Discontinuities in Size-strength Scaling Laws: Another Source of Wavy Size

Introduction: Previous studies with numerical collisional models have demonstrated that wave-like departures from traditional, linear power-law equilibrium size distributions can result from a small-size cutoff in the particle population [1–4]. At small sizes, a particle cutoff may result from the rapid removal of bodies by non-gravitational forces such as PoyntingRobertson drag or the Yarkovsky effect. Another mechanism inducing waves in an equilibrium size distribution is the distinct change in slope in a size-strength scaling law resulting from the transition from strength-scaling for bodies smaller than about 100–200 m in diameter to gravity-scaling for larger objects [5]. As the strengths of larger bodies increase relative to what they would have been had the strength-scaling law continued to larger sizes, their collision lifetimes increase and these “excess” objects represent an overabundant source of projectiles for larger objects. Thus, a wave propagates up the size distribution starting at the size corresponding to the minimum in the size-strength scaling law. In this study, I demonstrate that discontinuities in the size-strength scaling law, as might occur at small sizes when particles are no longer homogeneous over the scale of fragmentation, can also induce significant waves in evolved size distributions. Discontinuous Scaling Laws: Figure 1 shows four hypothetical scaling laws which define the critical specific energy, QD, for bodies from 100 μm to 200 km in diameter. QD is the energy per unit target mass required to fragment and disperse the target, leaving a largest remnant with 50% the mass of the original target. Scaling law H1 defines the ‘reference’ scaling law for this study, having a strength-scaling-like D size dependence for bodies smaller than D ≈ 100 m and a gravity-scaling-like D size dependence for bodies larger than D ≈ 100 m. At size D = 10 cm QD = 8100 J/kg, equivalent to the intermediate strength material of [6]. Scaling law H1 is similar to scaling law H2 in Fig. 3 of [5], with the critical specific energy a factor of ~1.5 smaller at all sizes. Here, scaling law H1 assumes that the same D size dependence of QD extends to the very smallest, dust-size particles. Most other studies have focused on the collisional evolution of comparatively larger objects and have not addressed the issue of the strength properties of very small particles. Scaling laws H2, H3, and H4 of this study are all identical to scaling law H1 for sizes larger than D = 1 cm. At smaller sizes, however, I allow for the possibility that the critical specific energy may change abruptly due to changing material properties at small size scales (see Discussion below). The actual size at which such a change in material properties occurs may be quite different than the 1-cm scale assumed for the convenience of this numerical study.