Up: Dust in brown dwarfs

Subsections

# 2 The equation of motion

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

 (1)

is the gravitational force, the radiative force due to absorption and scattering of photons, and the frictional force exerted by the surrounding gas via collisions. The frictional force depends on the relative velocity (drift velocity) between the dust particle and the gas . The gas is hereby considered as a hydrodynamic ensemble with velocity . is the density of the dust grain material.

## 2.1 Force of gravity

The gravitational force on the grain is given by

 (2)

where is the gravitational acceleration and the unit vector in radial direction. Because of the small extension of the atmospheres of brown dwarfs ( ), the radial distance r is about equal to the stellar radius , and becomes a constant.

The radiative force is given by the momentum transfer from the ambient radiation field to the grain due to absorption and scattering of photons

 (3)

where is the Eddington flux of the radiation field at wavelength and site . is a dimensionless extinction efficiency of the grain, which can be calculated from the optical properties of the grain material (real and imaginary part of the refractory index) by applying Mie theory, see e.g.Bohren Huffman (1983).

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

 (4)

For large grains, the radiative force is smaller than in the small particle limit and asymptotically scales as  a2. Therefore, we put the relation sign "'' in Eq. (4). The flux mean extinction coefficient is roughly given by a typical value of around the maximum of the stellar flux (see Table 1). Considering a wavelength interval from m to m, typical values of the extinction efficiency are found to vary between (crystalline, glassy materials like ) and (e.g.amorphous carbon, iron bearing solid materials like ), see Mutschke (1998), Tamanai (1998), Andersen (1999), Posch (1999) or, for an overview, see Woitke (1999).

 Red Giant: K, , 101 104 Brown Dwarf: K, , 101 104

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

 (5)

## 2.3 Force of friction

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

 (6)

Kn expresses the influence of inter-molecular collisions. In case (e.g.small gas densities), inter-molecular collisions are rare in the streaming flow. Consequently, the drag force results from a simple superposition of independent, elementary collisions where the velocity distribution of the impinging gas particles resembles a Maxwellian distribution characterised by the gas temperature T which is shifted in velocity space by ( free molecular flow).

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

 (7)

with the characteristic properties , , and . For large Reynolds numbers ( ) the flow around a dust particle is turbulent, whereas for small Reynolds numbers ( ) the flow is laminar (for a discussion on these limits see e.g.Großmann 1995).

In order to evaluate the kinematic viscosity of the gas we follow the considerations of Jeans (1967) for a mixture of ideal gases

 (8)

where ni, mi, ri and are the gas particle densities, masses, radii and thermal velocities, respectively. Assuming a mixture of H2 and He with particle ratio 5:1 and particle radii derived from friction experiments (  Å and  Å; Jeans 1967), we find the viscosity to be

 (9)

The mean free path entering into Eq. (6) is calculated backwards from Eq. (9), using the general relation in consideration of mean gas particles only, where   is the mean thermal velocity and is the mean molecular weight

 (10)

which corresponds to a value of the mean collisional cross section of the H2/He-mixture defined by

 (11)

of , which is more than three times larger as .

#### Free molecular flow ( ):

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,

 (12)

with abbreviations and . The error function is defined by . Equation (12) has the following asymptotic behaviour

 (13)

#### Viscous case ( ):

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

 (14)

where the drag coefficient is given by

 (15)

Comparable formulae can be found in (Huber Sommerfeld 1998), studying spherical coal particles in pipes, and in (Macek Polasek 2000), modelling the inverse problem of porous media in combustion engineering for elliptical particles. We note that Eqs. (12) and (14) are strictly valid only for perfectly rigid, spherical particles. Deviations from this ideal case, e.g.shape distortions of liquid grains, porosity and non-spherical shapes of solid grains, generally lead to an increase of the effective surface area and hence to a decrease of the gravitational fall speed as defined in Sect. 2.4 (for further details see Rossow 1978, p. 14). Such second order effects might roughly be accounted for by reducing the dust material density .

Using Eq. (7), the drag force in the viscous case according to Eq. (14) is found to have the following asymptotic behaviour

 (16)

which reveals the classical formulae for Stokes friction (laminar flow) and Newtonian friction (turbulent flow), respectively.

#### The general case:

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)

In order to arrive at a general formula for arbitrary Knudsen numbers we adopt a simple interpolation scheme

 (18)

where . We note that a Cunningham factor has been introduced in the literature (see e.g. Rossow 1978) in order to extrapolate the resulting formula for the final fall speed valid in the case (see Eq. (72)) into the regime. The constant is usually fixed by measurements. However, it is questionable whether the theoretically known friction law for the limiting case is revealed in this way. Our finding is consistent with (or if the Knudsen number is defined as , compare Eqs. (6) and (63)).

## 2.4 Equilibrium drift

 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=105cm s-2. A mass density of the dust grain material of (quartz - SiO2) is assumed.

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

 (19)

Equation (19) states an implicit definition for the gravitational fall speed (or, more precisely, the equilibrium drift velocity) . The equilibrium drift velocity is determined by , a, and T, and is always directed towards the centre of gravity, even within a horizontal gas flow. Whether or not this equilibrium state is reached in a realistic situation will be discussed in Sect. 2.5. In the general case, cannot be obtained from an analytical inversion of Eq. (19), but must be calculated by finding the root of Eq. (19) numerically, applying iterative methods.

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

 (20)

cm is the pressure scale height of the brown dwarf's atmosphere.

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).

## 2.5 Accelerated drift

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.

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

 (21)

This time-scale is to be compared with the other characteristical time-scales inherent in the ambient medium. A hydrodynamical time-scale results to be 10 s when the so-called micro-turbulence velocity cm s-1, introduced to fit otherwise unidentified line broadening effects, is considered on macroscopic scales cm. About the same value is found when mean convective velocities derived from mixing length theory are considered on microscopic scales  cm. Comparison to Fig. 2 shows that usually in brown dwarf atmospheres, unless very large grains with supersonic fall speeds are considered. This implies that the dust particles will reach their equilibrium drift velocity much faster than usual hydrodynamical changes occur.

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.

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