Thresholds for Particle Clumping by the Streaming Instability

The streaming instability (SI) is a mechanism to aerodynamically concentrate solids in protoplanetary disks and trigger the formation of planetesimals. The SI produces strong particle clumping if the ratio of solid to gas surface density-an effective metallicity-exceeds a critical value. This critical value depends on particle sizes and disk conditions such as radial drift-inducing pressure gradients and levels of turbulence. To quantify these thresholds, we perform a suite of vertically stratified SI simulations over a range of dust sizes and metallicities. We find a critical metallicity as low as 0.4% for the optimum particle sizes and standard radial pressure gradients (normalized value of pi = 0.05). This subsolar metallicity is lower than previous results, due to improved numerical methods and computational effort. We discover a sharp increase in the critical metallicity for small solids, when the dimensionless stopping time (Stokes number) is <= 0.01. We provide simple fits to the size-dependent SI clumping threshold, including generalizations to different disk models and levels of turbulence. We also find that linear, unstratified SI growth rates are a surprisingly poor predictor of particle clumping in nonlinear, stratified simulations, especially when the finite resolution of simulations is considered. Our results widen the parameter space for the SI to trigger planetesimal formation.