- 2.1 Force of gravity
- 2.2 Force of radiation
- 2.3 Force of friction
- 2.4 Equilibrium drift
- 2.5 Accelerated drift

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

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

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

2.2 Force of radiation

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

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

For large grains, the radiative force is smaller than in the small particle limit and asymptotically scales as

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

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

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

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

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

which corresponds to a value of the mean collisional cross section of the H

of , which is more than three times larger as .

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,

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

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

where the drag coefficient is given by

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

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

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

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

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

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

Equation (19) states an implicit definition for the gravitational fall speed (or, more precisely, the equilibrium drift velocity

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

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

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

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.

Copyright ESO 2003