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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)²φ^2/3γ, 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 0.1-micron-sized monomers, (a/R) ~ 0.1 is reasonable.

For aggregates, N-body simulations have been performed with particles containing up to 10⁶ 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 pre-collision 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 s₀ close to the smaller end of the size distribution, colliding with particle of sizes s_x, ranging from slightly smaller to much larger than s₀. Figure B.1 shows the minimum size of the fragments produced as a function of collider size s_x. The y-axis is normalized to the value obtained in equal-sized collisions (i.e. between two s₀ particles). For mass ratios below unity, s₀ acts as the target, and the mass of the largest fragment is assumed to equal m₀/2. For large mass ratios, s₀ acts as a projectile instead, and the largest fragment mass is set to m_x/ 2. Collisions at mass ratios above 10² are assumed to be erosive (Seizinger et al. 2013), with the largest fragment equalling m₀/2. 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 smin ≪ smax, and relating the largest fragment size to the heavier collider. The y-axis has been normalized with the value for smin in equal-mass collisions.
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