... grains[*]
In contrast, the radiative force generally dominates in the circumstellar envelopes of red giants. 
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... only[*]
Accordingly, the product of the characteristic numbers Kn and $Re_{\rm d}$ is given by ${Kn}\cdot Re_{\rm d}=3\vert\vec{v}_{\rm dr}\vert/{\bar{v}_{\rm th}}$. 
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... 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). 
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... scheme[*]
The interpolation coefficients in Eq. (18) do not sum up to 1, because at $Kn\!=\!Kn^{\rm cr}$ (where $\vec{F}_{\rm fric}^{\rm Sch}\!=\!\vec{F}_{\rm fric}^{\rm LBS}$) the true $\vec{F}_{\rm fric}$ is smaller than in both limiting cases. 
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... velocity[*]
Rossow (1978) uses the term "terminal velocity'' for $\vec{v}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}$, Ackermann ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Marley (2001) use "terminal fall speed''. 
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... hour[*]
Pushing these simple time-scale arguments forward, one could arrive at the wrong conclusion that brown dwarf's atmosphere with life-times $>\!10^9$ 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. 
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... temperatures[*]
We will not discuss coagulation caused by dust-dust collisions in this paper. According to Cooper ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$(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. 
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...))[*]
Schaaf (1963) provides an exact formula for the total collisional net heating rate $Q\!=\!Q_{\rm fric}\!-\!Q_{\rm coll}$ 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. 
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... case[*]
In the turbulent regime, the grains will be efficiently cooled by heat transport due to convection. 
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...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, Al2O3, MgSiO3) following stability arguments and considering a prescribed grain size distribution. 
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...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. 
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Copyright ESO 2003