2004MNRAS.348..589C

We review a range of statistical methods for analysing the structures of star clusters, and derive a new measure Q, 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 {bar}s (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) {bar}s decreases monotonically with α, from ∼0.8 to ∼0.6. Since {bar}s reflects all star positions, it implicitly incorporates edge effects. However, for fractal star clusters (with fractal dimension D between 1.5 and 3) {bar}s decreases monotonically with D (from ∼0.8 to ∼0.6). Hence {bar}s, 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 {bar}m [i.e. the mean length of the branches of the tree, divided by (N_totalA)^1/2/ (N_total - 1), where A is the area of the cluster and N_total 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 Q = {bar}s/ {bar}m does both quantify and distinguish between a smooth large-scale radial density gradient and multiscale (fractal) subclustering. IC348 has Q = 0.98 and ρ Ophiuchus has Q = 0.85, implying that both are centrally concentrated clusters with, respectively, α≃ 2.2±0.2 and α≃ 1.2 ± 0.3. Chamaeleon and IC2391 have Q = 0.67 and 0.66, respectively, implying mild substructure with a notional fractal dimension D ≃ 2.25±0.25. Taurus has even more substructure, with Q = 0.45 implying D '≃ 1.55±0.25. If the binaries in Taurus are treated as single systems, Q increases to 0.58 and D ' increases to 1.9±0.2.