NEGLECTED CLOUDS IN T AND Y DWARF ATMOSPHERES

1.1.1. L Dwarfs

1.1.2. T Dwarfs

As a brown dwarf continues to cool, it undergoes a significant transformation in its observed spectrum when it reaches an effective temperature of approximately 1400 K. Objects cooler than this transitional effective temperature begin to show methane absorption features in their near-infrared spectra and, when these features appear, are classified as T dwarfs (Burgasser et al. 2002; Kirkpatrick 2005). Within a small range of effective temperature, the iron and silicate clouds become dramatically less important. Marley et al. (2010) show that this transition could potentially be explained by the breaking up of these cloud layers into patchy clouds, but the details of the transition are still very much unknown. However, the recent discovery of highly photometrically variable early T dwarfs suggests that cloud patchiness may indeed play a role (Radigan et al. 2012; Artigau et al. 2009). Regardless, as the clouds dissipate, the atmospheric windows in the near-infrared clear. Flux emerges from deeper, hotter atmospheric layers, and the brown dwarf becomes much bluer in J − K color (see Figure 1).

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1.1.3. History of Modeling T Dwarfs

1.1.4. Y Dwarfs

Silicate, iron, and corundum, which are the condensates that dominate the cloud opacity in our L dwarf models, are not the only condensates that thermochemical models predict will form in substellar atmospheres as they cool. Other condensates will form at lower temperatures and add to the cloud opacity via the same physical processes that formed the iron and silicate cloud layers. In cool substellar atmospheres, Na₂S (sodium sulfide) has been predicted to play a potentially significant role (Lodders 1999; Lodders & Fegley 2006; Visscher et al. 2006). Other species expected to condense at these lower temperatures (roughly 600–1400 K) include Cr, MnS, ZnS, and KCl.

To our knowledge, none of these five condensates have been included in a brown dwarf atmosphere model before now. Marley (2000) estimated column optical depths for several of these species and recognized that Na₂S could be important at low T_eff but did not include this species in subsequent modeling because of lack of adequate optical constant data. Burrows et al. (2001, 2002) noted that Na₂S and KCl will condense in cool T dwarfs, but also noted that the indices of refraction are difficult to find. Helling and collaborators (Helling & Woitke 2006) also recognized that some of these species will form condensates in some cases but also did not compute model atmospheres that included this opacity source. Fortney (2005) noted that some of these species might be detectable in extrasolar planet transit spectra that probe a longer path length through the atmosphere.

To model cloudy T dwarf atmospheres, we modify the Ackerman & Marley (2001) cloud model. This model has successfully been used to model the effects of the iron, silicate, and corundum clouds on the spectra of L dwarfs (Saumon & Marley 2008; Stephens et al. 2009). Here, we do not include the opacity of iron, silicate, and corundum clouds; based on observed trends, we assume that the opacity of these clouds becomes negligible for the early T dwarfs. We instead include Cr, MnS, Na₂S, ZnS, and KCl.

The Ackerman & Marley (2001) approach avoids treating the highly uncertain microphysical processes that create clouds in brown dwarf and planetary atmospheres. Instead, it aims to balance the advection and diffusion of each species' vapor and condensate at each layer of the atmosphere. It balances the upward transport of vapor and condensate by turbulent mixing in the atmosphere with the downward transport of condensate by sedimentation. This balance is achieved using the equation

$$\begin{equation} -K_{zz} \frac{\partial q_t}{ \partial z} - f_{\rm{sed}} w_*q_c =0, \end{equation} \tag{ 1 }$$

where K_zz is the vertical eddy diffusion coefficient, q_t is the mixing ratio of condensate and vapor, q_c is the mixing ratio of condensate, w_* is the convective velocity scale, and f_sed is a parameter that describes the efficiency of sedimentation in the atmosphere.

This calculation provides the total amount of condensate at each layer of the atmosphere. The distribution of particle sizes at each level of the atmosphere is represented by a log-normal distribution in which the modal particle size is calculated using the sedimentation flux. A high sedimentation efficiency parameter f_sed results in vertically thinner clouds with larger particle sizes, whereas a lower f_sed results in more vertically extended clouds with smaller particles sizes. As a result, a higher f_sed corresponds to optically thinner clouds and a lower f_sed corresponds to optically thicker clouds.

The Ackerman & Marley (2001) cloud model code computes the available quantity of condensible gas above the cloud base by comparing the local gas abundance (accounting for upward transport by mixing via K_zz) to the local condensate vapor pressure p_vap. In cases where the formation of condensates does not proceed by homogeneous condensation, we nevertheless compute an equivalent vapor pressure curve as described in Section 2.4.2.

The cloud code is coupled to a one-dimensional atmosphere model that calculates the pressure–temperature profile of an atmosphere in radiative–convective equilibrium. The atmosphere models are described in McKay et al. (1989), Marley et al. (1996), Burrows et al. (1997), Marley & McKay (1999), Marley et al. (2002), Saumon & Marley (2008), and Fortney et al. (2008). This methodology has been successfully applied to modeling brown dwarfs with both cloudy and clear atmospheres (Marley et al. 1996, 2002; Burrows et al. 1997; Saumon et al. 2006, 2007; Leggett et al. 2007a, 2007b; Mainzer et al. 2007; Blake et al. 2007; Cushing et al. 2008; Geballe et al. 2009; Stephens et al. 2009).

In the atmosphere model, the thermal radiative transfer is determined using the "source function technique" presented in Toon et al. (1989). Within this method, it is possible to include Mie scattering of particles as an opacity source in each layer. Our opacity database for gases, described extensively in Freedman et al. (2008), includes all the important absorbers in the atmosphere. This opacity database includes two significant updates since Freedman et al. (2008), which are described in Saumon et al. (2012): a new molecular line list for ammonia (Yurchenko et al. 2011) and an improved treatment of the pressure-induced opacity of H₂ collisions (Richard et al. 2012).

Both the cloud model and the chemical equilibrium calculations (see Section 2.4) are coupled with the radiative transfer calculations and the pressure–temperature profile of the atmosphere; this means that a converged model will have a temperature structure that is self-consistent with the clouds and chemistry.

We calculate the effect of the model cloud distribution on the flux using Mie scattering theory to describe the cloud opacity. Assuming that particles are spherical and homogeneous, we calculate the scattering and absorption coefficients of each species for each of the particle sizes within the model. In order to make these scattering calculations, we need to understand the optical properties (the real and imaginary parts of the index of refraction) of each material.

The optical properties were found from a variety of diverse sources, summarized in Table 1. To calculate Mie scattering within the model atmosphere, we use a grid of optical properties at wavelengths from 0.268 to 227 μm. Where data were not available, we extrapolated the available data, following trends for similar known molecules.

The molecules with the most complete published optical properties are ZnS and KCl, both of which are obtained from Querry (1987), who tabulates the optical constants for 24 different minerals.

Optical properties for Cr are published in Stashchuk et al. (1984) from 0.26 to 15 μm. The optical properties from 15 μm to 227 μm were linearly extrapolated from these experimental data following the trend of the optical properties of Fe. Various extrapolations were tested; the choice of optical properties beyond 15 μm does not change the results of the calculations in any meaningful way.

Optical properties for MnS are published in Huffman & Wild (1967), from 0.09 to 13 μm. Optical properties from 15 μm to 227 μm are extrapolated, following the trends of the other two studied sulfide condensates ZnS and Na₂S.

The optical properties for Na₂S, the clouds with the largest optical depth, are combined from two different sources. Montaner et al. (1979) provide experimental data in the infrared, from 25 to 198 μm. Khachai et al. (2009) provide first principles calculations of the optical properties from 0.03 to 91 μm. In the region of overlap, the Montaner et al. (1979) laboratory values are used. The real and imaginary parts of the index of refraction are plotted in Figure 2.

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2.4.1. Gas-phase Chemistry

2.4.2. Cloud Condensation Chemistry

A simplified equilibrium condensation approach is used to calculate saturation vapor pressures and condensation curves (see Figure 3) for Cr, MnS, Na₂S, ZnS, and KCl as a function of pressure, temperature, and metallicity, based upon the more comprehensive thermochemical models of Lodders & Fegley (see Section 2.4.1) and Visscher et al. (2006, 2010). In each case, we consider condensation from the most abundant Cr-, Mn-, Na-, Zn-, and K-bearing gas phases at the cloud base as predicted by the chemical models. The relative mass of each cloud (relative to Na₂S) is listed in Table 3, assuming complete removal of available condensate material from the gas phase.

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Table 3. Abundances of Condensate-forming Species

Condensate	p*x Below Cloud Basea	Cloud Massb
Cr	p*Cr ≈ 8.87 × 10−7ptm	0.30
MnS	p*Mn ≈ 6.32 × 10−7ptm	0.36
Na2S	p*Na ≈ 3.97 × 10−6ptm	1.00
ZnS	p*Zn ≈ 8.45 × 10−8ptm	0.05
KCl	p*KCl ≈ 2.55 × 10−7ptm	0.12
Fe	p*Fe ≈ 5.78 × 10−5ptm	20.85

Notes. ^aWhere p*_x is the partial pressure (in bars) of each gas-phase species x below the predicted cloud base using solar-composition abundances from Lodders (2003), p_t is the total atmospheric pressure, and the metallicity factor m is defined by log m = [Fe/H]. ^bTotal condensate mass relative to the Na₂S cloud. Values for Fe shown for comparison.

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Chromium metal is the most refractory of the clouds considered here and condenses from monatomic Cr gas via the reaction

$$\begin{equation} {\rm Cr} = {\rm Cr(s)}, \end{equation} \tag{ 2 }$$

where "(s)" indicates a solid phase. The condensation condition for Cr-metal is defined by

$$\begin{equation} p_{\rm{Cr}}^* \ge p_{\rm{Cr}}^{\prime }, \end{equation} \tag{ 3 }$$

where p'_Cr is the saturation vapor pressure of Cr gas in equilibrium with Cr-metal and p*_Cr is the partial pressure of Cr below the cloud for a solar-composition gas (p*_Cr = q_Cr*p_t, where q*_Cr is the mole fraction abundance of Cr and p_t is the total atmospheric pressure). Upon condensation, the thermodynamic activity of Cr-metal is unity and the equilibrium constant (K_p) expression for reaction (3) can be written as

$$\begin{equation} p_{\rm{Cr}}^{\prime }= K_{p}^{-1}. \end{equation} \tag{ 4 }$$

Substituting for the temperature-dependent value of K_p, the saturation vapor pressure of Cr above the cloud base can be estimated using the expression

$$\begin{equation} \log p_{\rm{Cr}}^{\prime } \approx 7.490 - 20592/T, \end{equation} \tag{ 5 }$$

for T in Kelvin and p in bars. Below the cloud, we assume that Cr gas is approximately representative of the elemental Cr abundance in solar-composition gas (see Table 3):

$$\begin{equation} \log p_{\rm{Cr}}^* \approx -6.052 + \log p_t + [{\rm Fe/H}]. \end{equation} \tag{ 6 }$$

The condensation temperature as a function of the total atmospheric pressure (p_t) and metallicity can therefore be approximated by setting p*_Cr = p_Cr' and rearranging to give

$$\begin{equation} 10^{4}/T_{\rm{cond}}{\rm (Cr)} \approx 6.576 - 0.486 \log p_t - 0.486 [{\rm Fe/H}]. \end{equation} \tag{ 7 }$$

This expression yields a condensation temperature near ∼1520 K at 1 bar and solar metallicity (cf. Lodders & Fegley 2006), and shows that greater total pressures and/or metallicities will lead to higher condensation temperatures. Condensation of Cr-metal effectively removes gas-phase chromium from the atmosphere, and the abundances of Cr-bearing gases rapidly decrease with altitude above the cloud.

Our modeling of sulfide condensation chemistry follows that of Visscher et al. (2006), and the condensation reactions and temperature-dependent expressions presented here are taken from that study. The deepest sulfide cloud expected in brown dwarf atmospheres is MnS, which forms via the reaction

$$\begin{equation} {\rm H}_{2}{\rm S} + {\rm Mn} = {\rm MnS(s)} + {\rm H}_{2}. \end{equation} \tag{ 8 }$$

The formation of the MnS cloud is limited by the total manganese abundance, which is 2% of the sulfur abundance in a solar-composition gas. The condensation curve for MnS is thus derived by exploring the chemistry of monatomic Mn, which is the dominant Mn-bearing gas near the cloud base. Using results from Visscher et al. (2006), the saturation vapor pressure of Mn above the cloud is given by

$$\begin{equation} \log p_{\rm{Mn}}^{\prime } \approx 11.532 - 23810/T - [{\rm Fe/H}], \end{equation} \tag{ 9 }$$

where the metallicity dependence comes from H₂S (the dominant S-bearing gas) remaining in the gas phase above the MnS cloud base. By setting p*_Mn = p_Mn', the MnS condensation curve is approximated by Visscher et al. (2006):

$$\begin{equation} 10^{4}/T_{\rm{cond}}{\rm (MnS)} \approx 7.447 - 0.42 \log p_t - 0.84[{\rm Fe/H}], \end{equation} \tag{ 10 }$$

giving a condensation temperature near ∼1340 K at 1 bar in a solar-metallicity gas.

The Na₂S cloud is the most massive of the metal sulfide clouds expected to form in brown dwarf atmospheres because Na is more abundant than either Mn or Zn in a solar-composition gas (see Table 3). Sodium sulfide condenses via the net thermochemical reaction

$$\begin{equation} {\rm H}_{2}{\rm S} + 2{\rm Na} = {\rm Na}_{2}{\rm S(s)} + {\rm H}_{2}. \end{equation} \tag{ 11 }$$

The mass of the Na₂S cloud is limited by the elemental abundance of sodium, which is 13% of the abundance of sulfur in a solar-composition gas. Using results from Visscher et al. (2006), the saturation vapor pressure of Na above the cloud base is given by

$$\begin{equation} \log p_{\rm{Na}}^{\prime } \approx 8.550 - 13889/T - 0.50 [{\rm Fe/H}], \end{equation} \tag{ 12 }$$

where the metallicity dependence results from H₂S remaining in the gas phase above the Na₂S cloud and from the stoichiometry of Na and H₂S in the condensation reaction. The condensation temperature (where p*_Na = p_Na') is given by Visscher et al. (2006):

$$\begin{equation} 10^{4}/T_{\rm{cond}}({\rm Na}_{2}{\rm S}) \approx 10.045 - 0.72 \log p_t - 1.08 [{\rm Fe/H}], \end{equation} \tag{ 13 }$$

indicating condensation near ∼1000 K at 1 bar in a solar-metallicity gas.

The ZnS cloud layer forms via the reaction

$$\begin{equation} {\rm H}_{2}{\rm S} + {\rm Zn} = {\rm ZnS(s)} + {\rm H}_{2}, \end{equation} \tag{ 14 }$$

The formation of the ZnS cloud is limited by the total Zn abundance, which is 0.3% of the S abundance in a solar-composition gas. Using results from Visscher et al. (2006), the saturation vapor pressure of Zn over condensed ZnS is given by

$$\begin{equation} \log p_{\rm{Zn}}^{\prime } \approx 12.812 - 15873/T - [{\rm Fe/H}]. \end{equation} \tag{ 15 }$$

The condensation curve (where p*_Zn = p_Zn') is approximated by Visscher et al. (2006):

$$\begin{equation} 10^{4}/T_{\rm{cond}}{\rm (ZnS)} \approx 12.527 - 0.63 \log p_t - 1.26 [{\rm Fe/H}], \end{equation} \tag{ 16 }$$

giving a condensation temperature of ∼800 K at 1 bar in a solar-metallicity gas.

Our treatment of KCl condensation chemistry is similar to that for Cr-metal and the metal sulfides. With decreasing temperatures, KCl replaces neutral K as the dominant K-bearing gas in brown dwarf atmospheres (Lodders 1999; Lodders & Fegley 2006). The KCl cloud layer is thus expected to form via the net thermochemical reaction

$$\begin{equation} {\rm KCl} = {\rm KCl(s)}, \end{equation} \tag{ 17 }$$

and condenses as a solid over the range of conditions considered here. The vapor pressure of KCl above condensed KCl(s) is given by

$$\begin{equation} \log p_{\rm{KCl}}^{\prime } \approx 7.611 - 11382/T, \end{equation} \tag{ 18 }$$

derived from the equilibrium constant expression for the condensation reaction. The mass of the KCl cloud is limited by the total potassium abundance, which is 70% of the chlorine abundance in a solar-composition gas (Lodders 2003). Note that other K-bearing species may remain relatively abundant near cloud condensation temperatures, particularly at higher pressures (e.g., see Fegley & Lodders 1994 and Lodders 1999 for a more detailed discussion of chemical speciation). However, KCl is the dominant K-bearing gas near the cloud base for the relevant $P{\hbox {--}} T$ conditions expected in cool brown dwarf atmospheres (see Figure 3) over the range of metallicities (−0.5 to +0.5 dex) considered here. For simplicity, we therefore assume that KCl is approximately representative of the elemental K abundance below the cloud, given by

$$\begin{equation} \log p_{\rm{KCl}}^* \approx -6.593 + \log p_t + [{\rm Fe/H}]. \end{equation} \tag{ 19 }$$

The condensation temperature as a function of pressure and metallicity is estimated by setting p*_KCl = p_KCl' and rearranging to give

$$\begin{equation} 10^{4}/T_{\rm{cond}}{\rm (KCl)} \approx 12.479 - 0.879 \log p_t - 0.879 [{\rm Fe/H}], \end{equation} \tag{ 20 }$$

yielding a condensation temperature near ∼800 K at 1 bar in a solar-metallicity gas (cf. Lodders 1999; Lodders & Fegley 2006). In general, the condensation curve expressions demonstrate that condensation temperatures increase with total pressure, as illustrated in Figure 4. Furthermore, higher metallicities are expected to result in higher condensation temperatures and more massive cloud layers in brown dwarf atmospheres. In each case, the saturation vapor pressures of cloud-forming species rapidly decrease with altitude above the cloud layers.

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The Ackerman & Marley (2001) model is one method of several that have been applied to cloudy brown dwarf atmospheres. Helling et al. (2008) review various cloud modeling techniques and compare model predictions for various cases. The most important conceptual differences between these approaches lie in the assumptions of how condensed phases interact with the gas.

In the chemical equilibrium approach (e.g., Allard et al. 2001), condensed phases remain in contact with the gas phase and can continue to react with the gas even at temperatures well below the condensation temperature. As an example, when following this approach, Fe grains that first condense at temperatures of over 2000 K react with atmospheric H₂S to form FeS below 1000 K. In the condensation chemistry approach we employ here, the condensed phases are assumed to sediment out of the atmosphere and are not available to interact with gas phases at temperatures below the condensation temperature. Thus, Fe grains form a discrete cloud layer and do not react to form FeS. H₂S consequently remains in the gas phase and reacts to form condensates as outlined in Section 2.4.2. Jupiter is an excellent example of the applicability of this framework, as the presence of H₂S in the observable atmosphere is only possible because Fe is sequestered in a deep cloud layer, which prevents the formation of FeS that otherwise depletes other gas-phase S species (Fegley & Lodders 1994). The presence of alkali absorption in T dwarfs likewise demonstrates the applicability of condensation chemistry (Marley et al. 2002). A detailed comparison of true equilibrium condensation and cloud condensate removal from equilibrium can be found in Fegley & Lodders (1994), Lodders & Fegley (2006), and references therein.

A different approach is taken by Helling & Woitke (2006), who follow the trajectory of tiny seed particles of TiO₂ that are assumed to be emplaced high in the atmosphere and sink downward. As the seeds fall through the atmosphere they collect condensate material. In Helling & Woitke (2006) and numerous follow-up papers (Helling et al. 2008; Witte et al. 2009, 2011; de Kok et al. 2011), this group models the microphysics of grain growth given these conditions. Because the background atmosphere is not depleted of gaseous species until the grains fall through the atmosphere, a compositionally very different set of grains are formed. In particular, they predict "dirty" grains composed of layers of varying condensates.

A direct comparison between the predictions of the various cloud modeling schools is often difficult because of differing assumptions of elemental abundances and the background thermal profile. Modeling tests in which predictions of the various groups are compared to data would be illuminating, but this is far beyond the scope of the work reported here.

In order to calculate absolute magnitudes of the modeled brown dwarfs, we use the results of evolution models that determine the radius of a brown dwarf as it cools and contracts over its lifetime. We use the evolution models of Saumon & Marley (2008) with the surface boundary condition from cloudless atmospheres. Using a cloudless boundary condition instead of one consistent with these clouds changes the calculated magnitudes of the models very slightly, but does not change the overall trends or results.

To analyze the effect of these clouds, we generate a grid of 182 model atmospheres at effective temperatures and surface gravities spanning the full range of T dwarfs. We calculate pressure–temperature profiles and synthetic spectra for atmospheres from 400 to 1300 K (50–100 K increments), with log(g) (cgs) of 4.0, 4.5, 5.0, and 5.5, and cloud sedimentation efficiency parameter f_sed = 2, 3, 4, and 5. For this study, we use only solar metallicity composition. We then compare these, both photometrically and spectroscopically, to observed T dwarfs.

In Figure 3, we show the pressure–temperature profiles of models with effective temperatures of 400 K, 600 K, 900 K, and 1300 K. The surface gravity of the 400 K models is log g = 4.5; for the hotter models, log g = 5.0. We plot models with two different cloud sedimentation efficiencies that include only the Na₂S, MnS, ZnS, Cr, and KCl clouds. Because Na₂S and MnS are by far the most dominant cloud species (see Section 3.3), as a shorthand we refer to this collection of clouds as "sulfide clouds."

The condensation curves of major and minor species predicted to form by equilibrium chemistry are also plotted. The location of a given cloud base is expected to be where the pressure–temperature profile of the model atmosphere crosses the condensation curve. Each model crosses the condensation curve of each species at very different pressures and, to a lesser extent, temperatures, so we expect that the significance of the clouds will be strongly controlled by the effective temperature of the model.

The cloudy 400 K and 600 K models have two convection zones. All 900 and 1300 K models have a single deep convection zone.

For all models, it is clear that, as in previous cloudy models of L dwarfs, adding cloud opacity has a "blanketing" effect on the model, increasing the temperature of the atmosphere for a given atmospheric pressure. As the cloud becomes optically thicker, the entire pressure–temperature profile becomes hotter; thus, on a plot of pressure–temperature profiles such as Figure 3, increasing the cloud opacity moves the whole profile to the right.

In Figures 4 and 5, we show the spectra of the same example models at 1300, 900, 600, and 400 K: Figure 4 shows the wavelength-dependent brightness temperatures from these models, while Figure 5 shows the model fluxes computed from the top of the atmosphere. The brightness temperature gives some insight into the depth into the atmosphere probed at each wavelength. Flux from 0.8 to 1.3 μm comes from the deepest, hottest layers of the atmosphere. Clouds change the flux in this wavelength range by limiting the depth from which flux emerges; conversely, the clouds do not change the depth probed between ∼2 and 5 μm because the clouds form below the layers from which most of the flux is emerging. However, the hotter atmospheric temperatures at a given pressure (see Figure 3) lead to slightly higher fluxes at these wavelengths. Though not plotted here, flux in the mid-infrared also comes from above the cloud layer and is slightly higher because the entire pressure–temperature profile is hotter.

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For the hottest of these models (T_eff = 1300 K), the cloudy spectra look almost identical to the cloudless spectrum. This model is too hot to have much mass of these condensed species form in the photosphere. For a cooler model (T_eff = 900 K), the cloudy spectra look different from the cloudless spectrum. As we decrease the sedimentation efficiency f_sed in the model, the flux in Y and J bands decreases and the flux in K band increases.

For the coldest two models shown (T_eff = 400, 600 K), the cloudy spectra look dramatically different from the cloudless spectrum in the near-infrared; even the thinnest cloud considered here (f_sed = 5) causes the flux in Y and J to decrease by 50% and the flux in K to correspondingly increase. Decreasing the sedimentation efficiency enhances this effect.

Figure 6 shows the effect of the iron and silicate clouds on the spectra of models with the same effective temperatures (1300 K, 900 K, 600, and 400 K) and surface gravity. Note that unlike the sulfide clouds, these iron and silicate clouds substantially change the shape of the 1300 K and 900 K models by suppressing the flux in Y, J, and H and increasing the flux in K band and the mid-infrared. This strong effect at higher temperatures is due to the large amount of iron and silicate condensed in the visible atmosphere at those temperatures.

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Figure 7 shows the distribution of clouds in the model atmospheres for the three example cases. The locations of iron, silicate, and corundum clouds, using models that only include those clouds—the standard Saumon & Marley (2008) cloud configuration—are plotted for reference. The column optical depth is given by Equation 16 in Ackerman & Marley (2001), which calculates the cumulative geometric scattering optical depth by cloud particles through the atmosphere.

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For the 1300 K model, all of the sulfide clouds have tiny optical depths in the photosphere and do not affect the emergent spectra. The silicate and iron clouds would have significant optical depth (τ = 2–3) and would substantially change the emergent spectra.

For the 900 K model, all of the sulfide clouds have a column optical depth smaller than 1 in the photosphere. KCl and ZnS have tiny optical depth (τ < 2 × 10⁻²) and will not create an observable change in the spectrum. Na₂S and MnS have optical depth between 0.1 and 1 and will change the model spectra slightly.

For the 600 K model, Na₂S is the most important condensate opacity source. KCl has a small optical depth and ZnS has a negligible optical depth. This result is expected, based on the abundances of each species (see Table 3). The other two clouds, MnS and Cr, are below the near-infrared photosphere, so also do not change the spectrum. The silicate and iron clouds would also be below the photosphere.

Using our full grid of models, we can examine the importance of the Na₂S cloud as a function of T_eff and surface gravity. Figure 8 shows how the column optical depth of this cloud varies from 400 to 1300 K and log g from 4.0 to 5.0 for a constant value of the parameter f_sed. Moving to T_eff values below 1300 K, the Na₂S cloud grows in importance as it forms progressively deeper in the atmosphere, so that there is larger mass of condensate in the cloud. The maximum optical depth of the Na₂S cloud at pressure levels above the bottom of the 1–6 μm photosphere is largest for models with T_eff of 600 K. At lower T_eff, much of the cloud opacity is below the visible atmosphere. The optical depth within the photosphere is significant (τ ≳ 1) for models between 400 and 700 K.

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Figure 8 indicates that the cloud optical depth within the photosphere is largest for higher surface gravity atmospheres for a constant value of f_sed. However, this does not necessarily predict that higher gravity brown dwarfs will have thicker clouds than lower gravity brown dwarfs because the parameter f_sed is not necessarily independent of gravity.

In Figure 9, we plot the photometric colors in the near-infrared of all brown dwarfs with measured parallaxes and apparent J and K magnitude errors smaller than 0.2 mag (Dupuy & Liu 2012; Faherty et al. 2012). We also plot the calculated photometric colors of our suite of cloudless and cloudy models from 400 to 1300 K with several representative surface gravities. In Section 4.1, we discuss the general trends of the photometric colors of models at various effective temperatures and cloud sedimentation efficiencies. In Section 4.2, we compare our model results to the photometric observations.

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As discussed in Sections 3.2 and 3.3, clouds in T dwarf atmospheres tend to suppress the flux in Y and J bands and increase the flux in K band. The flux shift from J to K gives cloudy models larger (redder) J − K and J − H colors than cloudless models.

In Figure 9, the hottest cloudy models have nearly the same near-infrared colors as cloudless models. As we decrease the effective temperature of a cloudy model, more cloud material condenses; the model has a redder photometric color than a cloudless model with the same T_eff. If we reduce the sedimentation efficiency (f_sed) of the cloud, the cloud becomes optically thicker, and the model has a redder photometric color.

The upper panels, which show J − K photometric colors, show that our sulfide cloud models can easily reach the colors of red T dwarfs, with f_sed values of 4–5. The bulk of the T dwarf population is bluer than the f_sed = 5 model. However, the cooler T dwarfs are generally well matched by the models. In J − H, the color directly affected by cloud opacity limiting the depth to which one sees, the model colors are an excellent match to the data. The cloudy models are a much better match than the corresponding cloud-free models.

4.2.1. Expected Surface Gravity of T Dwarfs

4.2.2. Cloud Sedimentation Efficiency

4.2.3. WISE Color–Color Diagrams

Cushing et al. (2011) announced the discovery of six proposed Y dwarfs found by the Wide-field Infrared Survey Explorer (WISE) mission. Objects around the T-to-Y transition are cold enough to have NH₃ absorption features in the near-infrared.

We have obtained near-infrared photometry for two proposed Y0 dwarfs discovered using the WISE by Cushing et al. (2011). YJH was obtained for WISEP J140518.40+553421.5 and YJ for J154151.65−225025.2, using the near-infrared imager NIRI (Hodapp et al. 2003) on the Gemini-North telescope on Mauna Kea, Hawaii. The photometry is on the Mauna Kea Observatories (MKO) system (Tokunaga & Vacca 2005). The data were obtained via the Gemini queue program GN-2012A-Q-106 on 2012 February 10; the program aims to supplement and improve on the photometry presented in the discovery paper. Individual exposure times of 60 s were used at Y and J, and 30 s at H; a nine-position dither pattern with 10'' offsets was repeated as necessary for sufficient signal to noise. The total exposure time for WISEP J140518.40+553421.5 was 9 minutes at Y and J and 58.5 minutes at H; for WISEP J154151.65−225025.2 the total exposure time was 18 minutes at each of Y and J. Data were reduced in a standard fashion and flat-fielded with calibration lamps on the telescope. The UKIRT faint standards FS 133 and 136 were used for photometric calibration; J and H were taken from Leggett et al. (2006), and Y from the UKIRT online catalog (http://www.jach.hawaii.edu/UKIRT/astronomy/calib/phot_cal/fs_ZY_MKO_wfcam.dat).

The final reduced magnitudes are Y = 21.41 ± 0.10, J = 21.06 ± 0.06, and H = 21.41 ± 0.08 for WISEP J140518.40+553421.5; Y = 21.63 ± 0.13 and J = 21.12 ± 0.06 for WISEP J154151.65−225025.2.

Figure 10 shows how clouds will affect these cold objects. H − W2 is a useful temperature indicator for these objects, while J − H is sensitive to both the cloud structure and gravity. As the T_eff of the noncloudy models decreases from 800 to 500 K, the models become progressively bluer in J − H color. However, most of the proposed WISE Y dwarfs are redder than the cloudless model. The models that include the sulfide clouds match their colors better. This result tentatively suggests that for objects colder than T dwarfs, the sulfide clouds remain important. Of course, for objects with effective temperatures of ∼350 K, water will condense; at that point, H₂O clouds should contribute to the spectra (e.g., Burrows et al. 2003).

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We now compare model spectra to two relatively cold, red T dwarfs, Ross 458C and UGPS 0722–05. The near-infrared spectra of these two objects are not well matched by cloudless T dwarf spectra; by including our neglected clouds, which for these cool objects are dominated by the Na₂S cloud, we match their spectra more accurately.

We compare models to both near-infrared spectra and near- and mid-infrared photometry. As in previous studies of brown dwarfs (Cushing et al. 2008; Stephens et al. 2009), we find that in different bands, the observations are best fit by models of different parameters. In this study, we focus on finding models that fit the shape of the spectra in the near-infrared where clouds play a significant role.

4.3.1. Ross 458C

Ross 458C is a late-type T dwarf (T7–9) that is separated by over 1100 AU from a pair of M star primaries. It has anomalously red near-infrared colors (J − K =−0.21 ± 0.06). Burgasser et al. (2010) obtained spectroscopic observations with the FIRE spectrograph (Simcoe et al. 2008, 2010) on the Magellan Baade 6.5 m telescope at Las Campanas Observatory. They fit the spectrum using cloudless and cloudy models (which include only the opacity of the iron, silicate, and corundum clouds) and find that cloudy models fit significantly better than cloudless models. Burgasser et al. (2010) conclude that cloud opacity is necessary to reproduce the spectral data and invoke a reemergence of the iron and silicate clouds. We instead assume that the iron and silicate clouds are depleted, as we observe generally in other T dwarfs, and investigate the effect of the sulfide clouds.

In Figure 11, we show the FIRE spectrum and the best-fitting cloudy and cloudless models. We also show photometry in J, H, and K (Dupuy & Liu 2012), WISE photometry (Kirkpatrick et al. 2011), and Spitzer photometry (Burningham et al. 2011). The spectra used to generate these results differ somewhat from those in Burgasser et al. (2010) because we use models that include recent improvements to the opacity database (Saumon et al. 2012) for both ammonia and the pressure-induced opacity of H₂ collisions. All models are fit by eye to the observations.

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Like Burgasser et al. (2010), we find that clouds are essential to match the spectrum of Ross 458C. Figure 11 shows the best-fitting cloudless model and the two best-fitting cloudy models (one including the iron and silicate clouds and the other including the sulfide clouds). The cloudless model is a poor representation of the spectral data; the flux in Y and J is too high and the flux in H and K is too low. The cloudy models are better representations of the relative flux in each band.

Burgasser et al. (2010) found that the surface gravity of Ross 458C must be low (log g = 4.0) for models to match the observed spectrum. Likewise, we find that our best-fitting models have surface gravities of 4.0 (cloudless), 3.7 (silicate clouds), and 4.0 (sulfide clouds).

We conclude that we do not need to invoke a reemergence of iron and silicate clouds into the photosphere of Ross 458C to reproduce the observed spectrum. Instead, we are able to reproduce the spectrum using the sulfide clouds that are naturally expected to form in the photospheres of cool T dwarfs. Section 5.2 contains additional discussion on which cloud species we expect to be important.

The very red slope to the L-band spectrum of Ross 458C—much redder than all the models—is reminiscent of the behavior of some cloudy L dwarfs, including 2MASS2224 and DE 0255 (L3.5 and L9, respectively) and may be a signature of very small dust grains (Stephens et al. 2009).

The discrepancies at 4.5 μm are likely to be a result of non-equilibrium chemistry, which is not included in these models. This effect is discussed in more detail in Section 4.4.

4.3.2. UGPS J072227.51−054031.2

UGPS J072227.51−054031.2 (hereafter UGPS 0722–05) is a T9 or T10 dwarf with an effective temperature of approximately 500 K, discovered by Lucas et al. (2010). Previous spectral analysis with cloudless models has been unsuccessful at modeling the flux in the near-infrared in Y and J bands (Leggett et al. 2012).

In Figure 12, we plot the near-infrared spectra published in Lucas et al. (2010) and Leggett et al. (2012) with the cloudy and cloudless models that are fit by eye to be the closest representations of the data. We also show J, H, K, and Spitzer photometry (Lucas et al. 2010) and WISE photometry (Kirkpatrick et al. 2011). These models have similar temperatures and gravities to previous studies; Leggett et al. (2012) presented fits with T_eff between 492 and 550 K and log g = 3.52–5.0, whereas our fits have T_eff of 600 K (cloudless) and 500 K (with sulfide clouds) and both have log g = 4.5.

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Note that the flux in Y and J in cloudless models is systematically too high. The opacity of the sulfide clouds provides a natural mechanism to decrease the flux in Y and J and increase the flux in H and K bands. The match to the models is still not perfect. This may be due in part to incomplete line lists for methane; these cold objects have a significant amount of CH₄ in the atmosphere, that absorbs strongly in the near-infrared. The discrepancy at 4.5 μm is most likely due to absorption of CO as a result of non-equilibrium chemistry (see Section 4.4). Outstanding issues in T dwarf modeling are discussed in Section 5.3.

We note that both of our preferred sulfide cloud models predict brighter M-band (4.5 μm) photometry than is observed. This is likely a consequence of our neglect of non-equilibrium mixing of CO in this study. As first predicted by Fegley & Lodders (1996), such mixing is an important process in the atmospheres of brown dwarfs (Noll et al. 1997; Marley et al. 2002; Saumon et al. 2006; Stephens et al. 2009) as it is in solar system giants (Barshay & Lewis 1978) and the relative impact of the mixing increases with decreasing gravity (Hubeny & Burrows 2007; Barman et al. 2011). Absorption by excess atmospheric CO decreases the thermal emission in M-band and is likely responsible for the mismatches seen in Figures 11 and 12, particularly for the lower gravity Ross 458C.

The formation of the clouds considered in this study does not involve the species most affected by non-equilibrium chemistry such as CO and CH₄. The cloud models will therefore be only minimally affected by the changes in the pressure–temperature profile of the atmosphere due to the changes in gas opacity. However, the overall spectra of models will look quite different in regions where CO absorbs strongly, such as the 4.5 μm feature, so future, more comprehensive fits of sulfide cloud models to observations will have to include non-equilibrium models.

Clouds must form in brown dwarf atmospheres as they cool; there is no way to avoid the condensation of different species as the atmosphere reaches lower effective temperatures. In these models, we parameterize the opacity of clouds by creating a distribution of cloud material in the atmosphere that has a distribution of cloud particle sizes. Within the models, we can change those distributions. A cloud that sediments into a small number of large particles will settle into a thin layer and will not significantly change the emergent spectrum; the same cloud material organized into an extended cloud with small particle sizes will have a dramatic effect on the model spectrum.

For these reasons, we require a model of cloud particle sizes and distribution as well as the underlying chemistry. When we consider models that include new or different clouds, we do not change any of the underlying chemistry of condensation; we change the opacity of the condensate particles and in doing so modify the effect that the cloud formation has on emergent flux.

Burgasser et al. (2010) invoked the reemergence of silicate clouds to explain the spectrum of Ross 458C. We suggest instead that the initial emergence of sulfide clouds would have a similar effect on the spectrum and provide a more natural explanation for the results.

From observations, it is clear that the range in T dwarf colors just following the L/T transition is small; spectra of T dwarfs show no evidence that clouds still affect the emergent flux for objects slightly cooler than this transition. This observation suggests that the iron and silicate clouds have dissipated between 1400 and 1200 K (for typical field dwarfs) and are no longer important in T dwarf atmospheres. If iron and silicate clouds were sometimes important in T dwarf atmospheres, we would expect to see a population of relatively quite red objects at effective temperatures between that of Ross 458C and the L dwarfs; no brown dwarf with these properties has been observed.

As T dwarfs cool, the range in observed infrared colors increases; a population of red T dwarfs develops, which are redder in the near-infrared than cloudless models predict. Based on these observations, we favor a mechanism that cannot strongly alter T_eff ∼ 900–1200 K T dwarf atmospheres, but naturally reddens T_eff ≲ 800 K T dwarfs.

The emergence of sulfide clouds provides that natural explanation for this range in T dwarf colors at low effective temperatures. Just as the iron and silicate clouds condense in the photospheres of L dwarfs and change their observed spectra, the sulfide clouds begin to condense in the photospheres of T dwarfs with temperatures cooler than 900 K, changing their observed spectra.

We have not yet investigated whether the sulfide clouds will have identifiable spectral features that would confirm their presence in T dwarf atmospheres, but given the features in the sulfide indices of refraction (see Figure 2) these features would likely be in the mid-infrared.

There are several challenges in modeling T dwarfs that have not yet been addressed in these calculations. Because of the high densities in brown dwarf atmospheres, sodium and potassium bands at optical wavelengths are extremely pressure-broadened in T dwarf spectra (Tsuji et al. 1999; Burrows et al. 2000; Allard et al. 2005, 2007). The wings of these broadened bands extend into the near-infrared in Y and J bands, creating additional opacity at those wavelengths. For these calculations, we use the line broadening treatment outlined in Burrows et al. (2000), which is somewhat ad hoc and potentially creates some inaccuracies in the model flux in Y and J bands. A calculation of the molecular potentials for potassium and sodium in these high pressure environments, as is carried out in Allard et al. (2005, 2007), would improve the accuracy of these models.

Another challenge in modeling T dwarf spectra is the inadequacies of methane opacity calculations; methane is the only important gas-phase absorber with inadequate opacity measurements or calculations. Uncertainties in methane absorption bands create inaccuracies in T dwarf models, especially in the H band, where it is a very strong absorber (Saumon et al. 2012).

Sulfide clouds could form partial cloud layers with patchy clouds. One way to infer patchy cloud cover is to observe variability in photometric colors; variability can indicate high-contrast cloud features rotating in and out of view. Radigan et al. (2012) studied objects at the L/T transition and inferred that the iron and silicate clouds could be in the process of breaking up and forming patchy clouds in those atmospheres. A similar study of the variability of cool T dwarfs could reveal a similar physical process in sulfide clouds.

A larger number of high fidelity spectra of the coldest T dwarfs would help to constrain these cloudy models. Currently, there are a few objects with effective temperatures cooler than 700 K with moderate resolution spectra. A larger sample of objects would give us better statistics on the overall population of T dwarfs, with different surface gravities, metallicities, and cloud structures.

At cooler effective temperatures, water clouds will condense in brown dwarf atmospheres (Burrows et al. 2003). Oxygen is 300 times more abundant than sodium in a solar-composition gas and the silicate clouds only use 20% of the total oxygen in the atmosphere, so water clouds will be much more massive and important in shaping the emergent flux. As missions like WISE find colder objects (Kirkpatrick et al. 2011; Cushing et al. 2011) and these objects are observed spectroscopically, future models of brown dwarfs will need to include the condensation of these more volatile clouds.

Lodders (1999) and Lodders & Fegley (2006) predict that RbCl and CsCl will condense before the water clouds condense; however, the abundances of Cs and Rb are very low (Lodders 2003) so these clouds will be optically thin. If equilibrium conditions prevail, NH₄H₂PO₄ would also condense (Fegley & Lodders 1994; Visscher et al. 2006) with a mass similar to that of the Na₂S cloud. Whether NH₄H₂PO₄ condenses or P remains in the gas phase as PH₃ (as on Jupiter and Saturn) deserves further study to examine potential effects on the spectra of the coolest brown dwarfs.

Observations and models of T dwarfs provide a testbed to study planetary atmospheres. While brown dwarfs are more massive than planets, the atmospheres of T dwarfs have similar effective temperatures to those of young giant planets (Burrows et al. 1997; Fortney et al. 2008). The study of T dwarfs provides crucial tests of cloudy atmosphere models that will be applicable to directly imaged exoplanet atmospheres.

Cloud models designed originally for brown dwarfs are already being applied to exoplanets. Cloud models with iron and silicate clouds were originally developed to model L dwarf atmospheres; these models have been successfully applied to observations of the only directly imaged multiple planet system, HR 8799. Several studies of the HR 8799 planets have shown that the iron and silicate clouds play a significant role in their atmospheres (Marois et al. 2008; Barman et al. 2011; Galicher et al. 2011; Bowler et al. 2010; Currie et al. 2011; Madhusudhan et al. 2011; Marley et al. 2012).

As instruments like the Gemini Planet Imager and SPHERE begin to discover new planets in the next few years, we will be able to apply brown dwarf models to colder planetary atmospheres in which clouds will likely play a key role in shaping their spectra.