FINGERING CONVECTION AND CLOUDLESS MODELS FOR COOL BROWN DWARF ATMOSPHERES

Brown dwarfs with effective temperatures T_eff below ∼1000 K (T and Y spectral types) are of great interest to understand the physics of cool atmospheres and pave the way for future studies of cool exoplanets with the James Webb Space Telescope (JWST) and the European Extremely Large Telescope. Detailed spectra can now be obtained in the near-infrared (NIR) for these objects using, e.g., the Hubble Space Telescope or the Gemini NIR Spectrograph.

The reddening of the brown dwarf spectra with spectral types M and L (T_eff ≈ 2000 K) is thought to be the result of the condensation of silicate dusts and the subsequent cloud formation (Tsuji et al. 1996; Chabrier et al. 2000; Allard et al. 2001). At the L/T transition $({{T}_{{\rm eff}}}\approx 1300$ K), this reddening disappears with a sharp transition as a function of T_eff. Given the high amplitude variability observed at the L/T transition (e.g., Radigan 2014), this effect is interpreted as the clouds breaking up and forming holes at the photosphere (e.g., Marley et al. 2010). Yet, the spectral modeling of T and Y dwarfs remains a challenge because the IR colors predicted by the cloudless models below ∼1000 K appear too “blue” compared to observations. This additional reddening has recently been interpreted by Morley et al. (2012, 2014) as the emergence of sulfide and chloride clouds (MnS, Na₂S, and KCl). However, even with this cloudy approach, it appears difficult to obtain a good model of Y-dwarf spectra (see, for example, WISEPC J1217b in Leggett et al. 2015).

In the present paper, we show under which conditions the modeling of T/Y-dwarf spectra is possible with a cloudless model. In Section 2, we describe our one-dimensional (1D) numerical code ATMO and we apply it to T and Y dwarf atmospheres in Section 3. The spectral modeling of Y dwarfs is in good agreement with observations when vertical mixing is taken into account. For T dwarfs, good agreement is obtained when the temperature gradient in the atmosphere is reduced. We discuss these results in Section 4 and we suggest that the enhanced cooling in the deep layers of T dwarfs arises from the onset of fingering convection, resulting from the condensation of some species in very thin dust.

We have developed a 1D radiative/convective code ATMO solving for the pressure/temperature (PT) structure of an atmosphere, assuming a vertical energy balance between the internal heat flux and the radiative and convective fluxes, and vertical hydrostatic equilibrium. The convection is taken into account using mixing length theory as decribed in Henyey et al. (1965) and Gustafsson et al. (2008) with a mixing length of 1.5 times the local pressure scale height.

The PT structure is solved on a cartesian plane-parallel grid for a given effective temperature T_eff and surface gravity ${\rm log} \;g$. The chemistry takes into account ∼150 species at equilibrium with the condensation of ∼30 liquids/solids assuming elemental abundances from Caffau et al. (2011), and is solved by minimization of Gibbs free energy. The references of the thermodynamic data for the gas-phase species can be found in Venot et al. (2012), except for H₂ whose thermodynamic data were fitted from the Saumon–Chabrier equation of state (see Saumon et al. 1995). The data for the condensed species can be found in Robie & Waldbaum (1968) and Chase (1998). In the present models, we neglect rain-out processes and clouds. This approximation is critical to assess the abundances of some refractory elements such as Na, K, and S because of the different condensation paths that can happen depending on the thermal history of the atmosphere (e.g., NaAlSi₃O₈/KAlSi₃O₈ will not form if Al and Si are removed at higher temperatures and Na and K will condense in Na₂S and KCl). However in the spectral window of interest, the main absorbers such as H₂O, NH₃, CH₄, and CO are not affected by this hypothesis and their abundances can even be computed analytically (see Burrows & Sharp 1999). The code is coupled to the kinetic chemical network of Venot et al. (2012), which includes ∼1000 reversible reactions associated with ∼100 species based on C, H, N, and O elements. Vertical mixing is taken into account with a free parameter K_zz representing the vertical eddy diffusion coefficient (see Ackerman & Marley 2001). We perform a full chemical network run on the atmosphere by integrating the chemistry in time (usually up to 10¹⁰ s) and we reconverged the PT structure of the atmosphere on the fly until both the chemistry and the PT structure are in a steady state. The PT structure can be solved consistently with the out-of-equilibrium chemistry and can have a significant difference if a main absorber is quenched below the photosphere by the vertical mixing. We tested the kinetic chemical network by reproducing the results of Venot et al. (2012) for HD 209458b and HD 189733b. Our results are also in good agreement with Hubeny & Burrows (2007), especially considering the differences in elemental abundances and opacity line lists. For our case at T_eff = 1000 K, log g = 4.5, and log K_zz = 8.0, we predict a quenched abundance of CO at ∼2 $\times \;{{10}^{-4}}$ and of NH₃ at ∼7 $\times \;{{10}^{-6}}$, and the model with T_eff = 900 K, log g = 4.5 and log K_zz = 8.0 in Hubeny & Burrows (2007) indicates a CO abundance of $\sim 1.5\times {{10}^{-4}}$ (fast2 chemical timescale Yung et al. 1988) and a NH₃ abundance of $\sim 5\times {{10}^{-6}}$.

The line-by-line radiative transfer has already been described and tested in Amundsen et al. (2014). The differences in radiative flux and heating rate between our line by line code and the radiative scheme used in the global circulation model of the UK Met-Office, the Unifed Model (Amundsen et al. 2014) are generally of the order of a few percent. We use the same opacities for H₂–H₂, H₂–He, NH₃, H₂O, CO, TiO, and VO (see Table 1 in Amundsen et al. 2014). Methane (CH₄) has been updated with the new line list from the Exomol project (Yurchenko & Tennyson 2014) and we included the CO₂ opacity using the line list from Tashkun & Perevalov (2011) with the line broadening available in the literature (broadening coefficients from Thibault et al. 1992, 2000; Sharp & Burrows 2007; Padmanabhan et al. 2014). We have also calculated opacities due to the extremely pressure-broadened sodium and potassium doublets using the line profiles of Allard et al. (2007). We have used the correlated-k method described in Amundsen et al. (2014) to speed up the convergence and combined the k-coefficients by using the random overlap method for the mixture (Lacis & Oinas 1991). Then we compute the surface spectrum from the converged PT structure using the line by line radiative transfer. The radiative transfer module also includes Rayleigh scattering by H₂ and He, using the accelerated Λ-iteration technique described in Bendicho & Bueno (1995).

3.1. Y Dwarfs and NH3 Quenching

Saumon et al. (2006) have already demonstrated that ammonia is well below its chemical equilibrium abundance in T dwarfs based on the observed strength of th 9–11 μm absorption feature. It has also later been confirmed in the NIR lines of NH₃ (e.g., Saumon et al. 2012), and Cushing et al. (2011) confirmed the signature of vertical mixing in all but one of their Y dwarfs by the very absence of more detectable NH₃ absorption. It has been expected that absorption by ammonia in the Y band should be present, but so far the data have not identified such a signature (e.g., Figure 9 in Cushing et al. 2011). It has been also recently proposed by Leggett et al. (2015) that ammonia is depleted in the atmospheres of Y dwarfs. Using the Y dwarf WISEPC J1217b (Y0, Leggett et al. 2014) as a template, we show in the left panels of Figure 1 that indeed a low abundance of ammonia is required to reproduce the shape of the Y-band flux around 1 μm and the correct flux in the H band.

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Indeed, as shown in Figure 2, when vertical mixing is taken into account with a diffusion coefficient, the abundance profile of ammonia is quenched at a relatively deep level (50–100 bars), resulting in a depletion of ammonia by approximatively a factor 3, significantly reducing the absorption in the peaks in the Y band. We took a value of ${{K}_{{\rm zz}}}={{10}^{8}}$ cm² s⁻¹ for the eddy diffusion coefficient, which is within the high values explored by Hubeny & Burrows (2007); however the quenching of the NH₃/N₂ chemistry is relatively insensitive to the choice of K_zz (see Cushing et al. 2011). When we include the vertical mixing, the condensation of H₂O and NH₃ is ignored for simplicity, this approximation is not an issue because the difference with and without condensation at equilibrium is small at these effective temperatures (T_eff is shifted by ∼10 K). Importantly for the color–magnitude diagrams in the IR, the flux in the H band at 1.6 μm is increased and matches well the observed spectrum. As shown in Figure 1, this increase in the flux is sufficient to produce the reddening required in J–H to match the observations. Applying the same physical treatment to the cooler brown dwarf WISEPA J1541 (Y0.5, Cushing et al. 2011), we obtain the same significant improvement between observed and synthetic spectra, as shown in the bottom-left panel of Figure 1. Since we have included the complete chemical network of Venot et al. (2012) in ATMO, we also predict the quenching of CO and CO₂ (see Figure 2), which will impact the flux at 4.5 μm.

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3.2. T Dwarfs and Reduced Temperature Gradient

T dwarfs have bluer colors in J–H and $J-K$ than L dwarfs which indicates that the silicate and iron clouds should fall below the photosphere at the L/T transition. However, their spectra still present a reddening compared to cloudless models and Morley et al. (2012) proposed that another type of clouds such as sulfide clouds might be responsible for this residual reddening. We have calculated spectra for the T dwarfs ROSS 458C (Burgasser et al. 2010) and UGPS 0722-05 (Lucas et al. 2010) whose spectra are shown in right panels of Figure 1. As suggested by Burgasser et al. (2010), we used a supersolar metallicity ([M/H] = 0.3) for ROSS 458 C based on the measured metallicity of its two companions ROSS 458 A and B. Our cloudless models overestimate the flux in the Y and J bands, yielding significantly bluer colors in J–H/J–K compared to observations. Since the photosphere in the Y and J bands is deeper than the the photosphere in the H and K bands, reducing the temperature gradient in the atmosphere should decrease the $Y/J$ flux compared to the $H/K$ flux. We have tested this idea by artificially modifying the adiabatic index $\gamma ={{C}_{P}}/{{C}_{V}}$ of the atmosphere. The pressure/temperature profiles obtained with γ = 1.2 for ROSS 458 C and γ = 1.27 for UGPS 0722–05, respectively, are displayed in Figure 3, and the corresponding spectra in Figure 1. These models correctly reproduce the spectra and the colors in J–H/J–K without the need to invoke additional cloud opacity. As shown in Figure 3, the amplitude of the temperature reduction required to match the observed spectra decreases with the effective temperature. Therefore, for Y dwarfs we found that there is no need to adjust the adiabatic index with respect to its expected value (∼1.45). The relative importance of the effects of vertical mixing and the reduction of the temperature gradient can be seen on the color–magnitude diagrams in Figure 4.

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As shown in Section 3.1, a coherent cloudless model taking into account vertical mixing in the atmosphere is able to reproduce the NIR spectrum of Y dwarfs. Figure 4 shows the color–magnitude diagrams for various cool T and Y brown dwarfs in M_J versus J–H and M_Y versus Y–J. We have plotted a series of different models at constant ${\rm log} \;g=4.5$ as well as the cloudless and sulfide/H₂O cloud models of Morley et al. (2014). We display the models with and without vertical mixing and with and without modified adiabatic index to illustrate the relative impacts of these processes. Although the cloudy models do yield redder colors in J–H, providing a better match to the observed colors (see Figure 16 in Morley et al. 2014), they predict also a large reddening in Y–J which is incompatible with the observations. The problem is even worse at larger gravity (${\rm log} \;g=5$). In contrast, our cloudless models reproduce the correct J–H and Y–J colors for Y dwarfs, without any adjustable parameter. Furthermore, as shown in Section 3.1, cloudless models consistently taking into account departure from chemical equilibrium due to vertical mixing in the atmosphere nicely reproduce the NIR spectrum of Y dwarfs. Our cloudless models are shifted in J–H with respect to the ones of Morley et al. (2014) by nearly one magnitude at log g = 4.5 and we would get similar offsets if we compare our models at log g = 4.5 to their models at log g = 5.0. Part of the shift (0.3 mag) can be explained by changing in our calculations, (i) the elemental abundances to Lodders (2003), (ii) the line list for methane to the STDS line list (Wenger & Champion 1998), and (iii) the thermodynamical data of H₂ to the one of the JANAF database as used in Morley et al. (2014). The origin of the rest of the shift remains unknown but is probably due to differences in the opacities of the other molecules. It seems more likely, however, that such low-temperature, low-mass objects have gravities close to log g = 4.5 rather than log g = 5.0 (Chabrier & Baraffe 2000).

For T dwarfs, both our cloudless models and the cloudy models reproduce the J–H and Y–J colors. In the former case, as shown in Section 3.1, a good match of the spectrum requires a lower value of the atmospheric characteristic adiabatic index (a good match is also obtained in M_J versus J–K). This adjustment can thus reflect either the effect of clouds or the signature of an energy transport mechanism that reduces the temperature gradient in the atmosphere. Given the fact that for T dwarfs, iron and silicate clouds are expected to have settled below the phostosphere, as mentioned previously, we favor the second suggestion. We suggest that the cooling is due to the onset of fingering (thermohaline) convection in T dwarf deep atmospheric layers, similar to the one triggered by salt gradients in Earth oceans (e.g., Rahmstorf 2003). In the limit of “infinitely thin” dust condensation, we have computed the mean-molecular-weight gradient in the atmosphere, ${{\nabla }_{\mu }}$,⁶ assuming that all the condensates remain in the atmosphere in their “mono-molecular” form (e.g., assuming that equilibrium chemistry predicts a given abundance of solid MgSiO₃, we compute the mean molecular weight by keeping this abundance with the molecular weight of MgSiO₃). This mean-molecular-weight gradient should be fairly representative of ${{\nabla }_{\mu }}$ in the presence of small grains if the grain formation process does not significantly change throughout the atmosphere. As shown in Rosenblum et al. (2011; see also Leconte & Chabrier 2012), the value of the dimensionless parameter ${{R}_{0}}=({{\nabla }_{T}}-{{\nabla }_{{\rm ad}}})/{{\nabla }_{\mu }}$ determines the extent of the overturning convection zone (${{R}_{0}}\lt 1$). In case the mean molecular weight increases with height in a stable atmosphere (${{\nabla }_{\mu }}\lt 0$, and $({{\nabla }_{T}}-{{\nabla }_{{\rm ad}}})\lt 0),$ as in the present context, the possible existence of fingering convection is determined by the relation $1\lt {{R}_{0}}\lt 1/\tau$, where $\tau ={{\kappa }_{\mu }}/{{\kappa }_{T}}$ defines the inverse Lewis number, i.e., the ratio of the molecular diffusivity ${{\kappa }_{\mu }}$ to the radiative thermal diffusivity ${{\kappa }_{T}}=16\sigma {{T}^{3}}/(3\kappa {{\rho }^{2}}{{C}_{p}})$. This is the opposite situation to oscillatory double-diffusive convection, due to a positive molecular weight gradient and a destabilizing temperature gradient, suggested to occur in some giant planet interiors (Leconte & Chabrier 2012, 2013). Figure 5 shows the atmospheric profiles of $1/(\tau {{R}_{0}})$ for ROSS 458 C and WISEPA J1541, taking into account the molecular diffusivity of MgSiO₃(s) and assuming that the thermal diffusivity is dominated by the radiative processes (${{\kappa }_{T}}$ varies from $\sim {{10}^{-1}}$ cm² s⁻¹ at 1 kbar to $\sim 3\times {{10}^{5}}$ cm² s⁻¹ at 1 bar). As seen, fingering convection (i.e., $1/(\tau {{R}_{0}})\gt 1$) is predicted to occur in some parts of these atmospheres. The magnitude of the mean-molecular-weight gradient is modest: $|{{\nabla }_{\mu }}|\;\sim$ 10⁻⁴–10⁻⁵ for ROSS 458 C. Interestingly, while the extension of the fingering-convection zone is very large for ROSS 458 C, it remains relatively small for the Y dwarf WISEPA J1541. This is fully consistent with the fact that we need a lower adiabatic index for T dwarfs whereas this modification is not necessary for Y dwarfs.

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The transport mechanism needed to lower the temperature gradient has to be efficient to transport energies at the level of the internal flux of the object. The possibility that fingering convection could carry out such energy fluxes remains to be demonstrated. Also the mixing induced by the convective fingers will reduce the destabilizing mean-molecular-weight gradient. A consistent theory of fingering convection in our model is needed in order to assess the possibility to have a self-sustained steady-state energy flux from the convection. Nevertheless our results strongly suggest that fingering convection could occur in T dwarf atmospheres due to the destabilizing effect of dust condensation in the stably stratified part of the atmosphere, potentially impacting the energy transport in these objects compared to both warmer and cooler objects and leading to efficient cooling of their deep atmospheric layers. It can be expected that the increase of the convective energy transport will reduce the radiative flux in the corresponding region, hence reducing the temperature gradient in the atmosphere, as needed for the reddening in J–H.

In the present paper, we have shown the following.

• 1.  
  Cloudless atmosphere models can reproduce the observed infrared Y dwarf spectral energy distribution, provided vertical mixing and out-of-equilibrium chemistry are properly taken into account to correctly predict the quenching of ammonia at deep levels, modifying its abundance profile.
• 2.  
  Cloudless models for T dwarfs with a reduced temperature gradient in the atmosphere correctly reproduce observed fluxes and colors. Such a reduced gradient can be obtained by a modification of the adiabatic index in the atmosphere and could reflect either the effect of clouds or of another type of energy transport in the atmosphere. If clouds are effectively responsible for the reddening, the modification of the adiabatic index is an easy way to mimic the effect and could be used to better constrain cloud models that are currently used.
• 3.  
  We suggest that fingering convection could be responsible for such a reduced temperature gradient. We demonstrate that the condensation of very thin dust under typical T dwarf atmosphere conditions could trigger this mechanism and that the extent of this process decreases with decreasing effective temperature, essentially vanishing for Y dwarf atmosphere conditions.

Fundamental physical mechanisms such as atomic diffusion and hydrodynamical instability might thus take place in cold brown dwarf atmospheres and play a major role in their spectral evolution. Future observations of cool objects with SPHERE, GPI, and JWST combined with comparisons with the different models should enable us to distinguish between these two effects: presence of clouds or reduced temperature gradient in T dwarf atmospheres. Our ability to constrain which physics is indeed present in cool brown dwarf atmospheres will bear important consequences for the future understanding of cool exoplanet spectra.

We thank Adam Burgasser, Sandy Leggett, Philip Lucas, Mike Cushing, and Chas Beichman for providing their data. The calculations for this paper were performed on the DiRAC Complexity machine, jointly funded by STFC and the Large Facilities Capital Fund of BIS, and the University of Exeter Supercomputer, a DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of BIS, and the University of Exeter. This work is partly supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/20072013 grant Agreement No. 247060). Part of this work is supported by the Royal Society award WM090065 and the consolidated STFC grant ST/J001627/1. O.V. acknowledges support from the KU Leuven IDO project IDO/10/2013 and from the FWO Postdoctoral Fellowship programme.