Free Access
Volume 566, June 2014
Article Number L2
Number of page(s) 5
Section Letters
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 = ρagg/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 γagg 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 γagg ~ (a/R)2φ2/3γ, 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 st>sp. 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 sp ~ st, and erosion/cratering when stsp.

Appendix B.1: Catastrophic fragmentation

Since both particles are destroyed completely we may set Mfrag = mp + mt. For simplicity we will assume in this section that the change in surface energy is dominated by the fragments ΔUS = γAfrag. 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 Mfrag = κmp, with κ of the order of unity. For large mass ratios μmp. Furthermore, assuming that the change in surface energy is dominated by the new fragments, ΔUS = γAfrag, 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 sx, 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 sx. 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 mx/ 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.

thumbnail Fig. B.1

Minimum fragment size resulting from destructive unequal collisions, assuming sminsmax, 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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© ESO, 2014

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