Flat Layers: Dust Settling Equilibrium, Settling Times, and the Turbulence That ALMA's Thin Discs Require

Dust settles toward the midplane until turbulence stirs it back up, and the equilibrium layer thickness is a direct gauge of the turbulence. This paper computes the gauge and reads the observations with it. The settling equilibrium H_d/H = sqrt(alpha/(alpha + St)) gives layer thicknesses of 0.10 for alpha = 1e-3 and Stokes 0.1, 0.30 for alpha = 1e-2, and 0.032 for alpha = 1e-4. Settling itself is fast — 2.62e+02 years at 30 AU for Stokes 0.1 — so every observed layer is an equilibrium layer, not a work in progress. Inverting the gauge for the thin layers that edge-on disc observations report (H_d/H of 0.1 to 0.2 for millimetre grains at tens of AU) requires alpha of 1.0e-03 to 4.2e-03 for Stokes-0.1 grains, and 1.3e-03 for Stokes 0.03. The midplanes of real discs are therefore quiescent at the level planetesimal-formation thresholds require, a conclusion reached here from geometry alone. Computed from stated assumptions, not measured; the disc material sits at 4.4e-70 of the gravitational frame's density ceiling even at 100 AU.