Sulfate Aerosol Hazes and SO2 Gas as Constraints on Rocky Exoplanets' Surface Liquid Water

TL;DR

This paper proposes that the presence of sulfate aerosols and SO2 gas in an exoplanet's atmosphere can serve as indicators of the absence of significant surface liquid water, aiding in habitability assessments.

Contribution

It introduces a simple sulfur cycle model linking atmospheric sulfur compounds to surface water presence, providing new remote sensing indicators for exoplanet habitability.

Findings

Sulfate haze and SO2 levels are incompatible with >0.001 Earth oceans of surface water.

Detection of sulfate haze and SO2 suggests a lack of significant surface liquid water.

Modeling shows these indicators are robust under realistic planetary conditions.

Abstract

Despite surface liquid water's importance to habitability, observationally diagnosing its presence or absence on exoplanets is still an open problem. Inspired within the Solar System by the differing sulfur cycles on Venus and Earth, we investigate thick sulfate (H2SO4-H2O) aerosol haze and high trace mixing ratios of SO2 gas as observable atmospheric features whose sustained existence is linked to the near absence of surface liquid water. We examine the fundamentals of the sulfur cycle on a rocky planet with an ocean and an atmosphere in which the dominant forms of sulfur are SO2 gas and H2SO4-H2O aerosols (as on Earth and Venus). We build a simple but robust model of the wet, oxidized sulfur cycle to determine the critical amounts of sulfur in the atmosphere-ocean system required for detectable levels of SO2 and a detectable haze layer. We demonstrate that for physically realistic ocean pH values (pH > 6) and conservative assumptions on volcanic outgassing, chemistry, and aerosol microphysics, surface liquid water reservoirs with greater than 1e-3 Earth oceans are incompatible with a sustained observable H2SO4-H2O haze layer and sustained observable levels of SO2. Thus, we propose the observational detection of a H2SO4-H2O haze layer and of SO2 gas as two new remote indicators that a planet does not host significant surface liquid water.

Equations

$$u_{\text{ce}SO2}^{\ast }=\frac{f_{\text{ce}SO2}^{\ast }{p}_{\text{surf},\oplus }}{g_{\oplus }}\frac{{\mu }_{\text{ce}SO2}}{{\mu }_{\text{air,}\oplus }}$$

$$p_{\text{ce}SO2,\text{surf}}^{\ast }=u_{\text{ce}SO2}^{\ast }g\frac{{\mu }_{\text{air}}}{{\mu }_{\text{ce}SO2}}$$

$${\kappa }_{\text{e}}=\frac{3}{4}\frac{Q_{\text{e}}}{r{\rho }_{\text{aero}}},$$

$$\delta =u{\kappa }_{\text{e}}$$

$$M_{\text{ce}H2SO4}^{\ast }=u^{\ast }w4\pi R_{\text{P}}^2$$

$$N_{\text{ce}H2SO4}^{\ast }=\frac{M_{\text{ce}H2SO4}^{\ast }}{m_{\text{ce}H2SO4}}=\frac{16\pi }{3}\frac{{\delta }^{\ast }}{Q_{\text{e}}}r{\rho }_{\text{aero}}{R}_{\text{P}}^2\frac{w}{m_{\text{ce}H2SO4}}$$

$${\tau }_{\text{life}}=\min \{{\tau }_{\text{fall}},{\tau }_{\text{mix}}\}$$

$${\tau }_{\text{fall}}=\frac{z_{\text{fall}}}{v_{\text{Stokes}}}=\frac{9}{2}\frac{z_{\text{fall}}\eta }{r^{2}g{\rho }_{\text{aero}}{C}_{\text{C}}}.$$

$$\text{ce}SO2+O+M->SO3+M$$

$$\text{ce}SO2+OH+M->HSO3+M$$

$$\text{ce}SO3+H2O->H2SO4,$$

$${\tau }^{-1}=\frac{1}{4}\underset{{\lambda }_{\text{UV}}}{\int }q(\lambda )\sigma (\lambda ){\varphi }_{\gamma }(\lambda )\mathrm{d}\lambda$$

$$\frac{N_{\text{ce}H2SO4}}{{\tau }_{\text{life}}}=\frac{N_{\text{ce}SO2}}{{\tau }_{\text{ce}SO2\to \text{ce}H2SO4}}.$$

$$p_{\text{ce}SO2,\text{tropopause}}^{\ast }=N_{\text{ce}H2SO4}^{\ast }\frac{g{m}_{\text{ce}SO2}}{4\pi R_{\text{P}}^2}\frac{{\tau }_{\text{ce}SO2\to \text{ce}H2SO4}}{{\tau }_{\text{life}}}$$

$$\frac{p_{\text{ce}SO2,\text{tropopause}}}{p_{\text{tropopause}}}=\alpha \frac{p_{\text{ce}SO2,\text{surf}}}{p_{\text{surf}}}.$$

$$p_{\text{ce}SO2,\text{surf}}=K_{\text{H}}(T)[\text{ce}\text{ce}SO2(aq)],$$

Full text

exoplanet atmospheres — exoplanet surface characteristics — habitable planets

1 Introduction

Surface liquid water is considered essential to Earth-like life (e.g., Scalo et al.,, 2007; Kasting,, 2012), so determining whether a planet possesses liquid water is essential to constraining its habitability. As exoplanet detection and characterization techniques improve, observational constraints on the presence of surface water in the form of oceans will be a precursor to building an understanding of the occurrence rate of Earth-like planets and, ultimately, to building an understanding of the occurrence rate of Earth-like life. For many planets, the absence of surface oceans can be determined from straightforward physical arguments. Oceans require a surface temperature and pressure consistent with the stability of liquid water. These requirements limit possible ocean-hosting planet candidates to low-mass ($M\lesssim 10M_{\oplus}$) planets in the habitable zone (e.g., Kasting et al.,, 1993; Kopparapu et al.,, 2013).

While these fundamental requirements can be reasonably evaluated from a planet’s density and received stellar flux, evaluating whether rocky, habitable-zone planets actually possess surface oceans will be a substantial challenge, even for next-generation telescopes. While atmospheric spectra can identify water vapor in a planet’s atmosphere (e.g., Deming et al.,, 2013; Huitson et al.,, 2013; Fraine et al.,, 2014; Sing et al.,, 2016), such a detection says nothing conclusively about surface liquid water. Temporally resolved reflected light spectra can potentially identify the presence of liquid water via color variation due to dark ocean color (Cowan et al.,, 2009), polarized light variation due to ocean smoothness (Zugger et al.,, 2010), and/or ocean glint due to ocean reflectivity (Robinson et al.,, 2010). However, these methods all involve some combination of caveats, including still-unresolved potential false positives, decades-away instrumentation, and a reliance on ideal planetary conditions (Cowan et al.,, 2012). To systematically probe the presence of surface oceans on exoplanets, additional methods will unquestionably be needed. Here, we propose two such methods.

Though all existing ocean detection proposals exploit liquid water’s radiative properties, other planetary-scale implications of the presence of surface liquid water exist. A hydrosphere can substantially alter a planet’s chemistry, including—most notably here—its sulfur cycle. Previous studies of the observability of sulfur in exoplanet atmospheres suggest that sulfur has the capability to be observed in atmospheres in both aerosol and gas species (Kaltenegger and Sasselov,, 2010; Kaltenegger et al.,, 2010; Hu et al.,, 2013; Misra et al.,, 2015; Lincowski et al.,, 2018). In this paper, we test whether the observations of a permanent sulfate (\ceH2SO4-H2O) haze layer and atmospheric \ceSO2 can diagnose the absence of significant surface liquid water. (Note that we do not consider the inverse of this hypothesis—i.e., a lack of observable atmospheric sulfur implies the presence of an ocean—which is not particularly tenable.)

\ce

H2SO4-H2O aerosols with sufficient optical depth could be detected in exoplanet atmospheric spectra in the near future (Hu et al.,, 2013; Misra et al.,, 2015). Within the Solar System, Venus and Earth are examples of rocky planets with drastically different surface liquid water volumes and sulfur aerosol opacities. Venus has a global, optically thick \ceH2SO4-H2O aerosol layer (Knollenberg and Hunten,, 1980) and no surface liquid water. Earth, in contrast, only briefly hosts \ceH2SO4-H2O aerosol layers of significant optical depth after large volcanic eruptions (McCormick et al.,, 1995). Earth’s present inability to sustain an \ceH2SO4-H2O aerosol layer in its atmosphere can be directly tied to the presence of oceans, as we show in Section 3.

Beyond aerosols, \ceSO2 gas also has the potential to be detected at $\gtrsim$ 1 ppm mixing ratios (Kaltenegger and Sasselov,, 2010; Hu et al.,, 2013). Previous results have found that building up sulfur concentrations to this level with an Earth-like sulfur cycle requires implausibly high outgassing rates (Kaltenegger et al.,, 2010; Hu et al.,, 2013). We show in Section 3 that the presence of surface liquid water represents a fundamental barrier to maintaining high trace atmospheric sulfur gas levels in such circumstances.

The structure of this paper is as follows. In Section 2, we define the range of planetary atmospheres we consider. Section 3 describes our model of the sulfur cycle on a wet, oxidized world. Section LABEL:sec:results presents the key findings of our model on the incompatibility of observable \ceH2SO4-H2O aerosols and \ceSO2 gas with abundant surface liquid water. In Section 5, we discuss the implications and limitations of this study and make some suggestions for future work. Section 6 summarizes our conclusions.

2 The Sulfur Cycle on Wet, Oxidized Planets

In this paper, we model the sulfur cycle on wet, oxidized planets. We take a wet planet to be a planet with a reservoir of surface liquid water and an active hydrological cycle. We define an oxidized planet based on the atmospheric gas and aerosol sulfur species present, as discussed in further detail below.

Figure 1 summarizes the sulfur cycle on a wet planet. Sulfur gas (\ceSO2 or \ceH2S) is outgassed from melt transported from the interior via volcanism. In the ocean, this gas dissolves, and the aqueous sulfur species are dynamically mixed; in the lower atmosphere, the gas is dynamically mixed. Some gas reaches the upper atmosphere where a series of photochemical reactions produces sulfur gas that condenses into aerosols (\ceH2SO4-\ceH2O or \ceS8, depending on the atmospheric redox state). These aerosols are transported from the upper atmosphere to the lower atmosphere via gravitational settling or mixing on timescales that may depend on the size of the aerosol. Both sulfur gas and aerosols are removed from the lower atmosphere via their interactions with cloud-water and rainwater. Henry’s law dictates an equilibrium between sulfur gas in the atmosphere and aqueous dissolved sulfur species. Further reactions lead to the formation of sulfur minerals, which precipitate out of the ocean when they become saturated. These sediments may eventually be recycled in the interior depending on the planet’s tectonic regime.

Because sulfur can stably occupy a wide range of redox states (commonly -2, 0, +4, +6; or S(-II), S(0), S(IV), S(VI), respectively), the precise sulfur species involved in a planet’s cycle depend on the oxidization state of its interior, atmosphere, and ocean (e.g., Kasting et al.,, 1989; Pavlov and Kasting,, 2002; Johnston,, 2011; Hu et al.,, 2013). Generally, oxidized atmospheres have \ceSO2 gas and form aerosols composed of \ceH2SO4 and \ceH2O, whereas reduced atmospheres have \ceH2S gas and aerosols made of \ceS8 (Hu et al.,, 2013). A planetary system in a more intermediate redox state will have both types of sulfur species present, with ratios between them determined by the precise oxidation level (Hu et al.,, 2013).

In this work, we focus on the oxidized sulfur cycle, as (1) it is much more tightly coupled to planetary water inventory and (2) its aerosol photochemistry is better constrained by observations on Earth. We consider an oxidized atmosphere to be one in which most sulfur aerosols present are composed of \ceH2SO4-\ceH2O rather than \ceS8. The issue of how oxidized and reduced sulfur cycles can be distinguished observationally via transit spectra is discussed in Section 5.2.

3 Methods

As implied by Figure 1, the sulfur cycle involves atmospheric dynamics, photochemistry, microphysics, aqueous chemistry, and interior dynamics. Here, we aim to combine these components into a model that maximizes simplicity while retaining all of the most essential processes. Our emphasis on simplicity is driven by our focus on the implications of the sulfur cycle for the detection of an ocean. Furthermore, as we discuss throughout this section, the large uncertainties associated with many of these processes mean that more complicated models would be unlikely to give significantly more accurate results.

In our model, key parameters are constrained from basic physical arguments and Solar System analogs. For each parameter $x$, we consider both the best reasonable estimate ($x^{\text{b}}$) and the limiting scenario that promotes conditions for observable sulfur ($x^{\ell}$). The latter presents the most challenging conditions for our hypothesis that oceans are incompatible with sustained observable \ceSO2 or \ceH2SO4-H2O haze and thus represents the most stringent test of our hypothesis. Table LABEL:tab:parameters summarizes the two sets of parameter values.

Our model considers three reservoirs for surface sulfur: (1) an isothermal upper atmosphere (stratosphere), (2) a moist adiabatic lower atmosphere (troposphere), and (3) a surface liquid water layer (ocean). This basic atmospheric thermal structure arises from physical principles that are expected to be a reasonable approximation across a wide variety of planetary conditions (e.g., Pierrehumbert,, 2010). The sulfur species and phases that we include are \ceH2SO4 ($\ell$) and \ceSO2 (g) in the stratosphere; \ceSO2 (g) in the troposphere; and \ceSO2 (aq), \ceHSO3-(aq), and \ceSO3^2- (aq)—S(IV) species—in the ocean. The planetary parameters required are surface pressure $p_{\text{surf}}$, surface temperature $T_{\text{surf}}$, stratosphere temperature $T_{\text{strat}}$, atmospheric composition, planetary radius $R_{\text{P}}$, planetary mass $M_{\text{P}}$, and surface relative humidity (saturation of water vapor) $\text{RH}_{\text{surf}}$.

Our modeling approach begins with the critical \ceSO2 mixing ratio for detection of atmospheric \ceSO2 or the critical aerosol optical depth for observational detection of a haze layer. Working backward, we then calculate the critical number of sulfur atoms in the atmosphere and ocean system necessary for detection. Finally, we compare this critical value to the expected number of surface sulfur atoms to evaluate whether observable \ceSO2 buildup or observable haze formation is likely. In the following subsections, we discuss the various components of our model in detail.

3.1 Critical Atmospheric Sulfur for Observable \ceSO2

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SO2 gas is observable because it absorbs in characteristic bands in the infrared. In order to be observed, \ceSO2 must be present in high-enough concentrations such that its absorption lines have sufficient width and strength to be identified. Kaltenegger and Sasselov, (2010) find a constant \ceSO2 mixing ratio of 1-10 ppm observable via transit spectroscopy in an Earth-like atmosphere. We thus set the critical \ceSO2 mixing ratio for detection $f^{\ast}_{\ce{SO2}}$ to $f^{\ast\text{b}}_{\ce{SO2}}=1$ ppm in our reasonable case scenario and conservatively to $f^{\ast\ell}_{\ce{SO2}}=0.01$ ppm in our limiting case scenario. Hu et al., (2013) illustrate that photochemical loss of \ceSO2 can be a significant limiting factor in the lifetime of \ceSO2 in the atmosphere and thus can prevent high \ceSO2 buildup. Here, we ignore the photochemical loss of \ceSO2—again to conservatively estimate the minimum sulfur required for detection. As shown in Section LABEL:sec:results, we can draw strong conclusions even from this minimum \ceSO2 value, so there is little need to complicate this portion of our model with photochemistry. We discuss the photochemistry of \ceSO2 in the context of aerosol formation in Section 3.4. Finally, conceivably there may be exoplanet UV-photon-limited regimes where \ceSO2 is neither significantly photodissociated nor destroyed from reactions with reactive photochemical products, in which case this maximum estimate would be valid.

To first order, the amount of \ceSO2 in the light path, not \ceSO2’s relative abundance in the atmosphere, is what characterizes \ceSO2’s spectroscopic influence (Pierrehumbert,, 2010); mixing ratio is a convenient and traditional way of expressing abundance of trace gases, but its relationship to observability in transmission spectra will vary with total atmospheric pressure. To make Kaltenegger and Sasselov, (2010)’s \ceSO2 detection threshold more broadly applicable to non-Earth planetary conditions, we translate their critical mixing ratio of \ceSO2 for detection in an Earth-like atmosphere to a critical mass column (mass per unit area) of \ceSO2 $u^{\ast}_{\ce{SO2}}$:

[TABLE]

where $p_{\text{surf},\oplus}$ is the Earth’s surface pressure, $g_{\oplus}$ is the Earth’s surface gravity, $\mu_{\ce{SO2}}$ is the molar mass of \ceSO2, and $\mu_{\text{air,}\oplus}$ is the average molar mass of Earth air. Evaluating Equation (1) yields $u^{\ast\text{b}}_{\ce{SO2}}=2.3\times 10^{-2}$ kg m*-2* and $u^{\ast\ell}_{\ce{SO2}}=2.3\times 10^{-4}$ kg m*-2*. We then set the critical partial pressure of \ceSO2 at the surface $p_{\ce{SO2},\text{surf}}^{\ast}$ as

[TABLE]

where $g$ is the local surface gravity and $\mu_{\text{air}}$ is the average molar mass of the planet’s air. In Sections 3.2-3.6, we calculate the corresponding critical $p_{\ce{SO2},\text{surf}}^{\ast}$ for observable \ceH2SO4-H2O aerosols, before returning in Section 3.7 to the implications of this critical $p_{\ce{SO2},\text{surf}}^{\ast}$ for the sulfur budget of the ocean.

3.2 Aerosol Extinction

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H2SO4-H2O aerosols are observable because they extinguish (scatter and absorb) light. This extinction of light is most effective per unit mass of particle in the Mie scattering regime, where particles are the same order-of-magnitude size as the wavelength of light being scattered. The mass extinction coefficient $\kappa_{e}$ characterizes how effectively light is extinguished per unit particle mass as

[TABLE]

where $r$ is the average particle radius, $\rho_{\text{aero}}$ is the average particle density, and $Q_{\text{e}}$ is the particle extinction efficiency (Pierrehumbert,, 2010). Physically, $r$ and $\rho_{\text{aero}}$ are determined from the microphysics of \ceH2SO4-H2O aerosol formation. In contrast with water cloud formation, \ceH2SO4 and \ceH2O can condense directly from the gas phase to form liquid aerosol particles at physically realizable saturation levels (Seinfeld and Pandis,, 2012; Määttänen et al.,, 2018).

The ratio of \ceH2SO4 to \ceH2O by mass in an aerosol $w$ depends on both temperature and the ambient number densities of \ceH2SO4 and \ceH2O vapor (Määttänen et al.,, 2018). The ratio $w$ controls both $\rho_{\text{aero}}$ ($\rho_{\text{aero}}=w\rho_{\ce{H2SO4}}+(1-w)\rho_{\ce{H2O}}$) and the index of refraction of the aerosol, which impacts $Q_{\text{e}}$. Theoretically, more \ceH2SO4 gas is predicted to lead to aerosols with higher \ceH2SO4 concentrations (Määttänen et al.,, 2018); however, observational estimates for $w$ from \ceH2SO4-H2O aerosols in Earth’s stratosphere (a sulfur-poor environment) and Venus’ \ceH2SO4-H2O haze layer (a sulfur-rich environment) both give $w\approx 0.75$ (Turco et al.,, 1979; Russell et al.,, 1996; Seinfeld and Pandis,, 2012; Hansen and Hovenier,, 1974; Ragent et al.,, 1985). We thus set $w=0.75$ for both limiting and best conditions.

Once nucleated, aerosols grow rapidly via condensation if either \ceH2SO4 or \ceH2O gas is supersaturated (Turco et al.,, 1979; Seinfeld and Pandis,, 2012). The aerosols additionally grow via coagulation: diffusion and turbulence lead to sticking collisions between aerosols, which increase particle size and decrease particle number (e.g., Seinfeld and Pandis,, 2012). In both Earth’s and Venus’s atmospheres, \ceH2SO4-H2O aerosols tend to be relatively mono-dispersed in size (Knollenberg and Hunten,, 1980; Seinfeld and Pandis,, 2012), so describing their size distribution by a single average radius $r$ is a valid approximation.

We estimate a reasonable $r$ value from the \ceH2SO4-H2O aerosols that dominate Venus’ haze layer’s radiative properties as $r^{\text{b}}$ = 1 $\mu$m (Hansen and Hovenier,, 1974; Knollenberg and Hunten,, 1980). From the inverse dependence of $\kappa_{\text{e}}$ on $r$ in Equation (3), smaller $r$ values are more favorable for haze formation for a given number of atmospheric sulfur atoms. However, particles too small will be observationally indistinguishable from Rayleigh scattering by gas molecules. We therefore set $r^{\ell}$ by considering where $Q_{\text{e}}$ transitions from the Rayleigh scattering limit to the Mie scattering regime.

$Q_{\text{e}}$ is calculated with Mie theory from $r$, the composition of the particle (within the \ceH2SO4-H2O system, specified by $w$), the index of refraction of this given \ceH2SO4-H2O mixture (Palmer and Williams,, 1975), and the wavelength of incident light ($\lambda$) being attenuated (Bohren and Huffman,, 2008). We take $\lambda$ to be the peak of the planet’s host star’s spectrum, which is easily calculated from Wien’s displacement law given the star’s observed effective temperature. Figure 2 shows $Q_{\text{e}}$ as a function of $r$ for both Sun-like ($\lambda=0.556\ \mu$m) and M-dwarf stars (for illustrative purposes, $\lambda=1\ \mu$m) with the Rayleigh limit superimposed. Based on this calculation, we set $r^{\ell}$ = 0.1 $\mu$m for a Sun-like star and $r^{\ell}$ = 0.2 $\mu$m for an M dwarf.

We calculate the aerosol layer vertical path optical depth $\delta$ as

[TABLE]

where $u$ is the mass column of aerosol particles. We estimate the critical minimum optical depth value for an observable haze layer $\delta^{\ast}$ by simulating transmission spectra. This procedure is described in Section 3.6 after we finish outlining the structure of the rest of the atmosphere.

From the definition of mass column $u$, we can calculate the critical mass of \ceH2SO4 $M^{\ast}_{\ce{H2SO4}}$ as

[TABLE]

where $R_{\text{P}}$ is the radius of the planet. Putting $u^{\ast}$ in terms of $\delta^{\ast}$ and aerosol parameters with Equations (3) and (4), we then calculate the critical number of \ceH2SO4 molecules in aerosols needed to create an observable haze layer as

[TABLE]

where $m_{\ce{H2SO4}}$ is the mass of a molecule of \ceH2SO4.

3.3 Aerosol Sedimentation and Mixing

Once in the troposphere, \ceH2SO4-\ceH2O aerosols are ideal cloud condensation nuclei because of their high affinity for water, and they are rapidly removed by precipitation (Seinfeld and Pandis,, 2012). For a mean residence time $\tau_{\text{life}}$, \ceH2SO4 remains in the stratosphere as an aerosol contributing to the optical depth of the sulfur haze. We only consider \ceH2SO4 as radiatively relevant until it is transported to the tropopause—the boundary between the isothermal stratosphere and convecting troposphere—because of \ceH2SO4’s short lifetime in the troposphere.

The parameter $\tau_{\text{life}}$ depends on the size of the aerosol. If coagulation and condensation are effective and particles grow large enough, they will gravitationally settle out of the upper atmosphere. Otherwise, small particles will instead likely be removed by dynamic processes first due to their slow settling velocities. Thus, the average lifetime of an \ceH2SO4-H2O aerosol in the stratosphere is approximately

[TABLE]

where $\tau_{\text{fall}}$ is the gravitational settling timescale and $\tau_{\text{mix}}$ is the dynamic mixing timescale of exchange between the stratosphere and troposphere.

We calculate $\tau_{\text{fall}}$ from the Stokes velocity of the falling particle as

[TABLE]

Here, $g$ is planet surface gravity, $\eta$ is the dynamic viscosity of air, and $C_{\text{C}}$ is the Cunningham-Stokes correction factor for drag on small particles (Seinfeld and Pandis,, 2012). $z_{\text{fall}}$ is the average distance a sulfur aerosol must fall from its formation location to the tropopause. We conservatively estimate this parameter as the scale height of the stratosphere (i.e., $z_{\text{fall}}=RT_{\text{strat}}(\mu_{\text{air}}g)^{-1}$).

Calculating $\tau_{\text{mix}}$ from first principles is not possible because a general theory of stratosphere-troposphere mixing timescales as a function of external parameters does not yet exist. However, insights can be gained from studying Earth. Today, Earth’s strong stratospheric temperature inversion inhibits this stratosphere-troposphere exchange. Since radiatively generating an upper atmospheric temperature inversion stronger than Earth’s is difficult, Earth’s $\tau_{\text{mix}}$ is likely near an upper bound on stratosphere-troposphere mixing timescales. On the most observable M-dwarf planets, permanent day-night sides of planets due to tidal effects are predicated to generate strong winds (e.g., Showman et al.,, 2010; Showman and Polvani,, 2011) that will almost certainly reduce $\tau_{\text{mix}}$ from Earth values. Therefore, from Earth-based timescales, we estimate $\tau_{\text{mix}}$ = 1 year for all cases (Warneck,, 1999).

3.4 Aerosol Formation

The saturation vapor pressure of \ceH2SO4 at expected terrestrial planet stratosphere temperatures is extremely low ($\ll 10^{-6}$ Pa) (Kulmala and Laaksonen,, 1990), so all stratospheric \ceH2SO4 is effectively found condensed in aerosols rather than as a gas. The sulfur source for \ceH2SO4-H2O aerosols in oxidized atmospheres is \ceSO2 gas. Here, we define $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}$ as the timescale required to convert stratospheric \ceSO2 to \ceH2SO4. $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}$ is challenging to calculate precisely because \ceSO2 is oxidized to \ceH2SO4 through a series of photochemical reactions that are poorly constrained both in terms of reaction pathways and rates (Turco et al.,, 1979; Stockwell and Calvert,, 1983; Yung and DeMore,, 1998; Burkholder et al.,, 2015, etc.). The two main stages of the conversion of \ceSO2 to \ceH2SO4 are (1) \ceSO2 is converted to \ceSO3 via a photochemical product and (2) \ceSO3 and \ceH2O react to form \ceH2SO4 (Burkholder et al.,, 2015). For part 1 of the conversion, commonly proposed reactions are

[TABLE]

(Yung and DeMore,, 1998; Burkholder et al.,, 2015) or

[TABLE]

(Turco et al.,, 1979; Stockwell and Calvert,, 1983; Burkholder et al.,, 2015) followed by either

[TABLE]

(Turco et al.,, 1979; Stockwell and Calvert,, 1983; Burkholder et al.,, 2015). For part 2, the proposed reaction is

[TABLE]

but it is unclear whether this reaction proceeds as a two-body or three-body reaction kinetically (Seinfeld and Pandis,, 2012; Burkholder et al.,, 2015). Large uncertainties in reaction rates for these reactions are compounded by the reaction kinetics’ sensitivity to photochemical product (\ceOH, \ceO) concentrations, which require detailed knowledge of atmospheric composition for accurate results (Jacob,, 1999).

Despite these uncertainties, we can make simplifications because \ceSO2, \ceH2O, and stellar UV photons are essential for the conversion, regardless of the pathway. In particular, because photochemistry is the rate limiting step except potentially in extremely dry atmospheres, we can estimate a lower limit of $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}$ by considering how long it takes for the number of UV photons necessary for the reaction to proceed to strike the planet’s atmosphere.

For photolysis-driven reactions, the lifetime $\tau$ of the photolyzed species can be described by

[TABLE]

where $\lambda$ is wavelength of light, $q$ is the quantum yield or the probability that a photon will dissociate a molecule, $\sigma$ is the absorption cross section of the molecule, $\phi_{\gamma}$ is the flux of photons to a planet’s atmosphere per unit wavelength, and the factor of $\nicefrac{{1}}{{4}}$ accounts for day–night and stellar zenith angle averaging (Jacob,, 1999; Cronin,, 2014). To represent the most rapid conversion rate of \ceSO2 to \ceH2SO4, we set $q=1$ for all wavelengths (i.e., 100% probability that a photon will dissociate a molecule it strikes). At minimum, the \ceSO2 to \ceH2SO4 reaction requires one \ceO or \ceOH molecule. For weakly to strongly oxidized terrestrial planet atmospheres, this photochemical product will most likely be the result of the dissociation of \ceH2O, \ceO2, or \ceCO2. (Ostensibly, photodissociated \ceSO2 could also provide \ceO, but at the surface number densities of \ceSO2 we consider here, \ceSO2 is so optically thin that it cannot effectively absorb photons (Manatt and Lane,, 1993; Hamdy et al.,, 1991).) At each $\lambda$, we set $\sigma(\lambda)$ as the maximum absorption cross section from these three molecules, using data given in Chan et al., (1993), Mota et al., (2005), Lu et al., (2010), Yoshino et al., (1988), Kockarts, (1976), Brion et al., (1979), and Huestis and Berkowitz, (2010).

To set the wavelength for the upper limit of integration, we take the smallest minimum bond dissociation energy of \ceH2O, \ceO2, and \ceCO2 and convert it to a wavelength (via $E_{\text{dis}}=E_{\gamma}=h\nu=hc/\lambda$). A minimum energy of $8.2\times 10^{-19}$ J (deB. Darwent,, 1970) yields a maximum wavelength of 240 nm. We begin the integration from $\lambda=0$ nm for a maximum number of photons. In general, $\phi_{\gamma}$ will be determined from the host star’s spectrum and the distance of the planet from its host star. Here, we use the present-day flux of the Sun to Earth as measured by Thuillier et al., (2004) as a baseline $\phi_{\gamma}$ for a Sun-like star. The ratio of UV flux to stellar bolometric flux declines as stellar mass declines (due to lower effective temperatures and thus higher wavelength Planck peaks) while the ratio of extreme UV (EUV) flux to bolometric flux tends to increase for M dwarfs relative to Sun-like stars due to higher magnetic activity (Scalo et al.,, 2007; Shkolnik and Barman,, 2014; Schaefer et al.,, 2016; Wordsworth et al.,, 2018). The precise change in UV and EUV flux relative to bolometric flux for an M dwarf relative to the Sun will depend on the specific star and its age, but for illustrative purposes here, we assume a $10\times$ decrease in UV flux and a $10\times$ increase in EUV flux relative to the solar value. (Here, we define the EUV wavelength cutoff as $\lambda=91$ nm, following Wordsworth et al., (2018).) Plugging in values to Equation (12) then leads to $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}^{\ell}\approx$ 3.4 days for a Sun-like star and $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}^{\ell}\approx$ 2.5 days for an M dwarf. On modern Earth, $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}\approx 30$ days (Turco et al.,, 1979; Macdonald and Wordsworth,, 2017), which we take as $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}^{\text{b}}$.

Once we have estimated the lifetimes of \ceH2SO4 and \ceSO2 in the stratosphere, we can directly relate the steady-state $N_{\ce{H2SO4}}$ in aerosols to $N_{\ce{SO2}}$ in gas by assuming that the production rate of \ceH2SO4 is equal to its removal rate:

[TABLE]

From Equation (13) and the definition of pressure as force over area, $N_{\ce{SO2}}^{\ast}$ converted to a critical partial pressure of \ceSO2 ($p_{\ce{SO2}}^{\ast}$) at the tropopause is

[TABLE]

where $m_{\ce{SO2}}$ is the mass of a molecule of \ceSO2.

3.5 Lower Atmosphere Transport

On a wet, temperate planet, \ceSO2 faces no significant sinks below the photochemically active upper atmosphere until it encounters the water cloud layer in the troposphere. Once \ceSO2 encounters condensed water (as either rain or cloud droplets), the \ceSO2 dissolves and is subsequently rained out of the atmosphere (Giorgi and Chameides,, 1985; Seinfeld and Pandis,, 2012; Hu et al.,, 2013). The high effective solubility of \ceSO2 means that effectively all \ceSO2 at and below the cloud deck is removed every rainfall event (Giorgi and Chameides,, 1985). This process of wet deposition greatly limits the lifetime of \ceSO2 in the lower atmosphere and decreases the mixing ratio of \ceSO2 ($f_{\ce{SO2}}\equiv p_{\ce{SO2}}/p$) beyond the water cloud layer in the upper atmosphere relative to the surface.

In our model, we take the mixing ratio of \ceSO2 at the tropopause to be directly proportional to the mixing ratio of \ceSO2 at the surface, such that

[TABLE]

Here, $p$ is the total atmospheric pressure, and $\alpha$ is the change of mixing ratio between the surface and the upper atmosphere due to the effects of wet deposition. We do not yet know much about hydrological cycles on planets with less surface liquid water than Earth, so in the limiting case we completely ignore wet deposition and set $\alpha^{\ell}$ = 1. For the reasonable case, we assume $\alpha^{\text{b}}$ = 0.1, based on comparison with \ceSO2 vertical profiles measured on Earth (Georgii,, 1978; Meixner,, 1984).

We calculate a pressure-temperature profile from surface temperature $T_{\text{surf}}$ and pressure $p_{\text{surf}}$ assuming a dry adiabat until water vapor becomes saturated and then a moist adiabat until a specified (isothermal) stratospheric temperature $T_{\text{strat}}$ is reached, following the derivation of Wordsworth and Pierrehumbert, (2013). This calculation requires an assumed mixing ratio of water $f_{\ce{H2O}}\equiv p_{\ce{H2O}}/p$ at the surface. This mixing ratio is calculated from relative humidity at the surface $\text{RH}_{\text{surf}}$ and $T_{\text{surf}}$ via $f_{\ce{H2O}\text{,surf}}=\text{RH}_{\text{surf}}\times p_{\text{sat,\ce{H2O}}}(T_{\text{surf}})$ where $p_{\text{sat,\ce{H2O}}}(T_{\text{surf}})$ is the saturation pressure of water at temperature $T_{\text{surf}}$. We set $\text{RH}_{\text{surf}}^{\text{b}}$ = 0.77 like the Earth. Increasing $f_{\ce{H2O}}$ yields a higher tropopause, which results in a higher $p_{\ce{SO2},\text{surf}}$ for a given $p_{\ce{SO2},\text{tropopause}}$. Therefore, we set $\text{RH}_{\text{surf}}^{\ell}$ = 0 (yielding a dry adiabat atmospheric structure).

3.6 Aerosol Observability

Having reviewed the atmospheric conditions dictated by the presence of an observable \ceH2SO4-H2O haze layer, we can now return to the problem of how to calculate the critical optical depth $\delta^{\ast}$ for observation. We simulate the transmission spectra expected of a planet with a sulfur haze for a range of optical depths to determine at which $\delta$-value the hazy spectrum becomes distinct from the clear spectrum. The appendices of Morley et al., (2015, 2017) provide the details of our model for simulating transmission spectra, which uses the matrix prescription presented in Robinson, (2017). We calculate molecular cross sections as described in Freedman et al., (2008, 2014), including an updated water line list from Polyansky et al., (2018).

As inputs to our model, we require aerosol size and number densities, atmospheric gases number densities, and temperature as functions of pressure. From Equation (13), we calculate an \ceH2SO4-H2O aerosol number density profile, assuming exponential decay in aerosols from the tropopause until a parameterized cutoff height (due to lack of photochemical \ceH2SO4 production). We input atmospheric number densities assuming constant mixing ratios (except for \ceH2O) for a given atmospheric composition. The temperature-pressure profile follows a moist adiabat until it reaches an isothermal stratosphere. The mixing ratio of \ceH2O $f_{\ce{H2O}}$ is determined by the moist adiabat. $f_{\ce{H2O}}$ is set at the surface and is held constant until water vapor becomes saturated. Then, $f_{\ce{H2O}}$ evolves according to water’s saturation pressure until the tropopause, above which $f_{\ce{H2O}}$ remains constant.

Varying $\delta$ in these simulated transit spectra for an Earth-like atmosphere (see Section LABEL:subsec:res_obs_haze and Figure LABEL:fig:spectra) suggested that the critical minimum value for an observable haze layer is $\delta^{\ast}\approx 0.1$. Different atmospheres will yield different $\delta^{\ast}$ values, but likely not by orders of magnitude. The precise shape of a hazy planet’s transit spectrum is also somewhat sensitive to the photochemical haze cutoff height. Lower cutoff heights yield more traditionally haze-characteristic flat spectra (e.g., Kreidberg et al.,, 2014), and higher, less physical cutoff heights yield more structured—though still smoothed relative to a clear atmosphere—spectra; but the full range of possible haze cutoff values still yields spectra identifiable as hazy. In the absence of a detailed photochemical model, we choose to display results in Figure LABEL:fig:spectra with a cutoff height of two stratospheric scale heights (11.7 km), following Earth and Venus \ceH2SO4-H2O aerosol profile observations (e.g., Sekiya et al.,, 2016; Knollenberg and Hunten,, 1980).

3.7 Ocean Sulfur Storage

When surface liquid water is present, convection and the hydrological cycle act to bring the lower atmospheric and dissolved sulfur in the ocean into equilibrium rapidly. The partial pressure of \ceSO2 at the surface ($p_{\ce{SO2},\text{surf}}$) is held in equilibrium with the concentration of dissolved, aqueous \ceSO2 in the ocean [\ce\ceSO2 (aq)] via Henry’s law:

[TABLE]

where $K_{H}=6.96\times 10^{4}$ Pa L mol*-1* is the Henry’s law constant for \ceSO2 (Pierrehumbert,, 2010). Once dissolved, \ceSO2 reacts with the ambient water to form sulfurous acid, which dissociates to \ceH+, \ceHSO3-, and \ceSO3^2- ions:

4.3 Sulfur in the Atmosphere versus Ocean

We next translated our observable sulfur criteria into a critical amount of sulfur in the ocean-atmosphere system. We calculated the distribution of sulfur in an ocean between aqueous S(IV) species (\ceSO2(aq), \ceHSO3-, \ceSO3^2-) as a function of pH, assuming S(IV) saturation, in Figure 5. The fraction of S(IV) stored as \ceSO2—the only dissolved S(IV) species in direct equilibrium with the atmosphere via Equation (16)—exponentially declines as pH increases. For a modern-Earth ocean pH = 8.14, $5.3\times 10^{-6}$% of dissolved S(IV) is \ceSO2(aq), 10.3% is \ceHSO3-, and 89.7% is \ceSO3^2-.

From the distribution of S(IV) species, we then calculated the expected ratio of S(IV) sulfur in the atmosphere (\ceSO2) to S(IV) sulfur in the ocean (\ceSO2(aq), \ceHSO3-, \ceSO3^2-), again assuming saturation of aqueous S(IV) and atmosphere-ocean equilibrium. Figure 6 shows the preferential storage of S(IV) in the ocean over the atmosphere as a function of pH with varying total ocean mass. The storage capacity of the ocean linearly increases with increasing ocean mass and exponentially increases with increasing ocean pH. For Earth ocean pH and mass, the ratio of atmospheric to oceanic S(IV) is $2.13\times 10^{-9}$. (Note that this ratio is not actually observed in the modern ocean as the assumption of S(IV) saturation is violated because of the instability of aqueous S(IV).)

4.4 Observable Sulfur versus Ocean Parameters

Finally, we look at how the presence and characteristics of an ocean shape sulfur observability. First, we calculated conditions for atmospheric sulfur observability for a range of ocean pHs and masses using the best-guess model parameters given in Table LABEL:tab:parameters. Then, we calculated these conditions using the limiting parameters. For all inputs, we plot contours of the critical timescale for S(IV) decay ($\tau_{\ce{S(IV)}}^{\ast}$) necessary to have enough sulfur in the atmosphere-ocean system to sustain observable \ceSO2 or an observable \ceH2SO4-H2O aerosol layer versus ocean parameters of pH and mass. From present aqueous redox sulfur chemistry experimental results, we set $\tau_{\ce{S(IV)}}^{\ast}=0.1$ years $\approx 1$ month (white contour) as a reasonable timescale for aqueous S(IV) decay. For ocean pHs and total masses below this line, observable atmospheric sulfur is possible. Above this line, observable sulfur grows increasingly unlikely.

Low pH is most favorable for observable atmospheric sulfur in our results, but cations from surface weathering can act to buffer pH and likely maintain a pH near neutral (Grotzinger and Kasting,, 1993), particularly in low water content regimes with lots of exposed land. Empirically, paleo-pH estimates for both Earth and Mars seem to favor such a near-neutral regime (e.g., Halevy,, 2013; Grotzinger et al.,, 2014). From these considerations, we choose pH = 6 to report characteristic ocean sizes leading to observable atmospheric sulfur, given $\tau_{\ce{S(IV)}}^{\ast}=0.1$ years. We also choose, somewhat arbitrarily, to define significant surface liquid water as $10^{-3}$ Earth ocean masses or a global equivalent ocean layer of 2.75 m on an Earth-sized planet. For context, on Earth today, this amount of water is equivalent to about one third of the Mediterranean Sea (Eakins and Sharman,, 2010).

Figure 7 shows $\tau_{\ce{S(IV)}}^{\ast}$ versus ocean parameters for observing \ceSO2 with our best-guess parameter values. For surface liquid water content greater than $10^{-3}$ Earth ocean masses, observable \ceSO2 with reasonable model parameters requires $\tau_{\ce{S(IV)}}^{\ast}\gtrsim$ 10 years for even the lowest pHs most favorable for \ceSO2 buildup and $\tau_{\ce{S(IV)}}^{\ast}>$ 10,000 years for pH $>6$. Such long decay times are incompatible with present highest quoted values for decay in the literature, which are on the order of years (Ranjan et al.,, 2018). At pH = 6 and $\tau_{\ce{S(IV)}}^{\ast}=0.1$ years, observable \ceSO2 requires an “ocean” of less than $1\times 10^{-9}$ Earth’s ocean mass or a global equivalent layer of less than 3 $\mu$m.

Figure 8 shows $\tau_{\ce{S(IV)}}^{\ast}$ versus ocean parameters for observing an \ceH2SO4-H2O haze layer with our best guess parameter values. For surface liquid water content greater than $10^{-3}$ Earth ocean masses, observable haze formation with reasonable model parameters requires $\tau_{\ce{S(IV)}}^{\ast}\gtrsim$ 1 year for all pHs and, again, $\tau_{\ce{S(IV)}}^{\ast}>$10,000 years for pH$>6$. Again, these timescales are much, much longer than any $\tau_{\ce{S(IV)}}^{\ast}$ estimates currently present in the literature. At pH = 6 and $\tau_{\ce{S(IV)}}^{\ast}=0.1$ years, observable haze requires an ocean of less than $2.5\times 10^{-8}$ Earth’s ocean mass or a global equivalent layer of less than 68 $\mu$m.

Figures 9 and 10 show $\tau_{\ce{S(IV)}}^{\ast}$ versus ocean parameters for observing \ceSO2 and observing a haze layer, respectively, with our limiting parameter values. At pH = 6 and $\tau_{\ce{S(IV)}}^{\ast}=0.1$ years, observable \ceSO2 with these most stringent parameters still requires an ocean of less than $2\times 10^{-5}$ Earth’s ocean mass or a global equivalent layer less than 5.5 cm. We calculate that at pH = 6 and $\tau_{\ce{S(IV)}}^{\ast}=0.1$ years, an observable haze layer for a planet around an M-dwarf star requires an ocean of less than $1.3\times 10^{-3}$ Earth’s ocean mass or a global equivalent layer of 3.6 m; around a Sun-like star gives an ocean of less than $1.5\times 10^{-3}$ Earth’s ocean mass or a global equivalent layer of 4.1 m.

5 Discussion

5.1 Model Uncertainties

Robustly establishing sustained high trace mixing ratios of \ceSO2 and \ceH2SO4-H2O haze as indicators of limited surface liquid water will require experimental constraints on S(IV) decay pathways and kinetics. Better theoretical constraints on the upper bounds of sulfur outgassing rates and lower bounds of ocean pHs in water-limited regimes will also improve the robustness of these newly proposed observational constraints on surface liquid water.

However, in accounting for these poorly constrained inputs, we have concocted a worst-case scenario for each input, often physically inconsistent with each other (e.g., extremely high sulfur outgassing levels would not be consistent with extremely small aerosol particles microphysically). These limiting parameters are intended to push our model to its limits given the breadth of our unknown parameters, rather than just presenting likely values informed by predominantly Earth-based Solar System information. The incompatibility of observable \ceSO2 to below our threshold of significant liquid water by almost two orders of magnitude and the incompatibility of observable \ceH2SO4-H2O aerosols to very near our threshold of significant liquid water for these limiting conditions place the extremely strong constraints on surface liquid water for our best-guess model parameters into a larger context. Further, many of these model parameters will be more constrainable for individual systems with observations. Stellar spectrum observations would help constrain the timescale of \ceSO2 to \ceH2SO4 conversion $\tau_{\ce{SO2}\rightarrow\ce{H2SO4}}$, and observable stellar properties like rotation rate would help constrain stellar age, which would help place limits on the evolution of outgassing flux $\dot{N_{\ce{S}}}$.

Though in this paper we have emphasized the benefits of using a simple model for our immediate hypothesis test, in the future a hierarchy of models of varying complexities will be useful. Future work employing more complicated photochemistry, microphysics, aqueous chemistry, and interior dynamics models will allow for more detailed evaluations of the constrainability of surface liquid water from oxidized atmospheric sulfur observations in specific planetary scenarios. Such work would be particularly useful if it were combined with improved experimental constraints on key reaction rates in the atmosphere and ocean.

5.2 Oxidation State of the Atmosphere

Currently, the proposed atmospheric sulfur anti-ocean signatures are limited in scope to oxidized atmospheres. Theoretical predictions of the redox evolution of planets orbiting M dwarfs (our most observable targets) suggest such planets are more likely than those orbiting Sun-like stars to evolve toward more oxidized atmospheric conditions, given M-stars’ extended high EUV and UV flux period for their first gigayear of main sequence life (Shkolnik and Barman,, 2014) and the consequent higher potential for \ceH2O dissociation, \ceH2 escape, and \ceO2 buildup (Luger and Barnes,, 2015; Tian and Ida,, 2015; Wordsworth et al.,, 2018). Depending on planetary age and thermal evolution, this \ceO2 may be simply maintained in the atmosphere, or it may be absorbed into the mantle during a magma ocean phase and influence the oxidation state of the secondary outgassed atmosphere, but either way the planet is driven toward a more oxidized111We specify that we are referring here to bulk changes in atmospheric redox state as opposed to purely photochemical production of \ceO2 (e.g., Tian et al.,, 2014; Domagal-Goldman et al.,, 2014), which has been demonstrated to be unlikely given physically motivated planetary conditions (Harman et al., 2018a, ). atmosphere (Wordsworth et al.,, 2018). (Planets with Sun-like host stars can also undergo this process, although in this case its effectiveness is likely more dependent on the composition of the planet’s atmosphere (Catling et al.,, 2001; Wordsworth and Pierrehumbert,, 2013, 2014).) We can reasonably expect to encounter the oxidized planets that these methods apply to. Regardless, the study of the compatibility of buildup of sulfur gas and sulfur aerosol formation with liquid water could be extended to reduced atmospheres (featuring \ceH2S gas and \ceS8 aerosols) after better experimental characterization of the photochemical reactions that produce \ceS8 and the removal rates of associated dissolved products of \ceH2S.

Observationally determining the general oxidation state of a planet’s atmosphere will be possible from transit spectra in many cases. \ceH2-dominated atmospheres, which are strongly reducing, are identifiable from their extremely high scale heights due to low average molecular mass. \ceCO2 and \ceCH4 both have strong spectral features and have the potential to be identified even when not dominant atmospheric gases (Lincowski et al.,, 2018). While these carbon-bearing species can coexist to some extent, in terrestrial atmospheres, \ceCO2 is the dominant carbon molecule in oxidized atmospheres and \ceCH4 or \ceCO in reduced atmospheres (Hu et al.,, 2012). Measuring \ceCO2-to-\ceCH4 (or \ceCO) ratios can thus help to constrain an atmosphere’s redox state.

5.3 Identifying Haze as \ceH2SO4-H2O

Identifying haze particles as \ceH2SO4-H2O aerosols is challenging but tractable. While we leave explicit modeling of haze composition retrieval with stellar noise and specific instrument systematics to future work, we discuss the basic logistics of \ceH2SO4-H2O identification here.

\ce

H2SO4-\ceH2O aerosols are only weakly absorbing at visible wavelengths, so Mie scattering dominates their spectral effects. They therefore tend to lack distinctively characteristic spectral fingerprints (Hu et al.,, 2013). Our simulated transit spectra tests of aerosol cutoff height sensitivity suggest that if the aerosols were to extend throughout much of the otherwise optically thin upper atmosphere ($\gtrsim$ 5 scale heights), \ceH2SO4-H2O features could become identifiable. Such high extents would seem to be disfavored from \ceH2SO4 photochemical production and \ceSO2 vertical transport considerations, but more detailed photochemical modeling and simulated retrievals are required to determine whether this direct identification possibility is actually feasible. Regardless, \ceH2SO4-\ceH2O aerosols still have distinct radiative and formation properties when compared to other key spectra-flattening candidates: water clouds, organic photochemical hazes, and elemental sulfur hazes.

We tested the ability to distinguish both low-lying water clouds and higher ice clouds (with Earth-like characteristics) from an \ceH2SO4-H2O haze layer, but even at our assumed 100% cloud coverage, the \ceH2SO4-H2O haze is clearly distinguishable from \ceH2O clouds; the transit spectrum is probing higher optical depths than expected lower cloud peak heights, and even the higher ice clouds are too low to significantly flatten the transit spectrum. The latter two alternative spectra-flattening candidates—\ceS8 and photochemical haze—require reducing atmospheres to form. Identification of an atmosphere as oxidizing, as described in the preceding section, strongly disfavors these species’ formation. Distinguishing \ceH2SO4-\ceH2O aerosols from \ceS8 aerosols that form in reduced atmospheres could also be possible from a distinct spectral feature: \ceS8 aerosols’ radiative properties transition sharply between 0.3 $\mu$m and 0.5 $\mu$m from dominantly attenuating light via absorption to scattering (Hu et al.,, 2013); \ceH2SO4-H2O aerosols do not exhibit this feature.

Organic photochemical hazes are an active research topic (e.g., Hörst et al.,, 2018), but many studies suggest that they require \ceCH4/\ceCO2 ratios $\gtrsim$ 0.1 (DeWitt et al.,, 2009). Spectra indicating significant \ceCO2 and the absence of substantial mixing ratios of \ceCH4 would therefore further disfavor organic photochemical haze as the spectra-flattening agent. Additionally, organic photochemical hazes form as fractal aggregates (Bar-Nun et al.,, 1988) while \ceH2SO4-\ceH2O aerosols are spherical (Knollenberg and Hunten,, 1980). These distinct shapes have different radiative properties (Wolf and Toon,, 2010), which also have the potential to be recovered via inverse modeling (Ackerman and Marley,, 2001).

5.4 Life’s Impact on the Sulfur Cycle

We have neglected the influence of life on the sulfur cycle throughout this paper. Of course, constraining liquid water abundance is in the pursuit of finding life beyond Earth, so we would be remiss not to address life’s influence on the sulfur cycle as we have presented it here. The most important direct effect would be sulfur-consuming life’s capacity to exchange redox states of aqueous sulfur not expected from basic redox reactions (Johnston,, 2011; Kharecha et al.,, 2005). The thermodynamic instability of aqueous S(IV) species makes them an attractive microbial food source, so even if S(IV) is the waste product of one organism’s metabolism, it is likely to be rapidly consumed again and converted back to another redox state. The largest concern for our picture is the potential for metabolic chains to produce dissolved \ceH2S species, which will be in equilibrium with atmospheric \ceH2S via Henry’s law, that could be oxidized to \ceSO2 in the atmosphere and thus yield a new, unaccounted source of \ceSO2. However, this process would serve as an effective recycling term of sulfur that would be a fraction of outgassing. As a less than order of magnitude effect, this recycling is not likely to impact the conclusions of our study, and thus we do not expect these proposed liquid water constraining methods to be invalidated by the presence of life. We also neglect the possibility of intelligent life modifying its environment via sulfur products (e.g., Caldeira et al.,, 2013); we leave coupled ecosystem-sulfur cycle studies as an intriguing topic for future investigation.

5.5 Extensions to Observational Techniques beyond Transmission Spectroscopy

Finally, we have focused our efforts in this paper on observations of the sulfur cycle via transmission spectroscopy; however, our analysis could easily be extended to other observational techniques, notably reflected light spectroscopy given our interest in temperate planets. The impact of different observational techniques is in estimating the critical atmospheric sulfur mass that becomes observable. The technique of interest supplies a critical mass path $u^{\ast}$ or a critical aerosol optical depth $\delta^{\ast}$ at which \ceSO2 or \ceH2SO4-H2O aerosols, respectively, become observable. Once this calculation is complete, evaluating the feasibility of atmospheric sulfur buildup for a given amount of surface liquid water will be straightforward using our open-source sulfur model.

6 Conclusion

The presence of liquid water on an oxidized planet strongly influences its sulfur cycle—particularly the planet’s ability to sustain an optically thick \ceH2SO4-H2O haze layer or a high trace mixing ratio of \ceSO2 gas. Detectable levels of both \ceH2SO4-H2O aerosols and \ceSO2 gas require \ceSO2 in the upper atmosphere, but the presence of an ocean restricts the availability of \ceSO2 in the atmosphere. For expected ocean pHs, exponentially more \ceSO2 is stored in the ocean than in the atmosphere because of basic chemical properties of aqueous \ceSO2. Within the ocean, the dissolved products of \ceSO2 are thermodynamically unstable and thus short-lived. Recent outgassing must supply both the \ceSO2 present in the atmosphere necessary for observation and the accompanying amount of aqueous sulfur implied by the size of the ocean.

Via a quantitative model of this wet, oxidized sulfur cycle, we have shown that neither observable \ceH2SO4-H2O haze layers nor observable levels of \ceSO2 are likely compatible with significant surface liquid water ($\gtrsim 10^{-3}$ Earth ocean masses). Despite the uncertainties involved in modeling exoplanet processes, this incompatibility seems to persist even in the most extreme physical conditions to promote \ceSO2 buildup and haze formation. Thus, we propose the observational detection of \ceH2SO4-H2O haze and \ceSO2 gas as two new constraints on surface liquid water.

The code for our sulfur cycle model is available at https://github.com/kaitlyn-loftus/alien-sulfur-cycles. This work was supported by NASA grants 80NSSC18K0829 and NNX16AR86G. KL thanks Matthew Brennan, Junjie Dong, David Johnston, and Itay Halevy for helpful discussions on various aspects of sulfur chemistry. The authors also thank James Kasting for a productive review.