The Slow Grinder: Collisional Cascade Timescales for Debris Belts From First Principles

Debris discs are collisional cascades: belts of planetesimals grinding themselves to dust over billions of years, with the dust we see continuously resupplied and removed. The cascade's pace is set by the collision time of its largest bodies. This paper derives that time from first principles — a Dohnanyi size distribution normalised by the belt mass, the projectile population able to disrupt a 100 km body at the belt's collision speeds, and the resulting encounter rate — across belts of 0.1 to 10 Earth masses at 30 to 80 AU. The collision time is 9.4e+08 years for an Earth-mass belt at 30 AU, 8.2e+09 years at 50 AU and 6.0e+10 years at 80 AU, falling to 8.2e+08 years for a ten-Earth-mass belt at 50 AU. Low-mass belts therefore never process their largest bodies within a stellar lifetime — their cascades are fed from below and their dust is a trickle — while massive belts grind through their stock in well under a gigayear and fade. The corresponding initial fractional luminosities run from 2.0e-05 (Earth-mass, 50 AU) to 2.0e-04 (ten Earth masses), bracketing the brighter end of the observed debris population. The cascade clock, like the disc clocks before it, is set by ordinary collisions far below any gravitational bound (3.0e-69 of the frame's density ceiling at 50 AU). Computed from stated assumptions, not measured.