$\mathcal{Q}^{+}$: Characterising the structure of young star clusters

TL;DR

This paper introduces a new statistical algorithm to analyze and quantify the fractal sub-structure of young star clusters, characterizing their hierarchical complexity through three key parameters.

Contribution

The paper presents a novel algorithm that accurately constrains the fractal dimension, hierarchy levels, and density scaling of star clusters from observational data.

Findings

The algorithm reliably recovers 3D structure from 2D projections.

It quantifies the diversity of structures in real and simulated clusters.

The method advances understanding of star cluster formation and structure.

Abstract

Many young star clusters appear to be fractal, i.e. they appear to be concentrated in a nested hierarchy of clusters within clusters. We present a new algorithm for statistically analysing the distribution of stars to quantify the level of sub-structure. We suggest that, even at the simplest level, the internal structure of a fractal cluster requires the specification of three parameters. (i) The 3D fractal dimension, $\mathcal{D}$, measures the extent to which the clusters on one level of the nested hierarchy fill the volume of their parent cluster. (ii) The number of levels, $\mathcal{L}$, reflects the finite ratio between the linear size of the large root-cluster at the top of the hierarchy, and the smallest leaf-clusters at the bottom of the hierarchy. (iii) The volume-density scaling exponent, $\mathcal{C}=-\textrm{d}\ln[\delta n]/\textrm{d}\ln[L]$ measures the factor by which the excess density, $\delta n$, in a structure of scale $L$, exceeds that of the background formed by larger structures; it is similar, but not exactly equivalent, to the exponent in Larson's scaling relation between density and size for molecular clouds. We describe an algorithm which can be used to constrain the values of $({\cal D},{\cal L},{\cal C})$ and apply this method to artificial and observed clusters. We show that this algorithm is able to reliably describe the three dimensional structure of an artificial star cluster from the two dimensional projection, and quantify the varied structures observed in real and simulated clusters.