Volume 570, October 2014
|Number of page(s)||17|
|Published online||10 October 2014|
The skewed Maxwellian distribution, fskM (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, Vrel, 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 Tgas → 0 and that the relative velocity between a grain and the gas equals the drift velocity (i.e. v → Vdrift). 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
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. Graphitic-type materials consists of purely sp2 hybridised bonds C=C while diamond is of purely sp3 C-C bonds. An a-C(:H) material is somewhat in between the two materials and present a mix of hybridised sp3 and sp2 bonds with a significant H atom concentration. The binding energy, Eb, for this material can be expressed as (B.1)where EC − C (=3.6 eV) and EC = C (=6.3 eV) are the energy of the sp3 and sp2 bonds, respectively. The a-C(:H) binding energy then yields Eb = 6.5 eV.
As discussed by Tielens et al. (1994), the threshold vapourisation energy density, ϵv, is related to the binding energy and can be estimated as (B.2)where mp 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, vth) is (B.3)For a-C(:H) grains we then obtain a threshold vapourisation energy and velocity of ϵv = 5.2 × 1011 erg g-1 and vth = 20.3 × 105 cm s-1.
At threshold, the pressure of the shock in the solid, Pth,v, can be expressed as (B.4)where c0 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/a-C materials. We assume that c0 ≪ vth 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 a-C(:H) materials as for graphite. Jones et al. (1996) made a detailed estimation of these parameters for graphite obtaining Pth,v = 5.8 × 1012 dyne cm-2 and Pth,sha = 4.0 × 1010 dyne cm-2. Scaling these parameters, for a-C(:H) grains we then have Pth,sha = 2.0 × 1010 dyne cm-2. Furthermore, the threshold shattering pressure is a good estimate of the Young modulus of the material, EY, and therefore we can calculate the speed of sound in the medium as (B.5)obtaining c0 ≈ 1. We finally conclude that the assumption c0 ≪ vth is valid in this case.
© ESO, 2014
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