A&A 485, 547-560 (2008)
DOI: 10.1051/0004-6361:20078220

Dust in brown dwarfs and extra-solar planets

I. Chemical composition and spectral appearance of quasi-static cloud layers

Ch. Helling1 - P. Woitke2 - W.-F. Thi3


1 - SUPA, School of Physics & Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland, UK
2 - UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, Scotland, UK
3 - SUPA, Institute for Astronomy, The University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, Scotland, UK

Received 4 July 2007 / Accepted 21 March 2008

Abstract
Aims. Brown dwarfs are covered by dust cloud layers which cause inhomogeneous surface features and move below the observable $\tau=1$ level during the object's evolution. The cloud layers have a strong influence on the structure and spectral appearance of brown dwarfs and extra-solar planets, e.g. by providing high local opacities and by removing condensable elements from the atmosphere causing a sub-solar metalicity in the atmosphere. We aim at understanding the formation of cloud layers in quasi-static substellar atmospheres that consist of dirty grains composed of numerous small islands of different solid condensates.
Methods. The time-dependent description is a kinetic model describing nucleation, growth and evaporation. It is extended to treat gravitational settling and is applied to the static-stationary case of substellar model atmospheres. From the solution of the dust moments, we determine the grain size distribution function approximately which, together with the calculated material volume fractions, provides the basis for applying effective medium theory and Mie theory to calculate the opacities of the composite dust grains.
Results. The cloud particles in brown dwarfs and hot giant-gas planets are found to be small in the high atmospheric layers ( $a \approx 0.01$ $\mu $m), and are composed of a rich mixture of all considered condensates, in particular MgSiO3[s], Mg2SiO4[s] and SiO2[s]. As the particles settle downward, they increase in size and reach several 100 $\mu $m in the deepest layers. The more volatile parts of the grains evaporate and the particles stepwise purify to form composite particles of high-temperature condensates in the deeper layers, mainly made of Fe[s] and Al2O3[s]. The gas phase abundances of the elements involved in the dust formation process vary by orders of magnitudes throughout the atmosphere. The grain size distribution is found to be relatively broad in the upper atmospheric layers but strongly peaked in the deeper layers. This reflects the cessation of the nucleation process at intermediate heights. The spectral appearance of the cloud layers in the mid IR (7-$20~\mu$m) is close to a grey body with only weak broad features of a few percent, mainly caused by MgSiO3[s], and Mg2SiO4[s]. These features are, nevertheless, a fingerprint of the dust in the higher atmospheric layers that can be probed by observations.
Conclusions. Our models predict that the gas phase depletion is much weaker than phase-equilibrium calculations in the high atmospheric layers. Because of the low densities, the dust formation process is incomplete there, which results in considerable amounts of left-over elements that might produce stronger and broader neutral metallic lines.

Key words: stars: atmospheres - stars: low mass, brown dwarfs - methods: numerical - astrochemistry

1 Introduction

Dust in the form of small solid particles (grains) becomes an increasingly important component in understanding the nature of substellar objects with decreasing $T_{\rm eff}$, i.e. brown dwarfs and giant-gas planets. Observations have started to provide direct evidence for dust clouds covering brown dwarfs (Cushing et al. 2006) and extrasolar giant-gas planets (Richardson et al. 2007; Swain et al. 2007; Pont et al. 2008; Lecavelier des Etangs et al. 2008). The search for biosignatures in extraterrestrial planets becomes more complicated if such cloud layers cover the atmosphere and efficiently absorb in the wavelength region where e.g. the Earth vegetation's red edge spectroscopic features (600-1100 nm, Seager et al. 2005) or the extraterrestrial equivalents are situated. In fact, the development of life seems impossible below optically thick cloud layers, because the star light needs to reach the surface to create the necessary departures from thermodynamic equilibrium that allow for structure formation. Furthermore, abundances of molecules like O2and O3 as the carriers of spectral biosignatures will be strongly affected by the presence of cloud layers, because of chemical surface reactions and the element depletion due to dust formation in the atmosphere. Thus, the understanding of the details of cloud formation physics and chemistry is a major issue in modelling substellar atmospheres.

This paper presents a kinetic approach for modelling quasi-static atmospheres with stationary dust cloud layers including seed particle formation (nucleation), grain growth, gravitational settling, grain evaporation, and element conservation (Sects. 2 and 3). The model is a further development of the time-dependent description presented in Helling & Woitke (2006, Paper V) and is particularly suited to treat the formation of ``dirty'' dust grains (i.e. particles composed of numerous small islands of different solid condensates) in the framework of classical stellar atmospheres with consistent radiative transfer and convection (Dehn 2007; Helling et al. 2008). The results of the models (see Sect. 4) describe the vertical cloud structure and the amount of dust formed in the atmosphere of substellar objects as well as the amount of condensable elements left in the gas phase. The models provide further details such as the material composition of the cloud particles, the mean grain sizes and the size distributions as a function of atmospheric height. Section 5 demonstrates what spectral features such cloud layers made of dirty solid particles exhibit, and from which temperature and pressure level these features originate.

  
2 Nucleation, growth and evaporation of precipitating dirty grains

In Paper V, a kinetic description of the nucleation, growth and evaporation of dirty dust particles was developed by extending the time-dependent moment method of Gail & Sedlmayr (1988). The basic idea is that a ``dirty'' solid mantle will grow on top of the seed particles, because these seeds can only form at relatively low temperates ( $T \la 1400$ K) where the oxygen-rich gas is strongly supersaturated with respect to several solid materials. The dirty mantle is assumed to be composed of numerous small islands of different pure condensates. The formation of islands is supported by experiments in solid state physics (Ledieu et al. 2005, also Zinke-Allmang 1999) and by observations of coated terrestrial dust particles (Levin et al. 1996; Korhonen et al. 2003). Note that we consider dust formation by gas-solid reactions only and omit solid-solid reactions and lattice rearrangements inside the grains in our model.

Table 1: Chemical surface reactions r assumed to form the solid materials s. The efficiency of the reaction is limited by the collision rate of the key species, which has the lowest abundance among the reactants. The notation ${\scriptstyle{}^1\!\!/\!{}_2}$ in the rhs column means that only every second collision (and sticking) event initiates one reaction (see $\nu ^{\rm key}_r$ in Eqs. (4) and (10)). Data sources for the supersaturation ratios (and saturation vapour pressures): (1) Helling & Woitke (2006); (2) Nuth & Ferguson (2006); (3) Sharp & Huebner (1990).

In the following, we extend this description to include the effects of gravitational settling (drift, rain-out, precipitation) which is important to understand the long-term quasi-static structures of brown dwarf and gas-giant atmospheres. The challenge here is to properly account for the element conservation when dust particles consume certain elements from the gas phase at the sites of their formation, transport them via drift motions through the gas and finally release the elements by evaporation at other places. Furthermore, we want to abandon the assumption made in Paper V that the number of elements equals the number of condensates, because there are typically many more condensates than elements.

We consider the moments $L_{j}(\vec{x},t)$ [cmj g-1] (j = 0, 1, 2,...) of the dust volume distribution function $f(V,\vec{x},t)$ [cm-6] (for more details, see Paper V, Sect. 2.1), where $\vec{x}$ and t are space and time. V [cm3] is the volume of an individual dust particle. The total dust volume per cm3 stellar matter, $V_{\rm tot}$, is given by the 3rd dust moment as

 \begin{displaymath} \rho~L_3 = \int_{V_{\rm\ell}}^{\infty} f(V) V {\rm d}V = V_{\rm tot}\quad\left[\rm cm^3\ cm^{-3}\right], \end{displaymath} (1)

where $\rho $ [g cm-3] is the mass density and $V_{{\rm\ell}}$ is the lower integration boundary. In an analogous way, we define the volume ${V_{\rm s}}$ of a certain solid species s by

 \begin{displaymath} \rho L_3^{\rm s} = \int_{V_{\rm\ell}}^{\infty} f(V) V^{\rm ~s} {\rm d}V = {V_{\rm s}}\quad\left[\rm cm^3\ cm^{-3}\right], \end{displaymath} (2)

where $V^{\rm ~s}$ [cm3] is the sum of island volumes of material s in one individual dust particle. For simplicity, we assume that $V^{\rm ~s}/V={V_{\rm s}}/V_{\rm tot}$ is constant for all dust particles at a certain position in the atmosphere, i.e. we assume a unique volume composition of all grains at $(\vec{x},t)$, such that $V_{\rm tot}=\sum {V_{\rm s}}$ and $L_3 = \sum L_3^{\rm s}$.

By means of this assumption, it is possible to express the integrals that occur after integrating the master equation (Eq. (1) in Paper V) over size in terms of other moments. The results are the dust moment conservation equations. The change of the partial volume of the solid s can then be expressed analogously to Eq. (23) in Paper V

 
$\displaystyle \frac{{\rm\partial} }{{\rm\partial} t}\left(\rho L_3^{\rm s}\righ... ...d} V\Bigg) = {V_{{\rm\ell}}}^{\rm s} J_\star + \chi_{\rm net}^{\rm ~s}~\rho L_2$     (3)

where we have assumed that the Knudsen numbers are large ( ${\rm Kn} \gg 1$, compare Paper II) and that the drift velocities ${\vec v}_{\rm dr}(V)$ of the dust particles can be approximated by the equilibrium drift velocities ${\vec v}_{\rm dr}(V)$ (final fall speeds).

The source terms on the rhs of Eq. (3) describe the effects of nucleation, growth and evaporation of condensate s. $J_\star = J({V_{{\rm\ell}}}) = f({V_{{\rm\ell}}})~\frac{{\rm d}V}{{\rm d}t}\big\vert_{V=V_l}$ [s-1 cm-3] is the stationary nucleation rate (see Sect. 3.3). ${V_{{\rm\ell}}}^{\rm s}$ [cm3] is the volume occupied by condensate s in the seed particles when they enter the integration domain in size space. The net growth velocity of condensate s $\chi_{\rm net}^{\rm ~s}$ [cm s-1] (negative for evaporation) is given by (Eq. (24) in Paper V)

 \begin{displaymath}\chi_{\rm net}^{\rm ~s} = \sqrt[3]{36\pi}~\sum^R_{r=1} \frac{... ...ey}} \left(1 - \frac{1}{S_{\!r}~b^{\rm ~s}_{\rm surf}}\right). \end{displaymath} (4)

Here, r is an index for the chemical surface reactions (see Table 1), $\Delta V_r^{\rm ~s}$ is the volume increment of solid s by reaction r ( $\sum \Delta V_r^{\rm ~s} = \Delta V_{\rm r}$), $n_r^{\rm key}$ is the particle density of the key reactant, ${v}_r^{\rm rel}$ is its thermal relative velocity and $\alpha_r$ is the sticking coefficient of reaction r. $S_{\!r}$ is the reaction supersaturation ratio and $b^{\rm ~s}_{\rm surf} = V_{\rm tot}/V_{\rm s}$ is a b-factor which describes the probability of finding a surface of kind s on the total surface. Putting $b^{\rm ~s}_{\rm surf}$ independent of V, we assume that all grains at a certain point in the atmosphere have the same surface and volume composition, i.e. the grain material is a homogeneous mix of islands of different kinds (for more details see Paper V).

The divergence of the drift term in Eq. (3) is treated in the following way. Assuming large Knudsen numbers and subsonic drift velocities, the equilibrium drift velocity is given by ${\vec v^{\hspace{-0.8ex}^{\circ}}}_{\rm dr}(V) = -\vec{e}_z~a\sqrt{\pi}~g~\rho_{\rm d} /(2~\rho~c_{\rm T})$ (Eq. (63) in Paper II), where a is the particle radius, $\vec{e}_z$ the vertical unit vector (pointing upwards) and g the gravitational acceleration (downwards). $\rho_{\rm d} = \sum \rho_{\rm s}~{V_{\rm s}}/V_{\rm tot}$ is the dirty dust material density and $\rho_{\rm s}$ the material density of a pure condensate s. $c_T = \sqrt{2kT/\bar{\mu}}$ is the mean thermal velocity with T being the temperature, k the Boltzmann constant and $\bar{\mu}$ the mean molecular weight of the gas particles. Inserting this formula into the drift term in Eq. (3) yields

\begin{displaymath}\int_{V_{\rm\ell}}^{\infty} f(V) V^{\rm ~s} {\vec v^{\hspace{... ...} \frac{\rho_{\rm d}}{c_T} \frac{{V_{\rm s}}}{V_{\rm tot}} L_4 \end{displaymath} (5)

with the abbreviation $\xi_{\rm lKn}=\big(\frac{3}{4\pi}\big)^{1/3} \frac{\sqrt{\pi}}{2} g$ (note the difference to Eq. (3) in Paper III). Defining $L_4^{\rm s}=L_4~{V_{\rm s}}/V_{\rm tot}$, Eq. (3) in this paper can be rewritten as

 \begin{displaymath} \frac{{\rm\partial} }{{\rm\partial} t}\big(\rho L_3^{\rm s}\... ...partial} z} \left(\frac{\rho_{\rm d}}{c_T} L_4^{\rm s}\right). \end{displaymath} (6)

Equation (6) describes the evolution of the partial dust volume of solid s in space and time due to advection, nucleation, growth, evaporation and drift.

The third moment equation for the total dust volume $\rho L_3$ (Eq. (1) in Paper III) can be retrieved by summing the contributions from all condensates s as given by Eq. (6), because

\begin{displaymath}\sum_{\rm s} L_3^{\rm s} = L_3, \sum_{\rm s} L_4^{\rm s} = L_... ...t}, \sum_{\rm s} {V_{{\rm\ell}}}^{\rm s} = {V_{{\rm\ell}}}\ . \end{displaymath} (7)

2.1 Quasi-static, stationary case

In the case of a plane-parallel quasi-static stellar atmosphere ${\vec{v}_{\rm gas}} =0$ and the dust component is stationary $\frac{{\rm\partial} }{{\rm\partial} t}(\rho L_j) =0$, i.e. the lhs of the moment equations vanish. Introducing a convective mixing on time scale $\tau _{\rm mix}$[*], we have derived the following equations in Paper III for this case (see Eq. (7) in Paper III)

 \begin{displaymath} -\frac{\rm d}{{\rm d}z} \left(\frac{\rho_{\rm d}}{c_{\rm T}}... ...tar + \frac{j}{3}~\chi^{\rm net}~\rho L_{\rm j-1} \right)\cdot \end{displaymath} (8)

These are the moment equations with respect to the total dust volume for dirty grains and we use them for j=0, 1, 2. As outlined in (Eq. (9) in Paper III), $\tau _{\rm mix}$ is the timescale for mixing due to convective motion and overshoot which decreases rapidly above the convective layers with increasing height in the atmosphere. However, instead of using Eq. (8) with j=3 for one pure condensate, we have to use a set of equations for dirty grains, i.e. the third dust moment equations for all volume contributions, that is, one equation for each condensate s taken into account. From (Eq. (6)) we find

 \begin{displaymath} -\frac{\rm d}{{\rm d}z} \left(\frac{\rho_{\rm d}}{c_{\rm T}}... ...}}^{\rm s} J_\star + \chi_{\rm net}^{\rm ~s}~\rho L_2\right). \end{displaymath} (9)

Equations (8) for $j \in\ \{0,1,2\}$ and Eqs. (9) for $ {\rm s} \in \{1,2,~...~,S\}$ (S is the number of solid condensates taken into account) form a system of (S+3) ordinary differential equations for the unknowns $\{L_1,L_2,L_3,L_4^{\rm s}\}$.

The element conservation equations are not affected by the drift motion of the dust grains. Therefore, Eq. (29) in Paper V remains valid, from which we derived in the static stationary case (compare Eq. (8) in Paper III)

 
$\displaystyle \frac{n_{\langle{\rm H}\rangle}\left(\epsilon_i^0-\epsilon_i\righ... ...ha_r} {\nu_r^{\rm key}} \left(1-\frac{1}{S_{\!r}~b^{\rm ~s}_{\rm surf}}\right),$     (10)

where i enumerates the elements. ${N_{{\rm\ell}}}$ is the number of monomers in the seed particles when they enter the size integration domain and $\nu_{i,0}$ is the stoichiometric coefficient of the seeds (TiO2seeds: 1 for $i=\rm Ti$ and 2 for $i=\rm O$). $\nu_{i,{\rm s}}$ is the stoichiometric coefficient of element i in solid material s.

The element conservation equations (Eq. (10)) provide algebraic auxiliary conditions for the ODE system (Eqs. (8) and (9)) in the static stationary case, i.e. one first has to solve the system of non-linear algebraic Eqs. (10) for $\epsilon_i$ at given $\{L_2,L_4^{\rm s}\}$ (the dust volume composition $b^{\rm ~s}_{\rm surf}$ is known from $L_4^{\rm s}$) before the rhs of the ODE-equations can be calculated. Since $J_\star $, $n_r^{\rm key}$ and in particular $S_{\!r}$, however, depend strongly on $\epsilon_i$, this requires a complicated iterative procedure which creates the most problems in practise.

  
2.2 Grain size distribution function

The dust opacity calculations with effective medium theory and Mie theory (see Sect. 5) require the dust particle size distribution function $f(a)~\rm [cm^{-4}]$ at every depth in the atmosphere, where $a~\rm [cm]$ is the particle radius. This function is not a direct result of the dust moment method applied in this paper. Only the total dust particle number density $n_{\rm d}=\rho L_0$ and the mean particle size $\langle a\rangle=\sqrt[3]{3/(4\pi)}~L_1/L_0$ are direct results that have been used in Helling et al. (2006). In this paper, we reconstruct f(a) from the calculated dust moments Lj (j=1...4) in an approximate way. We want to avoid the zeroth moment, because it is only determined by a closure condition. The idea is to introduce a suitable functional formula for f(a) with a set of four free coefficients, and then determine these coefficients from the known dust moments. For further details, see Appendix A. Two possible functions for f(a) are discussed in Appendices A.1 and A.2.

2.3 Closure condition

Since L0 appears only on the rhs of Eq. (8) for j=0, we need a closure condition in the form L0=L0(L1,L2,L3,L4) to solve our ODE-system. In this paper, we use the results of the size distribution reconstruction technique explained in Appendix A in application to the double delta-peaked size distribution function and write the zeroth dust moment as

\begin{displaymath}\rho L_0 = N_1 + N_2, \end{displaymath} (11)

where the two dust particle densities N1 and $N_2~\rm [cm^{-3}]$ are introduced in Appendix A.1.

   
3 Material equations and input data

To solve our model equations, we need an atmospheric $(T, p, {v}_{\rm conv})$-structure and additional material equations to calculate the number densities of the key reactants $n_r^{\rm key}$, the nucleation rate $J_\star $, the reaction supersaturation ratios $S_{\!r}$, and the sticking probabilities $\alpha_r$.

3.1 Atmospheric (T,p)-structure

The approach of this paper is to investigate the behaviour of the dust component in quasi-static substellar atmospheres and to propose how our approach to treat the micro-physical processes of the formation of composite dust grains (see Sect. 2) can be included in the framework of stellar atmosphere codes.

For this purpose, it is sufficient to use a given $(T, p, {v}_{\rm conv})$ structure representing a brown dwarf or giant-gas atmosphere. The atmospheric structure used for our models are cond AMES atmosphere structures[*]. The feedback of the dust formation and the presence of the dust on the atmospheric structure is thereby neglected in this paper, although the model code to calculate the $(T, p, {v}_{\rm conv})$ structures did incorporate some dust modelling (Allard et al. 2001). Such a decoupled approach may not be entirely satisfying, but the understanding of the physics of cloud layers requires some basic studies, before the feedback mechanisms can be attacked in the frame of highly nonlinear stellar atmosphere codes. For consistent models of our dust treatment in the PHOENIX stellar atmosphere code including radiative transfer and convection see (Dehn 2007; Helling et al. 2008a).

3.2 Gas-phase chemistry

We calculate the particle densities of all gaseous species, including $n_r^{\rm key}$ as described in Paper III according to pressure, temperature and the calculated, depth-dependent element abundances $\epsilon_i$ in chemical equilibrium. For the well-mixed, deep element abundances  $\epsilon^0_i$ we use solar abundances according to the cond AMES input model. For those elements that are not included in the calculations (abundances are calculated for Mg, Si, Ti, O, Fe, Al, Ca, S), we put $\epsilon_i = \epsilon^0_i$.

   
3.3 Nucleation

The nucleation rate $J_\star $ is calculated for ${\rm (TiO_2)}_N$-clusters according to Eq. (34) in Paper V, applying the modified classical nucleation theory of Gail et al. (1984). We use the value of the surface tension $\sigma$ fitted to small cluster data by Jeong (2000) as outlined in Paper III.

3.4 Growth/evaporation: choice of solid material and condensation experiments

Several dozens solid species have been treated in phase equilibrium in the literature (Sharp & Huebner 1990; Fegley & Lodders 1994). We have taken into account 12 solids (TiO2[s], Al2O3[s], CaTiO3[s], Fe[s], FeO[s], FeS[s], Fe2O3[s], SiO[s], SiO2[s], MgO[s], MgSiO3[s], Mg2SiO4[s]) in phase-non-equilibrium to calculate the formation and composition of the dirty grains. Our selection is guided by the most stable condensates which yet have simple stoichiometric ratios that ensure that these solids can be easily built up from the gas phase. The selection covers the main element sinks during dust formation. Our choice is furthermore inspired by the following experiments.

SiO2[s], MgO[s], FeO[s], Fe2O3[s], MgSiO3[s], Mg2SiO4[s]: Vapour phase condensation experiments offer strong arguments against equilibrium assemblages even under controlled conditions in the terrestrial laboratory. Rietmeijer et al. (1999) have shown that from a Fe-Mg-SiO-H2O vapour only $\rm Mg_{\it x}Fe_{\it y}O_{\it z}$-condensates with simple stoichiometric ratios form during the condensation process in the laboratory, which is in accordance with some simple phase equilibrium calculations (e.g. Sharp & Huebner 1990), but is in contrast to well-established textbooks like Lewis (1997). Rietmeijer et al. (1999) report on the formation of the end-member oxides FeO[s], Fe2O3[s], MgO[s] and SiO2[s], and the fosterite olivine Mg2SiO4[s]. The appearance of SiO2[s] shows that kinetic factors can favour the formation of meta-stable states which is in contrast to phase equilibrium calculations.

Ferguson & Nuth (2006) revised the vapour pressure of SiO[s] and discuss implications for possible $\rm (SiO)_N$ nucleation. However, the experiments of Rietmeijer et al. (1999) which utilise an Fe-Mg-SiO-H2O vapour do not show any formation of SiO[s]. John (2002) pointed out that SiO[s] formation may proceed via SiO2[s] + Si[s] $\longrightarrow$ 2 SiO[s]. Since we are not yet able to treat solid-solid reactions, we must omit these processes in our model.

Fe[s], FeS[s]: We add Fe[s] as a high temperature condensate and FeS[s] as possible sulphur binding solid in accordance with the experimental findings by Kern et al. (1993) and Lauretta et al. (1996).

TiO2[s], Al2O3[s], CaTiO3[s]: ${\rm (TiO_2)}_N$acts as a nucleation species in our model (seed particle formation). For consistency, TiO2[s] must therefore also be included as a high-temperature solid compound to correctly account for the depletion of Ti from the gas phase. CaTiO3[s] is a very stable condensate and is included to study the consumption of Ca from the gas phase. According to our knowledge, no laboratory measurements on Ca-Ti oxides are available so far. The experiments by Kern et al. (1993) suggest the formation of Al2O3[s] as a high-temperature condensate.

Beckertt & Stolper (1994) have found that by melting a CaO-MgO-Al2O3-SiO2-TiO2 system, complex compounds like CaAl4O7, Ca3Ti2Al2Si2O14 and Ca3Al2Si4O14 can form. It seems logic to conclude that simpler compounds like MgO etc. need to form before more complex compounds can be built. It might be interesting to consider Earth-crust-like solids like CaSiO3 or CaCO3. CaSiO3 is very abundant on Earth but it forms only under high-pressure conditions inside the Earth crust at 104-106 bar (Rossi 2006, priv. com.). CaCO3 (carbonaceous calcite), as all other carbon bearing solids, can only form in an oxygen-rich environment if the carbon is not locked into CO molecules. Under the high pressure conditions in brown dwarf atmospheres, this release takes place only below $\approx$800 K under chemical equilibrium conditions. Toppani et al. (2005) have concluded that carbonates can only form in a H2O(g)-CO2(g)-rich, high-density region under conditions far away from thermal equilibrium.


Our selection of 12 solids is assumed to be formed by 60 chemical surface reactions (see Table 1). We assume for the sticking coefficient $\alpha_r=1$ due to the lack of data (see discussion in Paper V). The Gibbs free energy data of the solid compounds, from which the supersaturation ratios S are calculated, is mostly taken from Sharp & Huebner (1990) which were obtained for crystalline materials. The data for TiO2[s] is given in Paper III. The new vapour pressure data for SiO (Ferguson & Nuth 2006) is used. The reaction supersaturation ratios $S_{\!r}$ are calculated as outlined in Appendix B of Paper V.


  \begin{figure} \par\hspace*{1.5mm}\includegraphics[width=8.89cm]{8220Fig1a.eps}\\ [1mm] \hspace*{-2mm}\includegraphics[width=8.8cm]{8220Fig1b.eps}\end{figure} Figure 1: Calculated cloud structure for a model with $T_{\rm eff} = 1800$ K and $\log g = 5$ ( cond AMES). 1st panel: prescribed temperature T (solid) and mixing time scale $\tau _{\rm mix}$ (dashed). 2nd panel: nucleation rate $J_\star $ (solid) and net growth velocity $\chi _{\rm net}$ (dashed). 3rd panel: mean grain size $\langle a\rangle=\sqrt[3]{3/(4\pi)}~L_1/L_0$ (solid) and mean fall speed $\langle v^{\hspace{-0.8ex}^{\circ}}_{\rm dr}\rangle=\sqrt{\pi}~g\rho_{\rm d} \langle a\rangle~/~(2\rho c_T)$ (dashed). The 4th panel depicts the effective supersaturation ratios $S_{\rm eff}$ for all solids involved.
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4 Results

  
4.1 Cloud formation and cloud structure

The vertical structure of the dust cloud layer in a brown dwarf atmosphere is depicted in Fig. 1. The results are shown for a stellar parameter combination typically associated with the L dwarf regime.

The cloud structure results from a hierarchical dominance of nucleation (uppermost layers), growth and drift (intermediate layers), and evaporation (deepest layers). In the uppermost layers, small grains of size $\sim$0.01 $\mu $m form by nucleation and grow further as they settle in the atmosphere. The particles reach a maximum size of about 200 $\mu $m at the cloud base, shortly before they completely evaporate. The fall speed of the grains first decrease with increasing depth because of the increasing ambient gas densities, and then re-increase as the particles grow rapidly. Finally, they fall faster than they can grow (rain) and reach an approximately constant fall speed of a few m/s. Eventually, the grains enter the hotter atmospheric layers where they are no longer thermally stable. The grains shrink in size and dissolve into the surrounding hot and convective gas (the Schwarzschild pressure, where the atmosphere becomes convectively unstable, is 3.54 bar in this model). These results reflect the stationary character of the dust component in substellar atmospheres, where dusty material constantly forms at high altitudes and settles downward, and simultaneously, fresh uncondensed material is mixed up by convective motion and overshoot (see Ludwig et al. 2006, 2006; Young et al. 2003). These general results resemble well the results of Paper III, where only one sample dust species, TiO2[s], was considered.

However, in comparison to Paper III, there are new features resulting from the inclusion of more than one solid growth species and the consideration of the more abundant condensates. The net growth speed of the grains $\chi _{\rm net}$ is never determined by a single species alone, but results from a complicated superposition where the individual contributions $\chi_{\rm net}^{\rm s}$ can be positive (growth) or negative (evaporation). The small kinks in $\chi _{\rm net}$ (see 2nd panel in Fig. 1) result from the evaporation of one material which becomes thermally unstable at a certain temperature, in this case Mg2SiO4[s] at around 1800 K and Fe[s] at around 2000 K.

4.2 Vapour saturation in the atmosphere

In order to discuss deviations from phase-equilibrium, we consider the supersaturation ratio S, which usually indicates net growth for S > 1 and net evaporation for S < 1. However, for the complicated growth reactions listed in Table 1, where mostly at least 2 gas particles need to collide with the grain's surface (called ``type III reactions'' in Paper V), the supersaturation ratio is not unique, but reaction-dependent (see Appendix B of Paper V). In addition, the $b^{\rm ~s}_{\rm surf}$-factors are involved to account for the inequality of the active surfaces for growth and evaporation (see Paper V). The thermal stability of the solids is then better defined by $\chi_{\rm net}^{\rm ~s} = 0$ (see Eq. (4)). In order to discuss the saturation of the atmosphere, we define a unique effective supersaturation ratio $S_{\!\rm eff}$ for each solid as

 
$\displaystyle \sum^R_{r=1} \left(1 - \frac{1}{S_{\!r}~b^{\rm ~s}_{\rm surf}}\ri... ...lta V_r^{\rm ~s} n_r^{\rm key} {\rm v}_r^{\rm rel} \alpha_r} {\nu_r^{\rm key}}.$     (12)

These effective supersaturation ratios $S_{\!\rm eff}$ shown in Fig. 1 (4th panel) demonstrate that none of the considered solids obeys the condition of phase-equilibrium ( $S_{\rm eff}=1$) throughout the entire cloud layer in our model. In particular, the atmospheric gas is strongly supersaturated in the uppermost layers where the nucleation takes place. However, in contrast to $\rm TiO_2$[s] discussed in Paper III, the more abundant silicates and iron compounds reach a state of quasi-phase-equilibrium ( $S_{\rm eff} \ga 1$) already at p =0.05...1 bar whereas the rare condensates like TiO2[s], Al2O3[s], and CaTiO3[s] practically never obey the phase-equilibrium condition. This behaviour is caused by the different element depletion time scales being longest for the low abundant elements as we have demonstrated in Paper V (Sect. 4.4.3.). Since the observable molecular features are typically formed high in the atmosphere and even the broad dust features in the wavelength region 7-$20~\mu$m originate from layers $p \approx 0.2$...0.6 bar in this model (see Sect. 5.2), we conclude that the observable dust in brown dwarf atmospheres in not in phase-equilibrium with the gas.

Table 2: Typical dust volume composition $V_{\rm s}/V_{\rm tot}$, mass composition $M_{\rm s}/M_{\rm tot}$, and mean particles sizes $\langle a\rangle~[\mu$m] as a function of local temperature for models as depicted in Fig. 2. ($\nearrow $) - increasing in T-interval; ($\searrow $) - decreasing in T-interval.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{8220Fig2a.eps}\par\includegraphics[width=8.8cm,clip]{8220Fig2b.eps}\end{figure} Figure 2: Volume fractions $V_{\rm s}/V_{\rm tot}$ ( top) and mass fractions $M_{\rm s}/M_{\rm tot}$ ( bottom) of the various solid compounds for the model depicted in Fig. 1. The temperature scale follows the atmospheric stratification, being cool at the top and hot at the bottom.
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  \begin{figure} \par\includegraphics[width=8.8cm,clip]{8220Fig3a.eps}\\ \includegraphics[width=8.8cm,clip]{8220Fig3b.eps}\end{figure} Figure 3: Volume fractions $V_{\rm s}/V_{\rm tot}$ for different $T_{\rm eff}$ ( top) and different $\log~g$ ( bottom). Only the more abundant solid compounds are viewed, in contrast to Fig. 2.
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4.3 Chemical composition of the cloud particles

The chemical composition of the dust grains is expressed by the solid material volume fractions $V_{\rm s}/V_{\rm tot}$ (Fig. 2, top panel) and mass fractions $M_{\rm s}/M_{\rm tot} = V_{\rm s}~\rho_{\rm s}/(V_{\rm tot}~\rho_{\rm d})$ (lower panel). The dust material composition is mainly controlled by the temperature, and shows a constantly re-occurring pattern in models with different stellar parameter for brown dwarfs and gas giant-planets (see also Table 2). Once a solid becomes thermally unstable, the respective elements evaporate into the gas, making them available again to form other, more stable condensates. In this way, we find a complicated mix of all condensates high in the atmosphere with volume fractions resulting from kinetic constraints that favour the formation of the more abundant silicates and Fe-oxides. As these dirty particles settle in the atmosphere they stepwise purify, until only the most stable component parts like Al2O3[s], Fe[s], and CaTiO3[s] remain.

Therefore, two classes of solids can be distinguished: the high-temperature condensates Fe[s] and Al2O3[s] with some contributions of Ca-Ti-oxides in the deeper layers, and the medium-temperature condensates MgSiO3[s], Mg2SiO4[s] and SiO2[s] with some SiO[s] and FeS[s] in the upper layers.

As the temperature increases, SiO[s] starts to evaporate and SiO2[s] becomes the second most abundant solid material by volume fraction. Next, MgSiO3[s] evaporates which sets free some Mg and Si to form more Mg2SiO4[s] and SiO2[s], making SiO2[s] again the second most abundant solid. The transition from the medium-temperature to the high-temperature composition is characterised by the evaporation of Mg2SiO4[s], which leaves only Fe[s], Al2O3[s], TiO2[s] and CaTiO3[s] as stable condensates. As Fe[s] evaporates, the cloud base is soon reached where eventually the remaining Al-Ca-Ti oxides evaporate.

The volume fractions are important for the dust opacities (see Sect. 5), and in the observable layers the Mg-silicates turn out to be the most relevant condensates. For completeness we note, however, that the mass fraction $M_{\rm s}/M_{\rm tot}$ is dominated by Fe[s] from about 1200 K to 2000 K.

We recognise that the chemical composition of the dust grains does to some extent depend on the completeness of the selection of solids and on the availability of material quantities (e.g. the sticking coefficients as shown in Sect. 4.6. in Paper V). For example, the partial volume of Mg2SiO4[s] and MgSiO3[s] decreases if MgO[s] is included, and the partial volume of SiO2[s] decreases if SiO[s] is included. For this reason, the partial volume of SiO2[s] is smaller than described in Helling et al. (2006).


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{8220Fig4.eps}\\ \end{figure} Figure 4: Comparison of results for different stellar parameter. $1{\rm st}$ box: mass density $\rho $, 2nd box: nucleation rate $J_\star $, 3rd box: net growth velocity $\chi ^{\rm net}$ 4th box: mean particle size $\langle a\rangle$, 5th box: mean fall speed $\langle v^{\hspace{-0.8ex}^{\circ}}_{\rm dr}\rangle$. All quantities are plotted as function of temperature T. Dotted line: $T_{\rm eff}=1300$ K, $\log~g = 5$; solid line: $T_{\rm eff} = 1800$ K, $\log~g = 5$; dash-dotted line: $T_{\rm eff}=1300$ K, $\log~g = 3$; dashed line: $T_{\rm eff} = 1800$ K, $\log~g = 3$.
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4.4 Dependence on Teff and log g

Figure 3 shows the dependence of the dust material composition on $T_{\rm eff}$ and $\log~g$. If plotted against temperature, the volume fractions show a robust pattern for all calculated models (see Table 2). Only a slight shift to higher temperatures (deeper layers) can be noticed for lower $T_{\rm eff}$ (upper plot) and higher $\log~g$ (lower plot).

An increase of $\log~g$ results in a more compact atmosphere, i.e. generally higher pressures in the atmosphere. Since the dust sublimation temperatures increase with increasing pressure, the dust remains stable to even higher temperatures for larger $\log~g$. For the same reason, models with lower $T_{\rm eff}$ show dust at comparably higher temperatures. For lower $T_{\rm eff}$, a certain temperature is reached deeper inside the atmosphere, where the pressure is higher and, hence, the dust is more stable.

Figure 4 shows the dependence of the nucleation rate, the mean particle size and the mean fall speed on $T_{\rm eff}$ and $\log~g$. The nucleation zone typically lies between 600 K and 1400 K in all models. The maximum rate reaches higher values for higher $\log~g$ because of the higher gas densities. For lower $\log~g$, the nucleation maximum is more extended and shows a more complicated shape that is probably caused by the closer neighbourhood to the convective zone, which makes the up-mixing of fresh elements for nucleation more likely.

Concerning the mean particle size, all models show about the same small particles of order 0.01 $\mu $m high in the atmosphere, which grow to large particles between about 100 $\mu $ and 1000 $\mu $ in the deep layers. The maximum particle sizes reached at cloud base are larger for lower $\log~g$. As the nucleation zone ends, there is a zone of rapid grain growth around 1400 K to 1700 K. In this zone, most of the solid particles are actually ``mixed away'' according to our simple approach to treat the mixing by convective motion and overshoot. The small number of particles that stay, however, settle deeper into a denser environment where elements are mixed with higher efficiency due to the decreasing distance from the convection zone. The models show that this small number of growing particles is sufficient to maintain a state close to phase equilibrium with the gas for most elements, i.e. the growth is exhaustive. Thereby, the particles grow further along their way down the atmosphere while their number is ever decreasing due to lethal mixing.

The growth of the particles, however, is limited by their own fall speed which increases with size. The grains eventually reach a size where their residence time is so small that further growth becomes negligible. The residence time scale $\tau_{\rm sink} = v^{\hspace{-0.8ex}^{\circ}}_{\rm dr}/H_p$, however, depends on the atmospheric scale height Hp which is 100 times larger for the $\log~g = 3$models. Therefore, the dust particles have more time to grow in the low $\log~g$ models, producing larger sizes and larger fall speeds.

We note that the Knudsen numbers fall short of unity in the deeper layers, for particles larger than about 1 $\mu $m in the $\log~g = 5$ models and about 10 $\mu $m in the $\log~g = 3$models. This means, that the frictional force and the growth velocity should be calculated for the case of small Knudsen numbers in the deeper layers, making necessary a Knudsen number fall differentiation (see Paper II for details). Therefore, the results of this paper concerning the particle sizes in the deeper layers must be taken with care. The true fall speeds are probably higher there and the true growth velocities are probably smaller than our results, i.e. we expect that $\langle a\rangle$ remains smaller in the deeper layers compared to Fig. 4.


  \begin{figure} \par\includegraphics[width=8.8cm]{8220Fig5a.eps}\\ {\hspace*{1.5mm}\includegraphics[width=8.5cm]{8220Fig5b.eps}\hspace*{1.5mm}} \end{figure} Figure 5: Dust-to-gas mass ratios $\rho _{\rm d}/\rho _{\rm gas}$ and dust mass column densities $M_{\rm dust}$ for brown dwarf and giant-planets atmospheres. 1st panel: linear plot, 2nd panel: logarithmic plot. Dotted line: $T_{\rm eff}=1300$ K, $\log~g = 5$; solid line: $T_{\rm eff}=1800~$K, $\log~g = 5$; dash-dotted line: $T_{\rm eff}=1300$ K, $\log~g = 3$; dashed line: $T_{\rm eff}=1800~$K, $\log~g = 3$.
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Table 3: Maximum dust-to-gas ratio and temperature interval where the dust-to-gas ratio is larger than half maximum.


  \begin{figure} \par\includegraphics[width=8.8cm]{8220Fig6.eps}\end{figure} Figure 6: Element abundances in dust $\epsilon _{\rm d}$ (thick solid) and in the gas phase $\epsilon _{\rm i}$ (dashed). The solar values $\epsilon _{\rm i, Sun}$ (thin solid) and the linear dust-to-gas ratio $\rho _{\rm d}/\rho _{\rm gas}$ ( lowest panel) are shown for comparison. The stellar parameters are $T_{\rm eff} = 1800$ K and $\log~g = 5$ as in Fig. 1.
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  \begin{figure} \par\includegraphics[width=8.8cm,clip]{8220Fig7.eps}\end{figure} Figure 7: Element abundances involved in the dust formation process for two effective temperatures and two different gravities. Dotted blue: $T_{\rm eff}=1300$ K, $\log~g = 5$; solid green: $T_{\rm eff} = 1800$ K, $\log~g = 5$; dash-dotted blue: $T_{\rm eff}=1300$ K, $\log~g = 3$; dashed green: $T_{\rm eff} = 1800$ K, $\log~g = 3$.
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4.5 Dust-to-gas ratio

Figure 5 shows the dust-to-gas mass ratios $\rho _{\rm d}/\rho _{\rm gas}$ and the dust mass column densities $M_{\rm dust}=\int\rho_{\rm d}~{\rm d}z$ [g/cm2] as a function of pressure. The dust-to-gas ratio generally first increases inward log-linear, then reaches a plateau and finally decreases rapidly as the dust becomes thermally unstable. The upper plot shows the same quantities on a linear scale, further emphasising the truly dusty layers.

For the four models depicted in the previous figures, the dust-to-gas ratio reaches a constant maximum value between about 0.25% and $0.4\%$ in a temperature interval $T \in \rm [(1000$-1300) K ... (1600-1800) K] rather independent of $T_{\rm eff}$, where the temperature interval boundaries increase with increasing $\log~g$ (see Sect. 4.4). The width of the dusty temperature window is about 500 K to 600 K for the depicted models with $T_{\rm eff}=1300$ K and 1800 K (compare Table 3). This result is consistent with the simple dust approach made by Tsuji (2002), although the window is broader than assumed by Tsuji according to our findings. Similar maximum values are found even if we increase $T_{\rm eff}$ to 2200 K. However, if we increase $T_{\rm eff}$ further to 2500 K, the maximum dust-to-gas ratio drops by more than one order of magnitude (see Table 3). We conclude that $T_{\rm eff} \approx 2200$ K is the threshold value for truly dust-rich layers to occur in the models presented here.

The resulting dust-to-gas ratios demonstrate that the cloud layer is primarily attached to the local temperature. For lower $T_{\rm eff}$, the dust layer sinks deeper into the atmosphere and eventually disappears from the observable layers. Since the dust is then present in denser regions, the dust mass column density  $M_{\rm dust}$ increases.

4.6 Remaining gas-phase chemistry and metallicity

The remaining gas-phase composition depends on the amount of elements not locked up into dust grains. As a result of our model, phase-equilibrium is not valid in the upper layers (see Fig. 1, 4th panel) and should not be used to determine gas particle abundances. The remaining gas abundances $\epsilon _{\rm i}$ are strongly sub-solar in an extended layer above the cloud layer where the pressure drops by about 3 orders of magnitude (about 10 scale heights), see Figs. 6 and 7. Inside the cloud layer, the metal abundances increase with increasing atmospheric depth and finally reach even slightly larger than solar values at the cloud base, where those elements that have been locked up in grains in the upper layers are released by evaporation.

Figure 6 shows this phase lag between metal gas abundances and dust abundances clearly. Considering a path from the top to the bottom of the atmosphere, first the metals disappear, then the dust appears, then the metals reappear and finally the dust disappears (the sum of dust and gas abundances is not constant).

The calculation of the gas phase element abundances allows for a discussion of the metallicity in dust forming atmospheres (see Fig. 7), i.e. the logarithmic differences of the gas abundances with respect to the solar values. The metallicities are not the same - not even similar - for different elements. The strongest depletions occur for Ti, Mg and Si, for which the metallicities reach values as low as -6 to -7. The depletion of S and Ca is much less significant (>-0.5) in comparison. Concerning Ca, this is questionable because it is due to our limited choice of solids. There is only one Ca bearing solid species considered, CaTi3[s], and since Ti is less abundant than Ca, Ca cannot be locked up completely into grains. It is important to note, however, that according to our models, the gas depletion factors are kinetically limited by a maximum factor of about 106. Larger depletion factors simply cannot occur because the depletion timescale (see Paper V) would exceed the mixing timescale. In contrast, complete phase equilibrium would imply that the depletion factors can reach values as high as 1050 for low temperatures. Figure 7 also demonstrates that the metallicities change with altitude and depend on the stellar parameter.


  \begin{figure} \par\mbox{\includegraphics[width=8.8cm,clip]{8220Fig8a.eps} \incl... ...8220Fig8c.eps} \includegraphics[width=8.8cm,clip]{8220Fig8d.eps} } \end{figure} Figure 8: Grain size distribution function f(a,z) [cm-4] for a number of selected cloud altitudes. The broad distributions on the lhs of all plots correspond to high altitudes. In the left upper figure, the deeper distributions are very close to a $\delta $-function and not presentable anymore.
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A special feature of our models is that the metal abundances in the gas phase re-increase high above the cloud layer. In our model, all elements are constantly mixed upward, and although the mixing timescale is as long as $\tau_{\rm mix} = 10^{10}$ s in the highest layers, the timescale for a gaseous particle to find a surface to condense on exceeds this timescale, because the dust abundance drops steeply above the cloud layer. Consequently, the metal abundances asymptotically re-approach the solar values high above the cloud layer. The only exception is Ti, which can disappear from the gas phase via nucleation. This feature distinguishes our models from all other published models, and might open up a possibility to discriminate between the models by detailed line profile observations of neutral alkali and alkaline earth metal atoms. Our dust treatment results in deeper and broader resonance lines of Na I, and K I in the red part of the spectrum, which form high in the atmosphere (Johnas et al. 2007).

  
4.7 Grain size distribution

In Fig. 8 we show the results for the potential exponential grain size distribution function $f(a,z)\rm ~[cm^{-4}]$(see Appendix A.2) with parameters derived from the calculated dust moments $L_{\rm j}(z)$. The altitudes have been selected arbitrarily throughout the entire dust cloud for visualisation.

The evolution of the grain size distribution through the cloud layer is similar for models with the same $\log~g$. Considering the $\log~g = 5$ models (upper row in Fig. 8), the size distribution starts relatively broadly in the upper atmosphere where the nucleation is active, because the constant creation and simultaneous growth of the particles causes a broad distribution. Once the nucleation ceases, however, all particles merely shift in size space by a constant offset $\Delta a$ due to further growth, which means a narrowing in $\Delta(\log a)$. Since the particles grow by more than 4 orders of magnitude on their way down the atmosphere, the size distribution finally becomes strongly peaked.

In the $\log~g = 3$ models (lower panel in Fig. 8), the evolution of the size distribution function is more complicated. As shown in Fig. 4, the nucleation rate has a small shoulder as function of pressure on the left hand side. This feature leads to a narrowing of f(a) at first, followed by a re-widening of the size distribution function with increasing depth. The nucleation zone is closer to the convective zone and sometimes even overlaps with the convective zone in the $\log~g = 3$ models. Therefore, although the nucleation rate becomes tiny with increasing depth, it does not vanish completely until about 1600 K which still influences the size distribution and keeps it broad.

  
5 Simple radiative transfer modelling

5.1 Model description

We simulate the grain absorption features in two brown dwarf atmosphere cases ( $T_{\rm eff} = 1800$ K, $\log~g = 5$ and $T_{\rm eff}=1300$ K, $\log~g = 5$) and one example of a giant-gas planet atmosphere ( $T_{\rm eff}=1300$ K, $\log~g = 3$) using a simple radiative transfer code. The code considers the solid-state opacities similar to Helling et al. (2006)[*]. Grain opacities are calculated according to their size distribution f(a,z) [cm-4] and their solid material volume composition ${V_{\rm s}}(z)$ (Figs. 2, 8) using the effective medium theory according to Bruggeman (1935) to calculate the effective optical constants and Mie theory for spherical particles to calculate the extinction efficiencies $Q_{\rm ext}(a,\lambda)$. The main difference between the previous code and the present one is that the grains are here assumed to be distributed according to the potential exponential size distribution function (see Appendix A.2) determined from the calculated dust moments at height z, whereas in the previous model a $\delta $-function, $f(a,z)=\rho L_0(z)~\delta\big(a-\langle a\rangle(z)\big)$ representing the mean particle size $\langle a\rangle$, was assumed. The present code includes the 12 solid species discussed in the previous sections (see Table 4 for the references for the optical constants). The total grain extinction cross-section (see inner integral in Eq. (13)) is computed using a Gauss-Legendre quadrature integration over the entire grain size distribution function at a given atmospheric height.

Table 4: Reference for the optical constant of amorphous materials used in the radiative transfer modelling.


  \begin{figure} \mbox{\includegraphics[angle=-90,width=7cm,angle=180]{8220Fig9g.p... ...ps} \includegraphics[angle=-90,width=7cm,angle=180]{8220Fig9c.ps} } \end{figure} Figure 9: Left column: $T_{\rm eff} = 1800$ K, $\log~g = 5$; middle column: $T_{\rm eff}=1300$ K, $\log~g = 5$; right column: $T_{\rm eff}=1300$ K, $\log~g = 3$. 1st row: transmission spectra $F_\lambda~(T[z_0(\lambda)]) ~/B_\lambda(T_{\rm bb})$. 2nd and 3rd row: local temperature $T[z_0(\lambda )]$ and pressure $p[z_0(\lambda )]$, respectively, where optical depth reaches unity. 9.7 $\mu $m is the Si-O stretching mode of (SiO2[s], silicates), 16-19 $\mu $m the bending mode.
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We examine the radiation transfer in the 7 $\mu $m to 20 $\mu $m wavelength range, which encompasses the Si-O stretching mode (centred at 9.7 $\mu $m) and Si-O bending mode (around 18 $\mu $m) of crystalline silicates (Mg2SiO4 and MgSiO3) and quartz (SiO2). The model calculates the optical depth of the dust component as

 \begin{displaymath}\tau^{{\rm dust}}_\lambda(z) = \int\limits_0^z \int\limits_0^... ...rm ext}\Big(a,\lambda,{V_{\rm s}}(z')\Big)\;{\rm d}a~{\rm d}z' \end{displaymath} (13)

and determines the geometrical depth $z_0(\lambda)$ where $\tau^{{\rm dust}}_\lambda(z_0)=1$ for each wavelength. Furthermore, we replace the opacity of the solid species by vacuum to estimate their respective effects on the output spectrum. To calculate the transmission spectra (first row Fig. 9), we remove a blackbody emission as a continuum ( $B_{\lambda}(T_{\rm bb}$)) from the spectrum ( $F_{\lambda}$) as

\begin{displaymath}\mbox{transmission} = F_\lambda~\Big(T\big[z_0(\lambda)\big]\Big)~\Big/B_\lambda(T_{\rm bb}), \end{displaymath} (14)

i.e. all transmission curves for one model atmosphere are divided by the same black body continuum: ( $T_{\rm eff}$, $\log~g$; $T_{\rm bb}$) = {(1800 K, 5.0; 1310 K), (1300 K, 5.0; 1205 K), (1300 K, 3.0; 910 K)}.

  
5.2 Results

The absorption features can be distinguished from the continuum in all cases. But they amount to a maximum of 6% only at 9.7 $\mu $m in all three cases. The positions of the absorption features (9.7 $\mu $m and 17-18 $\mu $m) are typical of pure absorption by amorphous silicates, which implies that scattering by the larger grains in the size distribution does not contribute significantly to the total cross-sections. This result differs from the single grain size situation where scattering may modify the features, e.g. by shifting the peak absorptions.

The temperature and pressure at $z_0(\lambda)$, where $\tau^{{\rm dust}}_\lambda=1$, are plotted below the transmission spectra. The weak absorption reflects the shallow temperature and pressure variations in the $\tau^{{\rm dust}} \approx 1$region. The contribution of quartz is negligible, as testified by the same output spectrum whether quartz opacity (dash-dot line) is present or not. The two silicate species $\rm MgSiO_3$ and $\rm Mg_2SiO_4$contribute about equally to the output spectrum (dotted and dashed line).

Although grains exist even high in the atmosphere in this model, their number densities are too small to effect the opacity until the region of rapid grain growth is reached. The rapid grain growth implies a concomitant sudden rise in opacities. The phenomenon is particularly acute in the $(T_{{\rm eff}} = 1800~{\rm K}, \log~g = 5)$ model. Therefore, the absorption features probe in particular the cloud deck. The inclusion of gas opacities in the radiative transfer model will complicate the 7-20 $\mu $m spectra, definitively rendering the search for silicates features in brown dwarf atmospheres difficult.

6 Summary and conclusions

Based on a detailed micro-physical description of the formation of composite dust particles including nucleation, growth/evaporation and drift we have investigated the formation, chemical composition and spectral appearance of quasi-static cloud layers in quasi-static brown dwarf and gas-giant atmospheres. Our models show that

  
Appendix A: Grain size distribution function

In order to find the dust particle size distribution function $f(a)~\rm [cm^{-4}]$ we switch to another set of dust moments Kj where the integrals are performed in radius-space a (Dominik et al. 1986; Gauger et al. 1990) rather than in volume space V (Dominik et al. 1993). Substitution for $V=(4\pi a^3)/3$ yields

 \begin{displaymath}K_j = \int_{a_{\rm\ell}}^{\infty} f(a)~a^j~{\rm d}a = \bigg(\frac{3}{4\pi}\bigg)^{j/3}\!\rho L_j . \end{displaymath} (A.1)

As argued in the main text, we are left with four known properties of the size distribution function, namely K1, K2, K3 and K4. We will introduce a suitable functional formula for f(a) with a set of four free coefficients and will determine these coefficients from the known dust moments. This exercise is carried out for two functions in the following.

  
A.1 Double delta-peaked size distribution function

One option is to consider the superposition of two Dirac-functions

 \begin{displaymath}f(a) = N_1~\delta(a-a_1) + N_2~\delta(a-a_2), \end{displaymath} (A.2)

where $\delta $ is the Dirac-function, $N_1, N_2~\rm [cm^{-3}]$ are two dust particle densities and a1, a2 [cm] are two particle radii.

Assuming $a_1 > a_{\rm\ell}$ and $a_2 > a_{\rm\ell}$, where $a_{\rm\ell}$ is the lower integration size corresponding to $V_{{\rm\ell}}$, the moments are given by Kj = N1 a1 j + N2 a2 j, from which we can determine the four free coefficients. a1 is found to be the positive root of

a12 (K22 - K1 K3) + a1 (K1 K4 - K2 K3) + (K32 - K2 K4)  = 0 (A.3)

and the other free coefficients are given by
                          a2 = $\displaystyle \frac{a_1 K_2 - K_3}{a_1 K_1 - K_2}$ (A.4)
N1 = $\displaystyle \frac{\big(a_1 K_1 - K_2\big)^3} {\big(a_1 K_2 - K_3\big) \big(K_3 - 2a_1 K_2 + a_1^2 K_1\big)}$ (A.5)
N2 = $\displaystyle \frac{K_1 K_3 - K_2^2} {a_1\big(K_3 - 2a_1 K_2 + a_1^2 K_1\big)}\cdot$ (A.6)

  
A.2 Potential exponential size distribution function

Another, more continuous option is to consider the function

 \begin{displaymath}f(a) = a^B\!\exp\big(A - C~a\big), \end{displaymath} (A.7)

which, for positive coefficients A, B and C, is strictly positive with a maximum at B/C. Since there are only three coefficients in Eq. (A.7) we will determine them from K1, K2 and K3.

To find a simple analytical solution, we extend the integration in Eq. (A.1) to the interval $[-\infty~...+ \infty]$ which usually introduces only a small error, since f(a) is quickly vanishing for small a (see Fig. 8). The result is $K_j=\exp~(A+\ln\Gamma(B+1+j)-(B+1+j)\ln C)$, by which the free coefficients can be deduced:

                          B = $\displaystyle \frac{2 K_1 K_3 - 3 K_2^2}{K_2^2 -K_1 K_3}$ (A.8)
C = $\displaystyle (2 + B) \frac{K_1}{K_2}$ (A.9)
A = $\displaystyle \ln K_1 + (2 + B)\ln C - \ln\Gamma(2\!+\!B)$ (A.10)

where $\Gamma$ is the generalised factorial function with $\Gamma(n+1) = n!$ for $n\in\varmathbb{N}_0$ and $\Gamma(x+1) = x~\Gamma(x)$ for $x\in\varmathbb{R}^+$.

References

 
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