Issue 
A&A
Volume 558, October 2013



Article Number  A91  
Number of page(s)  21  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201321132  
Published online  14 October 2013 
Online material
Appendix A: Departure from the Cunningham velocity
The StokesCunningham velocity defined in Eq. (3)is derived under the assumption of low Reynolds number. Therefore it is not valid for turbulent flow and other expressions may be used when the Reynolds number increases. Here we derive better laws for intermediate and large Reynolds number.
Appendix A.1: Low Knudsen number
For small Reynolds number and small Knudsen number, the drag force exerted by a fluid on a sphere at rest is considered proportional to the kinetic energy of the fluid and the projected area of the sphere. The coefficient of proportionality, or drag coefficient, C_{D} is given by: (A.1)Then, equating gravity and drag forces leads to the settling velocity of a particle in an atmosphere: (A.2)Where ρ_{p} is the density of the particle. For small Reynolds numbers and high Knudsen number, C_{D} = 24 is constant and the settling velocity is the Stokes velocity. When increasing the Reynolds number, the non linear terms of the NavierStokes equation become important and C_{D} is no longer constant. We used tabulated values of the drag coefficient as a function of the Reynolds number given by Pruppacher & Klett (1978). We assume that C_{D} = 24 when the Reynolds number reaches 1 to stay consistent with Stokes flow and that C_{D} reaches its asymptotic value, C_{D} = 0.45, when N_{Re} ≡ 2R_{e} = 1000 and fit the relationship: (A.3)Then we follow the same method as Ackerman & Marley (2001). Noting that: (A.4)is independent of the velocity, we use the fit of Eq. (A.3) and extract the velocity: (A.5)
Appendix A.2: High Knudsen number
In the freemolecular regime, calculations have been made by Probstein (1968) leading to an expression for the drag coefficient: (A.6)
where s_{a} is the ratio of the object velocity over the thermal speed of the gas ( where is the sound speed) and erf is the error function.
For velocities much smaller than the sound speed, s_{a} → 0 and we can use an equivalent of the error function in 0: (A.7)Using this equation inside Eq. (A.6)and taking the limit s_{a} → 0, the term e^{−sa2} goes to 1 and the terms proportional to cancels out leading to: (A.8)For velocities much greater than the sound speed, the limit of Eq. (A.6)when s_{a} → ∞ is: (A.9)In order to simplify Eq. (A.6)we use the following expression for the drag coefficient at high Knudsen number: (A.10)Our approximation fits correctly the exact expression in the limit of low and high velocities. In between the difference to the exact expression is at most 30%. Replacing C_{D} by its value in Eq. (A.2)we obtain a second order equation for the velocity: (A.11)which leads to: (A.12)with . When the speed becomes small compared to the sound speed (V_{stokes} ≪ V_{T}) we obtain: (A.13)which is in good agreement with Eq. (3), derived for high Knudsen number and small Reynolds numbers.
Appendix A.3: Comparison with the Cunningham velocity
Figure A.1 shows the ratio of the Cunningham velocity (see Eq. (3)) to the ones we just derived. The difference is noticeable only for particles of the order of 100 μm at pressures less than the 10^{4} bar level and exceeding the 10 bar level. This difference is always less than one order of magnitude and concern only a tiny portion of the parameter space which has little relevance to our study (the largest particle sizes considered in our 3D models is 10 μm). Thus we decided to neglect this discrepancy in the main study.
Fig. A.1
Ratio of the Cunningham velocity to the more sophisticated model of the particle settling velocity considered in the Appendix. 

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