The statistical analysis of star clusters

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 (s) over bar (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 nproportional tor(-alpha) (with alpha between 0 and 2) (s) over bar decreases monotonically with alpha, from similar to0.8 to similar to0.6. Since (s) over bar reflects all star positions, it implicitly incorporates edge effects. However, for fractal star clusters (with fractal dimension D between 1.5 and 3) (s) over bar decreases monotonically with D (from similar to0.8 to similar to0.6). Hence (s) over bar, 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 (m) over bar [i.e. the mean length of the branches of the tree, divided by (N-total A)(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 = (m) over bar/(s) over bar does both quantify and distinguish between a smooth large-scale radial density gradient and multiscale (fractal) subclustering. IC348 has Q = 0.98 and rho Ophiuchus has Q = 0.85, implying that both are centrally concentrated clusters with, respectively, alphasimilar or equal to 2.2 +/- 0.2 and alphasimilar or equal to 1.2 +/- 0.3. Chamaeleon and IC2391 have Q = 0.67 and 0.66, respectively, implying mild substructure with a notional fractal dimension Dsimilar or equal to 2.25 +/- 0.25. Taurus has even more substructure, with Q = 0.45 implying D'similar or equal to 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.