The Race at the Heart of Planet Formation: Grain Growth Times Against Radial Drift Times From 1 to 100 AU

Dust in a protoplanetary disc must grow from micrometre grains to planetesimals before the gas drag that enables growth also drags it into the star. This paper computes both clocks on the same disc model. The growth e-folding time, 1/(Z Omega) for a dust-to-gas ratio Z = 0.01, is 15.9 years at 1 AU and 2.62e+03 years at 30 AU, and the climb from a micrometre to Stokes number 0.1 takes about ten e-folds, so the full growth time is 215 years at 1 AU, 5051 years at 10 AU and 21934 years at 30 AU. The drift time at Stokes 0.1 is 445 years at 1 AU but only 4445 years at 10 AU and 13335 years at 30 AU: growth wins narrowly at 1 AU and loses beyond a few AU. At Stokes number 1 the position is hopeless, with drift at 10 AU taking 880 years against 6210 years of growth, while at Stokes 0.01 drift at 30 AU takes 1.32e+05 years and small grains survive. The barrier is a window in Stokes space, roughly 0.03 to 1, that growth must cross faster than drift allows — in a smooth disc, it cannot, beyond the inner few AU. Midplane densities and shear rates sit at 1.4e-64 and 3.5e-40 of the gravitational frame's ceilings, so the barrier is purely aerodynamic. Computed from stated assumptions, not measured.