YOUNG STELLAR GROUPS AND THEIR MOST MASSIVE STARS

For each of the YSOs, we estimate the mass based on the spectral type. We assume a constant age of 1 Myr for all YSOs and follow the procedure outlined in Luhman et al. (2003), using a combination of models from Palla & Stahler (1999), Baraffe et al. (1998), and Chabrier et al. (2000) to estimate the masses. The assumption of a constant, 1 Myr age, as well as the exact stellar models chosen leads to uncertainties in the masses of order 50%. This is discussed in more detail in Appendix C.

In several parts of the following analysis, we compare the results for the four nearby star-forming regions with the large cluster encompassing the Trapezium in the Orion Nebula Cluster, to represent properties typical to young clusters. To make this comparison, we use the ONC1 data set from Hillenbrand (1997), adopting the masses and positions given there. We include all sources listed as having a 70% or higher probability of membership. This list contains 721 sources, of which we identify 410 of these as belonging to the Trapezium cluster (see Section 3 for our method of identifying groups and clusters). Of these 410 Trapezium cluster sources, 26 do not have estimated masses in Hillenbrand (1997); we assign each of these the median mass in the cluster (0.23 M_☉).

The groups we identify in Taurus, Lupus3, ChaI, and IC348 are shown in Figures 2 through 5, and their properties are summarized in Table 1. In the figures, the group members are indicated by blue circles (with the radius scaling linearly with mass), the red lines denote the MST structure, and the thin black circles denote nearby YSOs that do not belong to the group. The gray-scale images in the background indicate the present-day distribution of extinction based on stellar reddening. Displayed are extinction data from Froebrich et al. (2007) for Taurus, Dobashi et al. (2005) for ChaI, a combination of Teixeira et al. (2005) and Rowles & Froebrich (2009) for Lupus3, and Rowles & Froebrich (2009) for IC348.

As can be seen in Figures 2 through 5, the groups often lie near, but not on, regions of high extinction, suggesting the YSOs have accreted and/or blown away a similar distribution of gas from their immediate environs. In most cases, the mass of gas required to create the present-day stellar masses corresponds to a uniform gas and dust having A_V ≃ 3 mag, assuming a formation efficiency of 30%. This extinction due to smoothed-out stellar mass is approximately the difference between the maximum nearby extinction and the mean value of the extinction within the group.

The main group in IC348 is an exception, requiring roughly 10 times more material for the formation of the YSOs, which is substantially higher than the present-day nearby extinction. The Lupus3 group also shows a much larger difference in extinction than do most of the other groups, since the high-resolution extinction map from Teixeira et al. (2005) allows much larger peaks in extinction to be measured. At comparable resolutions to the other regions, however, the difference in extinction is similar to that in the other regions. At the same time, the total amount of material remaining in the area spanned by each group is usually insufficient, or barely sufficient, to form a group with mass equal to the mass in the group we observe.

In other words, the immediate areas in which we see the YSOs in the groups appear largely or entirely finished with the star formation process, but there are still reservoirs of material nearby which are capable of forming a significant number of new stars.

In Taurus, there is good correspondence between the groups we identify and those previously identified using other methods. The Gomez et al. (1993) Groups I through VI correspond to our Groups 1, 2, 7, 5, 6, and 4 (B209, L1495E, L1527, L1529, L1536, and L1551), respectively, with our remaining two groups (3/B213 and 8/L1517) corresponding to unnamed contours of higher stellar density in their Figure 8 (right of their Group III and the upper left of the figure respectively). Group I in Jones & Herbig (1979) corresponds roughly to our Groups 1 and 2 (B209 and L1495E), their Group IIa corresponds to our Group 5 (L1529), their surrounding Group II encompasses our Groups 6 and 7 (L1536 and L1527), and their Group III is our Group 4 (L1551). Our Group 3 (B213) lies between their Groups II and III with too few stars in their catalog to be classified as a group, and our Group 8 (L1517) lies beyond their catalog.

The main group in IC348 appears most similar to a small cluster, and has been studied in that vein by a variety of authors—see Herbst (2008) and references therein. The two largest groupings of stars in ChaI have also been viewed in a similar manner—see, for example, Luhman (2007).

As can be seen from the figures, some of the groups are irregularly shaped, particularly when the total number of members is small. Some groups appear filamentary, such as B213 in Taurus, unlike the conventional picture of a group, which tends to be more circular. Despite these irregularities, we find that the median member position (shown as white plus signs in Figures 2 through 5)¹ is usually a good representation of the apparent center of the group, although this becomes poorer in the case of the most filamentary groups, as illustrated in Taurus Groups 1 and 2 (Figure 2) versus Taurus Group 6 (Figure 3), for example. The mean member position is sometimes a worse descriptor of the group center than the median, as it is more likely to be skewed by one or two group members lying preferentially in one direction away from where most of the group is concentrated. The center of mass tends to lie between the median and mean positions. The centers of the groups as defined by the median positions of the group members are also given in Table 1.

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Using the masses estimated for each YSO, we examine the distribution of masses within each group. Figure 6 shows the distribution of masses within the main IC348 group. A prominent excess of sources with masses around 2–3 M_☉ is clearly evident; a similar trend is seen in many of the other groups, although at a much lower level, since the groups are smaller. This excess is unlikely real, and is at least in part caused by our adoption of a single age for all of the stars. Assuming a single age of 2 Myr instead of 1 Myr, for example, reduces the size of the excess (for a given spectral type, assuming an older age will reduce the mass estimated); allowing for an age spread would eliminate it completely (see, e.g., the IC348 mass distribution in Luhman et al. 2003). In order to keep our analysis simple, we maintain our assumption of a single age, recognizing that the details of the mass distributions in the groups are affected by this assumption. For our analysis, however, the detailed mass distribution is not important; we are concerned primarily with the rank of the masses (Sections 4.2–4.4), which is determined solely by the spectral type, as well as other broad properties (this section).

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Within each group, a significant fraction of the mass is concentrated in the few most massive members. This is illustrated in Figure 7 which shows the fraction of cumulative mass, f_M, as a function of the fraction of cumulative number of group members, f_N. While there is variation between the groups, some due to mass estimation uncertainties and some due to small number statistics, the overall trend is clear—half of the mass of the group is found within the 10%–30% most massive members. This property is similar to what would be expected for a cluster whose members follow the IMF. The black dashed line in Figure 7 shows the profile expected for the Kroupa IMF, using the formulation given in Weidner et al. (2010), within the mass range spanned by our groups. For comparison, we also calculated the relationship expected for a symmetric, log-normal mass function, given, e.g., by Chabrier (2005) and found a nearly identical relationship to the one shown for the IMF. Both describe the relationship seen in the observed groups reasonably well. It is notable that this property of a significant portion of the group mass being found in a relatively small fraction of the group members arises for groups whose most massive members have masses much greater than the typical mass for the group, but much less than that of O stars which dominate the largest clusters.

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Within clusters, the total cluster mass is correlated with either the mass of the most massive member or with the total number of members (Weidner et al. 2010). While the mass of the most massive member of each group, and to a lesser extent, the total mass in the group in our data set, are uncertain, both should be good to ∼50% or better. We compare the mass of the most massive member within each of our groups to the total group mass in Figure 8. In that figure, all of our groups are plotted (note that in Taurus, several of the groups have nearly identical values), along with the data in Weidner et al. (2010), using the new dynamical masses for the most massive members where available. Weidner et al. (2010) also derived an analytic expression for the most massive cluster member expected, based on a random sampling of the IMF, and setting the maximum mass possible for a star to be 150 M_☉; the dotted line in Figure 8 shows the approximate linear slope found by Weidner et al. (2010), assuming a Salpeter slope for the high-mass tail of the IMF. Regardless of the model fit to the relationship between the most massive star and total cluster masses, it is clear that our groups follow the same trend as the clusters do.

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On the other hand, the correlation between the maximum stellar mass and the total stellar mass for the groups studied here does not imply that these groups have a particular upper limit on the stellar mass for their model. The groups do not extend to high enough masses to show a reduction in slope, unlike the largest of the Weidner et al. (2010) clusters shown in Figure 8 which suggested the 150 M_☉ maximum stellar mass. At total cluster masses below 100 M_☉, Weidner et al. (2010) found their data to be consistent with a random sampling of the (Salpeter) IMF without the requirement of a maximum stellar mass.

The foregoing properties of the mass distributions in nearby groups suggest that similar physical processes are responsible for the mass distributions of stars in groups and in clusters, independent of their number of members.

Within each group, the most massive group member tends to lie close to the group center. We find that for the 14 groups, the median offset (or separation from the group center) of the most massive group member, O_1st, is 0.6 times the median offset for all of the members of the same group, O_med. Figure 9 shows the ratio of O_1st to O_med for all of the groups. (Note that in the few groups where there were two equal mass most massive members, the one closer to the center is used for the O_1st calculation and the other is used for O_2nd discussed below.) The vertical axis of Figure 9 shows the ratio of the mass of the most massive member, M_1st, and the median group member mass, M_med, indicating that in most groups, regardless of the value of M_1st/M_med, the most massive member of each group tends to lie nearer to the group center than typical. For comparison, the ONC1 cluster is also shown (see Section 2.2). These results are not sensitive to the method used to determine the group center—we find O_1st/O_med values which are nearly always less than one using the center of mass as the group center instead of the median position.

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As might be naively expected, a random distribution of group members tends to yield a ratio of offsets above and below one with roughly equal probability. We determine the ratio of offsets expected from a random uniform distribution by running 200,000 simulations for a variety of numbers of YSOs placed within a 2D circular or 3D spherical region. We measured the same ratio, O_1st/O_med, as described above. The vertical dotted lines in Figure 9 show the 25th and 75th percentile value of the ratio found in the random simulations for a uniform distribution of 25 randomly distributed YSOs within a 3D sphere. These values do not change substantially for either a 2D circular distribution or a different number of YSOs in the sample. We also ran similar tests on positions following a random 3D isothermal (probability ∝1/r²) profile and found similar results. The 3D isothermal distribution of O_1st/O_med values tends to be more peaked than the uniform distribution below O_1st/O_med = 1 and is shallower with a long tail for values above 1 (the tail actually extends beyond O_1st/O_med = 100, much larger than the range shown in Figure 9). The 25th and 50th percentile values for O_1st/O_med are nearly identical for both the isothermal and uniform distributions, while the 75th percentile value tends to be larger for the isothermal than the uniform distribution. The 75th percentile value varies with the level of truncation of the isothermal distribution (O_1st/O_med is ∼1.9 in the absence of truncation and decreases with the degree of truncation). Regardless of the distribution adopted, the observed groups have a more centrally located most massive member than would be expected from a random distribution of positions.

As a further test of whether these massive members are more centrally located than would be expected in a random distribution, we ran simulations where we kept the group members' positions the same, but randomized which mass belonged to each member. We did this for 10,000 trials for each group, and calculated the fraction of simulations where the most massive member had an offset ratio less than or equal to that which was observed. Where the mass segregation appears to extend to the second (and third) group member(s) (as discussed in the following subsection), we also computed the joint likelihood of having the two (or three) members with offset ratios smaller than or equal to those observed. Barring L1551 and ChaI-southwest, whose most massive members have large offset ratios, the probabilities found for the observed group mass configurations were small—at most, around a percent, and often lower. We also ran the same test on the three least massive group members and nearly always found substantially higher probabilities, since the least massive members do not show a central concentration.

Regarding the two outlying groups, L1551 and ChaI-southwest, it is clear that neither has the appearance of typical groups. ChaI-southwest has a linear morphology and does not contain any particularly massive stars (Figure 4), while L1551 (Figure 3) consists of a more typically shaped group (bottom right), connected to the most massive group members (top left) through a series of relatively widely spaced YSOs. Had the critical MST branch length been 5% smaller in L1551, the most massive group members would not have been considered group members, and the remaining group would show a much stronger central concentration of the most massive members. In most of the groups, the longest MST branches only connect a few low-mass YSOs to the rest of the group, and the overall group structure is little affected by their inclusion or exclusion, as discussed in more detail in Appendix D.1.

While neither L1551 nor ChaI-southwest has the appearance of traditional groups, this characteristic is insufficient to explain the different behavior between these two groups and the others. As can be seen in Figures 2 through 5, several other groups have a linear geometry (e.g., B213/Taurus Group 3) and loose spacings between members (e.g., L1527/Taurus Group 7; see also Table 1 for a comparison of median YSO spacings in each group) while still possessing a centrally located most massive member.

The central location of the most massive group member is suggestive of a more general property of mass segregation, where more massive members are progressively more centrally concentrated. Within larger clusters, mass segregation is often observed to varying extents. For example, Stolte et al. (2006) found evidence of mass segregation at all masses in NGC 3603, with the degree of segregation lessening in the lower mass bins, while Carpenter et al. (1997) found mass segregation only for the most massive members of the Monoceros R2 cluster. How do the groups which we analyze compare?

In clusters, mass segregation is measured in a variety of ways including the change in slope in the mass function (or luminosity function) with radius, the ratio of the number of high- and low-mass stars as a function of radius, and the mean radius of different masses of stars (see, e.g., Bonnell & Davies 1998; Ascenso et al. 2009, and references therein). There are too few YSOs in each of the groups we identify to be able to use any of the usual mass segregation measures. We can, however, extend the offset ratio measurement discussed in the previous section to look at the distribution of offsets from the group center as a function of mass rank.

Figure 10 shows the distribution of offset ratios for each of the three most massive members in the groups (top to bottom panels). The overplotted dashed, dash-dotted, and triple-dot-dashed lines show the distributions expected from our random simulations discussed in the previous section. The most massive group member (top panel) shows a large excess of sources with offset ratios smaller than expected from random sampling, but this excess is diminished considerably for the second most massive member (middle panel) and is gone completely for the third most massive member (bottom panel).

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A simple comparison between the distribution of observed offset ratios and those found in our random 3D simulations with a two-sided KS test gives probabilities of being drawn from the same parent sample of less that 0.2% for the most massive group member, and 62% and 50% for the second and third most massive members, respectively, for the uniform distribution, and probabilities of 0.1%, 47%, and 56% for the non-truncated isothermal distribution, with typically smaller probabilities for truncated isothermal distributions. Our largest source of error in the offset ratios is due to the definition of the center of the groups. Using instead the center of mass of each group to compute the offset ratios, we again find similar distributions. The two-sided KS test probabilities for the 3D uniform distribution are 0.2%, 22%, and 70% for the first, second, and third most massive members, respectively, and 0.2%, 10%, and 8% for the 3D isothermal distribution. The large change in the probabilities for the isothermal distribution is caused by a smaller minimum value of O_xth/O_med for the observed groups using the center of mass (instead of the median position). O_xth/O_med decreases sharply for the 3D isothermal distribution below 0.5, so the KS test is highly sensitive to how low the observed O_xth/O_med values extend.

Figure 17 illustrates the trend toward less-central locations with lower mass in an alternate manner, showing the mass of each source versus the order of the offsets in the group, visually confirming the statistic measures discussed above. It is interesting to note that the central location of the most massive member(s) in each group is not a function of the mass of the most massive member. In the ONC1 cluster, Hillenbrand & Hartmann (1998) found evidence for mass segregation extending down to 5 M_☉, and possibly beyond. Visually, a similar divide is seen for ONC1 (see extended version of Figure 17 in online journal). Quantitatively, the offset ratio O_1st/O_med is less than 1.1 for all members with masses above 4.8 M_☉; below this mass, the maximum of O_1st/O_med rapidly increases.

Another property which our groups share with larger clusters is a central concentration. Large clusters tend to have much smaller separations between members toward the center, coincident with where the most massive members tend to be located. While Figures 2 through 5 suggest that the most massive group members lie in or near zones with higher than average degrees of clustering within the groups, we can quantify this.

The measurement of the surface density of stars often provides a useful criterion for determining the clustering properties. Since our groups often have a very small number of sources, these measures are more vulnerable to errors from small number statistics, and so we follow a different approach; surface density measures are discussed briefly in Section 5.1. Here, we instead compute the radius, r_N, which encloses the N nearest sources for each group member. Group members which are located near the most clustered part of the group should show a sharper rise in a plot of N versus r_N than members located in a sparser part of the group. Figures 11 and 12 show the fraction of total group members, f_N versus f_rN, the radius r_N normalized to its maximum value in the group for all members of each group. For clarity, all group member profiles are not shown in the figures. Instead, the range in values spanned by all members is indicated by the light gray shading, while the median profile is overlaid as the dark gray line. The most massive group member is shown in black. These plots allow comparison of the local surface density for a given radius r_N, or for a given number of neighbors N, as a function of member mass.

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Figures 11 and 12 show that the most massive star is projected on a region of relatively high local surface density for a large fraction of the groups considered. The black line lies to the left of the gray line for the nine groups Taurus 1, 2, 3, and 6, Lupus3 1, ChaI 2 and 3, and IC348 1 and 2, and also for ONC1, the prototype of a large cluster with mass segregation. In contrast, the black line lies to the right of the gray line for two groups, Taurus 4 and ChaI 1. In the three groups Taurus 5, 7, and 8, the lines are too close to make a clear discrimination.

In combination with the result from Section 4.2, this result indicates that the most massive star in a group generally has a position which is central, and which is associated with a high surface density of lower mass stars.

The tendency for the most massive member of each group to lie in a region of higher than average group member surface density can be quantified further. As already discussed, the most massive group member tends to have a profile which lies to the left of the median profile in the preceding two figures. Put another way, for a fixed value of f_N, either f_rN or r_N is smaller for the most massive member than the median value. For each group, we compare r_N for the most massive member and the median value at fixed levels of f_N. Figure 13 shows r_N for the most massive member divided by the median value for f_N = 30% (left panel) and 40% (right panel) for each of the 14 groups (solid diamonds). For each group, the full range in range r_N normalized by the median value at f_N is indicated by the vertical line. Each vertical strip in Figure 13 therefore summarizes a horizontal cut along f_N = 30% (left) and 40% (right) in Figures 11 and 12, normalized to the median r_N at that f_N (i.e., the dark gray line would always lie at 1 in Figure 13).

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Plotted in this manner, it is easy to additionally compare the behavior of the most massive member of each group with the next two most massive members, shown in Figure 13 by the open diamonds. There is some scatter between the two panels, which is expected since the most massive member and median profiles shown in Figures 11 and 12 do not remain a fixed distance apart. The two panels shown in Figure 13 are, however, representative of the general trend—we made similar figures for values of f_N ranging from 25% to 50% in intervals of 5% and found similar results to the cases shown. Despite the scatter, it is easy to see from the plot that most of the most massive members (filled diamonds) lie near or below a value of one in most cases, i.e., they lie in locations which tend to be more clustered than typical for the group. The second and third most massive members only shown in Figure 13 follow this trend in only a limited number of groups.

The groups that we analyze tend to have properties scaled-down, but similar to those in larger clusters. In the literature, clusters are often defined and described using criteria based on the surface or volume density of sources. Lada & Lada (2003) defined clusters as having a minimum stellar density of 1 M_☉ pc⁻³ and greater than 35 members, in order to be resistant to quick dissolution. Jørgensen et al. (2008) found that protostars in the nearby Perseus and Ophiuchus molecular clouds tended to have denser clustered substructures and used a threshold of 1 M_☉ pc⁻³ to define loose associations and 25 M_☉ pc⁻³ to define tight associations within these. A minimum number of 35 members was again used to define a cluster; associations with smaller numbers were termed groups. Porras et al. (2003) similarly used a maximum of 30 members to define groups, with 31–100 members corresponding to small clusters and over 100 members corresponding to large clusters. Adams & Myers (2001) examined the transition between groups and clusters in more detail and argued that groups containing roughly 10–100 members evolve differently than large clusters—the groups tend to disperse quickly and are likely to be unaffected by supernovae or strong UV radiation (note the number of members is higher than in Lada & Lada 2003, as the role of gas in the cluster dissipation was also considered).

How do our groups compare? In terms of numbers of members, all but the main group in IC348 (with 186 members) fall easily within the Adams & Myers (2001) definition of a group, and many also do so with the N = 30–35 definition of Lada & Lada (2003), Jørgensen et al. (2008), and Porras et al. (2003). In terms of surface density, our groups tend to also lie below the standard cluster values.

Figures 11 and 12 show lines of constant surface density from 1 to 100 pc⁻² separated by factors of 10. In the Taurus and Lupus3 groups, values tend to range from a minimum of around 1 pc⁻² to a few times 10 pc⁻². The ChaI groups tend to have slightly larger surface densities (particularly ChaI-southwest/Group 1), while the IC348 groups have surface densities nearly 100 times larger. The embedded clusters in the sample of Gutermuth et al. (2009) tend to have much higher surface densities, typically peaking around a few hundred per square parsec. Assuming, as in Jørgensen et al. (2008), spherical symmetry and a typical YSO mass of 0.5 M_☉, the volume density thresholds of 1 and 25 M_☉ pc⁻³ correspond to roughly twice those values in number per square parsec for surface density. The groups in this paper are denser than their surroundings, but are generally much less dense than those considered in any of the above-cited works.

In most of the groups, we find that the single (or, for L1356, Lupus3-main, and IC348-main, several) most massive group member(s) are located near the group center, and near a region of higher than average surface density of sources. Mass segregation is often observed in large clusters, with some regions showing evidence of segregation at all masses, with the degree of segregation lessening in the lower mass bins (e.g., Stolte et al. 2006), while other regions appear to show mass segregation only for the most massive members (e.g., Carpenter et al. 1997). One complication in mass segregation measurements in clusters, particularly more distant clusters, is the observational bias due to crowding. Ascenso et al. (2009) argue that this bias, which leads to increasing levels of incompleteness in the lower mass objects at smaller cluster radii, may be partially or even fully responsible for the observed mass segregation in clusters. The central location of the most massive cluster members would appear to be robust, however, since these stars would be easily detectable in the outskirts of the clusters as well.

The young age of some of these clusters, coupled with the degree to which the most massive cluster members are concentrated in the center, has led to the argument that at least some of the observed mass segregation is primordial, rather than dynamical (e.g., Bonnell & Davies 1998). More recently, Moeckel & Bonnell (2009) have argued that the primordial mass segregation must be confined to only the most massive stars, as any initial amount of mass segregation in the lower mass population leads to an over-prediction of the mass segregation that should be currently observable in those clusters. Other work has questioned whether any primordial mass segregation is required (Allison et al. 2009) to match present-day observations of clusters.

In the groups we study, the mass segregation observed for the most massive members appears to require an early central concentration of these objects. The groups have ages around 1 Myr, while the crossing times are typically between 1 and 3 Myr, assuming a velocity dispersion of 1 km s⁻¹ (the approximate value found in the proper motion groups in Taurus in Luhman et al. 2009). Since there are too few group members for the scenario simulated in Allison et al. (2009) to be applicable, it seems that the most massive members of the groups must have formed near the group centers, rather than migrating there later. For the majority of their lifetime, the YSOs have been embedded in their natal gas; hence, the timescale for dynamical evolution can be estimated as

$$\begin{equation} t_{\rm relax} = \frac{N\epsilon ^{-2}}{10 {\rm ln}(N/\epsilon)} t_{\rm cross}, \end{equation} \tag{ 1 }$$

where N is the number of stars, is the star formation efficiency of the group, and t_cross is the crossing time of the group (Adams & Myers 2001). For a star formation efficiency value of 10%, the relaxation timescale is nearly five times the crossing time for the smallest groups, and becomes larger for both larger groups and lower star formation efficiencies; the relaxation timescale well exceeds 5 Myr for the groups under any reasonable assumption of the star formation efficiency. Given the ages estimated for the groups, the central locations of the most massive members cannot be wholly attributed to dynamical relaxation.

Some degree of early mass segregation appears to be consistent with models for massive star formation. In the competitive accretion scenario, protostars forming in the center of the cluster potential inhabit a higher density environment and hence accrete more mass than those formed at the cluster periphery (e.g., Bonnell et al. 2001). In the monolithic collapse scenario (McKee & Tan 2003), stars form out of quasi-equilibrium clumps. Clusters require several times the dynamical timescale to form, which enables mass segregation to occur in a greater amount than would be anticipated from a faster formation scenario (Tan et al. 2006). In the stationary accretion model of Myers (2009b), high-mass stars are only able to form in the densest environments, where the accretion rate is highest, while lower mass stars are able to form in the surrounding higher and lower density environments. It is unclear, however, whether these models can predict the clear transition between the location of the most massive few group members and the other group members, or, in some cases, operate at all for such small groups.

Association of the most massive star with a high surface density of lower mass stars tends to rule out a very simple cluster formation scenario, where the massive star accretes all the mass within a certain radius, causing a local minimum in the density of lower mass stars. Instead, it may be more realistic that a spherical zone around an accreting massive star feeds the massive star as well as numerous lower mass stars (Smith et al. 2009; Wang et al. 2010).

In their Herbig Ae/Be survey, Testi et al. (1999) found instances where the massive star appeared to be relatively isolated, as did Massi et al. (2003) in their survey of luminous IRAS sources in the Vela C and D molecular clouds. We also find a few instances of isolated massive stars in our catalogs—one A2 star in Taurus, one B6.5 and one F0 star in ChaI, one B4 star in Lupus3, and several A stars in IC348 which fall near but outside of the main group. These sources represent a much smaller fraction of isolated early-type sources than were found in the Testi et al. (1999) work, suggesting that isolated A and B stars are uncommon in young star-forming regions. Future observations may lower this fraction even further, either with evidence that these apparently isolated stars are interlopers or the discovery of more lower mass members nearby. The isolated early-type star in Lupus3, for example, lies well outside the main group, where there have likely been fewer surveys for low-mass members in the region.

We analyze the 352 Taurus members discussed in Luhman et al. (2009) and given in Table 7 of Luhman et al. (2010, hereafter L10). For the binary pair HD 28867A+C and B, we adopt the positions given in Walter et al. (2003); the L10 catalog names correspond to identical positions for both sources. Where the data exist, members were confirmed using proper motion data, as discussed in the appendix of Luhman et al. (2009). Otherwise, membership was based on a variety of observations including Ca ii emission, H α emission, X-ray data, spectral energy distributions, and/or spectra from optical through far-IR observations (e.g., Kenyon et al. 2008). Table 3 summarizes the data—the position, L10 name(s), spectral type, estimated mass (Section 2.1 and Appendix C), and the group (Section 3). Some of the spectral types are highly uncertain and were estimated based on the bolometric luminosity of the source, as indicated by footnotes in Table 3. In these cases, the spectral type and mass should be treated as being in the range given in the footnote.

The Spitzer Taurus team has also published a full catalog of YSOs in Taurus, including both Spitzer photometry of previously known members and new and candidate members based on Spitzer photometry, and in many cases, additional follow-up spectroscopy (Rebull et al. 2010, hereafter R10). For the analysis discussed in this paper, we use the L10 catalog because it spans a larger area of the cloud—the R10 catalog covers only the region mapped by Spitzer. Our results are largely unchanged when using the R10 catalog instead, as described in more detail in Appendix B.

We adopt a distance of 140 pc to the Taurus cloud, following Torres et al. (2007).

In ChaI, we analyze 237 sources whose properties are summarized in Table 4 (the same columns are used as in Table 3). This list includes the 226 known members of ChaI discussed in Luhman (2007); 215 of these sources are given in Table 6 of Luhman (2007), while the remaining 11 were excluded because they lacked accurate spectral types. These 11 sources are J11094192−7634584 and J11095505−7632409 from Table 5 of Luhman (2007), J11011926−7732383B from Table 1 of Luhman (2004b), and Ced110−IRS4, ISO86, Ced110−IRS6, ISO97, B35, IRN, ISO192, and ISO209 from Table 5 of Luhman (2004a). We also add the eight new members identified in Table 1 of Luhman & Muench (2008) and four new members identified in Table 4 of Luhman et al. (2008). Following the discussion in Luhman et al. (2008), ISO130 was excluded, as it is likely a galaxy, and sources J11183572−7935548, J11334926−7618399, J11404967−7459394, and J11432669−7804454 were removed as their proper motions indicate they are more likely to be members of η Cha than ChaI. Additionally, four new members were added: RXJ1129.2−7546, RXJ1108.8−7519a, RXJ1108.8−7519b, and Cha-MMS1, for which proper motion measurements indicate that they are likely ChaI members. As with the Taurus catalog, proper motion data were used where available to verify membership, otherwise, a variety of indicators of youth were used.

Following the discussion in Luhman (2008b), we adopt a distance of 160 pc to the region.

In Lupus3, we analyze 70 YSOs (Table 5) from the compilation of Comerón (2008). We include all of the sources in Comerón's Table 11 (well-known classical T Tauri stars from Thé 1962; Krautter et al. 1997; Hughes et al. 1994), Table 14 (additional low-mass members from Comerón et al. 2003), and Table 16 (possible low-mass members of Lupus3 from López Martí et al. 2005). The source list in Table 16 in Comerón (2008) gives less accurate positions than in López Martí et al. (2005), so we use the original data.

We do not include the sources given in Comerón's Table 13 (suspected Lupus3 members from Nakajima et al. 2000), as the survey from which these data originate only spans an R.A. of roughly 16:09:44 to 16:08:44 (in J2000), which is much smaller than the region spanned by the Lupus3 group. Inclusion of these sources could bias the clustering statistics due to their limited areal range (which is not centered on the apparent group center), and furthermore the spectral types of all of these sources are unknown. We also exclude the list of weak T Tauri stars in Lupus3, as these stars are thought to be older and not associated with the current groups (Comerón 2008).

We adopt a distance to Lupus3 of 200 pc, as recommended by Comerón (2008).

In IC348, we analyze a total of 363 sources whose properties are given in Table 6. This includes the 307 sources listed in Lada et al. (2006, Table 2) and the 41 sources identified in Muench et al. (2007, Table 1). We supplement this list with all other likely members with known spectral types: serendipitously discovered members 273 and 401 discussed in Appendix C of Muench et al. (2007), as well as source 30074, the companion of 166 (Luhman et al. 2005, and listed as 166B there), and 8078, the companion of 9078 (Luhman et al. 2003, listed as 78B and 78A, respectively, there). We use updated spectral classifications for 7 of the sources in the Lada et al. (2006) catalog (141, 174, 294, 334, 366, 1050, and 2103), and also add 11 sources with spectra (249, 250, 307, 313, 340, 1686, 1779, 1840, 6005, 10074, and 10095) recently obtained by K. Luhman (2010, private communication). Note that in the Muench et al. (2007) catalog, where there are multiple spectral classifications, we use the optical classification.

We adopt a distance of 300 pc, following the discussion in Herbst (2008).

The YSO catalogs of all four regions have good completeness. In Taurus, a comparison of X-ray and optical/IR survey data shows the catalog should be complete to ∼0.02 M_☉ for Class II and III stars and brown dwarfs. The completeness is good for Class I stars, but is difficult to determine for Class I brown dwarfs (later than ∼M6) due to confusion in Spitzer bands with faint red galaxies (Luhman et al. 2009). Well-known young protostars such as L1527−IRS1, L1521F, and IRAM 04191+1522 are included in the catalog. Other candidate Class 0/I protostars with very weak Spitzer fluxes may be missing, such as J041757.75+274105.5 (Barrado et al. 2009). Using the overly conservative estimate that all Class I brown dwarfs and Class 0 sources are missing from the catalog suggests only 6% of the total number of sources are missing from the catalog, using the ratio of classes of objects given in Evans et al. (2009).

López Martí et al. (2005) estimate they are complete in Lupus3 down to an R-band magnitude of 20 and and I-band magnitude of 19, which corresponds to ∼0.02 M_☉ at 1 Myr and 0.03 M_☉ at 5 Myr in the Chabrier et al. (2000) models, at a distance of 200 pc. The completeness level is not explicitly given in the Hughes et al. (1994) sample of T-Tauri stars. In the brown dwarf mass regime, the Lupus3 sample may be less complete than in Taurus; Comerón (2008) lists several additional studies with lists of candidate members that have not yet been spectroscopically confirmed. In particular, sources detected only in Spitzer have not yet had spectroscopic follow-up, so it is likely that most, if not all, Class I sources are not included in our analysis. Class I sources constitute only ∼4%–5% of the population in Lupus3 (Merín et al. 2008), however.

The ChaI catalog described in Luhman (2007) was found to be complete for masses above 0.01 M_☉ in regions where A_J ⩽ 1.4 mag. Spitzer data were not used for that catalog; hence, Class I sources are likely missing. Subsequent spectral surveys (Luhman et al. 2008; Luhman & Muench 2008) based on Spitzer data do include a limited number of Class I sources (four or five). If the Class I population in ChaI is similar to that in ChaII (Evans et al. 2009), then roughly one quarter of the Class I are currently included in the catalog, implying only ∼7.5% of the total sources are missing from the catalog.

In IC348, the central region covered in the Luhman et al. (2003) catalog was found to be complete to ∼0.03 M_☉ for A_V < 4 mag. The Spitzer data included in Muench et al. (2007) identified very few new members within the Luhman et al. (2003) survey bounds, and instead extended the catalog to larger distances from the center, with an estimated completeness of >80% for YSOs with H-band magnitudes of 16; this corresponds to a mass of ∼0.015 M_☉ at 1 Myr and 0.02 M_☉ at 5 Myr in the Chabrier et al. (2000) models, at a distance of 300 pc. In the Muench et al. (2007) data set, a total of 20 Class 0/I sources were identified, corresponding to ∼6% of the total. Other work (Jørgensen et al. 2008) suggests that the fraction of Class 0/I sources is ∼9% of the total IC348 population, which implies that only roughly 3% of the total sources are missing from the catalog.

The spectral types are typically uncertain to one subclass (K. Luhman 2010, private communication). As can be seen in Tables 3 through 6, however, there are some instances of uncertain or unknown spectral types in most of the regions, which are discussed below.

Sources with limits on their spectral types (e.g., less than K5) are assigned a spectral type equal to the limit for the purpose of estimating the mass (this affects three, four, and two sources in Lupus3, ChaI, and IC348, respectively). Sources with a range in spectral types given are assigned to the midpoint spectral type for the mass calculation (two K7–M0 sources in Lupus3). Finally, sources with completely unknown spectral types are assigned to have the median mass of YSOs in their region, in order to avoid bias in our later analysis (five, eleven, and seven sources in Lupus3, ChaI, and IC348, respectively).

Only some of the sources with uncertain spectral types fall within the groups we analyze. In Taurus, 16 of the 31 sources with spectral types estimated from bolometric luminosities (Appendix A) are group members; none are the most massive few members; hence, the uncertain spectral type has minimal impact on our analysis. All of the sources with uncertain spectral types in ChaI fall within ChaI-south (12 of 96 group members) and ChaI-north (three of 43 group members), and again do not have mass rankings within the top few group members. In Lupus3, none of the sources with uncertain spectral types fall within the group, and in IC348, only one falls within IC348-north.

The definition of the groups we identified relies on the value used for the critical MST branch length; larger values tend to increase the number of group members, while smaller values tend to decrease the number of members. Figure 15 shows our method for determining the critical branch length—the cumulative distribution of MST branch lengths is well described by a steep linear rise at small branch lengths, followed by a turnover and a shallow linear rise at large branch lengths. Figure 15 shows the data for ChaI; the other three regions show a similar trend. Following Gutermuth et al. (2009), we make linear fits to the two ends of the distribution, and define the critical branch length as the length at which the two best-fit lines intersect. For each of the four regions, we tested the range of possible critical values that could be fit, and found that the variation was less than 10% for all regions, and less than 5% in IC348.

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We examined the effect on the group definitions of a 10% larger or smaller critical branch length. One way to examine this is through a dendrogram (used recently for analyzing structure in 3D data cubes in Rosolowsky et al. 2008, for example). Figure 16 shows the dendrogram of the main group in Lupus3. The MST branch length connecting two sources is shown on the vertical axis at the connection of each pair of sources or previously connected nodes. The thick dashed horizontal line shows the critical branch length measured in Lupus3, while the two thin dashed lines indicate a range of ±10% around the critical branch length.

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View extended figure (13 panels)

In the Lupus3 group, most of the members are clearly tightly clustered with separations well below the critical branch length. An increase in the critical branch length of 10% would increase membership by only one late-type source and a decrease of 10% would result in the group decreasing by two or up to six late-type YSOs (90% of the critical branch length is very nearly equal to the branch length required to join four of the outlying group members to the rest of the group).

Most of the other groups show a similar behavior—an increase or decrease of the critical branch length by 10% changes the group membership by at most a handful of members (most often one or two). In the main group in IC348, the number of group members included/excluded by a variation in the critical branch length is larger, but still a small fraction (<5%) of the total number of members. The other exceptional group is L1536, also discussed in Appendix B—the critical branch length is only marginally larger than the branch length connecting two sub-groupings of YSOs, and would be split into two groups with a slightly smaller critical branch length, as occurred with our analysis using the R10 catalog. L1527 also shows a similar behavior although to a lesser extent—two smaller sub-groups would be lost if the critical branch were reduced by 5%–10%, but the structure as a whole is more robust to smaller perturbations in the critical branch length.

In terms of our analysis, the variation in the number of group members is less important than the impact on the relationship between the most massive group members and the rest of the group. In the Lupus3 group, as seen in Figure 16, the most massive group members lie in a much more highly clustered part of the group (as found in Section 4.4), and their relationship with the bulk of the group is little affected by the loss or gain of YSOs at the group outskirts. A similar result is found upon examination of dendrograms of most of the other groups and their nearest neighbors. As expected, the two groups whose most massive members are not centrally located (ChaI-southwest and L1551) are more liable to be excluded from the group structure if the critical branch length is decreased sufficiently. In L1536, if the group is split in half, each piece has a centrally located most massive member.

The MST algorithm identifies groups by linking members together through their closest neighbor, referred to as a "single linkage" technique for cluster (or group) identification. In fields outside of astronomy, the MST technique is often less popular than other linkage techniques which are better suited for identifying round clusters (E. Feigelson 2010, private communication). There are two other main classes of routines—"average linkage," which uses the distance of an object to the cluster center, and "complete linkage," which uses the distance of an object to the furthest cluster member (E. D. Feigelson & G. J. Babu 2011, in preparation); the latter is useful for identifying very concentrated clusters.

We experimented with both techniques to identify groups in our data set, using IDL's cluster_tree function. As with the MST or "single-linkage" technique, the maximum linkage length to define a group must still be chosen. We use the same method as we adopted for the MST, i.e., the critical length, measured using the cumulative distribution of lengths (discussed in more detail in Appendix D.1). Since the distance to a group's center or furthest member is larger than to the nearest neighbor, the critical lengths fit for the average and complete linkage techniques tend to be significantly larger than the value we found for the MST analysis. A (small) range of critical lengths can provide a good fit to the cumulative distribution; for reasons that will become apparent below, we use the largest critical length possible.

Using the most conservative complete linkage technique, only small groupings are identified in our data set. IC348, for example, is split into four groups, each with only a handful of members, which do not appear as visually distinct groupings. In our MST analysis, 182 members were found in the main group of IC348. In Taurus, only three groupings are identified; other visually striking groupings identified using independent methods (Section 3.2) are missed. We therefore conclude that these young nearby stellar groups are too sparse to be effectively identified using the complete linkage technique.

Using the less conservative average linkage technique, we have mixed results. In Lupus3 and Taurus, most visually striking groupings are identified. There is a good correspondence between these groups and the MST groups, although the average linkage groups tend to have fewer members. In Taurus, two of the eight groups identified with the MST (B213 and L1527) are no longer large enough to meet our minimum membership criterion (>10 members), and one MST-identified group becomes split in half (L1536; discussed in Appendix D.1 for a similar reason). In ChaI and IC348, however, we encounter a similar problem as found with the complete linkage technique—the groups identified are overly subdivided and do not appear visually distinct. IC348, for example, is split into eight groups, most of which border directly on several other groups and do not appear to be separate. The ChaI-south group identified using MSTs is similarly split into four groups with the average linkage technique. This fragmentation occurs despite the fact that we pushed the linkage length to the upper end of the range possible. In our data sets, it therefore appears that these other linkage techniques are overly sensitive to small-scale substructure perturbations, and are not optimal for identifying groups.

In Taurus and Lupus3, where the average linkage technique works, the most massive group members have a small offset from the center. The O_1st/O_med values found are slightly smaller than those measured with the MST groups.