MODEL ATMOSPHERES OF VERY LOW MASS STARS AND BROWN DWARFS

France Allard,¹ ­ Peter H. Hauschildt,² ­ David R. Alexander,¹ and ­ Sumner Starrfield³ ­

¹Department of Physics, Wichita State University, Wichita, Kansas 67260-0032; e-mail: allard@eureka.physics.twsu.edu ; dra@twsuvm.uc.twsu.edu

ABSTRACT	Section:	ChooseTop of pageABSTRACT <<INTRODUCTIONGENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIESGRAINSLINE ABSORPTIONCONVECTIONSTELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

Abstract  As progressively cooler stellar and substellar objects are discovered, the presence first of molecules and then of condensed particulates greatly complicates the understanding of their physical properties. Accurate model atmospheres that include these processes are the key to establishing their atmospheric parameters. They play a crucial role in determining structural characteristics by setting the surface conditions of model interiors and providing transformations to the various observational planes. They can reveal the spectroscopic properties of brown dwarfs and help establish their detectability. In this paper, we review the current state-of-the-art theory and modeling of the atmospheres of very low mass stars, including the coolest known M dwarfs, M subdwarfs, and brown dwarfs, i.e. T_eff 4,000 K and 4.0 [M/H] +0.0. We discuss ongoing efforts to incorporate molecular and grain opacities in cool stellar spectra, as well as the latest progress in (a) deriving the effective temperature scale of M dwarfs, (b) reproducing the lower main sequences of metal-poor subdwarfs in the halo and globular clusters, and (c) results of the models related to the search for brown dwarfs.

INTRODUCTION	Section:	ChooseTop of pageABSTRACTINTRODUCTION <<GENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIESGRAINSLINE ABSORPTIONCONVECTIONSTELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

The crop of extremely cool stars and substellar objects has been meager until very recently, when marked improvements in detection ability have finally started to yield a rich harvest. Stars with masses as low as 0.1M and white dwarfs as cool as 5000 K (Paresce et al 1995, Richer et al 1995, Cool et al 1996, Renzini et al 1996) have been resolved in nearby globular clusters using the Wide Field/Planetary Camera 2 on board the Hubble Space Telescope (HST). Charge-coupled device (CCD) astrometry has unveiled uncharted M dwarfs in the immediate vicinity of our Solar System (Henry et al 1995, Stone et al 1996, Tinney 1996). Growing numbers of halo carbon dwarfs have been discovered by deep multicolor CCD surveys (Heber et al 1993, Warren et al 1993, Deutsch 1994, Liebert et al 1994). Young deuterium-burning brown dwarfs have been identified in nearby open clusters (Rebolo et al 1995, Basri et al 1996, Rebolo et al 1996) and in the solar neighborhood (Thackrah et al 1997) using the Keck telescope. Cryogenic coronographic imaging and astrometric surveys are revealing cool, evolved brown dwarfs hiding in the solar neighborhood (Nakajima et al 1995, Oppenheimer et al 1995, Mazeh et al 1996, Williams et al 1997), and high signal/noise radial velocity and astrometric surveys are now sensitive to massive planets around nearby Sun-like stars (Mayor & Queloz 1995, Butler & Marcy 1996, Marcy & Butler 1996a, b, Butler et al 1997, Marcy et al 1997). These exciting developments, and those of the MACHO, EROS, and OGLE microlensing surveys (Aubourg 1995, Alcock et al 1996), will soon enable a reconstruction of the faint end of the galactic initial mass function and the determination of the baryonic fraction of the missing mass (Chabrier et al 1996b, Flynn et al 1996, Graff & Freese 1996). Very low mass (VLM) stars and brown dwarfs are probably the most numerous objects in the galaxy (Gould et al 1996, Méra et al 1996a). Nevertheless, until recently, little about their atmospheres, evolution, or spectral characteristics was clearly understood. The presence of both a wide variety of molecular absorbers (each with hundreds of thousands to millions of spectral lines) and numerous condensates greatly complicates accurate modeling of these cool stellar atmospheres. The extension of the convection zone to the outermost photospheric layers means that evolutionary models depend critically on accurate handling of the surface boundary. Recently, theoretical calculations of important molecular absorbers such as H₂O, TiO, CN, and CO, as well as grain opacities, have made it possible to generate greatly improved models of cool stellar and brown dwarf atmospheres, their high resolution spectra, and their evolution. In view of the latest progress both on the observational and theoretical fronts, it seems timely to summarize here our present understanding of the atmospheres and spectroscopic properties of cool VLM stars and substellar brown dwarfs. Previous reviews of these topics can be found in Liebert & Probst (1987), Stevenson (1991), Bessell & Stringfellow (1993), Burrows & Liebert (1993), Gustafsson & Jørgensen (1994).

GENERAL SPECTROSCOPIC PROPERTIES OF COOL DWARFS	Section:	ChooseTop of pageABSTRACTINTRODUCTIONGENERAL SPECTROSCOPIC PRO... <<A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIESGRAINSLINE ABSORPTIONCONVECTIONSTELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

A VLM star generally refers to a main sequence star with a spectral type ranging from mid K to late M and a mass from about 0.6 M to the hydrogen-burning minimum mass (0.0750.085 M, depending on metallicity). Such stars span a wide range of populations, from the youngest metal-rich M dwarfs in open clusters such as the Pleiades (Simons & Becklin 1992, Hambly et al 1993, Williams et al 1996, Zapatero Osorio et al 1997), the Hyades (Leggett & Hawkins 1989, Reid 1993, Bryja et al 1994, Leggett et al 1994), Ophiuchus (Cameron et al 1993), Persei (Zapatero Osorio et al 1996), and the galactic disk (Gliese & Jahreiss 1991), to the several billionyear-old metal-poor subdwarfs of the galactic halo (Green et al 1991, Monet et al 1992, Green & Margon 1994) and globular clusters (Paresce et al 1995, Richer et al 1995, Cool et al 1996, Renzini et al 1996). Even within the solar neighborhood, M dwarfs do not form a homogeneous sample of unique age and metallicity, but rather they span up to 10³ years in age and ±0.51.0 dex around the solar value in metallicity (Burrows & Liebert 1993).

In VLM star and brown dwarf atmospheres, most of the hydrogen is locked in H₂ and most of the carbon in CO, with excess oxygen bound in molecules such as TiO, VO, and H₂O. The energy distribution of a typical late-type M dwarf is entirely governed by the absorption of TiO and VO in the optical, and H₂O in the infrared, leaving no window of true continuum in the emergent spectrum. With their extremely low intrinsic faintness (10 ² 10 ⁵ L), in particular in the V bandpass, the painstaking spectral classification of the nearby stars aiming toward a complete census of the luminosity function is still in progress (Reid 1994, Kirkpatrick & Beichman 1995, Liebert et al 1995, Reid et al 1995a). Fortunately, the groundwork necessary to construct an effective temperature scale at the lower end of the main sequence has already been laid by Boeshaar (1976) in the visual (0.440.68 m), with (a) the first identification of CaOH bands at 0.540.556 m in dwarfs later than about M3.5 (these bands are excellent temperature indicators and good discriminants between field M dwarfs and background red giant stars); (b) the first report of a saturation of the visual TiO band strengths in M dwarfs later than M5; and (c) the introduction of the VO to TiO band strength index now being used to classify M dwarfs and substellar candidates later than M5 (Henry et al 1994, Kirkpatrick et al 1995, Martín et al 1996). Boeshaar's classifications soon were extended beyond even the limits of the classical Morgan & Keenan spectral sequence, i.e. to types M9.5>M10, by Kirkpatrick et al (1995) in the optical to near-infrared regime (0.651.5 m) and by Davidge & Boeshaar (1993), Jones et al (1994), Leggett et al (1996) in the near infrared (1.12.5 m).

Figure 1 summarizes a typical near-infrared spectral sequence of M dwarfs to brown dwarfs. The near-infrared water vapor bands become slowly stronger with the spectral type of M dwarfs. The CO overtones near 2.3 m (and 4.5 m, not shown in Figure 1) are still apparent, although much weaker than in late-type giant stars due to the stronger H₂O "continuum" in the dwarfs. At the hydrogen-burning limit, i.e. at a spectral type of M10.5 and a T_eff of about 2000 K (Baraffe et al 1995), the peculiar spectral distribution of GD 165B (Zuckerman & Becklin 1992, Kirkpatrick et al 1993a) suggests that all signs of the TiO bands disappear from the optical spectral distribution, leaving only atomic lines and perhaps VO bands (Davis 1994, Kirkpatrick et al 1995), CaH, CaOH, and/or FeH bands. As the effective temperature drops into the brown dwarf regime, methane (CH₄) features begin to appear (Tsuji et al 1995, Allard et al 1996, Marley et al 1996), and corundum (Al₂O₃), perovskite (CaTiO₃), iron, enstatite (MgSiO₃), and forsterite (Mg₂SiO₄) clouds may form, enhancing the carbon:oxygen abundance ratio and profoundly modifying the thermal structure and opacity of the photosphere (Sharp & Huebner 1990, Fegley & Lodders 1996).

View larger version (59K) Figure 1 A near-infrared spectral sequence of M dwarfs to brown dwarfs. The observed spectral distributions ( full lines) were obtained at UKIRT for the M dwarfs by Jones et al (1994), and for the brown dwarf Gl229B by Geballe et al (1996). A comparison to OS models with, from top to bottom, Teff = 3400, 3000, 2700, 2600, 2000, and 1000 K (F Allard & PH Hauschildt, in preparation) (dotted lines) reveals a growing overestimation of water vapor band strengths with decreasing mass. The peculiar optical spectrum of GD 165B forces an arbitrary choice of the model parameters (here set to those of a star at the hydrogen-burning limit) for this object.

The chemistry of cool dwarf atmospheres is, therefore, a complex nonlinear problem requiring a detailed knowledge of the concentration of atoms and molecules, which prevents a straightforward derivation of quantities such as excitation temperatures and metallicities from line ratios, as is possible for hotter stars. The most reliable way to estimate effective temperatures and metallicities of VLM stars and to identify substellar brown dwarfs is by a direct comparison of observed and model spectra.

A BRIEF HISTORY OF THE MODELS	Section:	ChooseTop of pageABSTRACTINTRODUCTIONGENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO... <<MOLECULAR OPACITIESGRAINSLINE ABSORPTIONCONVECTIONSTELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

Advances in atmospheric modeling of cool stars have been slowed by the twin bottlenecks of (a) incomplete molecular opacity data bases and (b) the inability to handle convection rigorously. Once these problems are addressed reasonably well, we still face other challenges: incorporating the effects of photospheric grain formation, chromospheres, magnetic fields, departures from local thermodynamic equilibrium, spatial variations in atmospheric structure due to starspots, cloud formation, and eventually weather patterns. Model atmospheres incorporating such processes have only become possible within the past two decades with the work of Mould (1975, 1976), Allard (1990), Kui (1991), Brett & Plez (1993), Allard & Hauschildt (1995b), Brett 1995a, b, Tsuji et al (1996a) for M dwarfs; Saumon et al (1994) for zero-metallicity subdwarfs; and Tsuji et al (1995, 1996b), Allard et al (1996), Marley et al (1996) for substellar brown dwarfs.

Mullan & Dermot (1987) have reviewed early efforts in modeling M dwarf atmospheres. Mould (1975, 1976) was the first to produce an extensive grid of convective M dwarf model atmospheres between 4750 and 3000 K. The models effectively combined the ATLAS code (Kurucz 1970), TiO band model opacities and chemical equilibrium by Tsuji (1966, 1973), H₂O opacities by Auman (1967), and a mixing-length treatment of convection (Bøhm-Vitense 1958, Kippenhan 1962). Mould also incorporated atomic line blanketing in the form of an Opacity Distribution Function (ODF; see Kurucz 1970, Mihalas 1978). However, the coarseness of his opacity grid kept him from adequately reproducing the observed spectral characteristics of the coolest M dwarfs.

It took another 15 years before model calculations finally broke the "3000-K barrier" in T_eff, with the work of Allard (1990), Kui (1991). Both adapted their model codes from that of Wehrse (1972), who had treated the more extreme atmospheric conditions of cool white dwarfs (T_eff 7000 K). Both authors also handled molecular opacity using band models and straight mean (SM) techniques that made it possible to includebeyond the dominant TiO and H₂O opacitiesa number of important molecular bands such as those of the hydrides (CaH, MgH, SiH, OH, CH), which are important in low-metallicity subdwarfs, as well as the red and infrared bands of VO (Keenan & Schroeder 1952) and CO, respectively, which act as sensitive temperature indicators (Henry et al 1994, Kirkpatrick et al 1995, Martín et al 1996). From the Allard (1990) grid, Kirkpatrick et al (1993b) derived a revised temperature sequence for M dwarfs that casts new light on traditional results based on blackbody methods. This new sequence yielded values of T_eff as much as 500-K higher at a given luminosity and shifted the positions of the late-type dwarfs in the HR diagram from cooling tracks to the blue side of theoretical lower main sequences (D'Antona & Mazzitelli 1985, Burrows et al 1989, 1993). This made it more likely that field late-type M dwarfs were hydrogen-burning stars rather than young, contracting, substellar brown dwarfs. Subsequent improvements to these modelssuch as the introduction of (a) laboratory oscillator strengths for the TiO bands (Davis et al 1986) instead of the smaller (by a factor of 23) empirically derived astrophysical values of Brett (1989, 1990) and (b) the FeH Wing-Ford bands near 0.98 m (Phillips et al 1987)allowed Allard (1994) to resolve most of the remaining discrepancies in the optical model spectra that had been pointed out by Kirkpatrick et al (1993b), Gustafsson & Jørgensen (1994), Jones et al (1994).

Despite the initial successes, comparison with observed near-infrared spectra uncovered another problem: The models failed to match the infrared spectrum governed by the water vapor opacity profile (Allard & Hauschildt 1995b, Bessell 1995, Tinney et al 1995). This situation is illustrated in Figure 1, which shows that the water bands are clearly too strong in the metal-rich models. The peak of the energy distribution of M dwarfs is located in the near infrared, at around 1 m. For brown dwarfs, most of the emitted flux emerges between 1 and 10 m. One difficulty in determining the quality of model spectra is due to telluric absorption in the Earth's atmosphere. Telluric water bands filter the light of these faint objects over most of the infrared range. While some near-infrared spectra from about 0.92.5 m can be obtained from ground-based facilities (e.g. the UKIRT spectra of Figure 1), these are unreliable in intervals where the water bands are strongest. A proper calibration of the measured fluxes becomes even more delicate for faint brown dwarfs in close binary systems (for an illustration of the uncertainties in calibrating the "K" band fluxes in the spectrum of Gl 229B, see e.g. Oppenheimer et al 1995, Matthews et al 1995, Geballe et al 1996). Beyond 2.5 m, the Earth's atmosphere is nearly opaque and red dwarfs must be observed with infrared space-based facilities such as the HST, NICMOS, ISO, and the planned SIRTF, NGST, and DARWIN missions. But while there remain uncertainties in the absolute calibration of ground-based spectrophotometry of faint M dwarfs, these cannot completely account for the observed flux discrepancy in the infrared spectra of M dwarfs. For example, Figure 1 indicates that the predicted H₂O bands grow in strength more rapidly with decreasing T_eff than those of observed M dwarfs (Kirkpatrick et al 1995). This comparison supports the conclusion that there are shortcomings in the models. One of those shortcomings is clearly the treatment of opacity in very cool atmospheres.

MOLECULAR OPACITIES	Section:	ChooseTop of pageABSTRACTINTRODUCTIONGENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIES <<GRAINSLINE ABSORPTIONCONVECTIONSTELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

Most of the molecules that play an important role in cool star atmospheres have been known since the early 1930s from the work of Russell (1934) and later De Jager & Neven (1957). Some of the most extensive studies of cool stellar atmosphere chemistry are by Vardya (1966), Morris & Wyller (1967), Tsuji (1973), Gurvich (1981) who published equilibrium constants for an extensive list of diatomic and polyatomic species. More recent studies such as those of Sauval & Tatum 1984, Rossi et al (1985), Irwin (1987, 1988), Cherchneff & Barker (1992), Neale & Tennyson (1995), Sharp & Huebner (1990) provide partition functions for most molecules directly, which allows for more flexible atmospheric calculations.

In the absence of detailed lists of transitions, or sometimes to cope with restricted computational facilities, atmospheric modelers often resort to band models or to average opacities such as the Just Overlapping Line Approximation (JOLA), SM, or ODF techniques, which approximate (by a continuum distribution) the absorption within a band or a predefined wavelength bin (Kurucz 1970, Mihalas 1978, Tsuji 1994). While computationally economical, they make the assumption that the rotational fine structure is smeared out; i.e. the lines overlap without being saturated. Such conditions are never truly met even for the strongest bands of TiO and H₂O in the densest of the VLM stellar atmospheres, and these methods tend to overestimate the resulting molecular blanketing by trapping photons that would have otherwise escaped from between the lines. A far more accurate account of molecular and atomic opacities in model atmospheres is achieved by applying an Opacity Sampling (OS) treatment of transitions lists on a prespecified fine grid of wavelengths (Peytremann 1974, Sneden et al 1976). This can be done either dynamically within the atmospheric calculations (Kurucz 1992b, Hauschildt et al 1992, Allard & Hauschildt 1995b) or be pretabulated as a function of pressure, temperature, isotopic ratios, and wavelengths (Plez et al 1992, Brett 1995a, Kipper et al 1996). While the advantage of a dynamical approach lies in the flexibility of handling depth-dependent mechanisms such as pressure broadening, departures from local thermodynamic equilibrium, microturbulence, and abundance variations, the more efficient pretabulation of the OS opacities gives the modeler freedom to incorporate his or her choice of complete line lists.

In an attempt to address the too-strong infrared water band problem in M dwarf models, Brett & Plez (1993), Allard et al (1994, 1996), Brett (1995a, b), and F Allard & PH Hauschildt (in preparation) used the OS treatment of molecular opacities to compute a new generation of M dwarf model atmospheres, which brought important breakthroughs in the understanding of M dwarf atmospheres. In the next sections, we summarize the most significant improvements in the treatment of opacities due to TiO and H₂O.

Optical Bands

The strengths of TiO bands define the optical (0.41.2 m) spectral distribution of late K to M stars. Together with the VO bands and a few other optical spectral features, they constitute the primary T_eff indicators in very cool stars. There currently exist three TiO line lists generated (a) from first principles by Collins & Faÿ (1974) and more recently extended for isotopic species and the system by Jørgensen (1994), (b) empirically from molecular levels assigned in laboratory experiments by Kurucz (1993), and (c) by Plez et al (1992). While substantial errors in the Kurucz (1993) line list have been acknowledged by the author, the Plez et al (1992), Jørgensen (1994) line lists lead to great improvements upon previous models based on SM treatment of opacities (Mould 1975, Kui 1991, Allard & Hauschildt 1995b) in the modeling of M dwarfs. Each TiO line list applied in an OS treatment of the opacities leads to better agreement with the observed optical absolute magnitudes of M dwarfs (see e.g. Brett 1995a, b, Chabrier et al 1996a). The new models also show excellent agreement with the measured parameters of the only two known M dwarfs in eclipsing binaries (see Section 9 below; also see Bessell 1991, 1995, Chabrier & Baraffe (1995).

Unfortunately, the TiO line lists of Plez et al (1992), Jørgensen (1994) give poor line positions and relative band strengths that prevent accurate high-resolution spectral syntheses of M dwarfs (Piskunov et al 1996, Schweitzer et al 1996). They also fail to reproduce the optical R-I colors of late-type dwarfs (F Allard & PH Hauschildt, in preparation), which may reflect either some remaining inaccuracies in the current estimates of the oscillator strengths (Davis et al 1986, Doverstal & Weijnitz 1992, Hedgecock et al 1995) or an incomplete account of VO or other opacity in the "R" bandpass. Indeed, despite the existence of a few spectroscopic studies of VO systems (Davis 1994, Merer et al 1987, Bauschlicher & Langhoff 1986), no list of transitions and oscillator strengths adequate for stellar atmosphere modeling is yet available for this important molecule. The Berkeley program has generated extensive line lists for FeH (Phillips et al 1987, Phillips & Davis 1993; see also Balfour & Klynning 1994), which, however, lack matching oscillator strengths. Moreover, the complexity of the FeH molecule has prevented theoretical models (Langhoff & Bauschlicher 1990, 1991) from reproducing the observed spectrum of FeH (Langhoff & Bauschlicher 1994). A similar situation also prevails for the electronic systems of CaOH (Bernath & Brazier 1985, Ziurys et al 1992, 1996), despite their importance as one of the strongest visual bands in the spectra of M-type dwarfs. Modelers have resorted to band models for most of these molecular systems (Brett 1989, 1990, Brett & Plez 1993, Allard & Hauschildt 1995b), which overestimate the resulting opacity and compromise both high-resolution spectral analysis and the determination of accurate atmospheric parameters. Fortunately, a new ab initio calculation of TiO is currently under way (SR Langhoff & CW Bauschlicher, Jr, in preparation), which should soon enable improved modeling of some aspects of cool M dwarfs.

H₂O Bands

In view of the initial success obtained with an OS treatment of the TiO opacities for the optical spectral distribution of M dwarfs, Alexander et al (1989) and later Plez et al (1992) developed an OS table of randomly distributed H₂O lines derived from line strength and line spacing data measured in the laboratory Ludwig 1971. (Brett & Plez 1993, Brett 1995a, b) then used the Plez et al (1992) table in their models of M dwarfs, but this treatment still failed to reproduce the infrared spectra and colors of M dwarfs (Bessell & Stringfellow 1993, Bessell 1995).

In retrospect, this result was to be expected because H₂O lines overlap more than those of TiO, so SM treatment of opacities is more appropriate for H₂O. Schryber et al (1995) therefore argued, based on results of their ab initio calculations for H₂O, that the H₂O laboratory cross sections obtained by Ludwig (1971), used by both groups in the form of either SM (Allard & Hauschildt 1995b) or OS (Plez et al 1992, Brett & Plez 1993, Brett 1995a, b) may be intrinsically overestimated when applied to gas hotter than about 1500 K. Theoretical lists of transitions that include "steam" or "hot" band transitionsbased on molecular levels assigned in laboratory experiments (semiempirical; e.g. Kurucz 1992a) or on a molecular model from first principles (ab initio; e.g. Miller et al 1994)are of far greater relevance for atmospheric calculations and are essential for an adequate account of molecular opacities in cool star and brown dwarf atmospheres. Over the past decade, efforts have converged in the development of improved theoretical opacity data for molecules of astrophysical interest with the creation of the Kurucz (1992a) and SCAN (Jørgensen 1992) data bases, and with the fruitful work of the University of the College of London (Miller et al 1994) and NASA Ames (Langhoff & Bauschlicher 1994) centers of quantum chemistry calculations.

Theoretical line lists for hot H₂O from three independent sources have recently been released by Jørgensen et al (1994), Miller et al (1994), Partridge & Schwenke (1997). The Miller et al (1994) list (6.2 million lines) uses a laboratory potential surface (Jensen 1989), while the Partridge & Schwenke (1997) list (300 million lines) uses a purely theoretical potential but the same computational approach. Both preliminary lists were computed up to J values of about 30; i.e. they do not include all the necessary hot or steam bands. The Jørgensen et al (1994) list (20 million transitions) on the other hand, while also based on the Jensen (1989) potential energy functions, was computed with the goal of completeness for the atmospheres of cool giants with some compromise on the treatment of the molecular binding. For example, they use a rigid rotator approximation with an a posteriori correction to the Hamiltonian. The three data sets lead to very different opacity profiles, with the Jørgensen et al and Partridge & Schwenke lists reproducing the results obtained previously with the Ludwig opacities. Only the Miller et al line list led to an improved fit of the infrared spectral distribution of M dwarfs (Allard et al 1994, Jones et al 1995, 1996, Leggett et al 1996), as well as to an excellent agreement of early-type M dwarfs with a whole new generation of evolutionary models that include improved non-gray surface boundary conditions (Baraffe et al 1995, 1997, Chabrier et al 1996a, Leggett et al 1996). However, none of the current H₂O line lists can explain the apparent saturation of the water vapor bands observed in the latest-type M dwarfs and illustrated in Figure 1. The cause of those discrepancies may therefore lie elsewhere, as is discussed in Section 5 below. A more accurate knowledge of the water vapor opacity profile is clearly needed and is now being addressed by the work of Viti et al (1995), Partridge & Schwenke (1997).

GRAINS	Section:	ChooseTop of pageABSTRACTINTRODUCTIONGENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIESGRAINS <<LINE ABSORPTIONCONVECTIONSTELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

The current generation of M dwarf model atmospheres (Brett 1995b, Allard et al 1996) does not include the condensation of molecules to grains. Condensation clearly must be included in the calculations as indicated by the work of Sharp & Huebner (1990), who report the abundance of condensates as a function of the gas conditions. If ZrO₂ one of the first condensates to appear at gas temperatures 2000 Kis not an important species in M dwarf atmospheres, the condensation of corundum at 1800 K and iron, VO, and enstatite at 1600 K most certainly affects the spectral distribution of late M dwarfs and brown dwarfs because of the large extinction of solid particles. The importance of condensation in the atmospheres of late-type M dwarfs and brown dwarfs has been confirmed by Tsuji et al (1996a, b), Fegley & Lodders (1996), who find large concentrations of such condensates in their model atmospheres.

The impact of condensation on the spectral distribution and atmosphere of a cool dwarf is to gradually deplete the gas phase abundance of titanium, iron, vanadium, and oxygen. If we ignore for the moment the opacity of the grains, the result is a more transparent optical spectral distribution because the TiO-, VO-, FeH-, and metal-line opacities decline with decreasing effective temperature of the star. This should be reflected by an observed saturation of these molecular bands in the latest-type M dwarfs and brown dwarfs, a behavior that is presently difficult to ascertain without accurate model atmospheres that incorporate the effects of condensation. Perhaps a confirmation can be found in the peculiar optical spectrum of the coolest known M dwarf, GD 165B, mentioned in Section 2 above. However, the true nature of GD 165B's atmosphere is uncertain because this object, the companion of an old pulsating DA white dwarf within an orbital distance of 128 AU (Becklin & Zuckerman 1988, Bergeron & McGraw 1990, Zuckerman & Becklin 1992, Bergeron et al 1993, Kirkpatrick et al 1993a), may be more metal-poor and/or more carbon-rich than other nearby stars as a result of the white dwarf's prior evolution.

Tsuji et al (1996a) were the first to calculate model atmospheres for M dwarfs and brown dwarfs including not only grain formation but also grain opacities, the so-called dusty models. Their results showed that including corundum, iron, and enstatite opacities, while assuming arbitrarily spherical grains with sizes set to 0.1 m, could heat the photospheric layers and change the overall structure of the atmosphere. The resulting dusty spectral distributions of late-type M dwarfs were redder with weaker molecular spectral features than models without grain opacities, and they were shown to reproduce the infrared broadband fluxes of the latest-type M dwarfs, including GD 165B. If confirmed, this greenhouse effect, caused by the presence of photospheric grains, may help explain the observed saturation of the near-infrared water vapor bands discussed in Section 4 and illustrated in Figure 1, as well , as well as perhaps the R-I colors (see Section 4.1) of late-type dwarfs, which the grainless models of Allard & Hauschildt (1995b), Brett (1995a, b) fail to reproduce. The calculations presented by Tsuji et al (1996a), however, are coarse, and a better treatment of both the molecular and grain opacities, as well as the formal inclusion of dust scattering in the solution of the radiative transfer equation, can be achieved.

Early attempts to compute the opacity of grains were made by Cameron & Pine (1973), Alexander 1975. More detailed calculations including the effects of chemical equilibrium calculations and grain-size distributions were reported by Alexander et al (1983), Pollack et al (1985). Alexander & Ferguson (1994a, b) have described the computation of the opacity of grains with the inclusion of equilibrium condensation abundances, the effects of the distribution of grain sizes, and the effect of grain shape through the continuous distribution of the ellipsoid model of Bohren & Huffman (1983). These calculations include the absorption and scattering due to magnesium silicates, iron, carbon, and silicon carbide grains for a wide range of chemical compositions down to temperatures of 700 K. The direct inclusion of the equilibrium calculations of Sharp & Huebner (1990) in the future will allow for more detailed treatment of the effects of trace condensates, lower temperature opacity sources, and the effects of different elemental abundances. The inclusion of high-temperature condensates such as Al₂O₃ and CaTiO₃ may have significant effects on the opacity in cool star atmospheres, even though their abundance is quite small because of the high absorption and scattering efficiency of grains. For lower temperatures, the optical effects of species such as FeS, Fe₃O₄, and H₂O need to be included. Pollack et al (1994) have produced opacities for water, ammonia, methane, and other low-temperature condensates. They assume complete condensation of all condensible species and extend the temperature range down to 300 K. These opacities offer an excellent basis for future brown dwarf and Jovian-type planet atmosphere calculations. However, the extinction caused by grains in a stellar atmosphere depends critically on the rate of grain formation and the resulting size distribution.

Moreover, constraints imposed by the lack of detection of cloud layers in Jupiter by the Galileo atmospheric probe (Isbell & Morse 1996, Keane et al 1996), and of any trace of scattering by grains in the evolved brown dwarf Gl 299B (Allard et al 1996, Tsuji et al 1996b), may imply an inhomogeneous vertical and/or horizontal distribution of the grains, such as scarce cloud distribution, gravitational settling, and sedimentation and rains of condensates in substellar dwarf atmospheres. While grains are likely to be destroyed by the radiative and convective heat in the inner layers of the atmosphere, the main effects of the sedimentation and rains of condensates should be a radial abundance gradient (Muchmore 1987, Guillot et al 1994) and a gradual depletion of the upper photosphere from its condensible elements over time.

The effect of grain formation and of its opacity on the atmospheric structure of M dwarf atmospheres will, therefore, not be fully understood until grain formation and time-dependent grain growth calculations incorporating the effects of sedimentation, diffusion, coagulation, and coalescence are included. Gail & Sedlmayr (1988), Dominik et al (1989) (see references therein) have developed a formalism to account for the phenomena in the outflows from cool giants (Beck et al 1992), supergiants (Seab & Snow 1989), and nova atmospheres (Beck et al 1995). Grain growth models have also been developed for the atmospheres of cool carbon-rich white dwarf (Zubko 1987) and Jovian planet (Rossow 1978, Dobrijevic et al 1992) atmospheres. However, as yet, no results have been obtained for oxygen-rich dwarf atmospheres.

LINE ABSORPTION	Section:	ChooseTop of pageABSTRACTINTRODUCTIONGENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIESGRAINSLINE ABSORPTION <<CONVECTIONSTELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

The contribution of atomic and ionic line transitions to photospheric opacities is relatively less important for M dwarfs than for cool giants and hotter stars. This result arises not only from the fact that molecular absorption bands dominate opacity, but also because the lower photospheric temperatures cause the number densities (N_i ) of atoms in higher excitation and ionization levels, such as those of the hydrogen Balmer series, to be quenched (N_i e ^{_i /kT} , where T is the gas temperature and _i is the excitation potential or ionization energy relative to the ground state). Moreover, the "locking" of elements into molecular compounds and further condensation of such elements to grains also reduces the available abundances of atomic species, as is the case for hydrogen, which is about 7085% H₂ in the photospheres of M dwarfs.

As a result, only the strongest resonance and subordinate lines, with the low excitation energy of mostly alkali and earth-alkali elements, prevail in the spectra of M dwarfs. Those lines can be very broad owing to van der Waals (vdW) pressure broadening, and they often contrast greatly with the narrow emission and weak absorption lines that originate in the chromospheric layers of active M stars (such as the Balmer series and the Ca H and K lines). Only a few of the atomic lines that are created in the photospheric layers can be detected within the haze of molecular lines and provide diagnostics of the photospheric parameters. Examples include the Na ID lines at 5889,5896 Å, as well as other Na I resonance transitions at 8183,8195 Å and 10746,10749,10835 Å and those of K I at 6911,6939, 7665,7699, 9950,9954, and 10480,10482,10487 Å. Lines of Rb I at 7950 Å and Ba I at 7911,7913 Å are also particularly strong (relative to the local continuum) in late-type M dwarfs and brown dwarf candidates.

Despite the relative scarcity of directly observable atomic lines in their spectra, an accurate modeling of M dwarf atmospheres nevertheless requires the use of a complete atomic line list that includes lines of ionized elements for a complete account of the opacity in the hotter layers (typically about 8000 K in M dwarfs) of the inner atmosphere. A failure to do so may result in atmospheric structures that are too cool globally, as the efficient convection zone assures the transfer of inner atmospheric heat to the outer photospheric layers. The most complete list of atomic transitions currently available is that by Kurucz (1994) and its revisions. Several other line lists, such as those generated from first principle model atoms of the Opacity Project (Seaton 1992, Seaton et al 1992) or semiempirically using atomic levels assigned in laboratory experiments (see Verner et al 1996 and revisions), are also available but are still too incomplete for the purpose of model atmosphere calculations.

Line Broadening Mechanisms

The high densities prevailing in VLM star atmospheres cause strong spectral lines to be significantly broadened. Because the gas temperatures are not high enough to sustain a significant amount of ionization, the electron and proton densities are much smaller than the densities of the most important neutral and molecular species. Consequently, the contribution of Stark broadening to the total damping constant is very small, even in stars with very low metallicities. The total thermal plus microturbulent line widths are always much smaller than the line width owing to vdW broadening:

(1)

which describes the interaction between two different, unpolarized neutral particles within the impact or static approximation, with _vdW the full-width half-maximum damping constant of the resulting Lorentz profile, v the relative velocity between perturber and absorber, and N_p the number density of perturbers. While the interaction constant C ₆ can be determined exactly for both the ground and excited states of a perturbed atom when the perturber is atomic hydrogen (Michelis 1976), no exact method has yet been developed for the case of collisions with the much slower molecular hydrogen perturbers that dominate the atmospheres of VLM stars, brown dwarfs, and Jovian-type planets (Guillot et al 1994). In those cases, the collisions are not instantaneous and the profiles not strictly Lorentzian (Kunde et al 1982, Goody & Yung 1989), but in the absence of accurate alternatives, modelers often resort to using the hydrogenic approximation formulated by Unsöld (1955) for collisions with neutral hydrogen with some ad hoc modifications:

(2)

where Z is the charge of the absorber, E the ionization energy (e.g E _H = 13.6 eV), and E_l and E_u the lower and upper level excitation energies of the absorber. Investigations by Weidemann (1955), for instance, showed that the values as calculated above are in good agreement with observed line widths for alkali metals but not for other elements, such as iron (Kusch 1958). This has led to the introduction of correction factors to the "classical" formula, which can range from 10 C ₆ in the Sun (Takeda et al 1996) to 10^1.8 C ₆ in white dwarfs for nonalkali-like species (Wehrse & Liebert 1980). No corrections are required for alkali elements. The Unsöld (1955) approximation, combined with this correction factor for non-alkali elements and with an explicit account of the different polarizabilities of each perturber ( _p/ _H) C ₆, where the subscript p refers to the perturber), leads to improved profiles that appear to describe well the atomic lines observed in late-type M dwarfs (Schweitzer et al 1996). The most abundant perturbers in M dwarfs and their polarizabilities are given by Weast (1988), Schweitzer et al (1996).

While the situation is poor for atomic line broadening, it is even worse for molecular lines, for which only a few sources and techniques exist (Lazarev & Pnomarev 1992, Kurucz 1993, Guillot et al 1994). Fortunately, individual molecular lines are usually not saturated, so that broadening is less important for them than for strong atomic lines. Moreover, molecular lines often overlap so strongly that their wings are completely masked (Schweitzer et al 1996), and only the Gaussian line cores of the strongest molecular transitions are observed. The atmospheres of VLM stars and brown dwarfs are therefore only weakly sensitive to the adopted value of the vdW damping constant in the bands of several of the most important molecular absorbers (e.g TiO and H₂O). This may, however, not be the case for some hydride bands and for the infrared CO overtones that show larger typical line spacings (Kui 1991, Davis 1994, Tsuji 1994).

CONVECTION	Section:	ChooseTop of pageABSTRACTINTRODUCTIONGENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIESGRAINSLINE ABSORPTIONCONVECTION <<STELLAR ACTIVITYTHE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited

The thin radiative skin above the convective region in an M dwarf determines the surface boundary conditions for the entire temperature structure of the fully convective photosphere and interior. This radiative zone is often limited to the outermost optically thin regions of the photosphere in early-type M dwarfs (Allard 1990, Kui 1991, Burrows et al 1993, Allard & Hauschildt 1995b): i.e. to optical depths below about 10 ³. Figure 2 illustrates how the outer atmosphere of a typical M dwarf is affected by the atomic and molecular opacities and convection. At such low optical depths ( = 10 ³ corresponds to log P _gas 3.8 in this model), the structure of the atmosphere is sensitive to the strong opacities of TiO and H₂O. Early-type M dwarf atmospheres, spectra, colors, and even their evolution are, therefore, very dependent upon elemental abundances and the treatments of molecular opacities and possibly convection (see Section 9 below; see Baraffe et al 1995 for an illustration of these effects). Early-type M dwarfs should serve as excellent stellar laboratories in which to study convection.

View larger version (49K) Figure 2 The influence of the molecular and atomic opacities and convection upon the atmospheric structure of a typical model atmosphere; here the T eff = 2800 K, log g = 5.0, and solar metallicity model of Allard & Hauschildt (1995b). A corresponding gray structure without convection (bold dot-dashed) is also shown for comparison. While the complete neglect of H2O opacities causes a dramatic cooling (by CO) of the atmosphere (long-dashed curve), uncertainties by a factor of two in the H2O opacity cross sections cause only negligible changes in the atmospheric structure (thin dot-dashed relative to dotted curve). A similar drop in the opacity cross section of TiO, however (thin short-dashed relative to dotted curve), causes a much more significant cooling of the atmosphere.

The standard mixing length theory (Bøhm-Vitense 1958, Kippenhan 1962, Mihalas 1978) used to model convective energy transport in stars is only a crude approximation. While nonlocal treatments of convection exist (for review, see Chan et al 1991, Gustafsson & Jørgensen 1994, Grossman 1996, Kim et al 1996) that may be better suited to the optically thin medium of cool stellar atmospheres, they are very computationally prohibitive and have not been applied to models of M dwarfs. Fortunately or sadly, depending on your point of view, the large opacities in M dwarfs mean convection is nearly adiabatic for values of the mixing length () comparable to the atmospheric pressure scale height. The atmospheres and synthetic spectra of M dwarfs therefore show very little sensitivity to changes in over the range typical of solar-type atmospheres; i.e. H_P = 1.2 to 2.2 (Brett 1995a, Baraffe et al 1997). Moreover, models indicate that the convection zone gradually retreats with decreasing mass in late-type M dwarfs because of their decreasing luminosity and with decreasing metallicity due to decreasing photospheric opacities (Allard 1990). In cool brown dwarf models such as those computed for Gl 229B, for example, the convection zone reaches no higher than optical depths of about unity (although models show signs of a second, separate convective layer closer to the surface in such cool objects). The spectroscopic and photometric properties of late-type M dwarfs, metal-poor subdwarfs, and possibly brown dwarfs are therefore relatively insensitive to the details of convection (see e.g. Brett 1995a). This means that standard mixing length approximations are probably suitable for these stars (which is good news for brown dwarf modelers), but that these same stars are not very good laboratories to study convection (which is bad news for convection modelers).

STELLAR ACTIVITY	Section:	ChooseTop of pageABSTRACTINTRODUCTIONGENERAL SPECTROSCOPIC PRO...A BRIEF HISTORY OF THE MO...MOLECULAR OPACITIESGRAINSLINE ABSORPTIONCONVECTIONSTELLAR ACTIVITY <<THE Teff SCALE OF M DWARF...THE DETECTABILITY OF BROW...THE METAL-DEFICIENT VLM S...CONCLUSIONSAcknowledgmentsLiterature Cited