A&A 399, 297-313 (2003)
DOI: 10.1051/0004-6361:20021734
P. Woitke^{1} - Ch. Helling^{1,2}
1 - Zentrum für Astronomie und Astrophysik, Technische
Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
2 -
Konrad-Zuse-Zenrum für Informationstechnik Berlin,
Takustraße 7, 14195 Berlin, Germany
Received 9 August 2002 / Accepted 19 November 2002
Abstract
In this paper, we quantify and discuss the physical and
surface chemical processes leading to the formation, temporal
evolution and sedimentation of dust grains in brown dwarf and giant gas
planet atmospheres: nucleation, growth, evaporation and gravitational
settling. Considering dust particles of arbitrary sizes in the
different hydrodynamical regimes (free molecular flow, laminar flow,
turbulent flow), we evaluate the equilibrium drift velocities (final
fall speeds) and the growth rates of the particles due to accretion
of molecules. We show that a depth-dependent maximum size of the
order of
exists, which depends on
the condensate and the stellar parameters, beyond which
gravitational settling is faster than growth. Larger particles can
probably not be formed and sustained in brown dwarf atmospheres. We
furthermore argue that the acceleration towards equilibrium drift is
always very fast and that the temperature increase of the grains due
to the release of latent heat during the growth process is
negligible. Based on these findings, we formulate the problem of
dust formation coupled to the local element depletion/enrichment of
the gas in brown dwarf atmospheres by means of a system of partial
differential equations. These equations state an extension of
the moment method developed by Gail
Sedlmayr (1988) with an
additional advective term to account for the effect of
size-dependent drift velocities of the grains. A dimensionless
analysis of the new equations reveals a hierarchy of nucleation
growth
drift
evaporation, which characterises the
life cycle of dust grains in brown dwarf atmospheres. The developed
moment equations can be included into hydrodynamics or classical
stellar atmosphere models. Applications of this description will be
presented in a forthcoming paper of this series.
Key words: stars: atmospheres - stars: low-mass, brown dwarfs - dust, extinction - molecular processes - methods: numerical
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 2001; Marley 2002; Tsuji 2002; Cooper 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 Mundt 2001a,b; Bailer-Jones 2002; Martín 2001; Eislöffel 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 2001, henceforth called Paper I). Three further points are to be mentioned:
(i) The stellar gravity ( ) is about a hundred times larger than in the earth's atmosphere and roughly 10^{4} to 10^{5} 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, . 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 CH_{4}, H_{2}O and CO_{2}, i.e.no CO-blocking), like in planetary atmospheres (e.g.Lodders Fegley 1994). Consequently, different chemical preconditions for the dust formation process are present as e.g.in M-type giants (Gail 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 . 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 Frisch 1983; MacGregor Stencel 1992; Krüger 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 (2001) and Sandin 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 1984; Gail Sedlmayr 1988). However, an unique velocity of the dust component is assumed. Assuming stationarity, Krüger (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, from the object's atmosphere, assuming that these elements have been consumed by dust formation guided by stability arguments (Burrows 1997; Burrows Sharp 1999; Saegers 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 Marley (2001) have extended these time-scale considerations to the turbulent regime for large dust Reynolds numbers by adopting various data fits. Cooper (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 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 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 ( ) and in the viscous case ( ). 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.
The trajectory
of a spherical dust particle of radius a and
mass
,
floating in a gaseous
environment like a stellar atmosphere, is determined by Newton's law
The gravitational force on the grain is given by
The radiative force is given by the momentum transfer from the ambient
radiation field to the grain due to absorption and scattering of photons
However, as the following rough estimation will demonstrate, the radiative
force is small compared to the other forces in brown dwarf atmospheres and can
be neglected. We simplify the integral in Eq. (3) by pulling out
the flux mean extinction efficiency
.
The
wavelength integrated Eddington flux is given by
,
where
is the Stefan-Boltzmann constant and
is the effective
temperature of the star. Considering furthermore the small particle limit
(SPL) of Mie theory
(Rayleigh limit), the extinction
efficiency is proportional to the grain size
,
and Eq. (3) results in
Red Giant: | ||
K, , | ||
10^{1} | ||
10^{4} | ||
Brown Dwarf: | ||
K, , | ||
10^{1} | ||
10^{4} |
Table 1 demonstrates that the radiative force on dust
grains in brown dwarf atmospheres is always much smaller than the
gravity, even in case of light but opaque grains^{}. Consequently, the radiative force can be
neglected, and the dust grain's equation of motion
(Eq. (1)) simplifies to
An unique description of the frictional force (drag force) is difficult to obtain for brown dwarf atmospheres. These difficulties arise from the fact that the behaviour of the gas flow around the moving dust grain changes qualitatively with changing grain size, changing drift velocity and/or changing thermodynamic state of the gas. There are transitions from freely impinging gas particles to a viscous flow, from a subsonic to a supersonic behaviour, and from a laminar flow to turbulence. The physical conditions in brown dwarf atmospheres are such that all transitions may possibly occur. Reliable physical descriptions of the drag force are only available in certain limited regimes and an unique description must be compiled from these special cases.
In order to quantify the behaviour of the streaming gas flow, the
following characteristic numbers are introduced: the Knudsen number Kn and
the dust Reynolds number
.
The Knudsen number Kn is
defined by the ratio of the mean free path length of the gas particles
to a typical dimension
of the gas flow
under consideration, here given by the diameter of the grain
In contrast, if
,
inter-molecular collisions are
frequent and the stream of gas particles colliding with the dust grain
becomes viscous. In this case, the drag force cannot be obtained via
reduction to elementary collisions. This is the regime of continuum
theory for which well-tested empirical formulae are available, e.g.
from engineering science (viscous case or slip flow), depending
on whether the gas flow around the dust grain is laminar or turbulent.
In order to characterise the transition from laminar (Stokes) friction
to turbulent (Newtonian) friction, the dust grain's Reynolds number
is introduced
In order to evaluate the kinematic viscosity of the gas
we follow the considerations of Jeans
(1967) for a mixture of ideal gases
(8) |
Considering the mechanics of rarefied gases, Schaaf (1963)
derives a formula for the drag force by freely impinging gas particles due to
elastic collisions, equally applicable in the subsonic as well as in the
supersonic range,
In this regime, continuum theory is valid for which well-tested
empirical formulae exist. Lain
(1999), carrying out
experimental studies on bubbly flows, arrive at the following
empirical expression for the drag force
Using Eq. (7), the drag force in the viscous case according to
Eq. (14) is found to have the following
asymptotic behaviour
For flows with an intermediate Knudsen number (
), so-called transitions flows, reliable expressions
for the drag force are difficult to obtain. We therefore define a
critical Knudsen number
where
equals
.
Considering the limiting cases of subsonic drift velocities and small
in Eqs. (13) and (16),
respectively, the result is exactly
(17) |
Figure 1: Contour plot of the equilibrium drift velocity [cm s^{-1}] as function of grain radius a and gas density at constant temperature K and gravitational acceleration g=10^{5}cm s^{-2}. A mass density of the dust grain material of (quartz - SiO_{2}) is assumed. | |
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Considering a dust particle of constant radius a floating in a gas at
constant thermodynamical conditions (,
T) and a constant
velocity
,
the particle will be accelerated until a
force equilibrium is reached, where the gravitational acceleration is
balanced by frictional deceleration
Figure 1 shows the resulting values of in a brown dwarf's atmosphere with . The equilibrium drift velocities roughly range in [10^{-4},10^{+6}] cm s^{-1} and are generally smaller for small particles and large densities. The small bendings of the contour lines around are no numerical artifacts but result from the measured re-increase of the drag coefficient of spherical particles between and (Eq. (15)), associated with the transition from laminar to turbulent friction.
Small dust particles can sustain longer in the respective atmospheric
layers, whereas large grains will "rain out'' sooner. Only dust
particles 100 m at gas densities 10
can hereby reach a drift velocity beyond the local
velocity of sound
.
However, such particles will remove themselves so
quickly from the respective atmospheric layers (
s) that this case seems very unlikely to be relevant
for any part of the atmosphere, unless there exists a physical process
(convective streams or atmospheric winds) which is capable to produce
supersonic upwinds. The time-scale for gravitational
settling is hereby defined as
Figure 1 demonstrates furthermore that even the smallest dust particles cannot sustain forever but will slowly sink into deeper layers. Assuming that the dust particles do not grow along their way down the atmosphere (which would increase their drift velocity), a m-particle starting in an atmospheric layer with g cm^{-3} needs about s (8 months) to pass one scale height. A dust particle with m needs only 1/4 hour^{}.
The destiny of those particles drifting inward is to finally reach an atmospheric layer where the temperature is high enough to evaporate them thermally. This sets free the elements the dust grains are composed off and thereby enriches the surrounding gas in this layer. Hence, the rain-out will tend to saturate the gas below the cloud base, where the "cloud base'' is identified with the level in the atmosphere where the dust grains are just thermodynamically stable (, see Sect. 3).
The actual relative velocity of the dust particle with respect to the gas, , can of course deviate from its equilibrium value defined in Sect. 2.4. It will only asymptotically reach for , if the parameters a, , and T are constant. However, considering a dust particle created in a brown dwarf atmosphere, the particle may grow by accretion of molecules ( , see Sect. 3) and the physical state of the surrounding gas may change with time (e.g. ) as the particle sinks into deeper layers of the atmosphere. Turbulence may furthermore create a time-dependent velocity field ( ), which provides an additional cause for temporal deviations between and .
Thus, an important question for the discussion of the dynamical behaviour of the dust component in brown dwarf atmosphere is, whether or not can be replaced by , at least approximately.
Figure 2: Contour plot of the acceleration time-scale towards equilibrium drift [s] as function of grain radius a and gas densities . Other parameters are the same as in Fig. 1. | |
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In order to discuss this question, we consider the dust particle
acceleration time-scale
towards equilibrium drift. Expressing
the dust particle's equation of motion (Eq. (5)) in terms
of the first-order differential equation
with
and
,
and
assuming small deviations
from the stability point
(where
), the temporal change of y is
.
Accordingly, the acceleration time-scale is given by
,
or
For a powerlaw dependence , Eqs. (20) and (21) result in , i.e.we find for . Since Sect. 3 will demonstrate that also the growth of the dust particles is slow in comparison to , we may conclude that the concept of equilibrium drift provides a good approximation for the description of the size-dependent relative velocities between dust and gas in brown dwarf atmospheres.
A fundamental process for the consideration of the time-dependent behaviour of the dust component in brown dwarf atmospheres is the growth of the dust particles by accretion of molecules. The respective reverse process (thermal evaporation, in view of more complex surface reactions also sometimes denoted by chemical sputtering) is important at high temperatures^{}. Considering the thermodynamical conditions in brown dwarf atmospheres, we are again faced with the problem of qualitative changes of the dynamical behaviour of the gas component due to different Knudsen numbers.
For large Knudsen numbers (lKn), gas molecules of all kinds are freely impinging onto the surface of the grain. Some of these dust-molecule collisions (sometimes a certain sequence of them) will initiate a chemical surface reaction which causes a growth step (or an evaporation step) of the dust particle. This case has been extensively studied in the circumstellar envelopes of AGB stars (Gail Sedlmayr 1988; Gauger 1990; Dominik 1993).
The accretion rate, expressed in terms of the increase of the
particle's volume
due to chemical surface
reactions for large Knudsen numbers is given by
The last term on the r.h.s. of Eq. (22) takes into
account the reverse chemical processes, namely the thermal evaporation
rates. It determines the sign of
and hence decides whether
the dust particle grows or shrinks.
is
a generalised supersaturation ratio of the surface reaction "r''
(Dominik
1993) where
is the particle density of the key
species in phase-equilibrium over the condensed dust material.
is not known a priori. In the case of simple surface
reactions, which transform
units of the solid material from the gaseous into the condensed phase (and vice versa), e.g.
(
,
)
or
(
,
), the generalised
supersaturation ratio
is related to the usual supersaturation ratio
of the dust grain
material
(Gauger
1990) by
(23) |
For small Knudsen numbers (sKn), the transport of gas molecules to the
surface of the grain (or the transport of evaporating molecules away
from the grain's surface) is not a simple free flight with thermal
velocity as assumed in Eq. (22), but is hindered by
inter-molecular collisions. Consequently, the grain growth and
evaporation is limited by the diffusion of molecules towards or away
from the grain's surface, considering growth or evaporation in the
laminar case, respectively (we disregard here convection as transport
process for the molecules, expected to occur in the turbulent case).
We consider the following particle conservation equation with a
diffusive transport term (see Landau & Lifschitz 1987, Eq. (38.2)
ff)
Considering the static case (
)
and assuming stationary
(
)
and spherical symmetry,
Eq. (24) results in
The solution of Eq. (28) with the boundary
conditions (29) and (30) is
Figure 3: Contour plot of the growth time-scale [s] as function of the grain radius a and the gas density at constant temperature K for quartz (SiO_{2}, ). We assume growth by accretion of the key species SiO with solar particle density and extreme supersaturation ( ). cm is estimated. | |
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For arbitrary Knudsen numbers we apply the same interpolation scheme
as outlined in Sect. 2.3. We define a critical Knudsen
number
by equating Eqs. (22) with (32)
(33) |
(34) |
(35) |
(37) |
The particle growth is found to be typically 3 orders of magnitude slower than the acceleration which allows us to assume instantaneous acceleration (equilibrium drift). For large Knudsen numbers, we find whereas for small Knudsen numbers the growth time-scale increases faster for larger grains and becomes density-independent, . Note that the influence of the drift velocities on the particle growth has not been considered in Eqs. (22) and (32) such that for supersonic drift velocities or large dust Reynolds numbers, the presented physical description is not valid.
An additional dashed line is depicted in Fig. 3, where . This line defines a maximum dust grain size in a brown dwarf atmosphere. For larger particles ( above the dashed line), the growth time-scale exceeds the time-scale for gravitational settling ( ) which means that such particles are already removed from the atmosphere before they can be formed. Consequently, such particles cannot exist. The maximum grain size varies between 1 m in the thin, outer atmospheric regions ( ) and 100 m in the dense, inner regions ( ). These values depend on the stellar parameters and the considered dust material density.
Note, that an absolute minimum of has been considered in Fig. 3 since extreme supersaturation ( ) and solar abundance of silicon in the gas phase have been assumed. In the case of an Si-depleted or nearly saturated gas (), becomes larger and the maximum particle radius becomes smaller. Furthermore, the values for are relatively independent of temperature, but will shift as in the free molecular flow case and in the laminar viscous case, remembering that in both cases and that (see also Eq. (81)).
Figure 3 demonstrates furthermore that dust particles moving with supersonic drift velocities cannot be expected in brown dwarf atmospheres. Similarly, the turbulent flow regime with dust Reynolds numbers is barely reached at very large densities. Therefore, we can conclude that for dust grains in brown dwarf atmospheres the subsonic free molecular flow and the laminar viscous flow are the important cases to be investigated.
The surface chemical reactions responsible for the growth of a dust
particle liberate the latent heat of condensation
[erg/g] which
causes a heating of the grain as
(39) |
A further heating process of the dust particle is given by the
friction caused by the motion relative to the gas,
In order to determine the dust temperature increase, we balance these heating
processes with the net energy losses due to radiation and due to inelastic
collisions. The net radiative cooling rate of a single dust grain is given by
The cooling due to inelastic collisions with gas particles, in particular with
H_{2}, depends again on the Knudsen number. For large Knudsen numbers (
)
the collisional cooling rate is given by
(46) |
The energy balance of a single dust grain is finally given by
(52) |
Figure 4: Contour plot of the temperature increase [K], due to the liberation of latent heat during grain growth and frictional heating, as function of the grain radius a and gas density at constant gas temperature K for quartz grains with the same parameters as in Fig. 3. We assume growth by accretion of the key species SiO with maximum particle density and extreme supersaturation ( ). The two dashed lines indicate where the two considered heating and cooling rates are equal. Above these lines, and , respectively. | |
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Figure 4 shows an example for the resulting temperature increase of quartz grains. We assume and again consider the explicit growth reaction (Eq. (38)) with a release of latent heat of eV per reaction at 1000 K, 5.61 eV at 1500 K and 5.49 eV at 2000 K (data reduced from the enthalpies of formation of the involved molecules and the solid, source: JANAF-tables, electronic version, Chase 1985).
For the sake of simplicity, we furthermore assume
(53) |
(54) |
Despite these simplifications, Fig. 4 clearly indicates that the warming of the dust grains due to the release of latent heat is negligible, being less than 3.5 K all over the relevant parts of the size-density-plane, where (compare Fig. 3). Here, we find that this heating is balanced by collisional cooling . Since both heating/cooling rates scale as for large Knudsen numbers and as for small Knudsen numbers, a constant value for tunes in for both cases, K for large Kn, and K for small Kn. Note that the calculated temperature differences are always an upper estimate. The actual temperature differences may be much smaller because we have assumed solar, undepleted abundances of Si in the gas phase and for the calculation of .
For larger particles (roughly at as defined in Sect. 3.1), the character of the energy balance of the dust particles changes. Here, the frictional heating due to the rapid relative motion and the radiative cooling dominate, i.e. . Much larger temperature deviations up to 10 000 K result in this case. However, as argued before, such large grains cannot be formed in brown dwarf atmospheres.
Thus, the resulting increase of the dust temperature is by far too small to reach the sublimation temperature , unless gas temperatures very close to are considered and, therefore, Eqs. (22) and (32) remain valid.
The physical and chemical processes discussed so far (nucleation, growth, evaporation, gravitational settling and element depletion/enrichment) occur simultaneously in the atmosphere and may be strongly coupled. Therefore, our aim in this section is to derive a consistent time-dependent description of the dust component in brown dwarf or giant gas planet atmospheres. We will derive a system of partial differential equations which describes the evolution of the dust component by means of the moments of its size distribution function. This idea was originally developed by Gail Sedlmayr (1988) and extended by Dominik (1993) to core-mantle and dirty grains. The resulting differential equations are supposed to be simple and includable into hydrodynamics or classical stellar atmosphere calculations. In contrast, we want to avoid an elaborate and time-consuming multi-component treatment of the dust component, e.g.by using discrete bins for the dust size distribution function with individual drift velocities in this paper.
Figure 5: Two surface chemical processes ( and ) populating or depopulating an infinitesimal dust grain volume interval . | |
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The master equation for dust particles
,
,
where
is the distribution function of
dust particles in volume space, is given by
(56) |
The maximum grain sizes to be expected in brown dwarf atmospheres (Fig. 3) allow us to concentrate on two major cases in the following, namely the subsonic free molecular flow and the laminar viscous flow.
For large Knudsen numbers, the chemical rates depicted in
Fig. 5 can be expressed according to Eq. (22):
Solving Eq. (19) with the frictional force according
to Eq. (13, subsonic case) the equilibrium drift
velocity is
= | (64) | ||
= | (65) |
The derivation of the moment equations for the case of small Knudsen numbers
is analogous to the previous subsection. We express the surface chemical
rates according to Eq. (32) by
= | (68) | ||
= | (69) | ||
= | (70) | ||
= | (71) |
= | (73) | ||
= | (74) |
For the sake of analysis and discussion, we transform the dust moment equations derived in Sects. 5.1 and 5.2 into their dimensionless form by introducing reference values as , , , , , , , , , following the procedure described in Paper I. The reference values are to be chosen according to the expected order of magnitude of the respective quantities and the length and time-scales under investigation. After this substitution, all quantities are dimensionless and can be compared by number.
This allows us to identify the leading terms in the equations, e.g.in the inner and outer regions of a brown dwarf atmosphere. The remaining constants (products of the reference values) can be summarised into characteristic numbers which provide an efficient way to describe the qualitative behaviour of the dust component.
The dimensionless dust moment equations for nucleation, growth,
evaporation, and equilibrium drift write for a subsonic free
molecular flow (
)
The following discussion is based on a typical structure of a brown dwarf atmosphere with solar abundances in the gas phase, i.e. neglecting the possible depletion due to dust formation (see Table 2). As underlying ()-structure we refer to a brown dwarf model atmosphere with K and which has been kindly provided by T. Tsuji (2002)^{}. As exemplary dust species we consider solid SiO_{2}(amorphous quartz), growing by the accretion of SiO and H_{2}O (Eq. (38)). Since the nucleation of SiO_{2} seems dubious (the monomer is rather unstable as a free molecule and hence not very abundant in the gas phase) we consider nucleation of TiO_{2}instead^{}.
Name | Characteristic | Value | ||||||
Number | inside | outside | ||||||
Mach number | ||||||||
Froude number | 0.842 | 0.495 | ||||||
Strouhal number | ||||||||
hydrodyn. Knudsen number | ||||||||
Knudsen number (Eq. (6)) | ... | |||||||
Drift number | ... | |||||||
combined drift number ( ) | ... | |||||||
combined drift number ( ) | ... | |||||||
: 1 | : 1 | |||||||
Sedlmaÿr number | : | : | ||||||
: | : | |||||||
: | : | |||||||
Damköhler no. of nucleation | 0 | 0 | ||||||
Damköhler no. of growth ( ) | 3.95 | |||||||
Damköhler no. of growth ( ) | ||||||||
Name | Physical Quantity | Reference Value | ||||||
inside | outside | |||||||
temperature | [K] | 1700 | ... | 1000 | ||||
density | [g/cm^{3}] | ... | ||||||
thermal pressure | [dyn/cm^{2}] | ... | ||||||
velocity of sound | [cm/s] | ... | ||||||
velocity | [cm/s] | |||||||
length | [cm] | 10^{+4} | 10^{+6} | 10^{+4} | 10^{+6} | |||
hydrodyn. time | [s] | |||||||
gravitational acceleration | [cm/s^{2}] | 10^{+5} | ||||||
mean particle radius | [cm] | 10^{-3} | ... | 10^{-6} | ||||
0th dust moment ( ) | [1/g] | ... | ||||||
nucleation rate | [1/s] | ... | ||||||
growth velocity ( , Eq. (67)) | [cm/s] | ... | ||||||
growth velocity ( , Eq. (76)) | [cm^{2}/s] | ... | ||||||
diffusion constant (Eq. (26)) | [cm^{2}/s] | ... | 2.43 | |||||
mean free path (Eq. (10)) | [cm] | ... | ||||||
total hydrogen number density | [1/cm^{3}] | ... | ||||||
molecular number density | [1/cm^{3}] | ... |
An analysis of the characteristic numbers in front of the source terms in Eqs. (77) and (78) (see Table 2) reveals a hierarchy of nucleation growth drift:
Name | Value | ||
dust material density | [g/cm^{3}] | 2.65 | |
monomer volume | [cm^{3}] | ||
lower dust grain radius | [cm] | ||
molecular radius | [cm] |
physical process | ||
time-derivative | ||
advective term | 1 | |
nucleation term | ||
growth term | ||
drift term |
: the importance of this process depends
on the considered dust moment j with the following associated mean dust quantity: dust particle density, dust size, dust surface area, dust mass density. |
Table 4 shows some dependencies of the combined characteristic numbers (the squared brackets in Eqs. (77), (78)), which provides scaling laws for the importance of the different processes in the different regimes:
In Table 2, we have assumed , i.e.we have considered time-scales of the order of , appropriate for disordered (e.g.turbulent) velocity fields, where the l.h.s. terms of Eqs. (77) and (78) are of comparable importance. However, if large-scale systematic motions are stable for a long time (e.g.a circulating thunderstorm or a stable convection roll), the system may reach a quasi-stationary situation where becomes much larger and hence . In this case, the first term on the l.h.s. of Eqs. (77) and (78) vanishes (see Table 4) whereas all other terms remain unaffected.
An interesting special case occurs if additionally , i.e.when the dust-forming system reaches the static case. In this case, also the advective terms in Eqs. (77) and (78) vanish and the source terms must balance each other. In the case, this means that the gain of dust by nucleation and growth must be balanced by the loss of dust due to rain-out, which means that the gas will be depleted. In the case, just the opposite is true, i.e.the loss by evaporation must be balanced by the gain of dust particles raining in from above. Consequently, the gas in such undersaturated layers will be enriched by the condensable elements liberated by the evaporating grains.
However, both control mechanisms (in the static limit) result in an efficient transport of condensable elements from the cool upper layers into the warm inner layers, which cannot last forever. We may conclude that if the brown dwarf's atmosphere is truly static for a long time, there is no other than the trivial solution for Eqs. (77) and (78) where the gas is saturated ( ) and dust-free ( ). This situation changes, however, if the brown dwarf's atmosphere is turbulent or, in particular, when it is convective. In that case, the replenishment of the atmosphere with fresh uncondensed gas from the deep interior will counteract the downward transport of condensable elements by the formation and gravitationally settling of dust grains. Simulations of this quasi-static balance will be the subject of the forthcoming paper in this series.
A dynamical modelling of the dust component in brown dwarf atmospheres
by means of a moment method as proposed in this paper - consistently
coupled to hydrodynamics, radiative transfer and element depletion in
the scope of hydrodynamical or classical stellar atmosphere
calculations - seems straightforward as soon as two major problems can
be solved:
One idea to construct such a closure condition has been developed by
Deufelhard
Wulkow (1989) and Wulkow (1992),
studying the kinetics of polyreaction systems. The size distribution
function f(V) is here approximated by a weight function
,
which describes the basic shape of f(V), and
modified by a sum of orthogonal polynomials
(k=0,1,2, ... ,n) as
In this paper, we have investigated the basic physical and chemical processes which are responsible for the formation, the temporal evolution and the precipitation of dust grains in brown dwarf atmospheres.
In contrast to other astronomical sites of effective dust formation, the dust particles are embedded in such a dense gas that the Knudsen numbers may fall short of unity. This requires a careful fall differentiation for the different hydrodynamical regimes: free molecular flow (subsonic and supersonic) and slip flow (laminar and turbulent case).
Compiling a general formula for the drag force from the different special cases, we have shown that the large gravity in brown dwarf atmospheres forces the dust particles to move with a considerably high downward drift velocity relative to the gas. The acceleration of the dust particles (on a time-scale ) towards the equilibrium drift velocity (final fall speed) results to be always much faster than any other considered process (nucleation, growth, hydrodynamical acceleration and sedimentation) such that an instantaneous acceleration of the particles to equilibrium drift can be assumed. In contrast, the outward acceleration of dust grains due to radiation pressure is completely negligible in brown dwarf atmospheres.
The large drift velocities are found to limit the residence time of the
forming dust grains and hence their maximum size
as
For small Knudsen numbers, the growth of the particles by accretion of molecules is limited by the diffusion of the molecules towards the grain surface, and the energy exchange with the surrounding gas is limited by heat conduction. The latter process co-works with the radiative gains and losses of the hot grains. According to our results, the release of latent heat during the growth does only lead to a small increase of the grain temperature (K) and has no particular influence on the growth rates.
Based on these findings, we have formulated a system of partial differential equations for the consistent physical description of the dust component in brown dwarf and giant gas planet atmospheres. These moment equations represent an unique tool to model the nucleation, growth and size-dependent equilibrium drift of the dust particles, and the element depletion/enrichment of the gas. We consider such a description as essential, because these processes occur simultaneously and are strongly coupled. The description allows for an inclusion into hydrodynamics or classical stellar atmosphere calculations, although a few unsolved questions still remain, e.g.a reliable closure condition and a clean Knudsen number fall differentiation.
A dimensionless analysis of the moment equations reveals the existence of the following three regimes associated with the formation of a cloud layer:
The life cycle of dust grains in brown dwarf atmospheres is finally completed by convective streams which mix up gas from the deep interior into the upper layers. On a large scale, we expect an intricate balance of this upward mixing of condensable elements by convection with the downward gravitational settling of the condensing dust grains, which will determine the large-scale structure of the element abundances in the atmosphere related to the observation of the various molecular features.
This work will be continued in the next paper of this series by solving the dust moment equations for the special case of a static atmosphere.
Acknowledgements
We thank T. Tsuji for contributing the hydrostatic reference model atmosphere and Akemi Tamanai for providing us with an electronic version of the optical constants of amorphous quartz. The anonymous referee is thanked for helpful comments, in particular concerning the fricional heating. We thank R. Klein for pointing out the work of Deufelhard Wulkow. This work has been supported by the DFG (Sonderforschungsbereich 555, Teilprojekt B8, and grand Se 420/19-1 and 19-2). Most of the literature search has been performed with the ADS system.