The 3D morphology of open clusters in the solar neighborhood III. Fractal dimension of open clusters

We analyzed the fractal dimension of open clusters using 3D spatial data from Gaia DR 3 for 93 open clusters from Pang et al. and 127 open clusters from Hunt & Reffert mainly within 500 pc. The box-counting method was adopted to calculate the fractal dimension of each cluster in three regions: the all-member region, r <= r(t) (inside the tidal radius), and r > r(t) (outside the tidal radius). In both the Pang and Hunt catalogs, the fractal dimensions are smaller for the regions r > r(t) than those for r <= r(t), indicating that the stellar distribution is more clumpy in the cluster outskirts. We classified cluster morphology based on the fractal dimension via the Gaussian mixture model. Our study shows that the fractal dimension can efficiently classify clusters in the Pang catalog into two groups. The fractal dimension of the clusters in the Pang catalog declines with age, which is attributed to the development of tidal tails. This is consistent with the expectations from the dynamical evolution of open clusters. We found strong evidence that the fractal dimension increases with cluster mass, which implies that higher-mass clusters are formed hierarchically from the mergers of lower-mass filamentary-type stellar groups. The transition of the fractal dimension for the spatial distribution of open clusters provides a useful tool to trace the Galactic star-forming structures, from the location of the Local Bubble within the solar neighborhood to the spiral arms across the Galaxy.