Line-of-sight kinematic studies indicate that many Galactic globular clusters have a significant degree of internal rotation. However, three-dimensional kinematics from a combination of proper motions and line-of-sight velocities are needed to unveil the role of angular momentum in the formation and evolution of these old stellar systems. Here we present the first quantitative study of internal rotation on the plane-of-the-sky for a large sample of globular clusters using proper motions from Gaia DR2. We detect signatures of rotation in the tangential component of proper motions for 11 out of 51 clusters at a $>$3-sigma confidence level, confirming the detection reported in Gaia collaboration et al. (2018) for 8 clusters, and additionally identify 11 GCs with a 2-sigma rotation detection. For the clusters with a detected global rotation, we construct the two-dimensional rotation maps and proper motion rotation curves, and we assess the relevance of rotation with respect to random motions ($V/sigmasim0.08-0.51$). We find evidence of a correlation between the degree of internal rotation and relaxation time, highlighting the importance of long-term dynamical evolution in shaping the clusters current properties. This is a strong indication that angular momentum must have played a fundamental role in the earliest phases of cluster formation. Finally, exploiting the spatial information of the rotation maps and a comparison with line-of-sight data, we provide an estimate of the inclination of the rotation axis for a subset of 8 clusters. Our work demonstrates the potential of Gaia data for internal kinematic studies of globular clusters and provides the first step to reconstruct their intrinsic three-dimensional structure.
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