Models of Stars, Brown Dwarfs and Exoplanets

France Allard, Derek Homeier, Bernd Freytag

Introduction

The modelling of the atmospheres of very low mass stars (hereafter VLMs) has evolved (as here illustrated with the development of the PHOENIX atmosphere code, which has allowed the detection of water vapour in extrasolar planets’ atmospheres by Barman et al. 2007, 2008) with the extension of computing capacities from an analytical treatment of the transfer equation using moments of the radiation field Allard (1990), to a line-by-line opacity sampling in spherical symmetry Allard et al. (1997); Hauschildt et al. (1999a, b), and more recently to 3D radiation transfer Seelmann et al. (2010). In parallel to detailed radiative transfer in an assumed static environment, hydrodynamical simulations have been developed to reach a realistic representation of the granulation and its induced line shifts for the sun and sun-like stars (see Freytag et al. 2011) by using a non-grey (multi-group binning of opacities) radiative transfer with a pure blackbody source function (scattering is neglected).

Molecular opacities

While earlier work has been developed for the study of red giant stars, the pioneering work on the modelling of VLM atmospheres has been provided by Mould (1975), Allard (1990) and Kui (1991) using a band model or Just Overlapping Line Approximation (JOLA) opacities developed by Kivel et al. (1952) and adapted for astrophysical use by Golden (1967). More realistic model atmospheres and synthetic spectra for VLMs, brown dwarfs and extrasolar planets have been made possible thanks to the development of accurate opacities calculated often ab initio for atmospheric layers where temperatures can reach 3000 K. The process of improvements was especially remarkable in the case of water vapour line lists. Indeed, water vapour has seen an important evolution through the years from band model approximations to straight means based on hot flames experiments, and then to ab initio computations. Nevertheless, the atmosphere models have failed to reproduce the strength of the water bands that shape the low resolution ( $R\leq$ 300) infrared spectral energy distributions of M dwarfs. At the lower temperatures of brown dwarfs methane and ammonia rival the effect of water. The discrepancies in the model synthetic spectra were therefore believed to be due to inaccurate or incomplete molecular opacities. In particular water vapour was suspected because the discrepancies were observed at infrared wavelengths in the relative brightnesses of the flux peaks between water vapour bands. As can be seen from Fig. 1 where the models are compared to the infrared spectrum of the M8e dwarf VB10, the water vapour opacity profile which shape this part of the spectrum has strongly changed over time with the improvement of computational capacities and a better knowledge of the interaction potential surface. And the most recent ab initio results confirm the earliest hot flames laboratory experiment results by Ludwig (1971). But in general, most opacity profiles produce an excess opacity (or lack of flux in the model) in the $K$ bandpass. Only the UCL1994 line list (due to incompleteness, and with much of its deviations cancelling out over the bandpasses) could produce seemingly correct $J-K_{s}$ colours.

The revised solar abundances

Model atmospheres for VLMs and in general for other stars assume scaled solar abundances for all heavy elements, with some enrichment of $\alpha$ -process elements (the result of a ”pollution” of the star-forming gas by the explosion of a supernova) when appropriate in the case of metal-depleted subdwarfs of the Galactic thick disk, halo and globular clusters. The revision of the solar abundances based on radiation hydrodynamical simulations of the solar atmosphere, on improvements in the quality of the spectroscopic observations of the Sun, and in its detailed line profile analysis by two separate groups using independent hydro codes and spectral synthesis codes (Asplund et al. 2009; Caffau et al. 2011) yield an oxygen reduction of 0.11 – 0.19 dex (up to 34%). compared to the previously used abundances of Grevesse et al. (1993). Since the overall SED of late K dwarfs, M dwarfs, brown dwarfs, and exoplanets is governed by oxygen compounds (TiO, VO in the optical and water vapour and CO in the infrared), the input elemental oxygen abundance used in the equation of state is of major importance. Fig. 2 shows an example of these effects for the optical and infrared SED of the M5.5 dwarf system Gl 866. However at other effective temperatures even stronger photometric effects can be seen, where the near-IR SED of different models diverges more (see Fig. 3). The comparison shows significant improvement compared to older models shown in Fig. 1, except for excess flux in the $H$ bandpass near 1.7 $\mu m$ due to incomplete FeH opacity data for this region. The comparison has particularly improved in the Wing Ford band of FeH near $0.99~{}{\mu}$ m, and in the VO bands thanks to line lists provided by B. Plez (GRAAL, Montpelier, France), although inaccurate or incomplete opacities are still affecting the models at optical wavelengths (e.g. the TiO line list by Langhoff 1997).

Cloud formation

One of the most important challenges in modelling these atmospheres (below 2600K) is the formation of clouds. Tsuji et al. (1996) had identified dust formation by recognising the condensation temperatures of hot dust grains (enstatite, forsterite, corundum: MgSiO3, Mg2SiO4, and Al2O3 crystals) to occur in the line-forming layers ( $\tau\approx 10^{-4}-10^{-2}$ ) of their atmospheres. The cloud composition, according to equilibrium chemistry, is going from zirconium oxide (ZrO2), to refractory ceramics (perovskite and corundum; CaTiO3, Al2O3), to silicates (e.g. forsterite; Mg2SiO4), to salts (CsCl, RbCl, NaCl), and finally to ices (H2O, NH3, NH4SH) as brown dwarfs cool down over time from M through L, T, and Y spectral types Allard et al. (2001); Lodders & Fegley (2006). This assumed (by Allard et al. 2001) sub-micron-sized crystal formation causes the weakening and vanishing of TiO and VO molecular bands (via CaTiO3, TiO2, and VO2 grains) from the optical spectra of late M and L dwarfs, revealing CrH and FeH bands otherwise hidden by the molecular pseudo-continuum, and the resonance doublets of alkali transitions which are only condensing onto salts in late-T dwarfs. The scattering effects of this fine dust is Rayleigh scattering which provides veiling to the optical SED of late-M and L dwarfs, while the greenhouse effect due to the dust cloud causes their infrared colours to become extremely red compared to those of hotter low mass stars. The upper atmosphere, above the cloud layers, is depleted from condensible material and significantly cooled down by the reduced or missing pseudo-continuum opacities.

One common approach has been to explore the limiting properties of cloud formation. One limit is the case where sedimentation or gravitational settling is assumed to be fully efficient such as the case B of Tsuji (2002), the AMES-Cond or condensed phase models of Allard et al. (2001), the clear case of Ackerman & Marley (2001) and the cloud-free case of Burrows et al. (2006). The other limit is the case where gravitational settling is assumed inefficient and dust, often only forsterite, forms in equilibrium with the gas phase such as the case A of Tsuji (2002), the AMES-Dusty or dusty models of Allard et al. (2001), the cloudy case of Ackerman & Marley (2001), or the case B of Burrows et al. (2006). These limiting cases of maximum dust content agree in describing the evolution of brown dwarfs from a molecular opacity governed SED towards a blackbody SED below 1500K. This description was suitable, at least in the case of the AMES-Dusty models, in reproducing the infrared colours of L dwarfs. The cloud-free limiting case on the other hand allowed to reproduce to some degree the colours of T dwarfs. Fig. 4 shows this situation for the AMES-Cond/Dusty limiting case models of Allard et al. (2001) compared with the effective temperatures estimates obtained by integration of the observed SED Golimowski et al. (2004); Vrba et al. (2004).

The purpose of a cloud model is therefore to go beyond these limiting cases and define the number density and size distribution of condensates as a function of depth in the atmosphere. The discovery of dust clouds in M dwarfs and brown dwarfs has therefore triggered the development of cloud models building up on pioneering work in the context of planetary atmospheres developed by Lewis (1969), Rossow (1978), and Lunine et al. (1989). The Lewis model is an updraft model (considering that condensation occurs in a gas bubble that is advected from deeper layers). By lack of knowledge of the velocity field and diffusion coefficient of condensates in the atmospheres of the planets of the solar system, Lewis simply assumed that the advection velocity is equal to the sedimentation velocity, thereby preserving condensible material in the condensation layers. This cloud model did not account for grain sizes. Rossow on the other hand developed characteristic timescales as a function of particle size for the main microphysical processes involved (condensation, coagulation, coalescence, and sedimentation). The intersections of these characteristic timescales gives an estimate of the condensate number densities and mean grain sizes. However, this model made several explicit assumptions concerning the efficiency of supersaturation, the coagulation, etc.

Helling et al. (2008a) have compared different cloud models and their impact on model atmospheres. Most cloud models define the cloud base as the evaporation layer provided by the equilibrium chemistry. In the unified cloud models of Tsuji (2002); Tsuji et al. (2004) a parametrization of the radial location of the cloud top by way of an adjustable parameter Tcrit was used. This choice permits to determine the cloud extension effects on the spectra of these objects but does not allow to reproduce the stellar-substellar transition with a unique value of Tcrit since the cloud extension depends on the atmospheric parameters.

Allard et al. (2003) using PHOENIX and the index of refraction of up to 40 condensible species, have applied the Rossow cloud model, ignoring coalescence and coagulation, and comparing the timescales of condensation, sedimentation and mixing (extrapolated from the convective velocities into the convectively stable layers), and assuming efficient nucleation (monomers equilibrium densities). The cloud model was then solved layer by layer inside out to account for the sequence of grain species formation as a function of cooling of the gas. But this version of the BT-Settl (with gravitational settling) models did not allow for the formation of enough dust in brown dwarf atmospheres due to a too conservative prescribed supersaturation value.

Ackerman & Marley (2001) have solved the particle diffusion problem of condensates assuming a parametrized sedimentation efficiency $f_{\rm sed}$ (constant through the atmosphere) and a mixing assumed constant and fixed to its maximum value (maximum of the inner convection zone). Saumon & Marley (2008) found that their models could not produce the color change with a single value of $f_{\rm sed}$ .

Helling et al. (2008b) use the PHOENIX code to compute the Drift-Phoenix models. The cloud model used, in the contrary to all other cases mentioned, studies the nucleation and growth of grains as they sediment down into the atmosphere. This cloud model determines the number density and size distribution of grains by 1D nucleation simulations, and the resulting distribution is read in by PHOENIX which computes the resulting opacities and radiative transfer. These models solve the nucleation problem but only for the assumed monomer types and have been successfully applied to fit the dusty atmospheres of L dwarfs, but the reversal in IR colours observed for the L/T transition could not be explained (Witte et al. 2011).

These RHD simulations allow an estimation of the diffusion processes bringing fresh condensible material from the hotter lower layers to the cloud forming layers. We have therefore updated our cloud model (BT-Settl models) to account for the mixing prescribed by the RHD simulations. Another important improvement concerns the supersaturation which as been computed rather than using the fixed conservative value recommended by Rossow. One can see from Fig. 4 that the late-type M and early-type L dwarfs behave as if dust is formed nearly in equilibrium with the gas phase with extremely red colours in some agreement with the BT-Dusty models. The BT-Settl models reproduce the main sequence down to the L-type brown dwarf regime, subjected in the $K$ bandpass to the greenhouse effect of dust clouds, before turning to the blue in the late-L and T dwarf regime as a result of methane formation in the $K$ bandpass. This constitutes a major improvement over previous models, and is promising that we can reach in the near futur a full explanation of the stellar substellar transition.

Diffusion has also been held responsible for deviations in ultracool atmospheres from gas phase chemical equilibrium (CE), as noted in early observations of T dwarfs showing an excess of carbon monoxide absorption (Noll et al. 1997; Griffith & Yelle 1999). More recently, carbon dioxide (Tsuji et al. 2011), which was not expected at such low temperatures, has been detected. Similarly, ammonia has been shown to be underabundant (Saumon et al. 2006). This is understood as the result of slowing down of crucial chemical reaction steps, so that some important molecules (CH4, NH3) would not have the time to form in equilibrium while undergoing mixing, whereas others (CO, CO2, N2) remain at enhanced abundances. The RHD simulations of Freytag et al. (2010) have allowed to understand the underlying mixing processes, obviating the need to describe them with an additional free parameter.

Applications to Exoplanet Science

These planets are typically found at several dozens of AU from the star, and since the observations are done in the infrared the non-irradiated models can even be used directly. Indeed, Barman et al. (2001) have shown that the effects of impinging radiation from a star on the planetary atmosphere are Rayleigh scattering of the stellar light by H2 molecules (or clouds if present) at optical wavelengths (below 1 $\mu$ m for solar type stars), while the impact on the interior and evolution properties becomes negligible for orbital distances exceeding 0.1 AU. Nevertheless, we are developing for 2012 irradiated models and the capacity to compute them via the PHOENIX simulator (see below).

Summary and Futur Prospects

Beyond cloud modelling and molecular opacities, model atmospheres for these objects require reaction rates for the most abundant molecules and/or most important absorbers. Furthermore, these atmospheres are composed of molecular hydrogen which constitute the main source of collisions. Also needed are therefore collision rates (by H2) and corresponding damping constants for the broadening molecular lines.

In order to say something about the spectral variability of VLMs, brown dwarfs and planets, 3D global or ”star in a box” RHD simulations with rotation will be required. This is our current project supported by the French “Agence Nationale de la Recherche” for the period 2010-2015.