Grain-size measurements in protoplanetary disks indicate fragile pebbles and low turbulence
Recent laboratory experiments have revealed that destructive collisions of icy dust particles may occur at much lower velocities than previously believed. These low fragmentation velocities push down the maximum grain size in collisional growth models. Motivated by the smooth radial distribution of pebble sizes inferred from ALMA/VLA multi-wavelength continuum analysis, we propose a concise model to explain this feature and aim to constrain the turbulence level at the midplane of protoplanetary disks. Our approach is built on the assumption that the fragmentation threshold is the primary barrier limiting pebble growth within pressure maxima. Consequently, the grain size at the ring location can provide direct insights into the turbulent velocity governing pebble collisions and, by extension, the turbulence level at the disk midplane. We validate this method using the Dustpy code, which simulates dust transport and coagulation. We apply our method to 7 disks, TW Hya, IM Lup, GM Aur, AS 209, HL Tau, HD 163296, and MWC 480, for which grain sizes have been measured from multi-wavelength continuum analysis. A common feature emerges from our analysis, with an overall low turbulence coefficient of $α\sim10^{-4}$ observed in five out of seven disks when taking fragmentation velocity $v_{\rm frag} = 1{\rm \,m\,s}^{-1}$. A higher fragmentation velocity would imply a turbulence coefficient significantly larger than the current observational constraints. IM Lup stands out with a relatively higher coefficient of $10^{-3}$. Notably, HL Tau exhibits an increasing trend in $α$ with distance, which supports enhanced turbulence at its outer disk region, possibly associated with the infalling streamer onto HL~Tau. The current (sub)mm pebble size constrained in disks implies low levels of turbulence, as well as fragile pebbles consistent with recent laboratory measurements.