We review a range of statistical methods for analysing the structures of star clusters, and derive a new measure , which both quantifies and distinguishes between a (relatively smooth) large-scale radial density gradient and multiscale (fractal) subclustering.
The distribution of separations p(s) is considered, and the normalized correlation length (i.e. the mean separation between stars, divided by the overall radius of the cluster) is shown to be a robust indicator of the extent to which a smooth cluster is centrally concentrated. For spherical clusters having volume-density n∝r−α (with α between 0 and 2) decreases monotonically with α, from ∼0.8 to ∼0.6. Since reflects all star positions, it implicitly incorporates edge effects. However, for fractal star clusters (with fractal dimension D between 1.5 and 3) decreases monotonically with D (from ∼0.8 to ∼0.6). Hence , on its own, can quantify, but cannot distinguish between, a smooth large-scale radial density gradient and multiscale (fractal) subclustering.
The minimal spanning tree (MST) is then considered, and it is shown that the normalized mean edge length [i.e. the mean length of the branches of the tree, divided by , where A is the area of the cluster and is the number of stars] can also quantify, but again cannot on its own distinguish between, a smooth large-scale radial density gradient and multiscale (fractal) subclustering.
However, the combination does both quantify and distinguish between a smooth large-scale radial density gradient and multiscale (fractal) subclustering. IC348 has and ρ Ophiuchus has , implying that both are centrally concentrated clusters with, respectively, α≃ 2.2 ± 0.2 and α≃ 1.2 ± 0.3. Chamaeleon and IC2391 have and 0.66, respectively, implying mild substructure with a notional fractal dimension D≃ 2.25 ± 0.25. Taurus has even more substructure, with implying D′≃ 1.55 ± 0.25. If the binaries in Taurus are treated as single systems, increases to 0.58 and D′ increases to 1.9 ± 0.2.