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Table 4

Definition of the variables used in this paper.

Variable Definition Unit
Input variables

α t Turbulence strength parameter
u f Fragmentation threshold velocity cm s-1
Σg Gas surface density g cm-2
Σd Dust surface density g cm-2
ξ Power-law index of the mass distribution of fragments, see Eq. (10)
T Mid-plane temperature K
ρ s Volume density of a dust particle g cm-3

a Grain size, a = (3   m/(4π   ρs))1/3 cm
Other variables α Slope of the number density distribution n(m) ∝ m − α, see Eq. (4)
a BT Eq. (37) cm
a 12 Eq. (40) cm
a max Eq. (43) cm
a L Left boundary of the bump function b(a), see Sect. 5, paragraph 3 cm
a P Peak size of the bump function b(a), see Sect. 5, paragraph 3 cm
a R Right boundary of the bump function b(a), see Sect. 5, paragraph 3 cm
a sett Eq. (27) cm
a inc Eq. (52) cm
b(a) Bump function, see Eq. (53)
C m1,m2 Collision kernel, see Eq. (3) cm3 s-1
Collision kernel including settling effects, see Eq. (3) cm2 s-1
c s Sound speed, see Eq. (41) cm s-1
δ i Slopes of the fit-function, see Table 3
δu Width of the transition between sticking and fragmentation, taken to be 0.2 uf cm s-1
ϵ Interpolation parameter, see Eq. (47)
γ Power-law index of the function h(m2/m1) for large ratios of m2/m1, as defined in Eq. (21)
J Empirical parametrization of the discontinuity, see Eq. (50)
K Integral defined in Eq. (7)
k b Boltzmann constant erg K-1
m Mass of the particle, m = 4π/3   ρs   a3 g
m 0 Monomer mass g
m 1 Mass above which particles fragment g
μ Mean molecular weight in proton masses, taken to be 2.3
m p Proton mass g
n(m) Number density distribution g cm-3
N(m) Vertically integrated number density distribution g cm-2
ν Degree of homogeneity of the kernel as defined in Eq. (3)
Re Reynolds number, see Eq. (38)
ρ s Density of a dust grain g cm-3
St Mid-plane Stokes number in the Epstein regime, see Eq. (26)
σ H2 Cross-section of molecular hydrogen cm2
u gas Mean square turbulent gas velocity, see Eq. (46) cm s-1
Relative velocities between monomers and grains of size ai, see Eq. (45) cm s-1
Relative velocities between grains of size ai, see Eq. (48) cm s-1
V Parameter used in the calculation of J, see Eq. (49) cm s-1
y a Equals 1.6, parameter from Ormel & Cuzzi (2007)

Notes. The variables grouped as “input variables” are the parameters of the fitting recipe in Sect. 5. “Other variables” summarizes the definitions of all other variables used in this recipe.

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