Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 2 / v / Solution

For equal logarithmic width $d\log D$, the number of bodies is $n(D)Dd\log D$. Their mass is thereforeThe mass removed by catastrophic planetesimal collisions per logarithmic bin per unit time scales asAssume a steady collisional cascade with supply only at the large end, loss only at the small end, negligible reassembly or cratering contributions, and self-similar fragmentation that transfers mass through comparable logarithmic intervals. Mass conservation requires the downward mass flux across every intermediate size boundary to be the same; otherwise mass would accumulate or deplete there. Under the scale-invariant bin-transfer assumption, this is proportional by a size-independent factor to ${\dot{M}}_{\mathrm{loss}}$, soThis is constant mass flux in a collisional cascade. Steady state alone fixes the net flux, while the identification with the destruction rate of each bin additionally uses the stated fragmentation assumptions.