The statistical analysis of star clusters

Authors


★ E-mail: Annabel.Cartwright@astro.cf.ac.uk

ABSTRACT

We review a range of statistical methods for analysing the structures of star clusters, and derive a new measure inline image, 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 inline image (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 nr−α (with α between 0 and 2) inline image decreases monotonically with α, from ∼0.8 to ∼0.6. Since inline image reflects all star positions, it implicitly incorporates edge effects. However, for fractal star clusters (with fractal dimension D between 1.5 and 3) inline image decreases monotonically with D (from ∼0.8 to ∼0.6). Hence inline image, 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 inline image[i.e. the mean length of the branches of the tree, divided by inline image, where A is the area of the cluster and inline image 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 inline image does both quantify and distinguish between a smooth large-scale radial density gradient and multiscale (fractal) subclustering. IC348 has inline image and ρ Ophiuchus has inline image, implying that both are centrally concentrated clusters with, respectively, α≃ 2.2 ± 0.2 and α≃ 1.2 ± 0.3. Chamaeleon and IC2391 have inline image and 0.66, respectively, implying mild substructure with a notional fractal dimension D≃ 2.25 ± 0.25. Taurus has even more substructure, with inline image implying D′≃ 1.55 ± 0.25. If the binaries in Taurus are treated as single systems, inline image increases to 0.58 and D′ increases to 1.9 ± 0.2.

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