1 Introduction

Brown dwarfs are the only stars known so far that are cool enough to host small solid particles or fluid droplets (henceforth called dust or dust grains) in their atmospheres. The dust has a strong influence on the opacity and hence on the structure of the atmosphere as well as on the spectral appearance of brown dwarfs (Allard ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$2001; Marley ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$2002; Tsuji 2002; Cooper ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$2002), e.g.by smoothing out molecular bands and thermalising the radiation, by increasing the temperature below optically thick cloud layers, and in particular, by affecting the element composition of the gas depth-dependently.

Furthermore, the dust component seems to be responsible for a wealth of variability phenomena recently observed (Bailer-Jones ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Mundt 2001a,b; Bailer-Jones 2002; Martín ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$2001; Eislöffel ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Scholz 2001). The observed light variations are partly non-periodic and, thus, cannot be explained solely by rotation and magnetic spots.

In order to study the atmospheres of these ultra-cool stars and giant gas planets, a consistent physical description of the formation, temporal evolution and gravitational settling of dust grains is required, which states a new fundamental problem to the classical theory of stellar atmospheres. A better physical description of the dust component is likely to provide the key not only to understand the variability of brown dwarfs, but also the structure of their atmospheres and the observations of brown dwarfs in general.

In comparison to other astronomical sites of effective dust formation (Sedlmayr 1994), the atmospheres of brown dwarfs provide special conditions for the dust formation process. The convection replenishes the gas in the upper layers with fresh uncondensed gas from the deep interior, probably in a non-continuous and spatially inhomogeneous way. The convection energises turbulence which creates strongly varying thermodynamical conditions on small scales, causing an inhomogeneous and time-dependent distribution of the dust (Helling ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$2001, henceforth called Paper I). Three further points are to be mentioned:

(i) The stellar gravity ( $\log~g\!\approx\!5$) is about a hundred times larger than in the earth's atmosphere and roughly 10⁴ to 10⁵ times larger than in the circumstellar envelopes of red giants. This strong gravity puts severe physical constraints on the dynamical behaviour of the forming dust component. Once formed from the gas phase, dust grains are immediately forced to sink downwards. The atmosphere can be expected to clean up from dust grains via gravitational settling on time-scales ranging in minutes to months, depending on the dust grain size. This is just the opposite as encountered in the circumstellar envelopes of red giants, where the forming dust grains are accelerated outwards due to radiation pressure.

(ii) The dust forming gas in brown dwarf atmospheres is very dense, $\rho\!\approx\!10^{-7}\!...~10^{-3}~\rm g~cm^{-3}$. On the one hand side, these high densities simplify a physical description of the dust formation process, since chemical equilibrium in the gas phase can be assumed. On the other hand side, the large densities lead to a quite different molecular composition of the gas (for example a simultaneous occurrence of CH₄, H₂O and CO₂, i.e.no CO-blocking), like in planetary atmospheres (e.g.Lodders ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Fegley 1994). Consequently, different chemical preconditions for the dust formation process are present as e.g.in M-type giants (Gail ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr 1998). Other nucleation species and other surface chemical reactions can be important for the dust growth process. Other solid compounds and even fluid phases can be stable in a brown dwarf atmosphere.

(iii) The high densities lead furthermore to a qualitatively different dynamical behaviour of the gas flow around a dust grain. The mean free path lengths of the gas particles can be smaller than a typical diameter of a grain, leading to small Knudsen numbers ${ Kn}\!<\!1$. This affects the range of applicability of certain physical descriptions at hand, concerning for example the drag force or the growth of a dust grain due to accretion of molecules from the gas phase, which may be limited by the diffusion of the molecules toward the grain's surface.

One of the key processes to understand the structure of planetary and brown dwarf atmospheres - including element depletion and weather-like features - is the dust sedimentation, which means a non-zero relative motion between the dust particles and the surrounding gas, known as the drift problem. Various approaches have been carried out to simulate the dynamics of dust/gas mixtures, e.g.in the circumstellar envelopes of late-type stars (Gilman 1972; Berruyer ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Frisch 1983; MacGregor ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Stencel 1992; Krüger ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1994, 1997). Mostly, two-fluid approaches have been applied, assuming a constant dust grain size, where nucleation is disregarded or assumed to be followed by an instantaneous growth to the mean particle size. Simis ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$(2001) and Sandin ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Höfner (2003) have relaxed this approach by allowing for a varying mean grain size, according to the results of a time-dependent treatment of the dust nucleation and growth according to (Gail ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1984; Gail ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr 1988). However, an unique velocity of the dust component is assumed. Assuming stationarity, Krüger ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$(1995) have developed a bin method for the 1D drift problem in stellar winds including a full time-dependent description of the dust component, which explicitly allows for a size-dependent drift velocity. This powerful approach has inspired Lüttke (2002) to develop an adaptive bin tracking algorithm where the evolution of each bin is followed in time and space, using the multi-grid method of Nowak (1993).

In the business of fitting the spectra of brown dwarfs and extra-solar gas planets, much simpler approaches have been adopted so far in order to study the effects of element depletion and dust sedimentation by gravitational settling. In the frame of static model atmospheres with frequency-dependent radiative transfer, the usual procedure is to remove heavy elements like Ti, Fe, Mg, $\ldots$ from the object's atmosphere, assuming that these elements have been consumed by dust formation guided by stability arguments (Burrows ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$1997; Burrows ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sharp 1999; Saegers ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sasselow 2000). Depending on the purpose of the model, dust formation is either simply disregarded, the dust is assumed to be fully present or to have rained out completely, leaving behind a saturated gas. An extensive time-scale study of dust formation and sedimentation for the atmospheres of Jupiter, Venus, and Mars has been presented by Rossow (1978). Ackermann ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Marley (2001) have extended these time-scale considerations to the turbulent regime for large dust Reynolds numbers by adopting various data fits. Cooper ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$(2002) have presented further time-scale arguments in consideration of an atmosphere with prescribed supersaturation to arrive at a maximum size of dust particles as function of depth, emphasising the influence of particles sizes on the resulting spectra. However, usually much simpler ad-hoc assumptions about the grain size distribution are made, e.g.relying on the size distribution function known from the interstellar medium (e.g.Allard ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$2001). Very recently, Tsuji (2002) has published photospheric models based on the assumption that the dust particles remain very small (smaller than the critical cluster size) such that the particles are continuously evaporating and re-forming. In this case, the problem of the gravitational settling does not occur. All these simple approaches allow for an easy use of up-to-date solid opacity data in the simulations, but a consistent theoretical description of the dust component is still not at hand.

In this paper, we aim at a solution of this new problem in stellar atmospheres. We formulate a physical description of the formation, the temporal evolution and the gravitational settling of dust grains in brown dwarf atmospheres, consistently coupled to the element consumption from the gas phase, by modifying and extending the moment method developed by Gail ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr (1988). This description is based on partial differential equations for the moments of the dust grain size distribution function in conservation form, which avoids an elaborate and time-consuming binning of the size distribution function, thus making a straightforward inclusion into hydrodynamics and classical stellar atmosphere calculations possible.

After the outline of the forces in the equation of motion, the concept of equilibrium drift is discussed in Sect. 2. Section 3 contains a physical description of the dust growth by accretion of molecules in the free molecular flow ( ${Kn}\!\gg\!1$) and in the viscous case ( ${Kn}\!\ll\!1$). Section 4 investigates the influence of the latent heat of condensation and the frictional heating due to particle drift on the growth process. In Sect. 5, our new description of the dust component by means of moment equations is developed. The character of these equations is discussed by analysing the corresponding dimensionless equations and characteristic numbers in Sect. 6. Section 7 comprises our conclusions and future aims.