Unveiling the Hierarchical Structure of Open Star Clusters: the Perseus Double Cluster

We introduce a new kinematic method to investigate the structure of open star clusters. We adopt a hierarchical clustering algorithm that uses the celestial coordinates and the proper motions of the stars in the field of view of the cluster to estimate a proxy of the pairwise binding energy of the stars and arrange them in a binary tree. The cluster substructures and their members are identified by trimming the tree at two thresholds, according to the $\sigma$-plateau method. Testing the algorithm on 100 mock catalogs shows that, on average, the membership of the identified clusters is $(91.5\pm 3.5)$\% complete and the fraction of unrelated stars is $(10.4\pm 2.0)$\%. We apply the algorithm to the stars in the field of view of the Perseus double cluster from the Data Release 2 of Gaia. This approach identifies a single structure, Sub1, that separates into two substructures, Sub1-1 and Sub1-2. These substructures coincide with $h$ Per and $\chi$ Per: the distributions of the proper motions and the color-magnitude diagrams of the members of Sub1-1 and Sub1-2 are fully consistent with those of $h$ Per and $\chi$ Per reported in the literature. These results suggest that our hierarchical clustering algorithm can be a powerful tool to unveil the complex kinematic information of star clusters.