We consider whether equilibrium size distributions from collisional cascades match the frequency of impactors derived from New Horizons crater counts on Charon ( ) .
Using an analytic model and a suite of numerical simulations , we demonstrate that collisional cascades generate wavy size distributions ; the morphology of the waves depends on the binding energy of solids Q _ { D } ^ { \star } and the collision velocity v _ { c } .
For an adopted minimum size of solids , r _ { min } = 1 \mu { m } , and collision velocity v _ { c } = 1–3 km~ { } s ^ { -1 } , the waves are rather insensitive to the gravitational component of Q _ { D } ^ { \star } .
If the bulk strength component of Q _ { D } ^ { \star } is Q _ { s } r ^ { e _ { s } } for particles with radius r , size distributions with small Q _ { s } are much wavier than those with large Q _ { s } ; systems with e _ { s } \approx - 0.4 have stronger waves than systems with e _ { s } \approx 0 .
Detailed comparisons with the New Horizons data suggest that a collisional cascade among solids with a bulk strength intermediate between weak ice ( e.g. , ) and normal ice ( e.g. , ) produces size distributions fairly similar to the size distribution of impactors on Charon .
If the surface density \Sigma of the protosolar nebula varies with semimajor axis a as \Sigma \approx 30 ~ { } { g~ { } cm ^ { -2 } } ( a / { 1 ~ { } au } ) ^ { -3 / 2 } , the time scale for a cascade to generate an approximate equilibrium is 100–300 Myr at 45 au and 10–30 Myr at 25 au .
Although it is necessary to perform more complete evolutionary calculations of the Kuiper belt , collisional cascades are a viable model for producing the size distribution of solids that impacted Charon throughout its history .