- ... grains
^{} - In contrast,
the radiative force generally dominates in the circumstellar
envelopes of red giants.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
only
^{} - Accordingly, the product of the characteristic numbers
*Kn*and is given by .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... particles
^{} - We note that such empirical formulae
cannot be derived from first principles without additional
assumptions about the physics of the microscopic interactions which
can only be introduced as phenomenological values
(Sedlmayr 1976).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... scheme
^{} - The interpolation
coefficients in Eq. (18) do not sum up to 1, because at
(where
)
the true
is
*smaller*than in both limiting cases.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
velocity
^{} - Rossow (1978) uses the term "terminal
velocity'' for
,
Ackermann
Marley (2001) use
"terminal fall speed''.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... hour
^{} - Pushing these simple time-scale arguments
forward, one could arrive at the wrong conclusion that brown dwarf's
atmosphere with life-times yrs must be completely
dust-free. The error in this conclusion is that brown dwarf's
atmospheres are not static, but turbulent, and the convection leads to a
non-continuous replenishment of the upper atmosphere with
condensable elements.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... temperatures
^{} - We will not discuss
coagulation caused by dust-dust collisions in this paper. According
to Cooper
(2002), coagulation (also denoted as "coalescence''
if the dust-dust collisions are caused by the size-dependent drift
velocities) generally operates on larger time-scales than the
growth. We note, however, that the coagulation remains the only
active physical process influencing the dust size distribution if
the supersaturation ratio
*S*is very close to unity, where growth and nucleation vanish.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...))
^{} - Schaaf (1963) provides an exact
formula for the total collisional net heating rate
for large Knudsen numbers, but
unfortunately we are not aware of a comparable expression for
small
*Kn*. We have checked that our approach is in agreement with the Schaaf formula, by order of magnitude, for large*Kn*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
case
^{} - In the turbulent regime, the grains will be efficiently
cooled by heat transport due to convection.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...2002)
^{} - The static model atmosphere results from
frequency-dependent radiative transfer calculations and mixing
length theory for the convective energy flux. Dust is included in
form of three species (Fe, Al
_{2}O_{3}, MgSiO_{3}) following stability arguments and considering a prescribed grain size distribution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...instead
^{} - We are mainly interested in the qualitative behaviour
of the dust component in this paper. The following estimates will be
very similar for other refractory dust materials composed of
abundant elements.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .