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

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, 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, L, reflects the finite ratio between the linear size of the large root-cluster at the top of the hierarchy, and the smallest leafclusters at the bottom of the hierarchy. (iii) The volume-density scaling exponent,measures the factor by which the excess density, δn, 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 (D, L, 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.

Modelling the structure of star clusters with fractional Brownian motion

“…Because these two properties are linked, the BF model is unable to reproduce naturally substructured clusters, like Cha I and Taurus. We note that Jaffa et al (2017) add extra parameters to the BF model in order to address this problem. Their model has a similar level of complexity as the FBM model, and can be viewed as an alternative to the work presented here.…”

On the spatial distributions of dense cores in Orion B

“…clustered) distributions, and has been extensively utilized (e.g. Schmeja & Klessen 2006;Bastian et al 2009;Cartwright 2009;Cartwright & Whitworth 2009;Gutermuth et al 2009;Ś anchez & Alfaro 2009;Lomax, Whitworth & Cartwright 2011;P a r k e r& Meyer 2012; Delgado et al 2013;P a r k e re ta l .2014; Jaffa, Whitworth & Lomax 2017;Dib, Schmeja & Parker 2018). It employs a graph theory approach by constructing a minimum spanning tree (MST), which connects all of the points in a given distribution via the shortest possible path with no closed loops.…”

“…Other more complex distributions can be used as a comparison, but this can lead to an almost infinite amount of parameter space to consider (Bate, Clarke & McCaughrean 1998;P a r k e r&M e y e r 2012; Jaffa, Whitworth & Lomax 2017). We therefore restrict our comparison to either box fractals as defined by Goodwin & Whitworth (2004) ;Cartwright & Whitworth (2004) or centrally concentrated clusters with different radial density profiles (Cartwright & Whitworth 2004;Cartwright 2009).…”

“…We show 100 realizations of each geometry. meaning does not always fully describe the detailed level of substructure in a star-forming region (Jaffa et al 2017). Furthermore, a box fractal with notional fractal dimension D = 1.6 will have a much higher (local) density than a fractal with D = 3.0 (Bate, Clarke & McCaughrean 1998;Parker, Goodwin & Allison 2011b) for the same number of points (see also Lomax et al 2011;.…”

Dynamical evolution of fractal structures in star-forming regions

“…However, m alone is unable to distinguish between substructured and smooth centrally concentrated regions (Goodwin & Whitworth 2004). To overcome this, the Qparameter was introduced by Cartwright & Whitworth (2004), and further developed by Cartwright (2009), Lomax et al (2011), and Jaffa et al (2017. The Q-parameter is calculated using equation 1:…”