© ESO, 2011

1. Introduction

It is commonly accepted that star formation does usually not occur in isolation but that a large majority of young stars – up to 90% – are part of a cluster (Lada & Lada 2003; Evans et al. 2009). The dynamical evolution of a star cluster leaves a variety of imprints in the phase space of its stellar population which are good tracers of the dynamical age of the cluster. This quantity is in particular interesting when compared to the physical age. A higher dynamical than physical age means that observable dynamical imprints did not have enough time to evolve dynamically and thus must have been present – at least partially – already at the beginning. This is usually known as primordial origin.

One of the most widely discussed aspects of the dynamical evolution of young star clusters is that of mass segregation. From theoretical work it is well known that this process is inevitably entangled with the dynamical evolution of a self-gravitating system of at least two different mass components (Spitzer 1969; Farouki et al. 1983; Spurzem & Takahashi 1995; Khalisi et al. 2007). Due to energy equipartition – hence via two-body encounters – the more massive particles tend to settle towards the cluster centre over time while the lower-mass particles are preferentially pushed to the outer parts. However, it is a much more challenging task to identify mass segregation observationally in real objects than theoretically from “clean” numerical simulations. This is even more severe for young star clusters that are usually still embedded in their natal gas and the dynamical and physical age of which is much more difficult to estimate.

However, the investigation of mass segregation in these young stellar systems is of particular interest for a deeper understanding of the star formation process. Three fundamental questions are part of the scientific discussion in this context: (i) do young star clusters (really) show mass segregation? (ii) What is the observed degree of mass segregation? (iii) Could the observed degree of mass segregation have developed dynamically or can it be only explained by primordial origin?

An investigation of these important aspects of the star formation process requires a solid tool that is at best independent of the method used to determine the stellar masses and independent of the geometry of the object, that provides an unambiguous measure and is equally well applicable to observational and numerical data. Note that also dynamical models have the equivalent problem to identify mass segregation clearly and quantitatively.

So far mainly four measures have been used to investigate mass segregation in (young) star clusters: i) the slope of the (differential) mass function in different annuli around the cluster centre (${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$; e.g. Richer et al. 1988; Bolte 1989; Hillenbrand 1997); ii) the slope of the cumulative mass function in different annuli around the cluster centre (${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{c}}$; e.g. Pandey et al. 1992; Hillenbrand & Hartmann 1998); iii) the characteristic radius of different mass-groups of stars (${\mathrm{ℳ}}_{\mathrm{R}}^{\mathrm{ch}}$; e.g. Farouki et al. 1983); and iv) the length of the minimum spanning tree (MST) of different mass-groups (${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$; developed by Allison et al. 2009b)¹.

Most of these methods suffer from several weaknesses. The first three, ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$, ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{c}}$, and ${\mathrm{ℳ}}_{\mathrm{R}}^{\mathrm{ch}}$, implicitly assume a spherical geometry and thus depend on the determination of some cluster centre. The first two of them introduce additional bias due to radial binning and uncertainties in deriving the slope of the mass function. Furthermore, ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$ suffers from uncertainties due to mass binning (see e.g. Stolte et al. 2006, for a comparison of ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$ and ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{c}}$ applied to observational data). There is a fundamental difference in the concept of the first and the last two methods: the former measure the mass distribution in different spatial volumes, the latter evaluate the spatial distribution of different sets of most massive stars. Consequently, ${\mathrm{ℳ}}_{\mathrm{R}}^{\mathrm{ch}}$ and ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ do not require a direct measure of stellar masses but only a qualitative criterion for correct ordering. This property is a huge advantage in the face of observational data. However, ${\mathrm{ℳ}}_{\mathrm{R}}^{\mathrm{ch}}$ has also a significant weakness: the characteristic radius of a small subgroup is very sensitive to the definition of the cluster centre. Hence this method is not viable for a low degree of mass segregation, i.e. a signature from only a few most massive stars.

In contrast, ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ does not show any of these disadvantages. However, it is also affected by quite low sensitivity that prevents unambiguous detection of weak mass segregation. To take advantage of the potential strength of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ we have developed an improved prescription for measuring mass segregation, in the following referred to as ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$, with significantly increased sensitivity. We present in this work our scheme and demonstrate its efficiency.

In Sect. 2 we describe our method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ for measuring mass segregation and discuss a scheme for setting up initially mass segregated star cluster models. In Sect. 3 we discuss various examples of the effectiveness of our scheme compared to previous methods and present a first numerical application. The conclusion and discussion mark the last section of this paper.

2. Methods

2.1. Geometrical minimum spanning tree Γ_MST

As a proxy for mass segregation we extend the method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ developed by Allison et al. (2009b) (see also Cartwright & Whitworth 2004; Schmeja & Klessen 2006). In summary, the authors use the minimum spanning tree (MST), the graph which connects all vertices within a given sample with the lowest possible sum of edges and no closed loops (Gower & Ross 1969). The length of the MST, l_MST, is a measure of the concentration or compactness of a given sample of vertices and is unique whereas its shape does not need to be. Mass segregation of a stellar system of size N is quantified by comparing l_MST of the n most massive stars, ${\mathit{l}}_{\mathrm{MST}}^{\mathrm{mass}}$, with the average l_MST of k sets of n random cluster stars, $\mathrm{⟨}{\mathit{l}}_{\mathrm{MST}}^{\mathrm{ref}}\mathrm{⟩}$, and its standard deviation, $\mathrm{\Delta }{\mathit{l}}_{\mathrm{MST}}^{\mathrm{ref}}$. The distribution of l_MST of the k random samples is indeed Gaussian. The ratios of these quantities, ${\mathrm{\Lambda }}_{\mathrm{MST}}\mathrm{=}\frac{\mathrm{⟨}{\mathit{l}}_{\mathrm{MST}}^{\mathrm{ref}}\mathrm{⟩}}{{\mathit{l}}_{\mathrm{MST}}^{\mathrm{mass}}}\mathit{,} \mathrm{\Delta }{\mathrm{\Lambda }}_{\mathrm{MST}}\mathrm{=}\frac{\mathrm{\Delta }{\mathit{l}}_{\mathrm{MST}}^{\mathrm{ref}}}{{\mathit{l}}_{\mathrm{MST}}^{\mathrm{mass}}}\mathit{,}$(1)provide a quantitative measure of the degree of mass segregation: the larger Λ_MST the more concentrated are the massive stars compared to the reference sample; the associated standard deviation ΔΛ_MST quantifies the significance of the result. To make this method work and comparable also with observational data all calculations are carried out in two dimensions, i.e. on a projection of the set of vertices.

We have chosen the number of random reference samples, k, such that a fraction p of the entire population of size N is covered independent of the sample size n: $\mathit{p}\mathrm{=}\mathrm{1}\mathrm{-}{\left(\frac{\mathit{N}\mathrm{-}\mathit{n}}{\mathit{N}}\right)}^{\mathit{k}} \mathrm{\Rightarrow } \mathit{k}\mathrm{=}\mathrm{\lceil }\frac{\ln \mathrm{(}\mathrm{1}\mathrm{-}\mathit{p}\mathrm{)}}{\ln \mathrm{(}\mathrm{1}\mathrm{-}\mathit{n}\mathit{/}\mathit{N}\mathrm{)}}\mathrm{\rceil }\mathit{,}$(2)where  ⌈x⌉  denotes the ceiling function² of x. The threshold has been set to p = 0.99.

Our method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ involves two crucial modifications of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ that make it computationally much more effective and boost its sensitivity. First, unlike Allison et al. (2009b) we do not calculate the separations between all possible pairs of stars but determine the MST (in two dimensions) in a three-step procedure: first we use a 2D Delaunay triangulation (from the software package GEOMPACK: Joe 1991) to construct a useful graph of stellar positions projected onto a plane, then we sort the edges of the triangles in ascending order, and finally adapt Kruskal’s algorithm (Kruskal 1956) with an efficient union-find-algorithm to construct the MST.

A Delaunay triangulation has the very useful property that the MST construction (in any dimension but for Euclidean distance) is its sub-graph (Preparata & Shamos 1985). The implementation of Joe (1991) involves a computational effort ${\mathit{t}}_{\mathrm{DT}}\mathrm{~}\mathrm{𝒪}\mathrm{(}\left|\mathit{E}\right|\mathrm{·}\log \left|\mathit{V}\right|\mathrm{)}\mathit{,}$(3)similar to the edge sorting by the quick-sort algorithm, ${\mathit{t}}_{\mathrm{QS}}\mathrm{~}\mathrm{𝒪}\mathrm{(}\left|\mathit{E}\right|\mathrm{·}\log \left|\mathit{E}\right|\mathrm{)}\mathit{,}$(4)where |E| is the number of edges and |V| the number of vertices. With two algorithmic “tricks” in the union-find algorithm (“union by rank” and “path compression”) we reduce its run-time to ${\mathit{t}}_{\mathrm{UF}}\mathrm{~}\mathrm{𝒪}\mathrm{(}\left|\mathit{E}\right|\mathrm{·}{\log }^{\mathrm{\ast }}\left|\mathit{V}\right|\mathrm{)}\mathit{,}$(5)where ${\log }^{\mathrm{\ast }}\mathrm{\left(}\mathit{n}\mathrm{\right)}\mathrm{=}\begin{array}{}\min \end{array}\{\mathit{s}\mathrm{\in }N\mathrm{|}\begin{array}{}\underset{\mathrm{􏽼}0\mathrm{􏽻􏽺}0\mathrm{􏽽}}{\log (\log \mathrm{\left(}\mathit{...}\log \mathrm{\right(}\mathit{n}\mathrm{\left)}\mathit{...}\mathrm{\right)})}\\ \mathit{s} \mathrm{times}\end{array}\mathrm{\le }\mathrm{1}\}$(6)and thus rather constant though in principle unlimited (Tarjan 1979). So the total computational effort is dominated by the Delaunay triangulation and sorting of edges and thus scales as ${\mathit{t}}_{\mathrm{tot}}\mathrm{~}\mathrm{𝒪}(\left|\mathit{V}\right|\mathrm{·}\log \mathrm{\left(}\left|\mathit{V}\right|\mathrm{\right)})\mathrm{=}\mathrm{𝒪}(\left|\mathit{E}\right|\mathrm{·}\log \mathrm{\left(}\left|\mathit{E}\right|\mathrm{\right)})\mathit{,}$(7)with . This is a significant improvement over the cost of the brute-force method, .

Second, we do not use directly the sum of the edges l_MST as a measure yet their geometric mean γ_MST, ${\mathit{\gamma }}_{\mathrm{MST}}\mathrm{=}(\underset{\mathit{i}\mathrm{=}\mathrm{1}}{\overset{\mathit{n}}{\mathrm{􏽙}}}{\mathit{e}}_{\mathit{i}}{)}^{\mathrm{1}\mathit{/}\mathit{n}}\mathrm{=}\exp \left[\frac{\mathrm{1}}{\mathit{n}}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{n}}\ln {\mathit{e}}_{\mathit{i}}\right]\mathit{,}$(8)and its associated geometric standard deviation Δγ_MST, $\mathrm{\Delta }{\mathit{\gamma }}_{\mathrm{MST}}\mathrm{=}\exp \left(\sqrt{\frac{{\sum }_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{n}}\mathrm{(}\ln {\mathit{e}}_{\mathit{i}}\mathrm{-}\ln {\mathit{\gamma }}_{\mathrm{MST}}{\mathrm{)}}^{\mathrm{2}}}{\mathit{n}}}\right)\mathit{,}$(9)where e_i are the n MST edges. Analogous to Λ_MST, we obtain the new measure Γ_MST via a proper normalisation: ${\mathrm{\Gamma }}_{\mathrm{MST}}\mathrm{=}\frac{{\mathit{\gamma }}_{\mathrm{MST}}^{\mathrm{ref}}}{{\mathit{\gamma }}_{\mathrm{MST}}^{\mathrm{mass}}}\mathit{,} \mathrm{\Delta }{\mathrm{\Gamma }}_{\mathrm{MST}}\mathrm{=}\mathrm{\Delta }{\mathit{\gamma }}_{\mathrm{MST}}^{\mathrm{ref}}\mathit{.}$(10)Note that the usage of a geometric mean involves a multiplicative standard deviation ΔΓ_MST, i.e. the upper and lower 1σ intervals are given by Γ_MST·(ΔΓ_MST)^±1.

The geometrical mean has two important properties that turn out to be very useful for our purpose of using the MST as a measure of mass segregation:

• 1.

  The n-th root implicitly involves an intermediate-pass that damps contributions from extreme edge lengths very effectively (i.e. it gives a lower weight to very short or very long edges). Hence the mean edge length of a compact configuration of even few stars will not be affected much by an “outlier”. We demonstrate this property via a simple case of n−1 edges with length l = e₀ and one very short “outlier” representing a tight binary with l = e₁ = ϵe₀, where ϵ ≪ 1 (e₁ and ϵ could be also very large here in principle). Then the geometric mean would yield $\mathrm{⟨}\mathit{l}\mathrm{⟩}\mathrm{=}{\mathrm{\left(}{\mathit{e}}_{\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}{\mathit{e}}_{\mathrm{1}}\mathrm{\right)}}^{\mathrm{1}\mathit{/}\mathit{n}}\mathrm{=}{\mathit{e}}_{\mathrm{0}}^{\mathit{n}}{\mathit{ϵ}}^{\mathrm{1}\mathit{/}\mathit{n}}\mathit{.}$(11)In any practical case (i.e. realistic star clusters models, observational data) the relevant binary separation will be at most 3 orders of magnitude smaller than the mean separation of single stars (or binary centre-of-mass)³. So with ϵ = 10^-3 we obtain even for a very small sub-sample of n = 5 stars a mean edge length  ⟨ l ⟩  ≈ 1/4e₀. This demonstrates the very effective damping of extreme values by the geometric mean.

• 2.

  The product of edges has the valuable property that common edges in the two samples of massive and reference stars, ${\mathit{\gamma }}_{\mathrm{MST}}^{\mathrm{mass}}$ and ${\mathit{\gamma }}_{\mathrm{MST}}^{\mathrm{ref}}$, are cancelled out by normalisation, Eq. (10). So it is only the disjoint set of edge lengths that determines Γ_MST and hence the degree of mass segregation. Compared to ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ our scheme ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ is thus much more robust when applied to stellar systems with a high binary fraction.

2.2. Primordial mass-segregation

A thorough test of our method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ requires the generation of predefined mass segregated stellar systems. For this purpose we found the method of Šubr et al. (2008) to be very useful. The strength of the method is the generation of a well defined degree of mass segregation that is controlled via one single parameter, S, and the creation of dynamically consistent models by modelling local potentials and velocities in a quasi-equilibrium state. The so-called index of mass segregation covers the range S ∈  [0,1), where S = 0 corresponds to an entirely unsegregated state, while the upper limit S = 1 marks the maximum possible segregation. In Fig. 1 we show two initially mass segregated models of 100 particles with S = 0.1 and S = 0.9 and their corresponding MST, respectively.

In summary, the authors find that a mean inter-particle potential of the form $\mathrm{⟨}{{\mathit{U}}^{\mathit{ij}}}^{\mathrm{⟩}}\mathrm{=}\mathrm{2}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{S}{\mathrm{)}}^{\mathrm{2}}⟨{\mathit{U}}_{\mathrm{tot}}⟩\hspace{0.17em}\frac{{\mathit{m}}_{\mathit{i}}\hspace{0.17em}{\mathit{m}}_{\mathit{j}}}{{\mathit{M}}_{\mathrm{c}}^{\mathrm{2}}}\hspace{0.17em}{\left(\frac{{\mathit{M}}_{\mathrm{sub}}^{\mathit{i}}\hspace{0.17em}{\mathit{M}}_{\mathrm{sub}}^{\mathit{j}}}{{\mathit{M}}_{\mathrm{c}}^{\mathrm{2}}}\right)}^{\mathrm{-}\mathit{S}}\mathit{,}$(12)using ordered subsets of stars, ${\mathit{M}}_{\mathrm{sub}}^{\mathit{i}}\mathrm{\equiv }\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathit{i}}\hspace{0.17em}{\mathit{m}}_{\mathit{j}}\mathit{,}{\mathit{m}}_{\mathrm{1}}\mathrm{\ge }{\mathit{m}}_{\mathrm{2}}\mathrm{\ge }\hspace{0.17em}\mathit{...}\hspace{0.17em}\mathrm{\ge }{\mathit{m}}_{\mathit{N}}\mathit{,}$(13)provides one of the simplest forms that characterise locally consistent mass segregation. This equation provides a constraint on the distribution function from which one can construct a cluster with the desired degree of mass segregation S by adding one-by-one the individual stars from a mass-ordered set. The underlying distribution function $\mathit{n}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{\propto }{\mathit{r}}^{\mathrm{2}}{\mathrm{\left(}{\mathit{r}}_{\mathrm{p}}^{\mathrm{2}}\mathrm{\right(}{\mathit{M}}_{\mathrm{sub}}^{\mathit{i}}\mathrm{)}\mathrm{+}{\mathit{r}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{-}\mathrm{5}\mathit{/}\mathrm{2}}$(14)with ${\mathit{r}}_{\mathrm{p}}\mathrm{\left(}{\mathit{M}}_{\mathrm{sub}}^{\mathit{i}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{3}\mathit{\pi }}{\mathrm{32}}\hspace{0.17em}\frac{\mathit{G}{\mathit{M}}_{\mathrm{c}}^{\mathrm{2}}}{\mathrm{\left|}⟨{\mathit{U}}_{\mathrm{tot}}⟩\mathrm{\right|}}\hspace{0.17em}\frac{\mathrm{1}}{\mathrm{1}\mathrm{-}\mathit{S}}{\left(\frac{{\mathit{M}}_{\mathrm{sub}}^{\mathit{i}}}{{\mathit{M}}_{\mathrm{c}}}\right)}^{\mathrm{2}\mathit{S}}\mathit{,}$(15)provides a good estimate of the real distribution function of a Plummer sphere with mass segregation index S.

The authors provide a numerical C-code plumix for generating the cluster according to the algorithm described in their paper on the AIfA web page: http://www.astro.uni-bonn.de

3. Results

In this section we provide examples showing the much better performance of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ than ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$. However, we will also compare with the more traditional method ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$ introduced in Sect. 1.

3.1. Special configurations

First, we will demonstrate the power of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ for some simple setups of artificial mass segregation as shown in Fig. 2.

The idea to improve ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ and develop ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ was in fact motivated by the goal to find a measure that reflects the optical impression of a lower or higher degree of mass segregation. The plots in Fig. 2 depict three different artificial configurations of the five most massive stars in a star cluster of 1 k members, designated “cross”, “ring”, and “clump”, from top to bottom. All these configurations are characterised by identical l_MST, i.e. according to ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ they represent the same degree of mass segregation. However, from an observer’s point of view the cross appears to show a lower degree of mass segregation than the ring (i.e. the latter appears more compact), while the clump would usually be interpreted as a highly segregated system with a peculiar outlier.

Translated into a consistent algorithm we aim at measuring the compactness of a stellar system by assigning a higher weight to the dominant configuration of stars and hence to overcome the “degeneracy” of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$. As already discussed in Sect. 2.1 this is mainly achieved by damping the contribution from “outliers”, i.e. single edges with extreme lengths e_i compared to the median of all edges, via the geometrical mean. Figure 3 demonstrates the effect of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$. While by construction Λ_MST (black dashed line) is identical for all three configurations, Γ_MST depends strongly on the degree of concentration of the dominant sample of stars. Using the five most massive stars for the calculation of Γ_MST the highly concentrated “clump” shows a roughly two times higher degree of mass segregation than the other two configurations. Also, it’s significance (i.e. standard deviation in relation to the mean) is about 1.5 times higher than in the case of Λ_MST.

Note that though only the five most massive stars form a centrally concentrated configuration, i.e. are mass-segregated, one obtains a signature of mass segregation for a sample of up to 20 most massive stars. In particular, Γ_MST of the ten most massive stars does even depend on the geometry of the configuration.

3.2. Initially mass-segregated clusters

Using the method of Šubr et al. (2008) to create initially mass-segregated star clusters (see Sect. 2.2) we have generated a set of low- to high-mass star clusters with 100, 1 k, 10 k, and 100 k stars, respectively. Their underlying mass function from Kroupa (2001) in the range 0.08−150 M_⊙ results in an average stellar mass of  ~0.6 M_⊙. So our clusters span the mass range from 60 M_⊙ to 60   k M_⊙ which basically represents the entire observed cluster population in our Galaxy. All clusters have been set up with initial degrees of mass segregation S =  {0.1,0.2,0.3,0.5,0.9} , each with ten configurations with different random seeds to reduce statistical uncertainties. However, in the following plots we show rescaled error bars that represent the statistical uncertainties of one single cluster.

In Figs. 4 and 5 we show a selection of the most relevant results. On the left-hand side we compare Γ_MST (“geometric”) and Λ_MST (“arithmetic”) for the 5 (red), 10 (green), 20 (blue), 50 (magenta), 100 (cyan), 200 (brown), 500 (orange), and 1000 (black) most massive stars. The error bars and line thickness mark the 1σ, 2σ, and 3σ uncertainties. The horizontal dotted lines mark integer values, the horizontal dashed line represents the unsegregated state Γ_MST = Λ_MST = 1. Star clusters of intermediate masses  ~1000 M_⊙ are well observed in our local neighbourhood. These objects are represented by our simulations with 1 k particles. From Fig. 4 we find that our measure of mass segregation, Γ_MST, detects a very low degree of mass segregation, S = 0.1, just above the 1σ-level, nearly independent of the number of most massive stars considered. The significance improves drastically for an intermediate degree of S = 0.3 to at least 3σ and even up to 4σ for the 20 to 100 most massive stars. Finally, for very strong mass segregation with S = 0.9 we obtain a very clear signature of more than 5σ for any number of (i.e. at least five) most massive stars.

These results are very much better than using Λ_MST which never approaches a significance of 3σ and – in particular – for S = 0.3 provides only a very weak 1σ significance compared to 3−4σ in the case of Γ_MST.

On the right-hand side we plot the corresponding mass functions for comparison with the traditional method ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$ to detect mass segregation. Here the mass function has been calculated in annuli with a radius r =  {1/4   r_h,1/2   r_h,r_h,r_c} , where r_h is the half-mass and r_c the total radius of the star cluster. It is clearly demonstrated that only very strong mass segregation with S ≈ 0.9 becomes evident via this method. In this case it is the mass range of stars around 10 M_⊙ that provides the strongest signature.

In contrast, our new measure Γ_MST is much more effective. In particular, the 10 to 20 most massive stars of a 1 k star cluster usually provide the clearest signature of mass segregation.

Qualitatively, we find the same results for more massive clusters of 10 k stars in Fig. 5: ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ provides the best measure of mass segregation by far. However, there are some important quantitative differences. First, a much lower degree of mass segregation can be detected for more massive clusters, e.g. for 10 k stars with S = 0.1 a significance of 3σ is reached. Second, for a similar significance mass segregation becomes most evident for a larger number of most massive stars, e.g. the 20 most massive stars for a 10 k system with S = 0.3 compared to the 10 most massive for a 1 k system with S = 0.9.

Both effects are in agreement with our expectations that in a more populous star cluster i) the same degree of mass segregation S will result in a higher relative concentration of the same number of most massive stars, i.e. Γ_MST becomes larger; and ii) the absolute number of stars that show the same relative concentration is larger, i.e. ΔΓ_MST becomes lower.

The interplay of these two properties explains the existence and dependence of the local maximum of the significance of Γ_MST: it peaks at the maximum ratio of Γ_MST and ΔΓ_MST and thus increases for larger cluster masses and lower degrees of mass segregation towards larger numbers of most massive stars.

From our entire set of initially mass-segregated clusters we find that a sample size of 10 to 20 most massive stars generally provides the clearest signature of mass segregation.

Note that from Figs. 4 and 5 we conclude that using ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$ the best estimate of mass segregation in star clusters from 1 k to 10 k members would be for masses at  ~10 M_⊙ in an annulus of radius r ≈ 1/4   r_h.

3.3. Mass-segregation in the ONC

We recall that one of our goals was to develop a method that equally well applies to data from simulations and observations and thus demonstrate the excellent performance of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ on observational data of the  ~1 Myr old Orion nebula cluster (ONC) obtained by Hillenbrand (1997). The sample contains 929 stars with mass estimates and so provides a robust test of the algorithm.

The resulting plot of Γ_MST for the 5, 10, 20, 50, 100, 200, 500 most massive, and all 929 stars in the upper part of Fig. 6 provides a clear signature of a significant concentration of the 20 most massive stars, much stronger than given by Λ_MST. Interestingly, our calculation of Λ_MST shows slightly stronger mass segregation than published by Allison et al. (2009a). Compared to the initially mass-segregated cluster models with 1 k particles presented in Fig. 4 the plot of Γ_MST for the ONC shows a more complex distribution: there is a sharp drop between the 10 and 20 most massive stars. The significance of the first two data points (≳4σ) corresponds to the model with S = 0.9, the  ≲3σ significance of the other resembles the much less segregated models with S ≤ 0.3.

In summary, i) mass segregation in the ONC at the current age  ~1 Myr is much stronger than estimated before; and ii) it is clearly detectable not only for the five most massive stars yet for the 20 most massive objects.

However, the particular features of the Γ_MST distribution require a more careful analysis. We do so by calculating Γ_MST of disjoint particle groups, i.e. the most massive 5, 6 to 10, 11 to 20, etc. stars. The corresponding plot at the bottom of Fig. 6 shows that it is only the five most massive stars that are really strongly mass segregated (with a significance  >4σ). Their configuration is so dominant that adding the next 15 most massive stars – which are not segregated – still shows a strong signature at the 3σ level, similar to the results in Sect. 3.1.

Hillenbrand (1997) argues that her sample of combined photometric and spectroscopic data appears completely representative of the ONC, showing in particular a uniform completeness with cluster radius. However, because observational data are usually biased by incompleteness this issue is addressed in Appendix A. We demonstrate that incompleteness has only a moderate effect on ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ and provide a simple prescription for recovering the original signature of mass segregation via the completeness function.

We conclude that all but the five most massive ONC stars do not show any signs of mass segregation. This result is in excellent agreement with Huff & Stahler (2006, Fig. 4. The five most massive stars’ extraordinary degree of mass segregation resembles their peculiar tight trapezium-like configuration (e.g. McCaughrean & Stauffer 1994; Hillenbrand & Hartmann 1998).

3.4. Dynamical evolution of mass-segregation

As a last application of our new method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ we analyse again numerical data yet here we trace the dynamical evolution of a star cluster model over time. The initial configuration is based on our ONC model as described in previous publications (e.g. Olczak et al. 2010). In short, it is a single star cluster with an isothermal initial density distribution, a Maxwellian velocity distribution, the mass function of Kroupa (2001) in the range 0.08−150 M_⊙, and an initial size of 2.5 pc. Note that we use a spherically symmetric model with a smooth density distribution without any substructure. However, in contrast to our basic equilibrium model here our stellar system is initially collapsing, starting from a virial ratio Q = 0.1. The simulations were carried out with Nbody6⁴ (Aarseth 2003) until a physical age of 5 Myr, corresponding to 13.5 N-body time units of the cluster.

In Fig. 7 we show the mean of three simulations of the same model with different random seeds used to generate individual positions, velocities, and masses. We plot Γ_MST and Λ_MST only for the 5, 10, 20, 50, and 500 most massive stars for reasons of visibility. The much better performance of our improved method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ compared to ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$ is evident. Though both measures do show similar maximum degrees of mass segregation Λ_MST is subject to large variations over time. Only for the five most massive stars does it show high significance over a longer period while for the 10 and 50 most massive stars even values lower than one are quite common.

In contrast, Γ_MST clearly increases over time with the steepest gradient within the first 2 Myr (or 5.5 N-body time units). It clearly separates the different mass groups, inversely correlated with the sample size. The much better robustness of Γ_MST reflects again its very efficient damping of “outliers”.

We note that mass segregation in a collapsing, intermediate-size stellar cluster of 1 k stars can occur very quickly, i.e. within only a few crossing times. This finding demonstrates that rapid mass segregation (in terms of dynamical time scale) does not require substructure as has been recently investigated by Allison et al. (2009a). However, a detailed investigation of dynamical mass segregation in young star clusters will be presented in an upcoming publication.

4. Conclusion and discussion

We have developed a new method, ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$, to measure mass segregation by significantly improving a previous approach of Allison et al. (2009a) based on the minimum spanning tree (MST), here referred to as ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$. Their method uses the normalised length of the MST of a given sample of stars, Λ_MST, as a measure of compactness.

Compared to “classical” measures of mass segregation such as the slope of the (differential) mass function in different annuli around the cluster centre (see Sect. 1), Λ_MST overcomes several substantial weaknesses. Our new method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ inherits all the advantages provided by ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Lambda }}$:

• independence of cluster geometry,

• no requirement of cluster centre,

• no requirement of quantitative measure of masses,

• one-parametric unique measure of mass segregation,

and adds three significant improvements:

• nearly linear computational efficiency,

• robustness against peculiar configurations, and

• high sensitivity.

This gain is based on two boosting ingredients:

• 1.

  the implementation of an efficient ) algorithm for thecalculation of the MST, and

• 2.

  the calculation of the geometrical mean of the connecting edges, Γ_MST, instead of just their sum, Λ_MST.

The high performance is achieved by construction of a graph via 2D Delaunay triangulation (from the software package GEOMPACK: Joe 1991), quick sorting of its edges, and an efficient numerical implementation of Kruskal’s algorithm (Kruskal 1956). The advantage of the geometrical mean is that it damps the contribution of “outliers” very efficiently and thus traces the concentration of the dominant stars of a configuration.

We have demonstrated the advantage of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ compared to other known and frequently used methods via various examples, both for numerical and observational data.

• 1.

  In general, using only the ten to twenty most massive stars Γ_MSTprovides a robust and sensitive measure of mass segregation forthe entire population of star clusters in our Galaxy.

• 2.

  In particular, very low degrees of mass segregation can be detected in massive clusters like NGC 3603 that consist of 10 k or more stars.

• 3.

  We have also confirmed the very strong mass segregation of the five most massive stars in the ONC as reported by Huff & Stahler (2006).

• 4.

  Incompleteness of observational data has only a moderate effect on ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ and corrections can be implemented easily similar to methods like ${\mathrm{ℳ}}_{\mathrm{MF}}^{\mathrm{d}}$.

However, we also stress three important aspects that have to be considered for a careful investigation of mass segregation via ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$:

• 1.

  Our conclusion that the 10 to 20 most massive stars generallyprovide the most sensitive measure of mass segregation has beenderived from model clusters with smooth mass segregation overthe entire sample. We caution that this is not to be expected if onlyspecific subsamples are affected by mass segregation(as shown for the five most massive stars of theONC in Sect. 3.3); this is of particular concern forstudies of primordial mass segregation. We thus generallyrecommend to use both the cumulative and the differentialanalysis of a set of mass groups to investigate mass segregation in(young) star clusters.

• 2.

  The signature of a small sub-sample can be strong enough to bias Γ_MST of a larger parent sample. To avoid wrong conclusions one should also calculate Γ_MST of disjoint samples, e.g. of the 5, 6 to 10, 11 to 20 etc. most massive stars.

• 3.

  The value of Γ_MST of a sub-sample is independent of its location with respect to the entire sample. It only reflects the compactness of this sub-sample, not its concentration towards some centre of the entire sample. So a large Γ_MST does not necessarily mean that the corresponding sub-sample is mass-segregated. Thus, in practice, in particular when dealing with observational data, the determination of some centre of the sample as reference point is still required. However, the value of Γ_MST remains independent of the reference point and so its advantage compared to other methods.

Our improved method ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ will be vital for future studies of mass segregation in stellar systems. Its robustness and sensitivity is ideal for tracing lower degrees and more complex types of mass segregation than before. This is of particular interest for the earliest stages of star formation that seem to form protoclusters of complex structure (e.g. Teixeira et al. 2006). Furthermore, it will also help to investigate the question whether mass segregation is observed at all in young star clusters (e.g. Ascenso et al. 2009). Finally, our method can be used in general for precise structure analysis of any type of stellar system from planetary debris disks to galaxy clusters.

As a first application of ${\mathrm{ℳ}}_{\mathrm{MST}}^{\mathrm{\Gamma }}$ to numerical simulations we have demonstrated that mass segregation in young star clusters can occur very quickly under dynamically cold initial conditions. Indeed we do expect young stellar systems to form under such conditions as a result of the gravitational collapse of their parent molecular cloud. Whether collapse still occurs in largely evolved young clusters like the ONC – the scenario corresponding to our simulation – is still unknown yet became favoured recently (e.g. Jones & Walker 1988; Kroupa 2000; Scally et al. 2005; Fűrész et al. 2008; Proszkow et al. 2009; Tobin et al. 2009). However, at least some fraction of the oldest stars must have formed during global collapse. Hence rapid mass segregation affects the structure of star forming regions at the earliest stages and could thus explain the observed mass segregation of the most massive stars in the ONC.

We will investigate this and other aspects of dynamical mass segregation in young star clusters under various conditions in an upcoming publication.

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Acknowledgments

C.O. and R.S. acknowledge support by NAOC CAS through the Silk Road Project, and by Global Networks and Mobility Program of the University of Heidelberg (ZUK 49/1 TP14.8 Spurzem). C.O. appreciates funding by the German Research Foundation (DFG), grant OL 350/1-1. R.S. is funded by the Chinese Academy of Sciences Visiting Professorship for Senior International Scientists, Grant Number 2009S1-5. We have partly used the special supercomputers at the Center of Information and Computing at National Astronomical Observatories, Chinese Academy of Sciences, funded by Ministry of Finance of Peoples Republic of China under the grant ZDY Z20082. We thank S. Aarseth for providing the highly sophisticated N-body code Nbody6 (and its GPU extension) and greatly appreciate his support.

References

Appendix A: Data incompleteness and Γ_MST

We provide an example on how incompleteness of observational data affects Γ_MST and present a simple prescription for recovering the original signature of mass segregation via the corresponding completeness function.

Here we use again the original sample of 929 stars in the ONC from Hillenbrand (1997) to generate a reduced sample via convolution with an artificial completeness function with radial and mass dependence of the form $\begin{array}{ccc}\mathit{c}\mathrm{\left(}\mathit{r,}{\mathit{r}}_{\mathrm{u}}\mathit{,p,}{\mathit{a}}_{\mathrm{l}}\mathit{,}{\mathit{a}}_{\mathrm{u}}\mathit{,m,}{\mathit{m}}_{\mathrm{l}}\mathit{,}{\mathit{m}}_{\mathrm{u}}\mathit{,}{\mathit{b}}_{\mathrm{l}}\mathit{,}{\mathit{b}}_{\mathrm{u}}\mathrm{\right)}& & \mathrm{=}\\ & & \mathrm{1}\mathrm{-}\exp \mathrm{\{}\mathrm{ℛ}\mathrm{\left(}\mathit{r,}{\mathit{r}}_{\mathrm{u}}\mathit{,p,}{\mathit{a}}_{\mathrm{l}}\mathit{,}{\mathit{a}}_{\mathrm{u}}\mathrm{\right)}{\left[\mathrm{ℳ}\mathrm{\left(}\mathit{m,}{\mathit{m}}_{\mathrm{l}}\mathit{,}{\mathit{m}}_{\mathrm{u}}\mathrm{\right)}\right]}^{\mathrm{ℛ}\mathrm{\left(}\mathit{r,}{\mathit{r}}_{\mathrm{u}}\mathit{,p,}{\mathit{b}}_{\mathrm{l}}\mathit{,}{\mathit{b}}_{\mathrm{u}}\mathrm{\right)}}\mathrm{\}}\mathit{,}\end{array}$(A.1)where $\begin{array}{ccc}& & \mathrm{ℛ}\mathrm{\left(}\mathit{r,}{\mathit{r}}_{\mathrm{u}}\mathit{,p,}{\mathit{q}}_{\mathrm{l}}\mathit{,}{\mathit{q}}_{\mathrm{u}}\mathrm{\right)}\mathrm{=}\mathrm{(}{\mathit{q}}_{\mathrm{u}}\mathrm{-}{\mathit{q}}_{\mathrm{l}}\mathrm{)}\mathrm{(}\mathit{r}\mathit{/}{\mathit{r}}_{\mathrm{u}}{\mathrm{)}}^{\mathit{p}}\mathrm{+}{\mathit{q}}_{\mathrm{l}}\mathit{,}\\ & & \mathrm{ℳ}\mathrm{\left(}\mathit{m,}{\mathit{m}}_{\mathrm{l}}\mathit{,}{\mathit{m}}_{\mathrm{u}}\mathrm{\right)}\mathrm{=}\frac{\log \mathrm{(}\mathit{m}\mathit{/}{\mathit{m}}_{\mathrm{l}}\mathrm{)}}{\log \mathrm{(}\mathit{m}\mathit{/}{\mathit{m}}_{\mathrm{u}}\mathrm{)}}\mathit{,}\end{array}$\arraycolsep1.75ptand the indices l and u denote lower and upper values, respectively.

The parameters used to simulate loss of data due to observational incompleteness are presented in Table A.1. The corresponding plot in Fig. A.1 shows the completeness as a function of stellar mass in dependence of radial position in the cluster. From right to left the graphs represent increasing radii in steps of 0.25 pc from the centre to 2.5 pc, the outer boundary of the ONC. The solid (red), dashed (blue), and dotted (black) lines are grouped in steps of 1 pc, respectively.

The functional form has been adopted such that it resembles observational incompleteness due to crowding in the dense and bright cluster centre and due to individual extinction over the entire cluster. It is expected that stars with masses  ≳10 M_⊙ are too bright to be missed while low-mass objects close to the cluster centre with it’s four luminous Trapezium stars remain mostly undetected.

To reconstruct a larger unbiased sample from the incomplete sample we have used the inverse completeness function in the following simple prescription:

• 1.

  Go to the next object i in the incomplete sample.

• 2.

  Calculate its completeness c_i using its radial position r_i and mass m_i.

• 3.

  If c_i > c_min:

  • (a)

    calculate the missing number of objects n_i = 1/c_i−1 with r_i and m_i,

  • (b)

    generate for each additional object j its

    • i.

      mass m_j = m_i(1 + λ), λ ∈  [−0.05,0.05]  a random number, and

    • ii.

      radial position r_j = r_i(1 + η), η ∈  [−0.1,0.1]  a random number.

In short, the parents’ individual completeness c_i determines the number of “clones” n_i with slightly varying properties. The threshold c_min is used to prevent massive cloning of rare objects that could introduce a significant bias. We found that a surprisingly low value of c_min = 0.2 is save.

We compare Γ_MST of the three samples – original, incomplete, and reconstructed – in Fig. A.2. The left-hand side shows Γ_MST of the 5, 10, 20, 50, 100, 200, 500 most massive, and all stars, i.e. cumulative mass groups; the right-hand side shows Γ_MST of the 5, 6 to 10, 11 to 20, 21 to 50, 51 to 100, 101 to 200, 201 to 500, and 500 to all most massive stars, i.e. differential mass groups. The top plot refers to the original sample of 929 ONC stars: it is the same data as on the left-hand side of Fig. 6. Below we plot Γ_MST of the incomplete sample that has been artificially reduced to 485 stars via Eq. (A.1). This removal of nearly half the cluster population – preferentially low-mass stars in the cluster centre – has a marginal effect on Γ_MST: only for the five and ten most massive stars a significant difference is observed. However, the differential plot reveals again a dominant contribution from the five most massive stars, Γ_MST of the 6−10 most massive remains nearly unchanged.

The bottom plot contains data of additional 345 cloned objects and so 830 stars in total – about 10% less than the original sample. The reconstruction procedure seems to work fairly well: Γ_MST of the five and ten most massive stars is indistinguishable from its original values. However, the next intervals up to the 200 most massive stars all show slightly lower degrees of mass segregation than originally. The reason is that the sample of even less massive stars (200−500 and 500−830 most massive stars) is marginally mass segregated (i.e. Γ_MST > 1 unlike in the original sample): these low-mass stars that make up the main part of the random reference sample were cloned preferentially closer to the cluster centre which reduced Γ_MST of the more massive stars.

However, considering the reconstruction of up to 80% individual incompleteness and nearly half of the entire incomplete sample, the result is very promising: the strong conclusion for the original sample that the five most massive stars in the ONC are highly mass segregated while all other stars do not show any signature of mass segregation remains unchanged for the reconstructed sample.

All Tables

All Figures