Issue 
A&A
Volume 566, June 2014



Article Number  L2  
Number of page(s)  5  
Section  Letters  
DOI  https://doi.org/10.1051/00046361/201423862  
Published online  05 June 2014 
Online material
Appendix A: Applicability to aggregates
For porous aggregates, the basic principles explored here are expected to hold, but some material properties have to be altered. First, aggregates have an internal filling factor φ = ρ_{agg}/ρ that is <1, and might be as low as 10^{4} in some extreme cases (Okuzumi et al. 2012; Kataoka et al. 2013). Second, the ’effective’ surface energy γ_{agg} will be smaller, since there is only limited contact between the aggregate’s constituents to begin with. Assuming the parent bodies are built up of spherical monomers, the effective surface energy can be estimated as γ_{agg} ~ (a/R)^{2}φ^{2/3}γ, where a and R denote the radius of the contact area shared by monomers, and the radius of the monomers themselves. The fraction (a/R) depends on the size of the monomers and the material properties, but for 0.1micronsized monomers, (a/R) ~ 0.1 is reasonable.
For aggregates, Nbody simulations have been performed with particles containing up to 10^{6} monomers, and values of η range from close to unity (Dominik & Tielens 1997; Wada et al. 2009), to several orders of magnitude less (Ringl et al. 2012), and depend on the employed contact model (Seizinger et al. 2013).
Appendix B: Collisions with different mass ratios
Here we extend the theory to collisions between “targets” and “projectiles” of arbitrary sizes s_{t}>s_{p}. Assuming the fragment distribution can be described as before, we can still use Eq. (2), while the precollision kinetic energy now equals (B.1)For a given collision velocity, we might then think of two cases, complete fragmentation when s_{p} ~ s_{t}, and erosion/cratering when s_{t} ≫ s_{p}.
Appendix B.1: Catastrophic fragmentation
Since both particles are destroyed completely we may set M_{frag} = m_{p} + m_{t}. For simplicity we will assume in this section that the change in surface energy is dominated by the fragments ΔU_{S} = γA_{frag}. Using the same definition for η as before and focussing on the α = 3.5 case, we obtain (B.2)
Appendix B.2: Erosion
When the mass ratio becomes very large, it is no longer realistic to assume the target is completely disrupted. Rather, such collisions result in erosion, and the eroded mass is typically of the order of the mass of the projectile (Schräpler & Blum 2011). Thus, we write M_{frag} = κm_{p}, with κ of the order of unity. For large mass ratios μ → m_{p}. Furthermore, assuming that the change in surface energy is dominated by the new fragments, ΔU_{S} = γA_{frag}, we obtain (B.3)Consider now a particle with a size s_{0} close to the smaller end of the size distribution, colliding with particle of sizes s_{x}, ranging from slightly smaller to much larger than s_{0}. Figure B.1 shows the minimum size of the fragments produced as a function of collider size s_{x}. The yaxis is normalized to the value obtained in equalsized collisions (i.e. between two s_{0} particles). For mass ratios below unity, s_{0} acts as the target, and the mass of the largest fragment is assumed to equal m_{0}/2. For large mass ratios, s_{0} acts as a projectile instead, and the largest fragment mass is set to m_{x}/ 2. Collisions at mass ratios above 10^{2} are assumed to be erosive (Seizinger et al. 2013), with the largest fragment equalling m_{0}/2. Since both the excavated mass, and the largest fragment mass, depend on the projectile, the curves in the erosive regime do not depend on the mass ratio directly. However, the size of the target does set an upper limit on κ.
Fig. B.1
Minimum fragment size resulting from destructive unequal collisions, assuming s_{min} ≪ s_{max}, and relating the largest fragment size to the heavier collider. The yaxis has been normalized with the value for s_{min} in equalmass collisions. 

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