A Search for In Situ Field OB Star Formation in the Small Magellanic Cloud

The degree to which massive stars can form in isolation provides an important discriminant between the two dominant theories of massive star formation: competitive accretion and core accretion. The competitive accretion model theorizes that cores accrete matter from a shared reservoir of gas (Zinnecker 1982). The core with the highest mass accretes the most matter because of its size and location at the center of the subcluster (Bonnell et al. 2001), while lower-mass cores accrete the remaining gas. Thus, this model requires that low-mass stars must form in the presence of massive stars, and vice versa (Bonnell et al. 2004), yielding a spectrum of stellar masses (e.g., Zinnecker 1982; Bonnell et al. 2001). This stipulation implies a relationship between the mass of the most massive star formed in the cluster m_max and the total mass of the cluster M_cl, by ${m}_{\max }\propto {M}_{\mathrm{cl}}^{2/3}$ (Bonnell et al. 2004).

In contrast, the core accretion model allows for occasional formation of isolated massive stars (Li et al. 2003; Krumholz et al. 2009). The model is a scaled-up version of low-mass star formation. It suggests that cores do not compete to accrete gas, and instead the amount of gas they accrete depends on the masses of the cores themselves before collapse (Shu et al. 1987). Clouds maintain their mass because fragmentation is prevented by heating from the accretion process (Krumholz & McKee 2008). Monolithic collapse could then finally happen for sufficiently dense clouds with high column densities (at least 1 g cm⁻²), forming massive stars. Thus, if OB stars are able to form in situ in the field, this would provide substantial evidence favoring the core accretion model, whereas competitive accretion requires all OB stars to form in clusters.

Oey et al. (2004) found that OB clusters and the H ii region luminosity function (e.g., Oey & Clarke 1998; Oey et al. 2003) follow a power-law distribution $\sim {N}_{* }^{-2}$ for the number of OB stars N_* per cluster. This power law extends to the extreme value of ${N}_{* }=1$, implying that OB stars with no other nearby massive stars appear as field stars, simply by populating the low end of the cluster mass function. These individual field OB stars may simply be the "tip of the iceberg" (TIB) on tiny, sparse clusters at the low-mass extreme that are difficult to detect. Lamb et al. (2010) provide evidence for the existence of such sparse clusters, or minimal O star groups, associated with field OB stars in the Small Magellanic Cloud (SMC). With observational data from the Hubble Space Telescope on eight SMC field OB stars, they find that three out of the eight are in sparse clusters with $\leqslant 10$ companion stars, each having masses of $1\mbox{--}4{M}_{\odot }$.

Additionally, the existence of these sparse clusters is consistent with the stochastic nature of the cluster mass function and the stellar initial mass function (IMF). Monte Carlo simulations show that tiny clusters with OB stars can occur if clusters are built stochastically by randomly sampling stars from a universal IMF, which implies that the maximum stellar mass in a cluster is independent of cluster mass (Parker & Goodwin 2007; Lamb et al. 2010). However, Monte Carlo simulations by Weidner & Kroupa (2006) tested various methods of populating clusters including a completely stochastic sampling. They found that clusters populated through random sampling do not fit observations of young clusters as well as a cluster populated through sorted sampling, in which stellar masses are sorted in ascending order and their sum is constrained to be the cluster mass. This would imply that clusters form in an organized fashion and is consistent with the relation ${m}_{\max }\propto {M}_{\mathrm{cl}}^{2/3}$.

An important test for star formation and cluster models is thus the existence of truly isolated, in situ OB star formation. We note that this is true whether or not "isolated" OB stars are binaries, since most binaries form from a single star-forming core. While it is currently almost impossible to determine whether any OB stars form in true isolation, some tantalizing observations exist. In the SMC, Oey et al. (2013) present a sample of 14 field OB stars that are strong candidates for in situ formation. These objects are found in the center of circular H ii regions, showing no bow shocks, implying supersonic motion, and having radial velocities matching those of the local Hi components. Five of these targets are extremely isolated. Also in the SMC, observations by Selier et al. (2011) show that the compact H ii region N33 is consistent with this object being a case of isolated massive-star formation. In the 30 Doradus region of the LMC, Bressert et al. (2012) identified 15 O stars as candidates for isolated formation. These stars are not in binary systems and show no evidence of clustering. Additionally, Oskinova et al. (2013) suggest that one of the most massive stars in the Milky Way, WR102ka, may have been born in isolation. It is not a runaway because it shows a circumstellar nebula with no bow shock, and there is no evidence of an associated star cluster. de Wit et al. (2004) found that 4% ± 2% (4–11 out of 193) of the Galactic O star population either cannot be traced to OB associations or have nonrunaway space velocities.

OB stars that are ejected from clusters could be runaway stars (>30 km s⁻¹) or walkaways, slower stars that are unbound but below the runaway threshold velocity. These are known to comprise a significant fraction of the field OB population (e.g., Blaauw 1961). Over the course of their lifetimes, these stars move far beyond their original birthplaces, and so by definition they are distinct from stars in clusters. There are two mechanisms for producing runaways and walkaways: dynamical ejection (Poveda et al. 1967) and binary supernova ejection (Blaauw 1961). The dynamical mechanism ejects stars primarily from unstable three- or four-body systems (Leonard & Duncan 1990) and is prevalent in dense cluster cores. In the binary supernova scenario, the primary star of a binary system explodes as a supernova, which ejects the secondary star by a slingshot release that may be combined with a supernova "kick."

The frequency of runaway OB stars is not well established, but it is generally believed to be large. Some estimates suggest that it is on the order of 50% of the field OB star population (e.g., Gies & Bolton 1986; de Wit et al. 2005), while others suggest that almost all field OB stars are runaways (e.g., Gies 1987; Gvaramadze et al. 2012). Recent work from our group using Gaia DR2 proper motions is consistent with runaways strongly dominating the field OB population in the SMC (Oey et al. 2018; Dorigo Jones et al. 2020).

Another way to generate field OB stars is a hybrid between in situ star formation and dynamical effects. The loss of stars from small clusters may result in some of these being much smaller than when they formed, and if an OB star is present, it would be observed as a TIB star, as described in Section 1.1. In the most extreme case, the OB star could be completely abandoned by its cohorts, although studies are needed to determine the likelihood of this scenario. Many clusters, especially at low mass, become unbound by gas expulsion and feedback not long after the stars form, a phenomenon dubbed "infant mortality" or "infant weight loss" (e.g., Lada & Lada 2003; Goodwin & Bastian 2006). However, Farias et al. (2018) suggest that gas expulsion may be more difficult than previously believed. Alternatively, Ward et al. (2020) find that the formation of smaller, unbound associations with OB stars may be relatively commonplace, even in lower density environments, thus supporting scenarios where massive stars are not all formed in bound clusters. There is evidence of this in the Cyg OB2 association, which has a high frequency of wide binaries that would be disrupted through dynamical encounters in clusters (Griffiths et al. 2018), but are possible if they form in unbound associations.

The FOF cluster-finding method uses a methodology similar to a minimum spanning tree algorithm. It identifies associated stars, or "friends," as those that are located within a given clustering length (l_c) of another member.

We use the FOF algorithm in this manner to search for faint companion stars in the OGLE-III images with $I\lt 19.0$ (M_I = 0.11, corresponding roughly to an A1 star), which might correspond to small clusters, of which our target OB stars are the TIBs. The FOF clustering length, l_c, is the value characteristic of the background stellar density. Since our target OB stars are in locations of varying background density, we define l_c specific to each field. To do this, we calculate the number of clusters found as a function of different clustering lengths in a given field. The value that yields the maximum number of clusters is adopted as the l_c of the field (Battinelli 1991). Our distribution of our final l_c for our target fields is shown in Figure 1.

Zoom In Zoom Out Reset image size

The Wing of the SMC has a much lower stellar density than the Bar, generating a long tail in the l_c distribution (Figure 1), to values much greater than what are likely to include bound stars in tiny clusters. We therefore adopt a maximum fixed l_c of 39 pixels (3.0 pc), which is 1σ above the median value, for any l_c above 50 pixels. This is on the order of the core radii for small clusters (Lada & Lada 2003).

An inherent weakness in the FOF algorithm is that it has the tendency to associate stars in a filamentary structure that are physically unrelated. Although some clusters show real filamentary structure, the fact that this algorithm identifies such formations in artificial data demonstrates that it could overestimate N_* (Schmeja 2011). An example of this type of structure is shown in Figure 2.

Zoom In Zoom Out Reset image size

We then apply the FOF algorithm, using the target OB stars as the origin for the algorithm. We evaluate the results using two criteria (Campana et al. 2008; Schmeja 2011), one based on the number of stars (N_*) that are found to be associated; and another based on the M-value, which is a parameter that also takes into account the separation of the identified associated stars (see below). For both of these tests, a larger value indicates a higher probability that the associated stars correspond to a real cluster.

However, some clusters do have a filamentary structure, and we cannot reliably distinguish cases that are real from those that are not. Therefore, we generate three realizations of random-field data sets for each of our targets to serve as controls. The random fields have the same size ($1000\times 1000\ {\mathrm{px}}^{2}\$) and the same number of stars as the observed field for our targets, but with the stars randomly placed. This also allows us to evaluate the potential effects of random Poisson noise on the observed data.

Our N_* distribution, as well as those from the random fields, is shown in Figure 3. We see that the observed distribution differs from the random fields at small N_* values, but they are otherwise remarkably similar.

Zoom In Zoom Out Reset image size

We use the Wilcoxon signed rank test for matched pairs to evaluate whether the observed and randomized distributions are statistically indistinguishable. This test is appropriate for two nonparametric data sets that are not independent, and is calculated from the difference between the pairs of data points for each field, the observed versus random field. The results are shown in Table 1. We adopt the conventional critical threshold of $p\lt 0.05$ for rejecting the null hypothesis that the two distributions originate from the same parent distribution. Comparing the N_* distributions, we obtain p-values that strongly indicate that the observed data set differs from the randomized ones, indicating the potential presence of some TIB clusters.

Table 1.  Statistical Test Results

Algorithm	Statistical Test	Data Set	p-Valuesa
FOF N*	Wilcoxon	Full Data vs. Random 1	0.001
		Full Data vs. Random 2	0.006
		Full Data vs. Random 3	2.2e-5
		Runaways vs. Random 1	0.010
		Runaways vs. Random 2	0.034
		Runaways vs. Random 3	0.020
		Nonrunaways vs. Random 1	0.003
		Nonrunaways vs. Random 2	0.044
		Nonrunaways vs. Random 3	0.002
	Anderson–Darling	Runaways vs. Nonrunaways	0.71
	Kolmogorov–Smirnov	Runaways vs. Nonrunaways	0.96
	Rosenbaum	Full Data vs. Random 1	0.25
		Full Data vs. Random 2	0.25
		Full Data vs. Random 3	0.5
		Runaways vs. Random 1	0.5
		Runaways vs. Random 2	0.5
		Runaways vs. Random 3	0.5
		Nonrunaways vs. Random 1	0.5
		Nonrunaways vs. Random 2	0.5
		Nonrunaways vs. Random 3	0.5
FOF M-Test	Wilcoxon	Full Data vs. Random 1	0.001
		Full Data vs. Random 2	0.010
		Full Data vs. Random 3	0.001
		Runaways vs. Random 1	0.006
		Runaways vs. Random 2	0.015
		Runaways vs. Random 3	0.003
		Nonrunaways vs. Random 1	0.058
		Nonrunaways vs. Random 2	0.14
		Nonrunaways vs. Random 3	0.001
	Anderson–Darling	Runaways vs. Nonrunaways	0.44
	Kolmogorov–Smirnov	Runaways vs. Nonrunaways	0.51
	Rosenbaum	Full Data vs. Random 1	0.5
		Full Data vs. Random 2	0.25
		Full Data vs. Random 3	0.5
		Runaways vs. Random 1	0.5
		Runaways vs. Random 2	0.5
		Runaways vs. Random 3	0.5
		Nonrunaways vs. Random 1	0.5
		Nonrunaways vs. Random 2	0.5
		Nonrunaways vs. Random 3	−0.5
NN Average	Wilcoxon	Full Data vs. Random 1	1.4e-5
		Full Data vs. Random 2	2.3e-5
		Full Data vs. Random 3	0.001
		Runaways vs. Random 1	0.028
		Runaways vs. Random 2	0.016
		Runaways vs. Random 3	0.020
		Nonrunaways vs. Random 1	5.8e-5
		Nonrunaways vs. Random 2	2.6e-4
		Nonrunaways vs. Random 3	0.013
	Anderson–Darling	Runaways vs. Nonrunaways	0.067
	Kolmogorov–Smirnov	Runaways vs. Nonrunaways	0.022
	Rosenbaum	Full Data vs. Random 1	0.25
		Full Data vs. Random 2	0.016
		Full Data vs. Random 3	0.5
		Runaways vs. Random 1	-0.25
		Runaways vs. Random 2	0.5
		Runaways vs. Random 3	−0.5
		Nonrunaways vs. Random 1	0.031
		Nonrunaways vs. Random 2	0.0078
		Nonrunaways vs. Random 3	0.5
NN Median	Wilcoxon	Full Data vs. Random 1	1e-5
		Full Data vs. Random 2	4.7e-5
		Full Data vs. Random 3	0.001
		Runaways vs. Random 1	0.020
		Runaways vs. Random 2	0.019
		Runaways vs. Random 3	0.023
		Nonrunaways vs. Random 1	7.5e-5
		Nonrunaways vs. Random 2	0.001
		Nonrunaways vs. Random 3	0.018
	Anderson–Darling	Runaways vs. Nonrunaways	0.090
	Kolmogorov–Smirnov	Runaways vs. Nonrunaways	0.035
	Rosenbaum	Full Data vs. Random 1	0.25
		Full Data vs. Random 2	0.016
		Full Data vs. Random 3	0.5
		Runaways vs. Random 1	−0.25
		Runaways vs. Random 2	0.5
		Runaways vs. Random 3	−0.5
		Nonrunaways vs. Random 1	0.031
		Nonrunaways vs. Random 2	0.0078
		Nonrunaways vs. Random 3	0.5

Notes.

^aA higher p-value indicates a higher probability for the null hypothesis that the two distributions originate from the same parent distribution. The negative p-values indicate comparisons for which the cluster-finding test favors the random data set. The bold p-values indicate statistically significant differences ($p\lt 0.05)$.

Download table as:  ASCIITypeset images: 1 2

We might expect that the tails of the distributions should be sensitive to the presence of TIB clusters, which would have higher positive N_* values and would skew both the median and the tail to higher values. The Rosenbaum test (Rosenbaum 1965) is optimized to evaluate the significance of differences in the tails of two distributions, by counting the number of data points between the highest values of the two samples and quantifying the significance of the difference. Using this test, we do not obtain a significant difference in spread, although the higher value does belong to our observed data set in each case, when compared to the random data sets. The results are given in Table 1 and show p-values higher than our threshold of 0.05, indicating that the tails of the observed data set versus the randomized ones do not differ statistically.

The second criterion we use to search for cluster candidates uses the M-values (Campana et al. 2008; Schmeja 2011), where

$$\begin{eqnarray}&&M=\displaystyle \frac{{l}_{c,\mathrm{SMC}}}{{l}_{c,\mathrm{field}}}{N}_{* }.\end{eqnarray} \tag{ 1 }$$

This takes the ratio of the average clustering length for all of the SMC fields (${l}_{c,\mathrm{SMC}}$) and the clustering length of a given field (${l}_{c,\mathrm{field}}$) and multiplies it by the number of stars associated by FOF for that particular field. Therefore, this test not only uses N_* but also takes into account the separation between these stars. A higher M-value corresponds to a larger number of stars that are closely spaced. The distribution of M-values is shown in Figure 4 for our observed data set and the random fields.

Zoom In Zoom Out Reset image size

We again find that the M-value distributions for our observed data and the random fields have different statistical test results. The Wilcoxon test p-values indicate that the observed and random data sets are statistically distinct, while the Rosenbaum test results do not show a significant difference in spread. It is possible that the Wilcoxon signal results from the presence of some tiny clusters that are too small to affect the Rosenbaum tests. On the other hand, this outcome could also be due to the background stars having positions that are not purely random. This is further discussed in Section 3.

NN is an algorithm that measures the stellar density (Σ_j) associated with a given target star. This is calculated by counting the number of stars enclosed within the radius to its jth nearest neighbor in 2D as shown in Equation (2) (e.g., Schmeja 2011):

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{j}=\displaystyle \frac{j-1}{{S}_{j}},\end{eqnarray} \tag{ 2 }$$

where j is the jth nearest neighbor and S_j is the area defined by the radius to the jth nearest neighbor.

We compare the resulting stellar density to the background density. Since the background density varies greatly across the SMC, we perform background density calculations for each target individually. The average background density (Σ_bg) is calculated from the total number of stars in the field N_t, the total number of stars N_j within S_j, and the area outside S_j, so as to not include the area within a potential cluster, as shown in Equation (3):

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{bg}}=\displaystyle \frac{{N}_{t}-{N}_{j}}{{S}_{t}-{S}_{j}},\end{eqnarray} \tag{ 3 }$$

where S_t is the total area of the field (1000 × 1000 px²). The Poisson error of Σ_bg is therefore

$$\begin{eqnarray}&&{\sigma }_{\mathrm{bg}}={{\rm{\Sigma }}}_{\mathrm{bg}}\times {N}_{\mathrm{bg},j}^{-1/2},\end{eqnarray} \tag{ 4 }$$

where ${N}_{\mathrm{bg},j}={{\rm{\Sigma }}}_{\mathrm{bg}}\times {S}_{j}$ is the number of background stars expected in area S_j. We caution that some of our fields may occasionally include external clusters or overdensities in the background, which would overestimate the background.

The NN algorithm works best for small clusters when Σ_j is averaged over a few values of j in the range $3\lt j\lt 20$ (Casertano & Hut 1985; Schmeja 2011); the lowest j values are sensitive to statistical fluctuations, while at high j values, the signal from small clusters becomes too diluted. We would need to select j values as low as 3 to probe within the cluster radius. However, after reviewing results for various ranges from j = 3 to j = 12, we find that statistical fluctuations are sufficiently damped around $j\gt 8$ (Figure 5).

Zoom In Zoom Out Reset image size

We therefore use the range j = 8–12 as the basis for our cluster-finding analysis. We calculate the difference between Σ_j and the background density Σ_bg in units of σ_bg for j = 8–12. We then obtain the average and median differences across these j values for each target field (Figure 6). Systems with higher Σ_j above the background Σ_bg are more likely to be physical clusters.

Zoom In Zoom Out Reset image size

We again compare our results with the random field data. Since NN is also calculated from a specific target star, we choose the star nearest the center in the random data sets as the origin of this algorithm. These results are plotted with our NN results in Figure 6.

The density distributions peak at slightly negative values because the density measurement is centered on a star rather than a random, starless point; it is caused by the fact that positions centered on stars are necessarily farther from the nearest star than random positions between them, which causes the stellar densities to be underestimated at the lowest j values (Casertano & Hut 1985).

The Wilcoxon and Rosenbaum tests comparing the observed and random-field data are given in Table 1. The statistical tests yield ambiguous results. While the Wilcoxon test shows a significant likelihood of TIB clusters being present, the Rosenbaum test for NN, like the previous results for FOF, find that our observed data are indistinguishable from two of the three random-field data sets. These contradicting statistical test results suggest that the observations are in a regime where TIB clusters are marginally detected, which is further discussed in Section 3.

Given the nondetection, or at best marginal detection, of any clusters by the above methods, we can improve our detection sensitivity for the aggregate sample by stacking the data for all fields. We measure the stellar density as a function of radius from each target star and then take the median of all of our target fields at each radial step. We do the same for the random fields. For these, the densities are measured by centering on the star closest to the center of the field, as before. The radial density profiles for the observed data and the random data sets are shown in Figure 7. The sawtooth pattern in the unsmoothed plots results from oversampling the relatively small number of discrete stars relative to the higher resolution pixel grid: the value of the stellar density associated with individual stars decreases with radius until additional stars are included within the target area. The random-field data show a trend of increasing stellar density with radius that flattens out around 60 px. This is again the statistical effect caused by selecting a star, rather than a truly random position, as the origin for the counting algorithm.

Zoom In Zoom Out Reset image size

The observed data do show an excess relative to the random fields at radii <60 px, suggesting the presence of a real aggregate enhancement that may be due to the presence of some TIB clusters. This will be discussed further in Section 5.

We know that a large fraction of the field OB stars are runaways, which would not be in TIB clusters. Oey et al. (2018) identified runaways with transverse velocities >30 km s⁻¹ from Gaia proper motions. Thus, we can evaluate the reliability of the cluster-finding algorithms by determining how many of the best cluster candidates are identified as runaways. For FOF, the best TIB candidates are those with the highest N_* and M-values, and for NN, they are the target fields with the highest overdensities relative to the background. We use the residual transverse velocities ${v}_{\mathrm{loc},\perp }$ that were measured by Oey et al. (2018) relative to the local velocity fields, adopting their runaway definition of ${v}_{\mathrm{loc},\perp }\gt 30$ km s⁻¹.

Of the top 20 TIB candidates from the FOF analysis, eight are runaways among the top N_* candidates, and seven are among the top M-test candidates. For the NN algorithm, nine and eight are among the top 20 candidates based on the median and average overdensities, respectively. These findings are summarized in Table 2, where these runaways are identified. Thus we see that on the order of one-half of even the top 20 TIB candidates for both FOF and NN are runaways. This is reasonably consistent with the 2/3 fraction of runaways in the RIOTS4 sample (Dorigo Jones et al. 2020). Although the Wilcoxon tests in Section 2 show a significant difference between the observed and random fields, the number of runaways among the best TIB candidates underscores the role of random density fluctuations in generating signals suggestive of TIB clusters by our algorithms.

The FOF algorithm shows significantly more runaways among the top five TIB candidates for both the N_* and M criteria than obtained by NN. Even three of the top five candidates identified by the N_* criterion are runaways. On the other hand, for both NN criteria, none of the top five candidates include known runaways. We caution that the Gaia measurement errors are relatively large (∼28 km s⁻¹), and there is moreover uncertainty regarding the proper motion for any individual object; since the RIOTS4 runaway threshold is 30 km s⁻¹, the measurement errors leave open the possibility that a significant fraction of nonrunaways are misidentified as runaways and could therefore be TIBs. However, note that this interpretation also depends on, and is consistent with, the difference between the FOF and NN results being due to the existence of a few real TIB clusters among the top candidates identified by NN.

Separating our sample into runaway and nonrunaway targets should strengthen the signal of any real TIBs among the latter. Thus, we compare the results of our cluster-finding algorithms for these subsamples in Figures 9–11. We also apply the Wilcoxon and Rosenbaum tests to compare the runaway and nonrunaway targets to their respective random fields, as well as to each other. Our results are shown in Table 1.

The Wilcoxon and Rosenbaum test results for the runaways are essentially identical to those for the full sample. For runaways, we would expect to not see any statistical differences from random fields. We therefore believe that the positive detections from the Wilcoxon test are not due to cluster detections, but instead result from other effects, like the possible nonrandom spatial distribution of field stars suggested earlier. This is consistent with the stacked field results for runaways, where at small radii they appear to have ambiguous, but slightly higher, densities.

Additionally, runaways show a significant fraction of targets found in lower density environments, as expected since they quickly move away from the dense cluster-forming environments where they originated. Figure 8 shows Σ_j and the corresponding background density, Σ_bg, as a function of R_j, the mean radius of the jth nearest neighbor. The nonrunaways on average have greater Σ_j and Σ_bg, while runaways on average have higher R_j values, indicating that runaways tend to be in target fields with lower stellar density. This is also reflected in Figure 9, wherein the peaks of the nonrunaway distributions are at larger values than those for the runaways.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For FOF, the nonrunaways do not show statistically significant differences from the runaways. They exhibit the same behavior that is seen for both the full and the runaway data sets, showing significant Wilcoxon test results but not Rosenbaum test results. For NN, however, the nonrunaways do show statistically significant differences from the runaways. In the NN density distribution comparisons with the random fields, the Wilcoxon results give p-values that are statistically significant and lower by an order of magnitude than those for the runaways. However, the Rosenbaum results are more ambiguous because only two NN comparisons with random fields show a statistically significant difference. This again demonstrates that the frequency of any TIB clusters in the observed data set is on the order of the random noise in our fields.

When compared to each other, the runaway and nonrunaway distributions also look noticeably different (Figure 9 and 10), confirming these trends. Since the runaway and nonrunaway data sets are independent of each other, we are able to use the Kolmogorov–Smirnov (K-S) test and also the Anderson–Darling (A-D) test when comparing these distributions. This version of the K-S test gives more weight to the tails of the distributions, which in our case are more sensitive to the detection of TIBs. The resulting p-values of the A-D and K-S tests are also shown in Table 1. For FOF, we have statistical test results similar to NN between the runaway and nonrunaway distributions of N_* and M-values (Figure 10); however, their A-D and K-S tests are unable to distinguish the runaways and nonrunaway distributions from each other, although we are able to see that nonrunaways are skewed toward higher values.

Zoom In Zoom Out Reset image size

Meanwhile, the NN data (Figure 9) do show evidence that the runaway and nonrunaway density distributions are different, with p-values of 0.022 and 0.035 in the K-S test, while the A-D results are close to the critical range, between p = 0.05 and 1.0. In Figure 9, runaways peak at lower values than the nonrunaways, and their maximum overdensities are less than those of the random-field data sets. On the other hand, nonrunaways have both higher peak and maximum values that are distinct from the random-field data.

Our stacked fields also show differences between our runaway and nonrunaway data sets (Figure 11). We find that the runaways show less variation from the random-field data sets, although they do appear to show, with some ambiguity, a significant density enhancement at the lowest radii that may be a product of the systematic errors similar to those found in NN. In contrast, the nonrunaways clearly show higher, and more centrally concentrated, densities than the nonrunaways. In Figure 11, we can see that these higher densities are present even when smoothing our distribution of observed overdensities. These higher densities may be due to the presence of TIB clusters and are likely the cause of the aggregate enhancement seen in Section 2.3. This is further discussed in Section 5.

Zoom In Zoom Out Reset image size

In summary, any signal of TIBs in our sample should be strengthened in nonrunaway data sets, while we expect runaway fields to behave more like random fields. Despite its positive detections, the FOF algorithm is not sensitive to the differences between nonrunaway and runaway fields. Therefore, its results are not a reliable indicator of the presence of TIBs in our sample and cannot be used to estimate the number of TIB clusters. On the other hand, the results given by NN and the stacked fields do show significant differences between the runaways and nonrunaways, which are consistent with the presence of a small, but real, number of TIB clusters. This is supported by the statistical differences between nonrunaway and runaway distributions given by their respective K-S and A-D test results. The relative effectiveness of NN and FOF is consistent with the analysis by Schmeja (2011), who find that NN is a superior cluster-finding algorithm when compared to other methods, including the minimum spanning tree, on which FOF is based.

Conversely to runaways, we can also examine objects that have been identified as field OB stars that formed in situ. Oey et al. (2013) identified 14 strong candidates in the RIOTS4 survey, based on their dense, symmetric H ii regions and radial velocities consistent with local H i systemic velocities. Of these, 10 are in our data set. We might expect that a few of these should be runaways, and we might expect some to be among the best TIB candidates from our cluster-finding algorithms.

We find that only two of the in situ targets ([M2002] SMC-70149, 71409) are among the top 20 FOF TIB candidates, and another two of them ([M2002] SMC-69598, 75984) are among the top NN candidates. Instead, five of the in situ candidates, including two in the top 20 for FOF, are identified as runaways. Images of these runaway-star fields are shown in Figure 12, and their kinematic information (Oey et al. 2018)⁸ is shown in Table 3. Two of these have low runaway velocities (targets [M2002] SMC-35491 and 71409) and are also consistent with being nonrunaways within the errors. However, others have velocities far above the runaway threshold of ${v}_{\mathrm{loc},\perp }\gt 30$ km s⁻¹ (targets [M2002] SMC-36514, 67334, and 70149).

Zoom In Zoom Out Reset image size

Table 3.  Kinematic Data for Isolated In Situ Candidates Identified as Runaways

Targeta	R.A. Velocityb	Error	Decl. Velocityb	Error	${v}_{\mathrm{loc},\perp }$ c	Error	Radial Velocityd	3D Velocitye	Error
	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)
35491	22	20	25	19	34	27	−23	40	29
36514	90	29	−51	24	103	38	−8	104	39
67334	119	24	−66	20	136	32	54	146	33
70149	82	29	−32	25	88	39	1	88	40
71409	28	26	19	15	33	31	⋯	⋯	⋯

Notes.

^aID from Massey (2002). ^bThe R.A. and decl. velocities are relative to local systemic velocity and are calculated using the R.A. and decl. velocities from Oey et al. (2018). ^cResidual transverse velocities from Oey et al. (2018). ^dFrom Lamb et al. (2016). The error in RV is on average 10 km s⁻¹. ^eSpace velocity relative to local frame.

Download table as:  ASCIITypeset image

The runaway frequency in this subsample is larger than what we find in Section 3.1. This is likely because the in situ candidate sample was selected to have no visual evidence of TIBs, and therefore these objects are more likely to be either runaways or candidates for isolated in situ star formation. Since we know the fraction of runaways in our sample is high (Oey et al. 2018; Dorigo Jones et al. 2020), this further enhances the likelihood that objects selected to appear isolated are runaways. Indeed, the fact that only half of the objects are confirmed runaways implies that the rest remain candidates for highly isolated, in situ star formation. Such objects would not be runaways or walkaways, nor would they show TIBs. For example, the H ii regions of targets [M2002] SMC-66415 and 69598 show "elephant trunks" pointing toward the targets, as shown in Figure 1 from Oey et al. (2013), which are difficult to explain if the objects originated far away.