  Collisions resulting in fragmentation are important in shaping the mass
spectrum of minor bodies in the asteroid belt, the Kuiper belt, and debris
disks. Models of fragmentation cascades typically find that in steady-state,
the solution for the particle mass distribution is a power law in the mass.
However, previous studies have typically assumed that the mass of the largest
fragment produced in a collision with just enough energy to shatter the target
and disperse half its mass to infinity is directly proportional to the target
mass. We show that if this assumption is not satisfied, then the power law
solution for the steady-state particle mass distribution is modified by a
multiplicative factor, which is a slowly varying function of the mass. We
derive analytic solutions for this correction factor and confirm our results
numerically. We find that this correction factor proves important when
extrapolating over many orders of magnitude in mass, such as when inferring the
number of large objects in a system based on infrared observations. In the
course of our work, we have also discovered an unrelated type of non-power law
behavior: waves can persist in the mass distribution of objects even in the
absence of upper or lower cutoffs to the mass distribution or breaks in the
strength law.
