Identifying Atmospheres on Rocky Exoplanets through Inferred High Albedo

The range of planetary temperatures we consider for this observational technique is limited to zero-albedo substellar temperatures between 410 and 1250 K by two theoretical calculations. We assume no heat redistribution, so the substellar temperature T_sub is related to the equilibrium temperature T_eq by the equation ${T}_{\mathrm{eq}}={T}_{\mathrm{sub}}{\left(\tfrac{1}{4}\right)}^{1/4}$. The range of substellar temperatures T_sub = 410–1250 K corresponds to equilibrium temperatures of 300–880 K.

The lower temperature limit is set by the runaway greenhouse limit at zero-age main-sequence luminosity. At temperatures lower than the greenhouse limit the planet's surface could include high-albedo salt flats or water-rich materials such as clays or granites, which would complicate the interpretation of a high-albedo detection. We base our estimate of the runaway greenhouse threshold on the calculations of Kopparapu et al. (2013). However, Yang et al. (2013) found that clouds on the substellar hemisphere could prevent a planet around an M dwarf from entering a runaway greenhouse state until stellar fluxes twice as large as those reported by Kopparapu et al. (2013). Therefore, we double the stellar fluxes of the Kopparapu et al. (2013) runaway greenhouse limit to conservatively account for clouds and other factors that may similarly delay the runaway greenhouse (Yang et al. 2013, 2014; Kodama et al. 2018). For all three planets we consider in this study, this method provides a lower substellar temperature limit of T_sub = 410 K. The conservative runaway greenhouse limit given by this calculation depends on the specific stellar and planetary parameters, but for M dwarfs it will be at stellar fluxes of ≈2300–2500 W m⁻².

The upper temperature limit is set by the rate at which rock can be partially devolatilized. At high enough temperatures, all components of the rock at the substellar point except corundum will vaporize, leaving behind a high-albedo calcium- and aluminum-rich surface made of materials such as Al₂O₃ (Kite et al. 2016). This high-albedo surface would again prevent distinction of a high-albedo atmosphere from a lower-albedo surface, so we limit our study to lower temperatures.

To derive the temperature at which rock devolatilization would impact the overall albedo, we use the MAGMA model of gas-melt chemical equilibrium to calculate the rate at which rock could be devolatilized, assuming a starting composition equal to that of the Earth's continental crust (Fegley & Cameron 1987; Schaefer & Fegley 2009; Kite et al. 2016). Continental crust devolatilizes faster than other possible starting rock compositions because it has a higher vapor pressure, so this choice of starting composition gives a conservative (i.e., low) estimate of the temperature at which devolatilization becomes significant. The MAGMA model outputs the pressure P of the rock vapor over the surface, which we convert to a flux F of rock from the dayside hemisphere to the nightside hemisphere using the equation

$$\begin{eqnarray}&&F=\displaystyle \frac{{c}_{s}P}{g},\end{eqnarray} \tag{ 1 }$$

where c_s is the sound speed and g is the gravitational acceleration. Here we assume that the wind speed of the rock vapor is equal to the sound speed. Models of tenuous vapor atmospheres, both for super-Earth exoplanets and for Jupiter's moon Io, indicate the presence of supersonic winds over a broad region of parameter space, with winds across the terminator typically 2–3 times the sound speed (Ingersoll et al. 1985; Castan & Menou 2011).

We then convert this flux of material over the terminator to a rate R of vaporization from the dayside hemisphere using the equation

$$\begin{eqnarray}&&R=\left(\displaystyle \frac{F}{\rho }\right)\left(\displaystyle \frac{2\pi {R}_{p}}{2\pi {R}_{p}^{2}}\right)=\displaystyle \frac{F}{{R}_{p}\rho },\end{eqnarray} \tag{ 2 }$$

where R_p is the planet radius and ρ is the density of the rock.

Figure 2 shows the rate of devolatilization as a function of temperature. We compare this rate to the rate of meteoritic gardening, which determines how quickly fresh, low-albedo material could be mixed from below the rock surface (Melosh 2011). We assume that impact gardening mixes regolith to a depth of order 1 m Gyr⁻¹ based on analogy to the solar system (Fassett et al. 2017; Warner et al. 2017), but our calculations are insensitive to the exact impact gardening rate to within an order of magnitude because the rock vapor pressure increases very rapidly with increasing substellar temperature (Figure 2). The rate of devolatilization will be slower than meteoritic gardening for substellar temperatures ≤1250 K. We find that this is approximately the temperature at the Roche lobe radius for an Earth-density planet orbiting a mid M dwarf, so all close-in planets around M dwarfs will be cool enough to avoid partial devolatilization.

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We investigate a variety of potential rock compositions with different albedo properties. We consider the eight compositions outlined in Hu et al. (2012): basaltic, clay, feldspathic, Fe-oxidized (50% nanophase hematite, 50% basalt), granitoid, ice-rich (50% ice and 50% basalt), metal-rich (FeS₂), and ultramafic. Hu et al. (2012) created reflectance spectra for three of these surfaces using laboratory measurements of rock powders, and for the other five surfaces used radiative-transfer modeling based on laboratory samples of component minerals. These model spectra assume relatively fine-grained rocks. Fine grains generally have higher albedos than coarse grains, so the albedos taken from Hu et al. (2012) represent conservative upper limits of the albedos of these eight compositions.

Figure 3 shows the albedos of these surfaces as a function of wavelength (Hu et al. 2012). The fundamental basis for the overall shapes of these spectra is that many rock-forming minerals have strong spectral slopes in the 0.4–5 μm range. We discuss the plausibility of these surfaces forming on terrestrial planets with T_sub = 410–1250 K in Section 4.1.

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For each rock composition, we determine the temperature of a hypothetical planet with a surface of that composition by setting its outgoing flux equal to the absorbed flux it receives from its star. The absorbed flux from the star as a function of wavelength, F_⋆(λ), is given by

$$\begin{eqnarray}&&{F}_{\star }(\lambda )=\left[{F}_{\lambda }(1-{\alpha }_{\lambda })\right]\left(\displaystyle \frac{{R}_{\star }^{2}}{{a}^{2}}\right),\end{eqnarray} \tag{ 3 }$$

where R_⋆ is the stellar radius, a is the distance from the planet to the star, ${\alpha }_{\lambda }$ is the planet's albedo as a function of wavelength, and F_λ is the spectral flux density in W m⁻³ from a PHOENIX model for the star (Husser et al. 2013). The flux emitted by the planet is approximated by

$$\begin{eqnarray}&&{F}_{p}(\lambda )=\pi {B}_{\lambda }({T}_{\mathrm{day}})(1-{\alpha }_{\lambda }),\end{eqnarray} \tag{ 4 }$$

where B_λ(T) is the flux emitted by a blackbody and T_day is the planet dayside brightness temperature, which is related to the substellar temperature T_sub through the equation ${T}_{\mathrm{day}}={\left(\tfrac{2}{3}\right)}^{1/4}{T}_{\mathrm{sub}}$. The factor of $\tfrac{2}{3}$ assumes zero heat redistribution and a bare rock surface (Hansen 2008).⁷ We also assume here that the planet is in 1:1 spin:orbit resonance, because we are considering hot worlds in close-in orbits around K and M dwarfs. Additionally, approximating the entire dayside as a single blackbody implicitly assumes that the dayside surface is completely covered by one rock type. We integrate these two equations over wavelength and iterate until they are equal to determine the planetary temperature.

For each of these planet surfaces, we sum the light reflected and emitted by the planet to get a planet spectrum as a function of wavelength. We use PandExo to simulate observations of these planets with JWST (Batalha et al. 2017). We simulate sets of five secondary eclipse observations using the Mid-Infrared Instrument's Low-Resolution Spectroscopy (MIRI LRS) slitless mode to observe between 5 and 12 μm. We integrate over this entire wavelength range to produce one broadband planetary flux measurement.

We calculate the planetary brightness temperature that would be inferred from these observations by inverting the equation

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{p}}{{F}_{\star }}={\left(\displaystyle \frac{{R}_{p}}{{R}_{\star }}\right)}^{2}{\int }_{\lambda =5\,\mu {\rm{m}}}^{12\,\mu {\rm{m}}}\left(\displaystyle \frac{{B}_{\lambda }({T}_{\mathrm{day},\mathrm{obs}})}{{B}_{\lambda }({T}_{\star })}\right)d\lambda ,\end{eqnarray} \tag{ 5 }$$

where $\tfrac{{F}_{p}}{{F}_{\star }}$ is the observed broadband planet-to-star flux ratio and ${T}_{\mathrm{day},\mathrm{obs}}$ is the planet dayside brightness temperature inferred from the observations. Note that this equation assumes that the planet's emissivity (_λ = 1 − α_λ) is unity. We make this assumption when interpreting our observed planet flux because the planet's emissivity cannot be known a priori. We approximate the star's flux as a blackbody for this calculation because the PHOENIX model spectra only extend to a wavelength of 5 μm. We then convert this temperature into an inferred planetary albedo using the equation

$$\begin{eqnarray}&&{T}_{\mathrm{day},\mathrm{obs}}={T}_{\star }\sqrt{\displaystyle \frac{{R}_{\star }}{a}}{\left[\displaystyle \frac{2}{3}(1-{\alpha }_{\mathrm{obs}})\right]}^{1/4},\end{eqnarray} \tag{ 6 }$$

where α_obs is the albedo inferred from the observations. This equation again assumes unit emissivity.

We calculate the inferred albedo for each of the eight planet surfaces for three planets: the canonical high signal-to-noise planet GJ 1132b (Berta-Thompson et al. 2015); TRAPPIST-1 b, which orbits a very small star and so is a relatively high-signal transiting planet with an equilibrium temperature near the lower end of our temperature range (Gillon et al. 2017); and the newly discovered TESS planet LHS 3844b, which is representative of the type of planets the TESS mission will continue to discover (Vanderspek et al. 2019). These three planets together span almost the entire temperature range from 410 to 1250 K. Table 1 lists the details of each planet we consider.

A key complication in relating an inferred albedo to the presence of an atmosphere is the difference between the inferred planet albedo and its actual Bond albedo. A realistic surface with a nonconstant albedo at the wavelengths where it emits radiation will have an albedo inferred from observations of planetary radiation that differs from its true Bond albedo (the actual percentage of starlight at shorter wavelengths that is reflected off the planet's surface). This is related to Kirchhoff's law of thermal radiation: a planet with an albedo that changes as a function of wavelength will emit relatively more or less light at certain wavelengths compared to a blackbody with a constant emissivity at all wavelengths.

Figure 4 demonstrates why inferred albedo differs from Bond albedo for a simple example where the albedo is a step function given by

$$\begin{eqnarray}{\alpha }_{\lambda }=\left\{\begin{array}{ll}0.5, & \lambda \lt 4\mu {\rm{m}}\\ 0.1, & \lambda \gt 4\mu {\rm{m}}\end{array}\right..\end{eqnarray} \tag{ 7 }$$

The lower panel of this figure shows the actual Bond albedo compared to the albedo inferred from observations with JWST/MIRI for a set of planets at different temperatures spanning the range we consider. For all of the planets, the step function means that the MIRI bandpass (5–12 μm) is at a lower albedo (and thus higher emissivity) than shorter wavelengths. Therefore, in order to satisfy energy balance, the planet must emit relatively more of its flux at the long wavelengths where the emissivity is higher than if it were emitting as a blackbody with a constant emissivity. This means the planet will appear to be at a higher temperature in the MIRI bandpass, and so the inferred albedo will be lower than the actual Bond albedo. For planets at higher temperatures, this effect is even stronger because more of the planet's emission is at low-emissivity short wavelengths. As a result, in order to satisfy energy balance, the relative amount emitted at longer wavelengths is even higher compared to a constant-emissivity blackbody.

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Figure 5 shows a comparison of the insolation flux-weighted albedos of the eight planetary surfaces at shorter wavelengths (0.1–3.5 μm, the range in which all three of the M dwarf stars we consider emit >90% of their flux) to their inferred albedos from broadband mid-infrared observations with JWST/MIRI. The error bars represent 1σ observational uncertainties for a set of five stacked secondary eclipses. In all cases the inferred albedo is lower than the actual Bond albedo because the spectra generally have higher albedos at shorter wavelengths and lower albedos at longer wavelengths.

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The surfaces can be grouped into two rough categories based on the shapes of their spectra (Figure 3). First, feldspathic, granitoid, ultramafic, and clay generally show a high albedo at short wavelengths, then an abrupt transition around 2–5 μm to low albedo at longer wavelengths (Figure 3). All of these surfaces also have a lower emissivity at 3–5 μm (the peak of the flux for a planet at the temperature of GJ 1132b if that planet emitted as a blackbody) than at 5–12 μm (the MIRI bandpass). This means they will emit a larger percentage of their flux in the MIRI bandpass, leading to a higher inferred temperature and a lower inferred albedo. The larger the increase from the 3–5 μm emissivity to the 5–12 μm emissivity, the lower the inferred albedo will be relative to the Bond albedo. Feldspathic surfaces have the largest difference in its emissivity in these two wavelength ranges, followed by clay, then granitoid, then ultramafic. Therefore, in this set of four surfaces, the feldspathic surface shows the largest deviation from the 1:1 line in Figure 5, and the ultramafic shows the smallest deviation.

The second category consists of ice-rich, basaltic, Fe-oxidized, and metal-rich surfaces. All of these surfaces can be approximated as a more gradual slope to lower albedos at longer wavelengths, without the sharp transition of the first four surfaces (Figure 3). Ice-rich is on the edge between the two categories, but its sharp drop-off is smaller and at shorter wavelengths. The metal-rich surface has an albedo that is close to constant, so its inferred albedo should be close to its Bond albedo and it should fall closest to the 1:1 line in Figure 5. The other three surfaces all have slightly higher emissivity at the MIRI wavelengths, so they should again all be slightly farther from the 1:1 line. Among those three surfaces, the Fe-oxidized surface has the smallest difference between its visible and MIRI emissivity, followed by ice-rich, then basaltic. So in this group of three surfaces, Fe-oxidized is closest to the 1:1 line and basaltic is farthest.

The stars GJ 1132 and LHS 3844 have nearly the same PHOENIX spectra because their effective temperatures only differ by 200 K. The Bond albedos for these two planets differ by <0.01. Therefore, the primary difference between GJ 1132b and LHS 3844b is that LHS 3844b is much warmer than GJ 1132b (zero-albedo, zero-redistribution T_day = 1024 K for LHS 3844b as opposed to 737 K for GJ 1132b), so that LHS 3844b emits its flux at slightly shorter wavelengths. The overall effect is that a smaller percentage of the planet's blackbody curve is in the high emissivity regions at longer wavelengths, so the warmer planet will emit relatively more at these wavelengths and the inferred albedos will appear even lower than for the case of GJ 1132b. For all of the surfaces except clay, the surface albedos at the peak of GJ 1132b's blackbody flux and that of LHS 3844b are within 0.04 of each other, so those surfaces all uniformly have lower inferred albedos. For the clay surface, the peak flux of LHS 3844b happens to be emitting in a region where the albedo is almost 0.2 lower than the surrounding parts of the spectrum. This means the peak flux for the clay surface is at a higher emissivity, so relatively less flux needs to be emitted at the MIRI wavelengths. Therefore the clay surface has a higher inferred albedo relative to the other surfaces compared to what it had for GJ 1132b.

The star TRAPPIST-1 is 700 K cooler than GJ 1132, but the difference in inferred albedos for TRAPPIST-1 b and GJ 1132b is again primarily due to the different planet temperatures (and not due to the difference in stellar effective temperature). TRAPPIST-1 b has a temperature of ≈470 K, which is cool enough that the peak of its flux is emitted in the MIRI bandpass. Therefore, the difference between the inferred albedo and the true Bond albedo simply depends on the difference between the emissivity at short wavelengths, where the starlight is absorbed, and at long wavelengths, where the planet emits its flux. A surface with a larger difference between its emissivity at wavelengths <3.5 μm and its emissivity at 5–12 μm will have a larger difference between its inferred and Bond albedos. Within the first category of surfaces, feldspathic has the largest difference between its short-wavelength and long-wavelength emissivities, followed by granitoid, ultramafic, and clay. Therefore, among these four surfaces, feldspathic has the largest difference between its inferred and Bond albedos. Similarly, the basaltic and metal-rich surfaces have the largest and smallest difference in emissivities among the second category of surfaces, so they have the largest and smallest difference between inferred and Bond albedos, respectively.

Our method of differentiating a bare rock surface from an atmosphere relies on the fact that bare surfaces have generally low albedos, so any high-albedo detection would have to come from an atmosphere. Our calculations indicate that observations of a bare rock surface would lead to inferring a lower albedo than that of the real surface. Additionally, despite the offset between Bond albedo and inferred albedo, low Bond albedo surfaces always lead to lower inferred albedos, and high Bond albedo surfaces always lead to higher inferred albedos. This means that the detection of a low secondary eclipse depth corresponding to a high inferred albedo above about 0.4 would unambiguously indicate an atmosphere. As discussed in Section 4.2, this conclusion is robust to considering processes that could darken or brighten the surface, which were not considered in Hu et al. (2012)

Given that an inferred high albedo can indicate the presence of an atmosphere, what can be said about that atmosphere? Although Rayleigh scattering alone can in principle cause an albedo >0.5, in practice a more likely cause of high albedo is clouds. For example, Venus's high albedo of 0.7 is due to clouds.

We calculated the albedos of clouds with a variety of properties to determine what types of atmospheres could be distinguished from bare rock surfaces on the basis of albedo alone. We constructed a grid of atmospheres with cloud column masses between 10⁻¹⁰ and 10¹ g cm⁻² and nonabsorbing cloud particles with radii between 10⁻¹ and 10^1.7 μm. From these parameters we calculated the optical depth (τ) using the equation

$$\begin{eqnarray}&&\tau =\displaystyle \frac{3{{Qm}}_{\mathrm{col}}}{4{\rho }_{p}{r}_{p}},\end{eqnarray} \tag{ 8 }$$

where Q is the scattering efficiency, m_col is the cloud column mass, and ρ_p and r_p are the particle density and radius, respectively (Pierrehumbert 2010). We then calculated the albedo (α) using the equation

$$\begin{eqnarray}&&\alpha ={\alpha }_{a}+\displaystyle \frac{(1-{\alpha }_{a}^{{\prime} })(1-{\alpha }_{a}){\alpha }_{g}}{1-{\alpha }_{a}^{{\prime} }{\alpha }_{g}},\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray}&&{\alpha }_{a}^{{\prime} }=\displaystyle \frac{(1-g)\tau }{1+(1-g)\tau },\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\alpha }_{a}=\displaystyle \frac{\tfrac{-1}{2}\beta +(1-\hat{g})\tau }{1+(1-\hat{g})\tau },\end{eqnarray} \tag{ 11 }$$

and $\beta =1-{e}^{-\tau }$ (Pierrehumbert 2010). In these equations, $\hat{g}$ is the asymmetry factor and α_g is the albedo of the rock surface. Mbarek & Kempton (2016) calculated equilibrium chemistry cloud compositions for secondary atmospheres at T = 410–1250 K, and several of these possible compositions (including K₂SO₄, KCl, and Na₂SO₄) have indices of refraction n_R ≈ 1.5 and n_I ≈ 0 (Querry 1987; Lide 2005). Therefore, we used values of Q and $\hat{g}$ for particles with indices of refraction n_R = 1.5 and n_I ≈ 0 (values range from 0.2 < Q < 4 and $0.2\lt \hat{g}\lt 0.8$).

Figure 6 shows a contour plot of potential albedos for GJ 1132b, assuming an ultramafic rock surface. The area above the black line is the region of parameter space where the total planet albedo is more than 2σ higher than the inferred albedo of the bare rock surface. A detection of an albedo higher than this value would suggest the presence of an atmosphere on this planet. For this surface, an atmosphere with a cloud column mass greater than 8 × 10⁻⁵–3 × 10⁻² g cm⁻² (τ > 0.8 − 5) would have a higher albedo. The red, magenta, and orange stars on Figure 6 indicate typical cloud parameters for Earth (Wallace & Hobbs 2006), Venus (Arney & Kane 2018), and Mars (Clancy et al. 2017), respectively. While clouds made of larger, Earth-sized particles would be harder to detect using our method, smaller-particle hazes such as those found in Venus's upper atmosphere would be detectable at lower cloud column masses. Previous work has found that it would take ≈10 or more transits to detect an atmosphere on an exoplanet with Venus-like hazes using transmission spectroscopy (Lustig-Yaeger et al. 2019). Our method provides a way to detect such hazy atmospheres more efficiently.

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Most of the surface types exhibit a similar behavior, with column masses greater than 4 × 10⁻⁵–5 × 10⁻² g cm⁻² (τ > 0.5–7) having high enough albedos to suggest an atmosphere. The minimum pressure required to support such high-albedo cloud layers is significantly smaller than the ≈1 bar required to transport heat. For example, Mars has regionally extensive high-albedo clouds made of both H₂O and CO₂ ice in a 6 mbar atmosphere (Smith 2008; Haberle et al. 2017). These clouds can be optically thick, especially near surface ice deposits.

Our calculation implicitly assumes that a large portion of the dayside is covered in clouds so that the disk-integrated dayside albedo is large. This is a reasonable assumption for the close-in terrestrial exoplanets we consider, because 3D global climate simulations of synchronously rotating planets exhibit upwelling and cloud cover over much of the dayside, with downwelling and clear skies confined to the nightside (Yang et al. 2013).

A high surface albedo can produce a low secondary eclipse depth, so using the secondary eclipse depth to screen for the presence of atmospheres will only work if there is a prior constraint on the distribution of possible surface albedos. We have two primary sources of information on the albedos of terrestrial planets: observations of rocky objects in the solar system and laboratory spectra of geologically plausible surfaces.

4.1.1. Surfaces Observed in the Solar System

Table 2 lists the albedos of several solar system bodies. The rocky bodies in the solar system generally have low albedos (Madden & Kaltenegger 2018). E-type asteroids⁸ are an interesting exception to the general trend of dark solar system rocks, with albedos >0.3. One of the brightest E-type asteroids, 44 Nysa, is listed in Table 2 for comparison (Takahashi et al. 2011). E-types are likely the source of enstatite chondrite meteorites. If this mapping between meteorite type and asteroid type is correct, then the cause of the high asteroidal albedo is that the rocks record very reducing conditions—so any iron gets reduced to Fe, and there is very little Fe²⁺ in the silicate (Fe and other transition metals being a big cause of the dark color of the most common silicate rocks). This matters because the reducing conditions are analogous to those expected for evaporated cores (which will be discussed further in Section 4.2.2) if enough hydrogen was originally present to overwhelm buffering by Fe-oxides. Although it is unclear what an evaporated core will look like because we have never imaged one, this redox similarity leads us to speculate that E-types may be the best solar system analog to the surface composition of evaporated cores.

Table 2.  Bond Albedos of Solar System Bodies (Takahashi et al. 2011; Madden & Kaltenegger 2018)

Body	Bond Albedo
Mercury	0.07
Moon	0.11
Mars	0.25
4 Vesta	0.18
1 Ceres	0.03
Io	0.6
44 Nysa	0.33

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The near-infrared reflectance spectrum of enstatite is shown in Figure 7. At the wavelengths where M dwarfs emit most of their light, enstatite has an albedo between 0.3 and 0.4 that is similar in shape and magnitude to that of ultramafic rock. Therefore, observations of an enstatite surface would likely lead to a similar inferred albedo as that of an ultramafic surface. Our method of calculating inferred albedo would still allow detection of a high-albedo atmosphere on a planet with an enstatite surface.

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Although sulfur species (mainly SO₂) are responsible for the high albedo of Io, SO₂ would not be condensed as a solid on the surface of the hot planets we consider here and sulfur would be liquid (for T > 427 K) and quite dark (Nelson et al. 1983).

The solar system contains only one world in a T > 410 K orbit—the ∼0.06 M_⊕ world Mercury. In principle exoplanet albedo measurements could be used to supplement solar system data, to build up an empirical prior on rocky planet surface albedo. However, so far direct measurements of rocky-exoplanet albedos are limited (Rouan et al. 2011; Sheets & Deming 2014, 2017; Jansen & Kipping 2018). Moreover, even to use the solar system data for exoplanets in hotter orbits we need to think about how the albedo would (or would not) be affected by processes at work on a hotter orbit. Therefore, our primary focus in this work is on laboratory spectra of hypothetical planet surface compositions, as well as processes that would make those compositions more or less likely.

4.1.2. Laboratory Spectra of Hypothetical Surfaces

There are several processes that could act to make the observed surfaces darker or brighter. We discuss these possibilities below.

4.2.1. Darkening Processes

4.2.2. Brightening Processes

Our method of using albedo to test for the presence of an atmosphere is complementary to that of Koll et al. (2019), who considered the possibility of detecting an atmosphere through measurements of reduced dayside thermal emission or heat redistribution. Figure 8 shows the relationship between these two methods of atmospheric detection. Colored contours on this plot indicate dayside effective temperatures for LHS 3844b for a variety of atmospheric pressures and surface albedos, while triangles indicate the albedos of surfaces we consider in this paper. Koll et al. (2019) present a method to detect an atmosphere that is thick enough to change the dayside temperature by greater than or equal to a certain amount (above/to the right of a given temperature contour in Figure 8), while our method allows detection of an atmosphere with an albedo higher than that of the most reflective plausible surface (above the dashed horizontal line in Figure 8).

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While Koll et al. (2019) find that an atmosphere thicker than about 1 bar will transport enough heat that its secondary eclipse depth will deviate from that of a bare rock, there are several ways to create a thinner atmosphere that has a high albedo. Any planet with plentiful surface condensables can make optically thick clouds at pressures well below those needed to shift heat between hemispheres. For example, Mars has a surface pressure of 6 mbar, but has CO₂ clouds, H₂O clouds, and dust storms, all of which can be optically thick at both visible and infrared wavelengths (Smith 2008; Clancy et al. 2017; Haberle et al. 2017). The top of the upper H₂SO₄ cloud deck on Venus is at a pressure level of ≈30 mbar (Arney & Kane 2018). Triton's clouds have an optical depth >0.1, and it is plausible that under a slightly different insolation Triton could make optically thick high-albedo clouds. Finally, sulfur hazes derived from volcanic sulfur can also be very reflective (Gao et al. 2017).