tokens: Schoonenberg Ormel 2017 ). Fig. 9 Revised critical solid abundance as a function of the dimensionless stopping time from Carrera et al. (2015) . The filled and open circles show our models with and without significant radial concentration of solids, respectively. The solid black line for and the dashed blue line for are the unmodified critical curve from Carrera et al. (2015) . The black solid line for shows the revised part of the critical curve, where the black dashed line is the extrapolation of it. For solid abundances above the black line (green region), spontaneous concentration of solid particles by the streaming instability can occur. Some discussion on our simplifying assumption of particles of the same size is in order here. It has been suggested by Bai Stone (2010a) that the particle-gas dynamics for a population of particles of various sizes is predominantly driven by the largest ones. They found a critical solid abundance of for particles of seven species ranged from to , and conjectured that it is the total abundance of the particles with that forms the criterion for triggering strong concentration of particles; that is, . However, if the dust coagulation is limited by the bouncing barrier, the largest particles in the distribution may only have ( Zsom et al. 2010 ; Dra ̧żkowska Dullemond 2014 ). In this case, the conjecture by Bai Stone (2010a) breaks down and it is not clear which size range of the particles would dictate the particle-gas dynamics in the system. On the other hand, when the dust coagulation is limited by the bouncing barrier, the distribution of the particle size tends to concentrate within about one order of magnitude near the barrier ( Zsom et al. 2010 ). Therefore, our assumption of particles of the same size might still assimilate this scenario. Finally, several open questions remain in concentrating small particles by the streaming instability. We have observed further sedimentation in dense filaments in our models, indicating that the velocity dispersion of the particles decreases in the dense regions. It remains to be seen if further dust growth could proceed inside such an environment, given the reduced relative velocities, and if the dust growth in turn could drive more spontaneous concentration of solid materials via the streaming instability. Moreover, the presence of the magnetic fields could render rich structure in the gaseous protoplanetary disks, and the disk mid-plane can be turbulent with various strengths at different locations ( Turner et al. 2014 ), which significantly affects the ability of solid materials to sediment onto the mid-plane (e.g., Okuzumi Hirose 2011 ; Zhu et al. 2015 ). This has nontrivial effects on the dust coagulation and the radial concentration of solids for planetesimal formation to occur ( Johansen et al. 2014 ; Testi et al. 2014 ). 6. Summary In this work, we revisit the condition for the spontaneous concentration of solid particles driven by the streaming instability
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