Open Access
Issue
A&A
Volume 629, September 2019
Article Number A135
Number of page(s) 11
Section Galactic structure, stellar clusters and populations
DOI https://doi.org/10.1051/0004-6361/201834080
Published online 17 September 2019

© S. Dib and T. Henning 2019

Licence Creative Commons
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The star formation process yields a large number of physical quantities and distribution functions that can be quantified by observations. Comparing these quantities and distributions between different star forming regions and with theoretical models and numerical simulations helps us gain insight into the relevance of various physical processes in different galactic environments. Some of the most studied quantities are the mass distribution of dense cores and of the stellar initial mass function in the early and emerging phases of the formation of stellar clusters (e.g., Johnstone & Bally 2006; Dib 2014; Hony et al. 2015; Dib et al. 2017). The spatial distribution of dense cores in the early phases of star formation, of protostars in the phases of a cluster’s buildup, and of stars in the (pseudo)gas-free phase of a young cluster, in conjunction with their masses and dynamics, can also encapsulate critical information on how clusters assemble and form in different environments.

The spatial distribution of stars in young and in evolved clusters has received a substantial amount of attention, both in observations (McNamara & Sekiguchi 1986; Sánchez & Alfaro 2009; Gouliermis et al. 2014; Parker & Alves de Oliveira 2017; Dib et al. 2018) and in numerical simulations of star forming regions (Schmeja & Klessen 2006; Lomax et al. 2011; Parker & Dale 2015; Gavagnin et al. 2017). However, quantifying the structure of star forming regions in the very early phases of star formation has remained elusive. This was principally due to the scarcity of observational data with the adequate spatial resolution to probe core masses in the stellar mass regime. With the advent of the Herschel Space Observatory (hereafter Herschel) and the Atacama Large Millimeter Array of radiotelescopes (ALMA), it is now possible to probe the spatial and mass distributions of dense structures in nearby star forming regions down to the mass regime of proto-brown dwarfs. The high sensitivity and spatial resolution of both Herschel and ALMA has provided unprecedented insight into quantities such as the dense core mass function (CMF) in the regions of Aquila, Taurus, Corona Australis, and W43 (André et al. 2010; Könyves et al. 2010, 2015; Marsh et al. 2016; Bresnahan et al. 2018; Motte et al. 2018). The published CMFs of these regions suggest, at the very least, a striking difference between the low mass star forming regions such as Aquila and Taurus, and regions of massive star formation, such as W43. When described by a power law of the form dN/dM ∝ Mα, the derived value of α for W43 is ≈1.9, which makes the CMF of W43 ostensibly shallower than in low mass star forming regions for which the derived values of α are close to 2.3 ( Motte et al. 1998, but see Sadavoy et al. 2010), or even steeper as in the case of the California Molecular Cloud (Zhang et al. 2018). Dib et al. (2008a) showed that the slope of the CMF is steeper for cores that are defined using molecular species that trace higher densities of the gas and that are associated with an increasing degree of gravitational boundedness of the cores. However, in the case of the aforementioned star forming regions, the differences in the slopes of the CMF cannot be attributed to differences in the choice of the density tracers as all of the cores in those regions are observed in the continuum submillimeter emission. Within individual star forming regions, it has now been established that there are environmental differences between the populations of cores that are found on and off the filamentary structure of the clouds (Polychroni et al. 2013; Olmi et al. 2016; Kainulainen et al. 2017; Bresnahan et al. 2018). This can be attributed to effects of early mass segregation or to an extended phase of gas accretion by cores that reside inside the filaments (Dib et al. 2010).

We complement the picture provided by the CMF of these different regions by analyzing the spatial distribution of their populations of dense cores. The structure and mass segregation of the cores in these regions lock important information on how star clusters form and evolve in different galactic environments. A number of studies have suggested a similarity between the structure of young clusters and of molecular clouds (Elmegreen et al. 2006; Gouliermis et al. 2014), while other works have argued that the similarity between the gas distribution and the distribution of stars can be quickly altered by the effects of rapid gas expulsion (Dib 2011; Dib et al. 2011, 2013) or by gas tidal shocking of the young star clusters by surrounding gas clouds (Kruijssen et al. 2012). The question of whether the most massive stars in young clusters are mass segregated with respect to the total population of stars is still highly debated and has received a significant amount of attention both in observational studies (Hillenbrand & Hartmann 1998; de Grijs et al. 2002; Gouliermis et al. 2004; Kerber & Santiago 2006; Chen et al. 2007; Bontemps et al. 2010a; Er et al. 2013; Habibi et al. 2013; Lim et al. 2013; Elmegreen et al. 2014; Yu et al. 2017; Kuhn et al. 2017; Dib et al. 2018; Moser et al. 2019) and in theoretical and numerical works (Bonnell et al. 2003; Dib et al. 2007a, 2008b; Dib 2007; Küpper et al. 2011; Olczak et al. 2011; Maschberger & Clarke 2011; Geller et al. 2013; Sills et al. 2018). The debate expands on whether the observed levels of mass segregation are primordial or due to dynamical interaction between stars in the clusters (Khalisi et al. 2007; Dib et al. 2018).

In this work, we measure the spatial distribution of dense cores in the regions of Aquila, Taurus, Corona Australis, and W43 using the 𝒬 parameter (Cartwright & Whitworth 2004). We also quantify the levels of mass segregation in those regions using a measurement of the mass segregation ratio following the methods of Allison et al. (2009) and Olczak et al. (2011). We explore how the structure of these star forming regions and their levels of mass segregation correlate with their star formation activity. The paper is organized as follows: in Sect. 2, we briefly present the data sets that are used in this study. In Sect. 3, we recall the basics of the methods used to quantify the structure parameter and mass segregation ratios. In Sect. 4, we present our results and in Sect. 5, we discuss them in light of previous work. In Sect. 6, we conclude.

2. The data set of star forming regions

We use the publicly available data of the following star forming regions: Aquila, the L1495 region in Taurus, Corona Australis, and the W43-MM1 region. Observations for the first three regions were performed in the framework of the Herschel Gould Belt Survey (André et al. 2010) with a wavelength coverage in the submm bands going from 70 to 500 μm. For all regions, the cores were extracted using the getsources algorithm (Men’shichikov et al. 2012). In Aquila, the area of the field covered by the Herschel observations is ≈11 deg2. The source extraction permitted the identification of 651 cores of which 446 are gravitationally bound and can be considered as prestellar cores (André et al. 2010; Bontemps et al. 2010b; Könyves et al. 2010, 2015). In the cloud Corona Australis (CrA), the field covered by the observations has a surface area of 29 deg2. When applied to the Herschel map of the region, getsources identified 163 cores, out of which 99 are prestellar core candidates (Bresnahan et al. 2018). in Taurus, we use the data related to the identification of dense cores around the L1495 sub-region which covers a surface area of ≈8 deg2 (Marsh et al. 2016). The application of getsources permitted the identification of 525 starless cores and 52 prestellar gravitationally bound cores. The dense cores in the W43 massive star forming region were extracted using getsources from the 1.3 mm maps using ALMA observations of the W43-MM1 region (Motte et al. 2018). For this region, getsources returned 131 cores. However, the observations do not distinguish yet whether these cores are unbound or whether they are prestellar in nature. As pointed out in Motte et al. (2018), the characterization of the gravitational boundedness of these cores requires more scrutiny1. For all three regions observed with Herschel, and given the multi-wavelength coverage of the Herschel instruments, the masses of cores are derived using an estimate of the dust temperature that is obtained by fitting the corresponding spectral energy distribution. In the case of W43, the mass estimates of cores obtained by Motte et al. (2018) were derived by combining all available continuum maps of the region and applying the PPMAP software (Marsh et al. 2015) in order to infer a surface density-weighted dust temperature. In general it is worth noting that for all regions considered here, the characterization of the gravitational boundedness of cores was performed using continuum emission data only. A determination of the temperature of the gas in the cores, and measurements of its dynamics, in and around the cores, are crucial in order to better assess their true degree of gravitational boundedness (e.g., Dib et al. 2007b).

3. Methods

We quantify the spatial distribution and mass segregation of dense cores using the same method as the one presented in Dib et al. (2018). Namely, we use two methods based on a minimum spanning tree (MST) which is the unique set of straight lines (“edges”) connecting a given set of points without closed loops, such that the sum of all edge lengths is a minimum (Kruskal 1956; Prim 1957; Gower & Ross 1969; Schmeja 2011).

3.1. The 𝒬 parameter

We characterize the spatial distribution of the cores with the 𝒬 parameter (Cartwright & Whitworth 2004) which is given by:

(1)

We recall that the 𝒬 parameter combines the normalized correlation length (i.e., the mean separation length between all cores) and the normalized2 mean edge length derived from the MST. The 𝒬 parameter allows us to distinguish between structures with a central density concentration and those that are more hierarchical with a fractal substructure. Large 𝒬 values (> 0.8) are associated with centrally condensed spatial distributions with radial density profiles of the type ρ(r)∝rα (with α >  0), while small 𝒬 values (< 0.8) indicate spatial distributions with a fractal substructure. 𝒬 is correlated with α for 𝒬 >  0.8 and anticorrelated with the fractal dimension D for 𝒬 <  0.8 (Cartwright & Whitworth 2004, in particular see Fig. 5 in their paper). A detailed description of the method, and in particular its implementation used in this study, is given in Schmeja & Klessen (2006). The 𝒬 parameter has been previously used to study the spatial distribution of dense cores and protostars in star forming regions (Guthermuth et al. 2009; Broekhoven-Fiene et al. 2014; Alfaro & Román-Zúñiga 2018; Parker 2018) and of stars in both young (Cartwright & Whitworth 2004; Schmeja & Klessen 2006; Schmeja et al. 2008) and evolved clusters (Gouliermis et al. 2012; Fernandes et al. 2012; Delgado et al. 2013; Kumar et al. 2014; Gregorio-Hetem et al. 2015; Dib et al. 2018).

3.2. Mass segregation ratio

Allison et al. (2009) introduced the mass segregation ratio (MSR), ΛMSR, as a measure to identify and quantify mass segregation in clusters. The method is based on the calculation of the length of the MST, lMST, which measures the compactness of a given sample of vertices in the MST. The mass segregation of a cluster is measured by comparing the value of lMST of the nMST most massive stars, , with the average lMST of k sets of nMST random stars, . The value of ΛMSR is then given by:

(2)

The method has been modified by Olczak et al. (2011) by using the geometric mean rather than the arithmetic mean in order to minimize the influence of outliers. This method works by constructing the MST for the nMST most massive stars and determining the mean edge length γmp. Then, we construct the MST of the same number of randomly selected stars from the entire sample and determine the mean edge length γrand. The value of the MSR following Olczak et al. (2011), ΓMSR, is then given by:

(3)

In this work, we compute both ΛMSR and ΓMSR. In each case, we use 100 subsamples in order to obtain the mean quantities and . A value of ΛMSR ≈ 1 (respectively ΓMSR ≈ 1) implies that both samples of stars (i.e., the most massive and the randomly selected) are distributed in a similar manner, whereas ΛMSR ≫ 1 (respectively ΓMSR ≫ 1) indicates mass segregation, and ΛMSR ≪ 1 (respectively ΓMSR ≪ 1) points to inverse mass segregation (i.e, the massive stars are more spread outwards than the rest).

4. Results

We derived the values of 𝒬, ΛMSR, and ΓMSR for the regions L1495 in Taurus, Aquila, CrA, and W43. For each region, we measured these quantities both for the total population of cores and for the subsample of gravitationally bound cores. The latter category includes cores that have been assessed as being prestellar (i.e., gravitationally bound) and cores that already harbor a protostar. The calculations for ΛMSR and ΓMSR are performed for values of nMST ranging from nMST = 3 to nMST = 100. The values we found for these different regions are summarized in Table 1. For ΛMSR and ΓMSR, we report in Table 1 their maximum values ( and , only if larger than 2) and the corresponding value of nMST at the location of the peak (given between brackets).

Table 1.

Structure parameters and mass segregation levels for the studied star forming regions.

The first observation that can be made is that the 𝒬 values of all regions are well below the transition value of 0.8, which qualifies all of these regions as having a hierarchical structure. These numbers quantify what is already visible by eye in Fig. 1. An interesting aspect is that the region with the smallest star formation activity, Taurus, displays the smallest value of the 𝒬 parameter, particularly for its population of gravitationally bound cores. The higher level of fractal structure in Taurus could be associated with a younger age of the region and/or with processes that are preserving the fractal structure of the cloud for a more extended period of the cloud’s lifetime. One of these physical agents could be stronger magnetic fields that are prevalent in Taurus (Chapman et al. 2011; Heyer & Brunt 2012). The 𝒬 parameters for the population of bound cores in these regions display the same behavior, with the bound cores in Taurus exhibiting stronger levels of substructure. The values of 𝒬 derived here for dense cores are markedly smaller than those found for open clusters including young clusters (with ages ≲10 − 12 Myr) which have 𝒬 values that lie in the range 0.65–0.8 (Dib et al. 2018).

thumbnail Fig. 1.

Spatial distribution of dense cores in the four star forming regions considered in this work. The sizes of cores have been scaled (with an arbitrary formula) according to their masses (i.e., larger sizes relate to more massive cores) in order to visually highlight the location of the most massive cores. The minimum spanning tree for the ensemble of the 6 and 50 most massive cores in each region are displayed with the yellow and purple lines, respectively.

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In terms of mass segregation, the regions of Aquila, CrA, and W43 show significant levels of mass segregation for the ensemble of the 6, 10, and 14 most massive cores, respectively (see Fig. 2 and Table 1). This corresponds to cores with masses in the range [6.2–19.7] M, [0.5–1.3] M, and [16–102] M, respectively. This is observed both in the ΛMSR and ΓMSR parameters. In contrast, the cores in Taurus do not show significant signs of mass segregation, within the uncertainties (reasoning in terms of ΓMSR instead of ΛMSR would imply very mild mass segregation in Taurus, within the 3-σ uncertainty levels)3. With the exception of Taurus, these results imply that mass segregation is primordial which means that massive cores, and possibly massive stars (given that current observations show that massive cores display little evidence of fragmentation; e.g., Bontemps et al. 2010a), tend to form in the denser parts of the star forming regions rather than fall into them by dynamical effects. The physical mechanism by which this occurs is likely related to the assembly process of the clouds and to the formation of a deeper gravitational potential in these dense regions. In a deep potential well, the accretion of gas by the cores and their mass growth is enhanced, leading to the observed levels of mass segregation (Dib et al. 2010). In contrast, in a region like Taurus with a shallower density profile, both the accretion and the relative motions of cores might be smaller due to magnetic support, and the effects of mass segregation are absent or delayed. A similar conclusion has been made for dense cores observed in Serpens South by Plunkett et al. (2018)4.

thumbnail Fig. 2.

Mass segregation ratios ΛMSR (Allison et al. 2009, top row) and ΓMSR (Olczak et al. 2011, bottom row) as a function of the number of most massive cores used in computing them, nMST. The values of ΛMSR and ΓMSR are calculated for the entire population of cores in the star forming regions (left column) and for the populations of bound cores (prestellar cores and cores with a protostar). A version of this figure that includes estimates of the uncertainties on ΛMSR and ΓMSR for each individual region is shown in Fig. B.1.

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In order to check the results of the MST method, we also performed an analysis of the mass segregation levels in those four star forming region using a different approach. The method is based on evaluating whether more massive cores reside in regions of higher surface density of cores, similar to what has been presented in Maschberger & Clarke (2011), Kirk et al. (2016), and Lane et al. (2016). While the two methods do not reflect the same meaning about what mass segregation is, the findings from this method (i.e., surface density–core mass relation) lend support to our findings using the MST-based approach (see Appendix A for more details and discussion). Estimating the effects of mass segregation could be affected by issues related to incompleteness and crowding as discussed by Ascenso et al. (2009) for the case of massive clusters. For all regions that are part of the Gould Belt survey and in observations of W43-MM1, the incompleteness limit is estimated by injecting synthetic sources that are randomly distributed across the high column density parts of the regions (see Könyves et al. 2015; Motte et al. 2018). When the CMFs of these different regions are compared for core masses above the 90% completeness limit, differences between those regions persist (for example in the CMF). It remains to be seen how this picture can be affected when synthetic cores that are injected in the maps are assigned specific spatial distributions and mass segregation levels, and whether these spatial distributions and mass segregation levels can also be recovered. Our intuition is that this would play only a minor effect, since the completeness limit in those regions (apart from W43) is well below the turnover point in each of the CMFs.

We explore whether the value of the 𝒬 parameter, both for the total population of cores and for the gravitationally bound ones is related to the star formation activity of the clouds, represented by the surface density of star formation, ΣSFR. For Taurus, the value of ΣSFR was derived using YSO counts from Rebull et al. (2010) and an AV >  2 gas mass from Pineda et al. (2010). This yields a value of ≈0.14 M yr−1 kpc−2 (Heidermann et al. 2010). For CrA, Heidermann et al. (2010) derived a value of ΣSFR = 3.43 M yr−1 kpc−2 and for Aquila, Evans et al. (2014) obtained a value of 1.8 M yr−1 kpc−2 which is also computed down to an extinction limit of AV = 2. The uncertainty from these SFR measurement using YSOs counts is dominated by the uncertainty on the ages of the class II phase which has been assumed by Heidermann et al. to be 50% (i.e., 2 ± 1 Myr). Using data from Louvet et al. 2014, we calculate a value of ΣSFR in W43-MM1 of ≈1043 M yr−1 kpc−2 (see Table 3 in Louvet et al. for the Ridge region)5. Figure 3 (top panel) displays the correlation between the structure parameter, 𝒬 for the entire and bound populations of cores (full and dashed lines, respectively) versus ΣSFR. A correlation is observed in which clouds that are more hierarchical in nature are associated with lower values of ΣSFR. The star formation activity is also correlated with the population of the most massive cores in each region that are found to be strongly mass segregated. Figure 3 (bottom panel) displays the correlation between ΣSFR and the ratio of the number of massive cores that are most mass segregated to the total number of cores in the star forming regions ηsegreg = Nsegreg/Ntot. A clear correlation is observed by which a higher star formation activity in the cloud is associated with a larger fraction of mass segregated massive cores6.

thumbnail Fig. 3.

Correlation between the structure parameter, 𝒬 for the entire and bound populations of cores (dashed and full lines, respectively) versus the surface density of star formation ΣSFR (top panel) and the correlation between the ratio of the most segregated massive cores to the total number of cores and the ΣSFR. The horizontal error bars are the 50% uncertainties estimates that are likely to be affecting the measurements of ΣSFR. The values of 𝒬 = 0.78 and 0.71 are the ones measured by Dib et al. (2018) for the populations of open clusters and young clusters in the Milky Way, respectively. The estimate of ΣSFR for W43 is made using a different approach than for the other three regions (see text for details). The lines connecting the different sets of points are merely shown to guide the eye.

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The data points in Fig. 3 (top and bottom) are only snapshots of the star forming regions at a given time in their evolution. Dib et al. (2018) showed that the 𝒬 parameter distribution for Galactic open cluster peaks at ≈0.78 and the one for young Galactic clusters at ≈0.71. Thus, one would expect that a significant fraction of protocluster forming regions to approach this limit. The correlation between 𝒬 and ΣSFR can either be the reflection of a time evolution effect (i.e., regions becoming more centrally condensed as they become more gravitationally bound and which is accompanied by a higher SFR) or otherwise represent an imprint of the cloud formation process. One expects that if a star forming region contracts globally under its own self-gravity, it will likely move in Fig. 3 (top) toward higher 𝒬 and ΣSFR values. However, for regions such as Aquila and CrA, the ascent along this track will be halted before they can reach substantially higher values of 𝒬 and ΣSFR. This is primarily due to the fact that dense cores have finite lifetimes that are shorter than the regions crossing times. For the L1495 region in Taurus, this time-evolution scenario is even more problematic, as it starts from both a low 𝒬 value and a low ΣSFR and it is likely that Taurus, if it manages to form any cluster(s), these clusters will simply populate the low 𝒬 tail in the distribution measured by Dib et al. (2018). Thus, we favor a scenario in which the currently observed 𝒬–ΣSFR correlation is an imprint of the cloud assembly process. When gas is expelled from each of the region proto-subclusters, the evolution toward a more centrally condensed structure with higher 𝒬 values that are measured for open clusters (Dib et al. 2018) can proceed in the gas-free phase without being accompanied by an increase in the ΣSFR. Therefore, we attribute the region-to-region variations of 𝒬 and ΛMSR and the correlation between these quantities and the star formation activity of the clouds to differences in the physical conditions that have been imprinted on them by the large scale environment at the time they started to form.

It is important to point out that the correlations observed in Fig. 3 are obtained using only 4 star forming regions. A larger body of observational data is needed before we can derive a more robust quantitative dependence between ΣSFR and the structure and mass segregation of dense cores in star forming regions. Nonetheless, these results offer a novel way for looking at the interplay between the structure of molecular clouds and their star formation rates.

5. Comparison to previous work and discussion

A few studies on individual star forming regions by other group corroborate our findings. Alfaro & Román-Zúñiga (2018) analyzed the structure and mass segregation of dense cores in the quiescent Pipe nebula cloud which is characterized by a very low star formation activity (Lada et al. 2008). While in our case we chose to present an analysis of 𝒬 and ΛMSR (and ΓMSR) for the total and bound population of dense cores, Alfaro & Román-Zúñiga (2018) took a different approach and measured these parameters for cores selected in different bins of mass and gas density. For the Pipe nebula cloud, they found 𝒬 values that are smaller for dense cores/peaks selected at either lower masses or lower gas volume densities on the order of ≈0.4, similar to what we have measured for Taurus. They also found that the ΛMSR values are relatively insensitive to the choice of the range of these physical parameters. Using data from the James Clerk Maxwell Telescope (JCMT) Gould Belt Survey, Kirk et al. (2016) applied both an MST based method (on the fluxes of cores rather than their masses) and the surface density-core mass method (see Appendix B) and found evidence of primordial flux/mass segregation in the three subregions of Orion B (L1622, NGC 2023/2024, and NGC 2068/2071). Lane et al. (2016) performed a very similar analysis to Kirk et al. for the entire Orion A cloud and reached similar conclusions. It is noteworthy that Kirk et al. (2016) reported very high values of 𝒬 for the three sub-regions in Orion B, namely 𝒬 = 1.18, 0.99 and 0.91 for L1622, NGC 2023/2024, and NGC 2068/2071, respectively. This is puzzling, given that in all three sub-regions, there is clear visual evidence of substructure. Parker (2018) re-analyzed the data of Kirk et al. (2016). He reported values of 𝒬 that are < 0.8 (𝒬 = 0.72, 0.65, and 0.71 for the sub-regions L1622, NGC 2068/2071, and NGC 20223/2024, respectively), and also variations in the values of ΛMSR between those sub-regions with one sub-region (NGC 2023/2024) displaying strong levels of mass segregation (ΛMSR ≈ 28, for the 4 most massive cores), another one (NGC 2068/2071) mild levels of mass segregation (ΛMSR ≈ 2) and the third region, L1622, showing no mass-segregation. Parker (2018) argued (see Appendix A in his paper) that the large 𝒬 values obtained by Kirk et al. (2016) are due to choices these authors made in how they normalized the quantities and (see discussion above in Sect. 3.1). We do not attempt to place the results of Alfaro & Román-Zúñiga (2018) and Parker (2018) on Fig. 3. This is primarily due to the fact that the core selection for the Pipe and Orion B clouds were performed using different core extraction methods/algorithms (the CLUMPFIND and FellWalker algorithms, respectively) and a quantitative comparison using inhomogeneous data sets can lead to misleading conclusions. Nonetheless, the results of these two studies are broadly consistent with our findings in that the low star forming Pipe molecular cloud has a small 𝒬 parameter, similar to the one we derived for Taurus, while the more intensely star forming regions in the Orion B cloud have a higher 𝒬 values. Stutz & Gould (2016) showed that dense cores in Orion A are well connected, both spatially and dynamically, to the integral shaped filament and the underlying gas structure which sets the gravitational potential of the region. Their approach is also another way of looking at the issue of whether there is a connection between the morphology of a star forming region, as traced by its population of dense cores, and the star formation activity in the region.

The existence of substructure in star forming regions is a direct consequence of turbulent fragmentation. When turbulence is injected into the cloud on large physical scales (i.e., scales equal or larger than than cloud scale), a natural consequence of the turbulent cascade is the formation of a network of compressed, post-shock regions of different sizes (Dib et al. 2007b; Federrath et al. 2010; Burkhart et al. 2012; Padoan et al. 2014). Furthermore, as these substructures continue to fragment to smaller scales, forming dense cores, a non-zero level of mass-segregation can be expected, as more massive cores are statistically more likely to form in more massive substructures due to the availability of a larger mass reservoir (Padoan & Nordlund 2002). In supersonic clouds that are magnetically subcritical such as Taurus7, star formation, which is mediated by ambipolar diffusion, proceeds at a slower pace due to effects of magnetic pressure which prevents substructure from merging efficiently (Nakamura & Li 2005; Dib et al. 2007b). Thus, a higher level of substructure in the clouds could be indicative of either a young age for the region and/or of the existence of a strong magnetic support.

With regard to the question of whether mass segregation is primordial (i.e., set at the dense core phase) or induced by the dynamical evolution of an initially sub-structured cluster, Domínguez et al. (2017) argued that a young cluster will quickly settle into a state where the most massive stars are mass segregated, regardless of the existence/absence of mass segregation within its different levels of substructure. On the other hand, Dib et al. (2010) argued that mild levels of mass segregation that can be generated by turbulent fragmentation could be significantly enhanced by gas accretion onto the cores. Dib et al. (2007a) proposed that another efficient channel for imprinting a significant level of primordial mass segregation is by the coalescence of cores. The process is more efficient in the densest regions of protocluster clouds where cores are more closely packed, and it may be the dominant mode of star formation in the center of massive starburst clusters such as Arches and NGC 3603.

6. Conclusion

We have analyzed the spatial distribution and mass segregation of dense cores in a number of nearby (the L1495 region in Taurus, CrA, and Aquila) and more distant (W43) star forming regions. The observations for these regions were performed using Herschel for the first three regions, and ALMA for W43. Cores were extracted in those four regions with a similar technique using the getsources algorithm (Men’shichikov et al. 2012). For each region, we quantify the spatial distribution of the cores using the 𝒬 parameter and the mass segregation ratios ΛMSR. Our analysis indicates that the spatial distribution of dense cores in those regions is very substructured. Given the relatively short lifetimes of most of these cores, it is thus very likely that the emerging young clusters in those regions will inherit the same level of substructure that is found for their parent core population. With the exception of Taurus, we observe mild-to-significant levels of mass segregation for the ensemble of the 6, 10, and 14 most massive cores in Aquila, CrA, and W43, respectively.

We show, for the first time, that the spatial distribution of dense cores is positively correlated with the star formation activity in the clouds, represented by the surface density of the star formation rate, ΣSFR. Regions that have the lowest star formation activity (i.e., Taurus) are those that display higher level of substructure and no signs of mass segregation of the cores. In contrast, regions with an intense star formation activity (W43) have a higher 𝒬 value which is indicative of a larger role by gravity in assembling a more centrally condensed structure. We also show that the ratio of the most massive cores that display the strongest levels of mass segregation to the total number of cores is also correlated with the ΣSFR. Our findings could be affected, to a limited extent, by effects of time evolution, with substructures merging due to gravity and forming a more centrally condensed distribution. However, we argue that the correlations we find are likely due to a dependence on the physical conditions in the clouds and which have been imprinted on them by the large scale environment at the time these clouds started to form. The analysis of a larger sample of clouds will allow us to shed more light on the interplay between star formation in clouds and the spatial distribution of dense cores, and help us better understand how star clusters form and evolve.


1

Since W43 is located at a much greater distance than the other 3 regions (i.e., at a distance of ≈5.5 kpc, Zhang et al. 2014), it is possible that the dense cores in W43 may be fragmented down to scales that are not yet resolved by the ALMA observations. The current spatial resolution in these observations is ≈0.01 pc and should normally be sufficient to resolve low mass cores. However, it is possible that the low mass cores in W43 may be more confined and have smaller sizes in the higher density and pressure environment of W43 than those found in nearby star forming regions. Another aspect is whether the filtering of large scale emission by the interferometric observations can remove a certain population of cores. The ALMA baselines used for the observations in the W43 range from 13 m to 1045 m, which makes it sensitive, at the distance of W43, to emission on all scales from ≈9.6 × 10−3 pc to ≈0.31 pc (Motte et al. 2018). The upper limit of these scales is large enough to encompass the entire region. One can also notice that cores are still detected in an evenly distributed way and are not restricted to the higher emission regions that are located in the central region and the 2 filamentary structures.

2

In this work, we follow Schmeja & Klessen (2006) when defining both normalization terms for and . For , the normalization term is given by (Marcelpoil 1993) where ACH is the area of the convex hull encompassing all cores, and N is the number of cores. The term is normalized by a radius of a circle, RCH, whose area is equal to ACH. Other authors use different normalizations. Cartwright & Whitworth (2004) and Parker (2018) normalize by , where A is the area of a circle whose radius, R, is the distance from the furthest point from the structure center. These authors normalize by dividing it by R. Kirk et al. (2016) also use the convex hull area to normalize , but they use the distance between the center of the convex hull and the most distant point from this center, RCH, max to normalize . More details on the normalization adopted in this work can be found in Appendix B in Schmeja & Klessen (2006). A discussion of the effects of the different normalizations is presented in Parker (2018).

3

Figure 2 is reproduced in Fig. B.1 for each star forming region, individually, including the error bars on the measurements of ΛMSR and ΓMSR.

4

Instead of using the position of the peak in the ΛMSR − nMST figure to measure what are the cores that are the most mass segregated, Plunkett et al. (2018) look at all nMST cores for which ΛMSR ≳ 1 (within the error bars) to asses the relevance of mass segregation.

5

The measurement of the SFR in W43 is performed differently than in the other regions, due to the absence of YSO counts in this region. The SFR is assumed to be given by the estimated total stellar mass that is to be found in cores, divided by a typical timescale for individual star formation of tSF = 0.2 M (assumed as an mean value over all core masses). The stellar mass that forms in each core is taken as being equal to one third of the mass of the core. Louvet et al. (2014) do not provide uncertainties over the SFR estimate in W43 but it is not unreasonable to think that an uncertainty of 50% seems plausible. This accounts for uncertainties in the core-to-star(s) formation efficiency and to uncertainties over the value of tSF. We can apply the same methodology to derive the ΣSFR for Taurus, Aquila, and CrA. The total mass in pre-stellar cores (i.e., bound cores) in those three regions is 31 M, 15.1 M, and 470 M, respectively. Adopting the same core-to (single) star efficiency of 0.3 and dividing by both tSF and the surface areas of each region listed in Heidermann et al., the corresponding values of ΣSFR for Taurus, Aquila, and CrA are ≈0.18, 3.9 and 7 M yr−1 kpc−2, respectively.

6

Moser et al. (2019) measured the 𝒬 parameter for the sample of 35 massive protostars in the IRDC G028.37+00.07 and obtained a value of 0.66. They also estimated the SFR in this IRDC to be 1.6 × 10−3M yr−1. Adopting a distance of 5 kpc and a surface area of 184 pc2 (estimated from data in Simon et al. 2006) yields ΣSFR ≈ 8.7 M yr−1 kpc−2. This places this region at the top right of CrA in Fig. 3 and in good agreement with the trends observed for the other regions.

7

Hildebrand et al. (2009) and Chapman et al. (2011) report values of the mass-to magnetic flux ratio of μ = 0.1 ± 0.02 and 0.32 ± 0.02, respectively, where μ is expressed in units of the critical value for collapse (Nakano & Nakamura 1978). In W43, μ is highly supercritical and is found to be much in excess of unity across the entire region (Cortes et al. 2016); i.e., in the range ≈10 − 60), and in Aquila, the available data about the magnetic field strength (Sofue & Nakanishi 2017) and mean column density Könyves et al. (2015) suggest that μ falls in the range 5 − 7 using the formula μ = 7.6 × 10−21(N(H2)/Blos (Troland & Crutcher 2008). There is no available information about the magnetic field strength in CrA.

8

We varied j between 5 and 8 and saw no qualitative difference in the surface density–mass plot for the different regions.

Acknowledgments

S. D. would like to thank Stefan Schmeja for insightful discussions on issues related to the characterization of structures using the minimum spanning tree method. We thank Vera Könyves, Thomas Nony, Davide Elia, Mark Heyer, Sylvain Bontemps, and Jouni Kainulainen for useful discussions on the star forming regions studied in this work. We also thank the anonymous referee for useful comments. This research has made use of data from the Herschel Gould Belt survey project and data from the ALMA-IMF project. S. D. acknowledges support from the french ANR and the german DFG through the project “GENESIS” (ANR-16-CE92-0035-01/DFG1591/2-1).

References

Appendix A: Measuring mass segregation using the surface density–mass plot

Here, we test a different method aimed at characterizing the level of mass segregation in the four star forming regions considered in this work. The method is based on inspecting the relationship between the local surface density of cores around each core and the core mass. Such an approach have been employed in the context of stellar clusters (e.g., Casertano & Hut 1985; Maschberger & Clarke 2011) and also in the context of galaxy groups (e.g., Dariush et al. 2016). The local surface density is calculated as being:

(A.1)

where rj is the distance to the jth nearest neighbor to the core. We use a value of j = 6. Casertano & Hut (1985) showed that, for a number of discrete objects between 30 and 1000, as in our case, this value is a good compromise between estimating the local densities and the number of density fluctuations in the overall structure8. Figure A.1 displays the surface density of cores versus core mass for the four star forming regions considered in this work. The vertical dashed lines in all subsets show the 90% completeness limit. The limits for completeness are taken from Könyves et al. (2015) for Aquila, Motte et al. (2018) for W43, and Bresnahan et al. (2018) for CrA. The threshold value for completeness in the L1495 region of Taurus has not been reported in Marsh et al. (2016). However, given that Taurus and CrA are at nearly identical distances (≈140 pc and ≈130 pc, respectively, we adopt a conservative value 0.02 M, only slightly higher than the value of 0.015 M in CrA. The full line in each subset of the figure shows the median values of Σ6 calculated in 11 bins of log(M), and the red line displays the median value calculated in bins that contain the same number of cores. Before discussing the results of the Σ6 − Mcore method for the individual star forming region, it is important to stress that the two method are inherently different. The MST-based method uses information about the positions and masses of all cores, whereas the second method uses only the local positions of the j-nearest cores around each core. As such, the Σ6 − Mcore can be used to determine whether massive cores reside in higher density environments, but by design, the method would be unable to specify whether massive cores are spatially mass segregated (i.e., if they have a more compact configuration than the rest of the population of cores). Results from the two methods could be affected differently by the effect of completeness. Since the completeness limit is higher in higher density environments, more low mass cores could be missed in the direct vicinity of massive cores, thus forcing the method to seek further out the j nearest neighboring cores. The MST-based method would be less affected by this as it uses the positions of a much larger sample of cores that is distributed across the entire region. What can make the two methods give similar results (but not necessarily with the same level of sharpness) is the hierarchical nature of molecular clouds which is essentially the consequence of turbulent fragmentation. The largest number of objects, including the most massive objects, will statistically be found in the largest structures, so when mass segregation is present, the two methods will indicate that massive cores are more closely packed and that they reside in higher density environments. The self-similar nature of molecular clouds (and of young clusters) can be affected by additional processes such as strong magnetic fields and in the case of young stellar clusters by stellar ejections.

thumbnail Fig. A.1.

Surface density of cores around each core versus its mass. In calculating the surface density around each core, we use the circular area enclosed within the distance to its sixth nearest neighbor. The vertical dashed lines in all subsets show the 90% completeness limit. The full line represents the median value of Σ6 for bins with a fixed value of log M (black line) and for a variable bin size which contains an equal number of objects (red line). The median values of Σ6 are shown for values of Mcore that are above the completeness limit in each region.

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In the case of the four star forming regions studied in the work, the Σ6 − Mcore plots tend to reflect the discussion above. As with the MST-based method, the Σ6 − Mcore plot for Taurus shows that all cores, including massive cores, reside in environments with a (broad) range of surface densities and that there is no correlation between Σ6 and Mcore. In contrast, in the three other regions, high mass cores reside preferentially in regions of higher surface density. In the CrA region, there is a noticeable increase of Σ6 with increasing core mass. This is the region that shows the strongest signal in ΛMSR (see Fig. 2). For Aquila, the increase in Σ6 with core mass is milder, but this is still reflected by an increase of Σ6 with Mcore by a factor of ≈4. In the case, of W43, the result from the Σ6 − Mcore is less conclusive. In this region, all massive cores (i.e., cores with masses ≳7 M) reside in high density environments, however, there is no clear increase of Σ6 with Mcore. This could be due to the more significant effects of completeness around the most massive cores, as discussed above. Finally, it is also important to realize that the results of this method, albeit being here in good qualitative agreement with results based on the MST method, they do suffer from a poor definition of the surface density of cores. Since the calculation of the area is based on the circular area enclosed within the jth neighbor, the method inherently can become less accurate for massive objects that may be surrounded by many low mass objects that reside in filamentary structures. A more accurate estimate of the surface area is needed. One possibility is to use the convex hull area, but such details are beyond the scope of this paper and are left for future work.

Appendix B: Uncertainties on the measurement of the mass segregation ratio

We have omitted showing the uncertainties associated with the measurement of the mass segregation ratios in Fig. 2 for the sake of clarity. While the discussion on how the uncertainties on both ΛMSR and ΓMSR are measured has been described in detail in Dib et al. (2018) and in the original works (Allison et al. 2009; Olczak et al. 2011), we provide here a reminder of the procedure and include the error bars for each of the regions studied in this work, individually. The uncertainty on ΛMSR is given by:

(B.1)

where is the standard deviation from the k random sets (we use k = 100 random sets). The standard deviation of ΓMSR is given by:

(B.2)

In Fig. B.1, we reproduce Fig. 2 for each star forming region, individually and show the associated error bars on ΛMSR and ΓMSR in the (ΛMSR, ΓMSR)−nMST plots. The inclusion of these error bars do not affect our conclusions.

thumbnail Fig. B.1.

Same as Fig. 2 but for the 4 regions shown separately and including the error bars on ΛMSR and ΓMSR.

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All Tables

Table 1.

Structure parameters and mass segregation levels for the studied star forming regions.

All Figures

thumbnail Fig. 1.

Spatial distribution of dense cores in the four star forming regions considered in this work. The sizes of cores have been scaled (with an arbitrary formula) according to their masses (i.e., larger sizes relate to more massive cores) in order to visually highlight the location of the most massive cores. The minimum spanning tree for the ensemble of the 6 and 50 most massive cores in each region are displayed with the yellow and purple lines, respectively.

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In the text
thumbnail Fig. 2.

Mass segregation ratios ΛMSR (Allison et al. 2009, top row) and ΓMSR (Olczak et al. 2011, bottom row) as a function of the number of most massive cores used in computing them, nMST. The values of ΛMSR and ΓMSR are calculated for the entire population of cores in the star forming regions (left column) and for the populations of bound cores (prestellar cores and cores with a protostar). A version of this figure that includes estimates of the uncertainties on ΛMSR and ΓMSR for each individual region is shown in Fig. B.1.

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In the text
thumbnail Fig. 3.

Correlation between the structure parameter, 𝒬 for the entire and bound populations of cores (dashed and full lines, respectively) versus the surface density of star formation ΣSFR (top panel) and the correlation between the ratio of the most segregated massive cores to the total number of cores and the ΣSFR. The horizontal error bars are the 50% uncertainties estimates that are likely to be affecting the measurements of ΣSFR. The values of 𝒬 = 0.78 and 0.71 are the ones measured by Dib et al. (2018) for the populations of open clusters and young clusters in the Milky Way, respectively. The estimate of ΣSFR for W43 is made using a different approach than for the other three regions (see text for details). The lines connecting the different sets of points are merely shown to guide the eye.

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In the text
thumbnail Fig. A.1.

Surface density of cores around each core versus its mass. In calculating the surface density around each core, we use the circular area enclosed within the distance to its sixth nearest neighbor. The vertical dashed lines in all subsets show the 90% completeness limit. The full line represents the median value of Σ6 for bins with a fixed value of log M (black line) and for a variable bin size which contains an equal number of objects (red line). The median values of Σ6 are shown for values of Mcore that are above the completeness limit in each region.

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In the text
thumbnail Fig. B.1.

Same as Fig. 2 but for the 4 regions shown separately and including the error bars on ΛMSR and ΓMSR.

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In the text

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