_{v}},\,\,\mathrm{dilute}\,\mathrm{limit}.\end{array}\right.\end{eqnarray} \tag{ 26 }$$

Our solution in Equation (25) has a similar form to Pierrehumbert's (2010) 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 T₀ where our approximation of Clausius–Clapeyron will be most accurate. For the runaway greenhouse T₀ 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 matches 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 (Figure 4). The reference temperature T₀ is independent of the background gas, so the greater value of ${\mathrm{OLR}}_{\infty }$ for N₂ relative to H₂ depends entirely on τ₀. In turn, background gases only affect τ through the temperature scale height H_T. The ratio of optical thicknesses in the dilute versus the steam limit is

$$\begin{eqnarray}&&\displaystyle \frac{{\tau }_{\mathrm{dilute}}}{{\tau }_{\mathrm{steam}}}=\displaystyle \frac{{H}_{T}^{\mathrm{dilute}}}{{H}_{T}^{\mathrm{steam}}},\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&=\ \displaystyle \frac{{c}_{p}{T}_{0}}{{L}_{v}},\end{eqnarray} \tag{ 28 }$$

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_pT₀/L_v ∼ 0.1, whereas in H₂ it 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 (Figure 4). Moreover, as we showed in Section 3.2, H₂'s impact as a background gas is essentially unique. Even if we consider condensable gases other than H₂O, we are not able to find a likely combination for which c_pT₀/L_v > 1 without H₂ as the background gas.⁶ 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 ${c}_{p}{T}_{0}/{L}_{v}\gt 1$ for all these condensable 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.

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 (Figure 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 ≪ 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

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{dT}}{{dp}}\right|}_{\mathrm{dilute}}\approx \displaystyle \frac{{R}_{d}}{{c}_{p}}\displaystyle \frac{T}{p}\times \left(1-\displaystyle \frac{{R}_{d}}{{c}_{p}}{\left(\displaystyle \frac{{L}_{v}}{{R}_{v}T}\right)}^{2}\displaystyle \frac{{e}^{* }}{p}\right).\end{eqnarray} \tag{ 29 }$$

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

$$\begin{eqnarray}&&{H}_{T}=-T\displaystyle \frac{{dz}}{{dp}}\displaystyle \frac{{dp}}{{dT}},\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&=\ \displaystyle \frac{{c}_{p}T}{g}\left(1+\displaystyle \frac{R}{{c}_{p}}{\left(\displaystyle \frac{{L}_{v}}{{R}_{v}T}\right)}^{2}\displaystyle \frac{{e}^{* }}{p}\right).\end{eqnarray} \tag{ 31 }$$

Here the factor R_d/c_p × (L/(R_vT))² × 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 τ. 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 τ, and thus decreases OLR. Importantly, the moistening feedback is largely independent of background gas because the latter 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 ≈ 2/7 for N₂ or H₂). This feedback explains why, after an atmosphere hits the dilute runaway, further warming then leads to an 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* ≪ 1. We find

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{dp}}{{dz}}\right|}_{\mathrm{steam}}=-\bar{\rho }g,\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&\approx \ {\rho }_{v}g\times \left(1+\displaystyle \frac{{R}_{d}-{R}_{v}}{{R}_{d}}\displaystyle \frac{{p}_{d}}{{e}^{* }}\right).\end{eqnarray} \tag{ 33 }$$

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

$$\begin{eqnarray}&&{H}_{T}=\displaystyle \frac{{L}_{v}}{g}\left(1+\displaystyle \frac{{R}_{d}-{R}_{v}}{{R}_{d}}\displaystyle \frac{{p}_{d}}{{e}^{* }}\right).\end{eqnarray} \tag{ 34 }$$

The sign of (R_d − R_v)/R_d × p_d/e* depends on the MMW of the background gas, so a background gas can either increase or decrease H_T and τ near the steam limit. For N₂ in H₂O, ${R}_{d}\lt {R}_{v}$, so N₂ has a net drying effect. However, as a steam atmosphere warms, this drying effect weakens and τ 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 τ 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 toward 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 (Figure 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 Figures 4 and 5(b)).

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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

$$\begin{eqnarray}&&{\tau }_{{LW}}({T}_{1})\gt 1.\end{eqnarray} \tag{ 35 }$$

Here T₁ is the surface temperature necessary so that the overlying atmospheric column contains enough water vapor to be optically thick. For example, with a gray absorption cross-section of κ_v = 0.01 m² kg⁻¹, T₁ ∼ 310 K in a thick N₂ background and T₁ ∼ 270 K in a thick H₂ background. In general T₁ depends on the detailed radiative properties of the condensing gas. However, it turns out that several condensable greenhouse gases start to become optically thick roughly around surface temperatures corresponding to their triple point (see the supplementary information in Koll & Cronin 2018). In the following we therefore approximate T₁ as the triple point temperature of each condensible gas.⁷

To sustain the dilute runaway an atmosphere needs to have sufficient background gas, otherwise it only experiences a transient overshoot/undershoot (Figure 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:

$$\begin{eqnarray}&&{p}_{s,\mathrm{dry}}\gg \displaystyle \frac{{R}_{d}}{{c}_{p}}{\left(\displaystyle \frac{{L}_{v}}{{R}_{v}{T}_{1}}\right)}^{2}{e}^{* }({T}_{1}).\end{eqnarray} \tag{ 36 }$$

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 × p_d/e*, which leads to the following threshold:

$$\begin{eqnarray}&&{p}_{s,\mathrm{dry}}\ll \displaystyle \frac{{R}_{d}}{{R}_{d}-{R}_{v}}{e}^{* }({T}_{1}).\end{eqnarray} \tag{ 37 }$$

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 condensable-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 2 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's 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.

A comparison between Figures 1 and 4 shows that H₂ atmospheres undershoot the steam limit as our gray model predicts. However, unlike our gray 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 gray and spectral calculations arises from the opacity of the background gas. Whereas our gray calculations assume that any background H₂ is transparent in the IR, 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 gray 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, κ_v = 10⁻² m² kg⁻¹ and κ_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 those with a transparent background (Figure 6). Further warming then results in the familiar asymptote of OLR toward the steam limit.

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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.