When material in a debris disc starts breaking up, it enters the “collisional cascade”. This describes the process by which larger bodies undergo collisions and break into smaller bodies, which then undergo further collisions and break into even smaller bodies, and so on. In steady state, and assuming scale invariance, this leads to a welldefined size distribution of colliding bodies in the cascade. This is approximately a power law n(s)ds ∝s−pds, (1) where n(s)ds is the number of bodies with radii between s and s + ds and p = 3.5 for an ideal infinite collisional cascade (Dohnanyi, 1969)4. The steep index of s−3.5 means that smaller bodies vastly outnumber larger ones. It also means that the total surface area (and hence thermal emission) of the disc is dominated by the smallest bodies, whilst the total mass is dominated by the largest bodies. Real discs may not exactly follow the size distribution of Equation 1 because the assumption of scale invariance is broken: at the larger end, the largest bodies may follow the primordial distribution, because these bodies take longer to enter the cascade (e.g. Krivov and Wyatt 2021). At the smaller end, the distribution is cut off by the blowout size, which causes “waves” to appear in the distribution of small grains. At intermediate sizes (around 100 m), a change in bodies’ strength from being dominated by material
4This can also be expressed in integral form, usually as a survival function N(≥s) ∝s−q with q = p −1; this is the form shown in Figure 3 below. The diameter D = 2s may also be used instead of the radius.

strength to gravity induces further deviations, recently detected in observations of Main Belt Asteroids (Burdanov et al, 2024). There may be further deviations from the simple powerlaw, for example if larger bodies undergo cratering collisions that break off relatively small fragments without significantly disrupting the body (e.g. Th´ebault and Augereau 2007; Desch and Jackson 2021). Simulations of collisions with different parameters show that the size distribution of fragments from a single collision can be convex or concave (Durda et al, 2007), and a steep power law distribution can persist in some collision families such as the Eurybates family (Marschall et al, 2022). Nonetheless, in collisionally evolved populations such deviations tend to relax, and then Equation 1 is still a reasonable approximation of the size distribution. Simulations of the long-term collisional evolution of populations of Solar System bodies broadly reproduce power-law distributions of sizes. Figure 3 shows the simulated size distribution of the main Asteroid Belt (Bottke et al, 2020) and the Kuiper Belt’s scattered disc (“destabilized population” in Figure 3; Bottke et al 2023), early in the Solar System’s lifetime. These distributions roughly follow the simple form of Equation 1 at small sizes, but are shallower and wavy at larger sizes because of the dependence on changing strength at ∼100 m. The size distribution of the two populations differ, because they do not follow the same disruption law. Kuiper Belt Objects (KBOs) are easier to break up than asteroids, because they have different material compositions and bulk densities; the weakest KBO body (i.e., the easiest to break up) would have a diameter of roughly 20 m, compared to 200 m for an asteroid (Bottke et al, 2023). Using debiased NeoWISE survey results, Bauer et al (2017) found that the size distribution of Jupiter Family Comets have a similar differential size distribution (∝D−3.3) to an ideal collisional cascade, while Long-Period Comets have a much flatter size distribution (D−2), which may reflect different histories of collisional or insolation processing between the two populations.