How Much Dust Is Enough? A Settling-Based Threshold for Streaming Concentration, Tested Against Measured Disc Dust Abundances

Streaming instability is the leading mechanism for concentrating pebbles into gravitationally collapsing clouds, but it engages strongly only when the midplane dust-to-gas ratio approaches unity. This paper derives the required vertically integrated dust abundance from that condition and the standard settling equilibrium, Z_req = sqrt(alpha/(alpha + St)), and tests it against measured disc dust abundances. For turbulence alpha = 1e-3, the requirement is Z = 0.302 at Stokes number 0.01, 0.100 at Stokes 0.1, and 0.032 at Stokes 1 — everywhere far above the Z of 0.005 to 0.02 that disc surveys measure globally. At the measured Z = 0.01, the midplane ratio reaches only 0.10 for Stokes 0.1 at alpha = 1e-3. The plain conclusion is that smooth discs of solar composition do not trigger strong streaming concentration: either turbulence must be as low as alpha = 1e-4, where the requirement at Stokes 0.3 relaxes to 0.018, or dust must first be concentrated by pressure traps, whose local enhancements supply the missing factor. Planetesimal formation by this route is therefore a local, not a global, event. Computed from stated assumptions, not measured.