Issue 
A&A
Volume 570, October 2014



Article Number  A32  
Number of page(s)  17  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201424368  
Published online  10 October 2014 
Online material
Appendix A: Properties of the skewed Maxwellian distribution
The skewed Maxwellian distribution, f_{skM} (see Sect. 3.3), is a velocity probability distribution with integral normalised to unity. In order to prove this property, we integrate over relative velocities, V_{rel}, from 0 km s^{1} to + ∞ km s^{1}. (A.1)We split the two terms in brackets into two different integrals. In the first term we make the following change of variables , while in the second term . This leads to (A.2)We now demonstrate that the skewed Maxwellian distribution, under the approximation that the thermal motion of the gas ions is negligible with respect to their drift motion, tends to the a Dirac delta probability. In this case we can assume that T_{gas} → 0 and that the relative velocity between a grain and the gas equals the drift velocity (i.e. v → V_{drift}). This then implies (A.3)and (A.4)Making the following change of variables (A.5)the skewed Maxwellian distribution then results As desired, the skewed Maxwellian distribution in this case tends
to a Dirac delta probability distribution, i.e. the case for the inertial sputtering described by Tielens et al. (1994), see Sect. 3.3.
Appendix B: Shattering and vapourisation parameters for aC(:H) grains
As shown by Tielens et al. (1994) and Jones et al. (1996), the shattering and vapourisation of colliding grains can be modelled in the framework of shocks in solids. The first step to estimate the relevant parameters involved in the model is to estimate the average binding energy of the target material. Graphitictype materials consists of purely sp^{2} hybridised bonds C=C while diamond is of purely sp^{3} CC bonds. An aC(:H) material is somewhat in between the two materials and present a mix of hybridised sp^{3} and sp^{2} bonds with a significant H atom concentration. The binding energy, E_{b}, for this material can be expressed as (B.1)where E_{C − C} (=3.6 eV) and E_{C = C} (=6.3 eV) are the energy of the sp^{3} and sp^{2} bonds, respectively. The aC(:H) binding energy then yields E_{b} = 6.5 eV.
As discussed by Tielens et al. (1994), the threshold vapourisation energy density, ϵ_{v}, is related to the binding energy and can be estimated as (B.2)where m_{p} is the proton mass and M the carbon atomic mass. Then the minimum relative velocity between the colliding grains for vapourisation (i.e. the threshold vapourisation velocity, v_{th}) is (B.3)For aC(:H) grains we then obtain a threshold vapourisation energy and velocity of ϵ_{v} = 5.2 × 10^{11} erg g^{1} and v_{th} = 20.3 × 10^{5} cm s^{1}.
At threshold, the pressure of the shock in the solid, P_{th,v}, can be expressed as (B.4)where c_{0} is the speed of sound in the target, ρ its density and s a parameter introduced by McQueen et al. (1970) and estimated to be s ≈ 1.9 for graphite/aC materials. We assume that c_{0} ≪ v_{th} and confirm the validity of this assumption below.
In order to estimate the corresponding parameters for the shattering process, we assume that the threshold vapourisation and shattering pressure scale for aC(:H) materials as for graphite. Jones et al. (1996) made a detailed estimation of these parameters for graphite obtaining P_{th,v} = 5.8 × 10^{12} dyne cm^{2} and P_{th,sha} = 4.0 × 10^{10} dyne cm^{2}. Scaling these parameters, for aC(:H) grains we then have P_{th,sha} = 2.0 × 10^{10} dyne cm^{2}. Furthermore, the threshold shattering pressure is a good estimate of the Young modulus of the material, E_{Y}, and therefore we can calculate the speed of sound in the medium as (B.5)obtaining c_{0} ≈ 1. We finally conclude that the assumption c_{0} ≪ v_{th} is valid in this case.
© ESO, 2014
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