  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 sub-Solar 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 $\leq 0.01$. 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 non-linear, stratified simulations, especially when the
finite resolution of simulations is considered. Our results widen the parameter
space for the SI to trigger planetesimal formation.
