A Search for In-Situ Field OB Star Formation in the Small Magellanic Cloud
Irene Vargas-Salazar, M. S. Oey, Jesse R. Barnes, Xinyi Chen, N. Castro, Kaitlin M. Kratter, Timothy A. Faerber
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A Search for In-Situ Field OB Star Formation in the Small Magellanic Cloud
Irene Vargas-Salazar, M. S. Oey, Jesse R. Barnes,
1, 2
Xinyi Chen,
1, 3
N. Castro, Kaitlin M. Kratter, andTimothy A. Faerber
1, 6 University of Michigan, 1085 S. University, Ann Arbor, MI 48109, USA Present address: Private Present address: Yale University, New Haven, CT 06520, USA Leibniz-Institut fr Astrophysik An der Sternwarte, 16 D-14482, Potsdam, Germany University of Arizona, Tucson, AZ 85721, USA Present address: Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden (Accepted to ApJ)
ABSTRACTWhether any OB stars form in isolation is a question central to theories of massive star formation.To address this, we search for tiny, sparse clusters around 210 field OB stars from the Runaways andIsolated O-Type Star Spectroscopic Survey of the SMC (RIOTS4), using friends-of-friends (FOF) andnearest neighbors (NN) algorithms. We also stack the target fields to evaluate the presence of anaggregate density enhancement. Using several statistical tests, we compare these observations withthree random-field datasets, and we also compare the known runaways to non-runaways. We find thatthe local environments of non-runaways show higher aggregate central densities than for runaways,implying the presence of some “tips-of-iceberg” (TIB) clusters. We find that the frequency of thesetiny clusters is low, ∼ −
5% of our sample. This fraction is much lower than some previous estimates,but is consistent with field OB stars being almost entirely runaway and walkaway stars. The lack ofTIB clusters implies that such objects either evaporate on short timescales, or do not form, implyinga higher cluster lower-mass limit and consistent with a relationship between maximum stellar mass( m max ) and the mass of the cluster ( M cl ). On the other hand, we also cannot rule out that some OBstars may form in highly isolated conditions. Our results set strong constraints on the formation ofmassive stars in relative isolation. Keywords: massive stars — field stars — Small Magellanic Cloud — star clusters — open star clusters— star formation — runaway stars — galaxy stellar content — initial mass function —multiple star evolution — OB associations — OB stars — stellar populations — young starclusters INTRODUCTIONRoberts (1957) first examined the question of whetherall massive stars form in clusters or whether a significantnumber might form in isolation as field stars. Based onthe limited data of that era, he concluded that OB starsrarely, if ever, form in the field. Additionally, there is ev-idence to suggest that star formation occurs in unboundassociations of OB stars (Ward et al. 2020; Griffiths et al.2018). However, while it is widely accepted that most
Corresponding author: Irene [email protected] massive stars form in clusters or associations (e.g., Lada& Lada 2003; Zinnecker & Yorke 2007; Elmegreen 1985),it is well known that a significant population of massivestars is also found in sparse, field environments. Thereported frequency of field OB stars varies, dependingon how the “field” is defined, but it is on the order of20 – 30% (e.g., Oey et al. 2004; Gies 1987).In spite of their significant numbers, the nature andorigin of the field massive stars has been unclear. Pre-vious investigations on the cluster mass function implythat many, if not most, field OB stars formed in situ (e.g., Oey et al. 2004; Lamb et al. 2010). On the otherhand, large populations of runaway OB stars are alsoknown to exist, and may dominate the field population a r X i v : . [ a s t r o - ph . S R ] S e p Vargas-Salazar et al. (e.g., Oey et al. 2018; de Wit et al. 2005; Renzo et al.2019). 1.1.
In-Situ Field OB Star Formation
The degree to which massive stars can form in iso-lation provides an important discriminant between thetwo dominant theories of massive star formation: com-petitive accretion and core accretion. The competitiveaccretion model theorizes that cores accrete matter froma shared reservoir of gas (Zinnecker 1982). The core withthe highest mass accretes the most matter due to itssize and location at the center of the sub-cluster (Bon-nell et al. 2001) while lower mass cores accrete the re-maining gas. Thus, this model requires that low-massstars must form in the presence of massive stars, andvice versa (Bonnell et al. 2004), yielding a spectrum ofstellar masses (e.g., Zinnecker 1982; Bonnell et al. 2001).This stipulation implies a relationship between the massof the most massive star formed in the cluster m max ,and the total mass of the cluster M cl , by m max ∝ M / (Bonnell et al. 2004).In contrast, the core accretion model allows for oc-casional formation of isolated massive stars (Krumholzet al. 2009; Li et al. 2003). The model is a scaled-up ver-sion of low-mass star formation. It suggests that coresdo not compete to accrete gas and instead, the amountof gas they accrete depends on the masses of the coresthemselves before collapse (Shu et al. 1987). Cloudsmaintain their mass because fragmentation is preventedby heating from the accretion process (Krumholz & Mc-Kee 2008). Monolithic collapse could then finally hap-pen for sufficiently dense clouds with high column den-sities (at least 1 g/cm ), forming massive stars. Thus, ifOB stars are able to form in situ in the field, this wouldprovide substantial evidence favoring the core accretionmodel, whereas competitive accretion requires all OBstars to form in clusters.Oey et al. (2004) found that OB clusters, and the Hii region luminosity function, (e.g., Oey & Clarke 1998;Oey et al. 2003), follow a power-law distribution ∼ N − ∗ for the number of OB stars N ∗ per cluster. This powerlaw extends to the extreme value of N ∗ = 1, implyingthat OB stars with no other nearby massive stars ap-pear as field stars, simply by populating the low end ofthe cluster mass function. These individual, field OBstars may simply be the “tip of the iceberg” (TIB) ontiny, sparse clusters at the low-mass extreme that aredifficult to detect. Lamb et al. (2010) provide evidencefor the existence of such sparse clusters, or minimal Ostar groups, associated with field OB stars in the SmallMagellanic Cloud (SMC). With observational data fromthe Hubble Space Telescope on 8 SMC field OB stars, they find that 3 out of the 8 are in sparse clusters with ≤
10 companion stars, each having masses of 1 − M (cid:12) .Additionally, the existence of these sparse clusters isconsistent with stochastic nature of the cluster massfunction and the stellar initial mass function (IMF).Monte Carlo simulations show that tiny clusters withOB stars can occur if clusters are built stochastically byrandomly sampling stars from a universal IMF, whichimplies that the maximum stellar mass in a cluster isindependent of cluster mass (Lamb et al. 2010; Parker& Goodwin 2007). However, Monte Carlo simulationsby Weidner & Kroupa (2006) tested various methodsof populating clusters including a completely stochasticsampling. They found that clusters populated throughrandom sampling do not fit observations of young clus-ters as well as a cluster populated through sorted sam-pling in which stellar masses are sorted in ascendingorder and their sum is constrained to be the clus-ter mass. This would imply that clusters form in anorganized fashion and is consistent with the relation m max ∝ M / .An important test for star formation and cluster mod-els is thus the existence of truly isolated, in situ OB starformation. We note that this is true whether or not ”iso-lated” OB stars are binaries, since most binaries formfrom a single star-forming core. While it is currentlyalmost impossible to determine whether any OB starsform in true isolation, some tantalizing observations ex-ist. In the SMC, Oey et al. (2013) present a sampleof 14 field OB stars that are strong candidates for insitu formation. These objects are found in the center ofcircular
Hii regions, showing no bow shocks implying su-personic motion, and having radial velocities matchingthose of the local Hi components. Five of these targetsare extremely isolated. Also in the SMC, observationsby Selier et al. (2011) show that the compact Hii regionN33 is consistent with this object being a case of iso-lated massive-star formation. In the 30 Doradus regionof the LMC, Bressert et al. (2012) identified 15 O stars ascandidates for isolated formation. These stars are notin binary systems and show no evidence of clustering.Additionally, Oskinova et al. (2013) suggests that oneof the most massive stars in the Milky Way, WR102ka,may have been born in isolation. It is not a runaway,since it shows a circumstellar nebula with no bow shock,and there is no evidence of an associated star cluster. deWit et al. (2004) found that 4% ±
2% (4-11 out of 193) ofthe Galactic O-star population either cannot be tracedto OB associations, or have non-runwaway space veloc-ities. 1.2.
Runaway and Walkaway OB Stars n-Situ OB Star Formation in SMC >
30 km s − ) or walkaways, slower starsthat are unbound, but below the runaway threshold ve-locity. These are known to comprise a significant frac-tion of the field OB population (e.g., Blaauw 1961).Over the course of their lifetimes, these stars move farbeyond their original birthplaces, and so by definition,they are distinct from stars in clusters. There are twomechanisms for producing runaways and walkaways: dy-namical ejection (Poveda et al. 1967) and binary super-nova ejection (Blaauw 1961). The dynamical mechanismejects stars primarily from unstable three- or four-bodysystems (Leonard & Duncan 1990) and is prevalent indense cluster cores. In the binary supernova scenario,the primary star of a binary system explodes as a su-pernova which ejects the secondary star by a slingshotrelease that may be combined with a supernova “kick”.The frequency of runaway OB stars is not well estab-lished, but it is generally believed to be large. Someestimates suggest that it is on the order of 50% of thefield OB star population (e.g., de Wit et al. 2005; Gies& Bolton 1986), while others suggest that almost allfield OB stars are runaways (e.g., Gies 1987; Gvara-madze et al. 2012). Recent work from our group using Gaia
DR2 proper motions is consistent with runawaysstrongly dominating the field OB population in the SMC(Oey et al. 2018; Dorigo Jones et al. 2020).1.3.
Remnants of Evaporated Clusters
Another way to generate field OB stars is a hybrid be-tween in situ star formation and dynamical effects. Theloss of stars from small clusters may result in some ofthese being much smaller than when they formed, andif an OB star is present, it would be observed as a TIBstar, as described in Section 1.1. In the most extremecase, the OB star could be completely abandoned by itscohorts, although studies are needed to determine thelikelihood of this scenario. Many clusters, especially atlow mass, become unbound due to gas expulsion andfeedback not long after the stars form, a phenomenondubbed “infant mortality” or “infant weight loss” (e.g.,Lada & Lada 2003; Goodwin & Bastian 2006). However,Farias et al. (2018) suggest that gas expulsion may bemore difficult than previously believed. Alternatively,Ward et al. (2020) find that the formation of smaller,unbound associations with OB stars may be relativelycommonplace, even in lower density environments, thussupporting scenarios where massive stars are not allformed in bound clusters. There is evidence of this inthe Cyg OB2 association, which has a high frequency ofwide binaries that would be disrupted through dynami- cal encounters in clusters (Griffiths et al. 2018), but arepossible if they form in unbound associations. A SEARCH FOR FIELD OB STAR FORMATIONTo understand the contribution, if any, of in situ
OB star formation to the field massive star population,we search for small clusters associated with field OBstars in the SMC, to establish and quantify their exis-tence. The SMC offers a complete sample of field OBstars in an external galaxy, and is located at a well-determined distance of 60 kpc (Harries et al. 2003). Weemploy two different cluster-finding algorithms, friends-of-friends (FOF) (Battinelli 1991) and nearest neighbors(NN) (Schmeja 2011), and we also examine the stackedfields around the target OB stars for an aggregate den-sity enhancement.We note that OB stars have a high multiplicity frac-tion. This implies that TIB stars in small clusters alsomay be binaries or multiples, which are difficult to dis-cern. Studies have shown that field massive stars alsomay have significant binarity, from about half the binaryfrequency of those in clusters (Stone 1981; Gies 1987) tofrequencies on the order of those in clusters (Mason et al.2009; Lamb et al. 2016).To carry out our analysis, we use the Runaways andIsolated O-Type Star Spectroscopic Survey of the SMC(RIOTS4 Lamb et al. 2016), which identifies a uniform,statistically complete sample of field massive stars inthe SMC. RIOTS4 represents the field-star subset ofOB star candidates that were photometrically identifiedby Oey et al. (2004) from the Massey (2002) survey ofthe SMC, which covers the star-forming expanse of thegalaxy. Field stars were differentiated from cluster starsby identifying stars that are at least 28 pc away fromany other OB candidates in the analysis of Oey et al.(2004). The RIOTS4 field stars are all spectroscopicallyconfirmed OB stars (Lamb et al. 2016), and they repre-sent ∼
28% of the total SMC OB population (Oey et al.2004). Lamb et al. (2016) find a binary frequency of (cid:38)
60% in this sample.To search for small, sparse clusters associated withthe field OB stars, we require deep stellar imaging oftheir fields. The Optical Gravitational Lensing Exper-iment (OGLE-III Udalski et al. 2008) has accumulated I -band photometry on the SMC for many years. OGLE-III uses the 1.3-m Warsaw Telescope at Las Campanas.Each CCD image is 35 ×
35 arcmin with a scale of 0.26arcsec/pixel.To carry out our cluster-finding algorithms, we re-quire high-quality astrometry of all the stars near ourtarget OB stars. Given the crowded fields in the SMCBar region, we therefore performed PSF-fitting photom- Vargas-Salazar et al. etry and astrometry on 1000 × (76 ×
76 pc )subframes centered on the target stars. We used theDAOphot software in IRAF applied to the OGLE-III I -band images. The image PSFs are generally on theorder of 3.0 pixels, or 0.23 pc FWHM with variationon the order of 10%. Within a 200-pixel (15-pc) radiusof the target, each field was carefully vetted with bothautomatic and manual identification of the stellar ob-jects, to optimize the sample completeness. We thencalibrated the photometry for the entire field using thepublished OGLE-III photometry (Udalski et al. 2008).Our photometric errors indicate that the data have ex-cellent completeness for I < .
0, and we applied thiscutoff to our dataset.Our final sample comprises 210 field OB stars. Thereare fewer stars in our sample than in the original RI-OTS4 survey for several reasons. Firstly, the OGLE-III survey excludes the eastern-most region of the SMCWing. Secondly, targets that are <
200 pixels from theedge of the OGLE-III CCD frames were discarded, sincethe cluster-finding algorithms rely on complete spatialdistribution of stars near the target. In addition, thereare targets for which we were unable to carry out theastrometry and photometry due to technical issues re-lated to field placement within the frame, for example,the presence of an extremely bright, foreground starwithin the field, and spatial distortions near the detectoredge. Finally, 6 stars (M2002-SMC 11802, 38893, 42654,46022, 62981, and 67029; Massey 2002) in the RIOTS4sample were inadvertently included even though theydid not meet the criterion of being 28 pc from anotherOB candidate; these were also deleted from our sample. IRAF was distributed by the National Optical Astronomy Ob-servatory, which was managed by the Association of Universitiesfor Research in Astronomy (AURA) under a cooperative agree-ment with the National Science Foundation.
Figure 1.
The distribution of clustering length l c for starshaving I < . l c = 39 px is adopted forfields with l c >
50 px.
Friends-of-Friends
The friends-of-friends cluster-finding method uses sim-ilar methodology as a minimum spanning tree algorithm.It identifies associated stars, or “friends”, as those thatare located within a given clustering length ( l c ) of an-other member.We use the FOF algorithm in this manner to searchfor faint companion stars in the OGLE-III images with I < . M I = 0 .
11, corresponding roughly to an A1star), which might correspond to small clusters of whichour target OB stars are the TIBs. The FOF clusteringlength, l c , is the value characteristic of the backgroundstellar density. Since our target OB stars are in locationsof varying background density, we define l c specific toeach field. To do this, we calculate the number of clus-ters found as a function of different clustering lengths ina given field. The value that yields the maximum num-ber of clusters is adopted as l c of the field (Battinelli1991). Our distribution of our final l c for our targetfields is shown in Figure 1.The Wing of the SMC has a much lower stellar densitythan the Bar, generating a long tail in the l c distribution(Figure 1), to values much greater than what are likelyto include bound stars in tiny clusters. We thereforeadopt a maximum fixed l c of 39 pixels (3.0 pc), which is1- σ 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 thatit has the tendency to associate stars in a filamentarystructure that are physically unrelated. Although someclusters show real filamentary structure, the fact thatthis algorithm identifies such formations in artificial data n-Situ OB Star Formation in SMC N ∗ (Schmeja2011). An example of this type of structure is shown inFigure 2. Figure 2.
An example of a filamentary string-like structurethat occurs with FOF. Target 4424 is shown by the greencircle.
We then apply the FOF algorithm, using the targetOB stars as the origin for the algorithm. We evaluatethe results using two criteria (Schmeja 2011; Campanaet al. 2008), one based on the number of stars ( N ∗ ) thatare found to be associated; and another based on the M -value, which is a parameter that also takes into accountthe separation of the identified associated stars (see be-low). For both of these tests, a larger value indicates ahigher probability that the associated stars correspondto a real cluster.However, some clusters do have filamentary structure,and we cannot reliably distinguish cases that are realfrom those that are not. Therefore, we generate threerealizations of random-field datasets for each of our tar-gets to serve as controls. The random fields have thesame size (1000 × ) and the same number ofstars as the observed field for our targets, but with thestars randomly placed. This also allows us to evaluatethe potential effects of random Poisson noise on the ob-served data.Our N ∗ distribution, as well as those from the randomfields, is shown in Figure 3. We see that the observeddistribution differs from the random fields at small N ∗ values, but they are otherwise remarkably similar. We use the Wilcoxon signed rank test for matchedpairs to evaluate whether the observed and randomizeddistributions are statistically indistinguishable. This Figure 3.
The distribution of N ∗ , the number of stars asso-ciated by FOF for stars with I < .
0. Our observed datasetis plotted in black, and the random fields are colored. TheWilcoxon test indicates that our observed data and randomfields are distinct, while our Rosenbaum test results revealambiguous results. test is appropriate for two non-parametric datasets thatare not independent, and is calculated from the differ-ence between the pairs of data points for each field, theobserved vs random field. The results are shown inTable 1. We adopt the conventional critical thresholdof p < .
05 for rejecting the null hypothesis that thetwo distributions originate from the same parent distri-bution. Comparing the N ∗ distributions, we obtain p -values that strongly indicate that the observed datasetdiffers from the randomized ones, indicating the poten-tial presence of some TIB clusters.We might expect that the tails of the distributionsshould be sensitive to the presence of TIB clusters, whichwould have higher positive N ∗ values, and would skewboth the median and the tail to higher values. TheRosenbaum test (Rosenbaum 1965) is optimized to eval-uate the significance of differences in the tails of twodistributions, by counting the number of data pointsbetween the highest values of the two samples and quan-tifying the significance of the difference. Using this test,we do not obtain a significant difference in spread, al-though the higher value does belong to our observeddataset in each case, when compared to the random datasets. The results are given in Table 1 and show p -valueshigher than our threshold of 0.05, indicating that thetails of the observed data set vs the randomized ones donot differ statistically. Vargas-Salazar et al.
Table 1 . Statistical Test ResultsAlgorithm Statistical Test Dataset p -values a FOF N ∗ Wilcoxon Full Data vs Random 1
Full Data vs Random 2
Full Data vs Random 3
Runaways vs Random 1
Runaways vs Random 2
Runaways vs Random 3
Non-Runaways vs Random 1
Non-Runaways vs Random 2
Non-Runaways vs Random 3
Anderson-Darling Runaways vs non-Runaways 0.71Kolmogorov-Smirnov Runaways vs Non-Runaways 0.96Rosenbaum Full Data vs Random 1 0.25Full Data vs Random 2 0.25Full Data vs Random 3 0.5Runaways vs Random 1 0.5Runaways vs Random 2 0.5Runaways vs Random 3 0.5Non-Runaways vs Random 1 0.5Non-Runaways vs Random 2 0.5Non-Runaways vs Random 3 0.5FOF M-Test Wilcoxon Full Data vs Random 1
Full Data vs Random 2
Full Data vs Random 3
Runaways vs Random 1
Runaways vs Random 2
Runaways vs Random 3
Non-Runaways vs Random 1 0.058Non-Runaways vs Random 2 0.14Non-Runaways vs Random 3
Anderson-Darling Runaways vs non-Runaways 0.44Kolmogorov-Smirnov Runaways vs Non-Runaways 0.51Rosenbaum Full Data vs Random 1 0.5Full Data vs Random 2 0.25Full Data vs Random 3 0.5Runaways vs Random 1 0.5Runaways vs Random 2 0.5Runaways vs Random 3 0.5Non-Runaways vs Random 1 0.5Non-Runaways vs Random 2 0.5
Table 1 continued n-Situ OB Star Formation in SMC Table 1 (continued)
Algorithm Statistical Test Dataset p -values a Non-Runaways vs Random 3 -0.5NN Average Wilcoxon Full Data vs Random 1
Full Data vs Random 2
Full Data vs Random 3
Runaways vs Random 1
Runaways vs Random 2
Runaways vs Random 3
Non-Runaways vs Random 1
Non-Runaways vs Random 2
Non-Runaways vs Random 3
Anderson-Darling Runaways vs Non-Runaways 0.067Kolmogorov-Smirnov Runaways vs Non-Runaways
Rosenbaum Full Data vs Random 1 0.25Full Data vs Random 2
Full Data vs Random 3 0.5Runaways vs Random 1 -0.25Runaways vs Random 2 0.5Runaways vs Random 3 -0.5Non-Runaways vs Random 1
Non-Runaways vs Random 2
Non-Runaways vs Random 3 0.5NN Median Wilcoxon Full Data vs Random 1
Full Data vs Random 2
Full Data Random 3
Runaways vs Random 1
Runaways vs Random 2
Runaways vs Random 3
Non-Runaways vs Random 1
Non-Runaways vs Random 2
Non-Runaways vs Random 3
Anderson-Darling Runaways vs Non-Runaways 0.090Kolmogorov-Smirnov Runaways vs Non-Runaways
Rosenbaum Full Data vs Random 1 0.25Full Data vs Random 2
Full Data vs Random 3 0.5Runaways vs Random 1 -0.25Runaways vs Random 2 0.5Runaways vs Random 3 -0.5Non-Runaways vs Random 1
Non-Runaways vs Random 2
Table 1 continued
Vargas-Salazar et al.
Table 1 (continued)
Algorithm Statistical Test Dataset p -values a Non-Runaways vs Random 3 0.5 a A higher p -value indicates a higher probability for the null hypothesis that the twodistributions originate from the same parent distribution. The negative p -values indicatecomparisons for which the cluster-finding test favors the random dataset. The bold p -values indicate statistically significant differences ( p < . Figure 4.
The distributions of M -values for FOF appliedto stars with I < .
0. Our observed dataset is plotted inblack while the random fields are colored. These results aresimilar to our N ∗ results. The second criterion we use to search for cluster candi-dates uses the M -values (Schmeja 2011; Campana et al.2008), where: M = l c, SMC l c, field N ∗ . (1)This takes the ratio of the average clustering length forall the SMC fields ( l c, SMC ) and the clustering lengthof a given field ( l c, field ) and multiplies it by the num-ber of stars associated by FOF for that particular field.Therefore, this test not only uses N ∗ , but also takes intoaccount the separation between these stars. A higher M -value corresponds to a larger number of stars that areclosely spaced together. The distribution of M valuesis shown in Figure 4 for our observed dataset and therandom fields.We again find that the M -value distributions for ourobserved data and the random fields have different sta-tistical test results. The Wilcoxon test p -values indicatethat the observed and random datasets are statisticallydistinct while the Rosenbaum test results do not showa significant difference in spread. It is possible that theWilcoxon signal results from presence of some tiny clus- ters that are too small to affect the Rosenbaum tests. Onthe other hand, this outcome could also be due to thebackground stars having positions that are not purelyrandom. This is further discussed in Section 3.2.2. Nearest Neighbors
NN is an algorithm that measures the stellar density(Σ j ) associated with a given target star. This is calcu-lated by counting the number of stars enclosed withinthe radius to its j th nearest neighbor in 2D as shown inequation 2 (e.g., Schmeja 2011):Σ j = j − S j , (2)where j is the j th nearest neighbor and S j is the areadefined by the radius to the j th nearest neighbor. Figure 5.
Average σ bg fluctuations as a function of j ,showing that statistical fluctuations produce less noise athigher j values. We compare the resulting stellar density to the back-ground density. Since the background density variesgreatly across the SMC, we perform background den-sity calculations for each target individually. The av-erage background density (Σ bg ) is calculated from thetotal number of stars in the field N t , the total numberof stars N j within S j , and the area outside S j , so as tonot include the area within a potential cluster, as shownin equation 3: Σ bg = N t − N j S t − S j , (3) n-Situ OB Star Formation in SMC S t is the total area of the field (1000 × ).The Poisson error of Σ bg is therefore, σ bg = Σ bg × N − / bg,j , (4)where N bg,j = Σ bg × S j is the number of backgroundstars expected in area S j . We caution that some of ourfields may occasionally include external clusters or over-densities in the background, which would overestimatethe background.The NN algorithm works best for small clusters whenΣ j is averaged over a few values of j in the range 3 < j <
20 (Schmeja 2011; Casertano & Hut 1985); the lowest j values are sensitive to statistical fluctuations, whileat high j values, the signal from small clusters becomestoo diluted. We would need to select j values as lowas 3 to probe within the cluster radius. However, afterreviewing results for various ranges from j = 3 to j =12, we find that statistical fluctuations are sufficientlydamped around j > j = 8 −
12 as the basis forour cluster-finding analysis. We calculate the differencebetween Σ j and the background density Σ bg in units of σ bg for j = 8 −
12. We then obtain the average andmedian differences across these j values for each tar-get field (Figure 6). Systems with higher Σ j above thebackground Σ bg are more likely to be physical clusters.We again compare our results with the random fielddata. Since NN is also calculated from a specific tar-get star, we choose the star nearest to the center in therandom datasets as the origin of this algorithm. Theseresults are plotted together with our NN results in Fig-ure 6.The density distributions peak at slightly negative val-ues because the density measurement is centered on astar, rather than a random, star-less point; it is causedby the fact that positions centered on stars are necessar-ily farther from the nearest star than random positionsbetween them, which causes the stellar densities to beunderestimated at the lowest j values (Casertano & Hut1985).The Wilcoxon and Rosenbaum tests comparing theobserved and random-field data are given in Table 1.The statistical tests yield ambiguous results. While theWilcoxon test shows a significant likelihood of TIB clus-ters being present, the Rosenbaum test for NN, like theprevious results for FOF, find that our observed data areindistinguishable from 2 of the 3 random-field datasets.These contradicting statistical test results suggest thatthe observations are in a regime where TIB clusters aremarginally detected, which is further discussed below inSection 3. Figure 6.
Our NN results for the average (bottom) andmedian (top) overdensities for j = 8 to 12 as well as the threerandom datasets. Our observed data are shown in blackwhile the random-field datasets are colored as shown. TheWilcoxon test indicates that our observed data and randomfields are distinct, while two out of 3 of our Rosenbaum testresults indicate the contrary. Stacked Fields
Given the non-detection, or at best, marginal detec-tion, of any clusters by the above methods, we can im-prove our detection sensitivity for the aggregate sampleby stacking the data for all the fields. We measure thestellar density as a function of radius from each targetstar and then take the median of all of our target fieldsat each radial step. We do the same for the randomfields. For these, the densities are measured by center-ing on the star closest to the center of the field, as before.The radial density profiles for the observed data and therandom datasets are shown in Figure 7. The sawtoothpattern in the unsmoothed plots result from oversam-pling the relatively small number of discrete stars rel-ative to the higher resolution pixel grid: the value ofthe stellar density associated with individual stars de-creases with radius until additional stars are includedwithin the target area. The random-field data show atrend of increasing stellar density with radius that flat-tens out around 60 px. This is again the statistical effect0
Vargas-Salazar et al.
Figure 7.
Comparison of stacked target fields with stacked random datasets. This plot shows the stellar density as a functionof radius from the target for our observed data (solid blue), along with those for our three random data sets (dashed lines).It is binned by 1 (left) and by 10 (right). The observed data show a modest excess relative to the random fields at radii <
60 px suggesting the presence of a real aggregate enhancement. The sawtooth pattern in the unsmoothed plot are due toundersampling of the data (see text). caused by selecting a star, rather than a truly randomposition, as the origin for the counting algorithm.The observed data do show an excess relative to therandom fields at radii <
60 px, suggesting the presenceof a real aggregate enhancement that may be due to thepresence of some TIB clusters. This will be discussedfurther below in Section 5. SUBPOPULATIONS3.1.
Runaways and Non-runaways
We know that a large fraction of the field OB starsare runaways, which would not be in TIB clusters. Oeyet al. (2018) identified runaways with transverse veloc-ities >
30 km s − from Gaia proper motions. Thus,we can evaluate the reliability of the cluster-finding al-gorithms by determining how many of the best clustercandidates are identified as runaways. For FOF, thebest TIB candidates are those with the highest N ∗ and M -values, and for NN, they are the target fields with thehighest overdensities relative to the background. We usethe residual transverse velocities v loc , ⊥ that were mea-sured by Oey et al. (2018) relative to the local velocityfields, adopting their runaway definition of v loc , ⊥ > − .Of the top 20 TIB candidates from the FOF analy-sis, 8 are runaways among the top N ∗ candidates, and7 among the top M -test candidates. For the NN algo-rithm, 9 and 8 are among the top 20 candidates based on the median and average overdensities, respectively.These findings are summarized in Table 2, where theserunaways are identified. Thus we see that on the orderof half of even the top 20 TIB candidates for both FOFand NN are runaways. This is reasonably consistentwith the 2/3 fraction of runaways in the RIOTS4 sam-ple (Dorigo Jones et al. 2020). Although the Wilcoxontests in Section 2 show a significant difference betweenthe observed and random fields, the number of runawaysamong the best TIB candidates underscores the role ofrandom density fluctuations in generating signals sug-gestive of TIB clusters by our algorithms.The FOF algorithm shows significantly more run-aways among the top 5 TIB candidates for both the N ∗ and M criteria than obtained by NN. Even 3 of the top5 candidates identified by the N ∗ criterion are runaways.On the other hand, for both NN criteria, none of the top5 candidates include known runaways. We caution thatthe Gaia measurement errors are relatively large ( ∼ − ), and there is moreover uncertainty regardingthe proper motion for any individual object; since theRIOTS4 runaway threshold is 30 km s − , the measure-ment errors leave open the possibility that a significantfraction of non-runaways are mis-identified as runaways,and could therefore be TIBs. However, note that thisinterpretation also depends on, and is consistent with,the difference between the FOF and NN results beingdue to the existence of a few real TIB clusters amongthe top candidates identified by NN. n-Situ OB Star Formation in SMC Table 2 . Runaways Among Top TIB Candidates a Target b FOF N ∗ FOF M -value NN Average NN Median3815 · · · ◦ · · · ◦ · · · · · · ◦ ◦ · · · ◦ ◦ ◦ · · · ◦ · · · · · · • (cid:70) · · · · · · · · · · · · ◦ ◦ · · · · · · ◦ ◦ ◦ · · · · · · · · · • · · · · · · · · · (cid:70) ◦ · · · · · · • · · · ◦ ◦ (cid:70) ◦ · · · · · · ◦ · · · · · · · · · (cid:70) (cid:70) • • · · · · · · • • · · · · · · • • a Open circles, filled circles, and stars correspond to objects identifiedamong the top 20, 10, and 5 TIB candidates, respectively. b ID from Massey (2002)
Separating our sample into runaway and non-runawaytargets should strengthen the signal of any real TIBsamong the latter. Thus, we compare the results of ourcluster-finding algorithms for these subsamples below inFigures 9, 10 and 11. We also apply the Wilcoxon andthe Rosenbaum tests to compare the runaway and non-runaway targets to their respective random fields, as wellas to each other. Our results are shown in Table 1.The Wilcoxon and Rosenbaum test results for the run-aways are essentially identical to those for the full sam-ple. For runaways, we would expect to not see any sta-tistical differences from random fields. We therefore be-lieve that the positive detections from the Wilcoxon testare not due to cluster detections, but instead result fromother effects, like the possible non-random spatial distri-bution of field stars suggested earlier. This is consistentwith the stacked field results for runaways, where atsmall radii they appear to have ambiguous, but slightlyhigher, densities.Additionally, runaways show a significant fraction oftargets found in lower density environments, as expectedsince they quickly move away from dense, cluster form- ing environments where they originated. Figure 8 showsΣ j and the corresponding background density, Σ bg , as afunction of R j , the mean radius of the j th nearest neigh-bor. The non-runaways on average have greater Σ j andΣ bg while runaways on average have higher R j values,indicating that runaways tend to be in target fields withlower stellar density. This is also reflected in Figure 9wherein the peaks of the non-runaway distributions areat larger values than those for the runaways.For FOF, the non-runaways do not show statisticallysignificant differences from the runaways. They exhibitthe same behavior that is seen for both the full and therunaway datasets, showing significant Wilcoxon test re-sults but not Rosenbaum test results. For NN, however,the non-runaways do show statistically significant differ-ences from the runaways. In the NN density distribu-tion comparisons with the random fields, the Wilcoxonresults give p -values that are statistically significant andlower by an order of magnitude than those for the run-aways. However, the Rosenbaum results are more am-biguous because only two NN comparisons with randomfields show a statistically significant difference. Thisagain demonstrates that the frequency of any TIB clus-2 Vargas-Salazar et al.
Figure 8.
Distribution of various NN values for our runaway and non-runaway (NR) data sets: the median Σ j (upper left), themedian σ bg (upper right) and median Σ bg (lower left) as a function of median R j values. The lower right plot is of the median R j values as a function of j . On average, the runaway data set shows higher values for R j , but lower Σ j and Σ bg values, thanthe non-runaways, indicating that they are found in lower density environments. ters in the observed dataset is on the order of the randomnoise in our fields.When compared to each other, the runaway andnon-runaway distributions also look noticeably different(Figures 9 and 10), confirming these trends. Since therunaway and non-runaway datasets are independent ofeach other, we are able to use the Kolmogorov-Smirnov(KS) test and also the Anderson-Darling (AD) test whencomparing these distributions. This version of the KStest gives more weight to the tails of the distributions,which in our case are more sensitive to the detection ofTIBs. The resulting p -values of the AD and KS tests arealso shown in Table 1. For FOF, we have similar statis-tical test results to NN between the runaways and non-runaways distributions of N ∗ and M -values (Figure 10)however, their AD and KS test are unable to distinguishthe runaways and non-runaway distributions from eachother, although we are able to see that non-runawaysare skewed towards higher values.Meanwhile, the NN data (Figure 9) do show evidencethat the runaway and non-runaway density distributionsare different, with p -values of 0.022 and 0.035 in the KS test; while the AD results are close to the critical range,between p = 0 .
05 and 1.0. In Figure 9, runaways peakat lower values than the non-runaways, and their max-imum overdensities are less than those of the randomfield datasets. On the other hand, non-runaways haveboth higher peak and maximum values that are distinctfrom the random-field data.Our stacked fields also show differences between ourrunaway and non-runaway datasets (Figure 11). We findthat the runaways show less variation from the random-field datasets, although they do appear to show, withsome ambiguity, a significant density enhancement atthe lowest radii that may be a product of the system-atic errors similar to those found in NN. In contrast, thenon-runaways clearly show higher, and more centrallyconcentrated, densities than the non-runaways. In Fig-ure 11, we can see that these higher densities are presenteven when smoothing our distribution of observed over-densities. These higher densities may be due to the pres-ence of TIB clusters, and are likely the cause of the ag-gregate enhancement seen in Section 2.3 before. This isfurther discussed below in Section 5. n-Situ OB Star Formation in SMC Figure 9.
NN density distributions comparing our observed data for runaway (top) and non-runaway targets (middle) withtheir respective fields in the random datasets. The observed data are shown in black while the three random datasets arecolored as shown. The left column shows the median overdensities for j = 8 to 12, and right column shows the averages. Thebottom row compares the runaway (stars) and non-runaway (lines) distributions normalized with respect to their total number.The Rosenbaum and Wilcoxon tests give contradicting results on whether our runaway targets are distinct from a randomdistribution. However, for non-runaway targets, these tests show strong statistical differences from a random distribution. Thenon-runaway and runaway distributions are also statistically different from each other in their KS test. In summary, any signal of TIBs in our sample shouldbe strengthened in non-runaway datasets, while we ex-pect runaway fields to behave more like random fields.Despite its positive detections, the FOF algorithm is notsensitive to the differences between non-runaway andrunaway fields. Therefore, its results are not a reliable indicator of the presence of TIBS in our sample, andcannot be used to estimate the number of TIB clusters.On the other hand, the results given by NN and thestacked fields do show significant differences betweenthe runaways and non-runaways, which are consistentwith the presence of a small, but real, number of TIB4
Vargas-Salazar et al.
Figure 10.
FOF results comparing our observed data for runaway (top) and non-runaway targets (middle) with theirrespective fields in the random dataset. The bottom row compares the runaway and non-runaway distributions normalizedwith respect to their total number. The left column shows the N ∗ distribution, while the right column shows the M -valuedistribution. Our observed data are in black, while the random dataset is in blue. In the bottom row, non-runaway andrunaway normalized distributions are shown with different hatching as shown. The Wilcoxon test identifies the runaway andnon-runaway distributions as distinct from random distributions, but the Rosenbaum does not. The AD and KS tests are unableto distinguish the runaway and non-runaway distributions from each other. clusters. This is supported by the statistical differencesbetween non-runaway and runaway distributions givenby their respective KS and AD test results. The rel-ative effectiveness of NN and FOF is consistent withthe analysis by Schmeja (2011), who find that NN isa superior cluster-finding algorithm when compared to other methods, including the minimum spanning tree,on which FOF is based.3.2. In-Situ Field OB Stars
Conversely to runaways, we can also examine objectsthat have been identified as field OB stars that formed n-Situ OB Star Formation in SMC Figure 11.
Plots of observed overdensities to those for random fields, as a function of radius, for our runaway targets (top) andnon-runaway targets (bottom). They are binned by 1 (left) and by 10 (right). Our runaway targets appear to have ambiguouslyhigher densities at very small radii that are not due to real density enhancements. However, non-runaway targets do showcentrally concentrated densities. in situ . Oey et al. (2013) identified 14 strong candidatesin the RIOTS4 survey, based on their dense, symmetric
Hii regions and radial velocities consistent with local Hi systemic velocities. Of these, 10 are in our dataset.We might expect that few of these should be runaways, and we might expect some to be among the best TIBcandidates from our cluster-finding algorithms. Table 3 . Kinematic Data for Isolated
In-Situ
Candidates Identified as RunawaysTarget a RA Velocity b err DEC Velocity b err v loc, ⊥ c err Radial Velocity d
3D Velocity e errkm/s km/s km/s km/s km/s km/s km/s km/s km/s35491 22 20 25 19 34 27 -23 40 29 Table 3 continued Vargas-Salazar et al.
Table 3 (continued)
Target a RA Velocity b err DEC Velocity b err v loc, ⊥ c err Radial Velocity d
3D Velocity e errkm/s km/s km/s km/s km/s km/s km/s km/s km/s36514 90 29 -51 24 103 38 -8 104 3967334 119 24 -66 20 136 32 54 146 3370149 82 29 -32 25 88 39 1 88 4071409 28 26 19 15 33 31 · · · · · · · · · a ID from Massey (2002) b The RA and DEC velocities are relative to local systemic velocity and are calculated using the RA and DEC velocities fromOey et al. (2018). c Residual transverse velocities from Oey et al. (2018). d From Lamb et al. (2016). The error in RV is on average 10 km s − . e Space velocity relative to local frame.
We find that only 2 of the in situ targets ([M2002]SMC-70149, 71409) are among the top 20 FOF TIB can-didates, and another 2 of them ([M2002] SMC-69598,75984) are among the top NN candidates. Instead, 5of the in situ candidates, including two in the top 20for FOF, are identified as runaways. Images of theserunaway-star fields are shown in Figure 12, and theirkinematic information (Oey et al. 2018) is shown in Ta-ble 3. Two of these have low runaway velocities (targets[M2002] SMC-35491 and 71409), and are also consistentwith being non-runaways within the errors. However,others have velocities far above the runaway threshold of v loc, ⊥ >
30 km s − (targets [M2002] SMC-36514, 67334and 70149).The runaway frequency in this subsample is largerthan what we find in Section 3.1. This is likely be-cause the in situ candidate sample was selected to haveno visual evidence of TIBs, and therefore these objectsare more likely to be either runaways or candidates forisolated in situ star formation. Since we know the frac-tion of runaways in our sample is high (Oey et al. 2018;Dorigo Jones et al. 2020), this further enhances the like-lihood that objects selected to appear isolated are run-aways. Indeed, the fact that only half of the objects areconfirmed runaways implies that the rest remain candi-dates for highly isolated, in situ star formation. Suchobjects would not be runaways or walkaways, nor wouldthey show TIBs. For example, the Hii 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 theobjects originated far away. FRACTION OF TIB CLUSTERS IN THE SMCThe non-runaway sample shows slightly positive skewsin the NN stellar density distributions. These are sta-tistically significant, and, as argued above, consistentwith the possible presence of TIB clusters. If there areany real clusters in our sample, we can estimate theirpotential number by determining the excess induced bythe positive skew of the observed distributions relativeto those of the random data. We obtain the excess num-ber of target fields having values beyond the midpointbetween the medians of our observed data and the ran-dom dataset. Table 4 summarizes our estimates for thepercentage of TIB clusters identified in our full sampleand subsamples. These estimates assume Poisson errorsand do not include any systematic effects.When estimating their frequencies, we obtain rela-tively large values in both runaway and non-runawaydistributions, as shown in Table 4. For the runaways,we do not believe that this excess corresponds to truecluster detections, since our targets were selected to befar from any OB association and runaways are unlikelyto originate as TIBs, and instead believe that these arecaused by systematic effects. These could, for example,be due to the background stars having positions thatare not purely random. As noted above, Figure 9 showsthat the runaway density distribution matches those ofthe random fields. Instead, the positive runaway detec-tions likely originate from objects that have moved intothe line of sight toward a density enhancement. n-Situ OB Star Formation in SMC Figure 12.
The 5 in situ candidates that were classified as runaways in our own study. North is to the right and East is up.On the left side are the H α images and on the right are the I -band images. In the H α images, the target is highlighted by thecircle. On the right, the green arrow shows the target’s v loc, ⊥ (Table 3). Their kinematic data is shown in Table 3. Vargas-Salazar et al.
Figure 12. - continued.
Table 4 . Estimated Percentage of TIB Clusters a Full Sample Non-Runaways Runaways Full Data Non-Runaways% % % Subtracted b Subtracted b NN Average vs Random 1 15 ± . ± . ± . ± . ± ± . ± . . ± . . ± . ± ± . ± . . ± . . ± . . ± ± . ± ± . . ± . ± Table 4 continued In Table 1 of Oey et al. (2018), columns 11 and 13 correspondto systemic RA and Dec velocities, respectively, of the local fieldsfor each target star. n-Situ OB Star Formation in SMC Table 4 (continued)
Full Sample Non-Runaways Runaways Full Data Non-Runaways% % % Subtracted b Subtracted b NN Median vs Random 2 15 ± . ± . ± . . ± . ± ± . ± . ± . . ± . . ± . a “NN Average” or “Median” specifies the method of combining the NN stellar density calculations for j = 8 − b Columns 5 and 6 show the estimated frequencies obtained by subtracting the runaway frequencies from those ofthe full sample, and non-runaway samples, respectively.
As shown earlier, half of the top 20 potential clus-ter candidates from NN were actually runaways (Table2), showing that it is possible for runaways to appearto be in TIB clusters due to happenstance. We takethe percentage of positive detections for runaways to bethe percentage of TIBs that arise by chance. To obtainan estimate for the TIB cluster frequency, we thereforesubtract this frequency of false positives from the rawestimates from the non-runaways and the full sample.The final resulting estimates are also shown in Table 4,which support the existence of a low fraction of TIBs inour sample.In general, the final, corrected TIB frequency esti-mates in Table 4 for non-runaways are 2 – 3 times thatfor the full sample. This is consistent with the bulkof these estimates representing real clusters, since non-runaways are about half the sample (101 out of 210 tar-gets).We show the top 10 cluster candidates identified byNN in Table 5, and their images in Figure 13. A few ofthese candidates visually appear to be in clusters. How-ever, the rest of them have ambiguous status, again con-sistent with the scenario that the TIB cluster frequencyis small, and of marginal statistical significance. Amongthe top 10 candidates identified by both NN algorithms,4 are not included in the top 20 FOF candidates. In-terestingly, three of these objects, [M2002] SMC-81646,75984 and 69598 are in less dense fields (Figure 13).We note in Figure 13 that some of our field OB starshave companions associated by NN that appear brighterthan our targets in I -band. A few of these red andyellow stars may be evolved massive stars, implying thatour target OB star may not always be strongly isolatedfrom other high-mass stars, since our selection criteriaare based only on separation from luminous blue stars(Lamb et al. 2016). For example, in at least one case,[M2002] SMC-81646, the bright companion is a likelySMC red supergiant, based on its radial velocity (Massey2002; Massey & Olsen 2003). Another object, [M2002] SMC-58947, has a possible yellow supergiant companion( B − V = 0 .
65; Massey 2002), but this could also be aforeground G star at a distance of 380 pc, which places it270 pc below the Galactic plane ( b = − . ◦ ). The othertwo cases in Figure 13, [M2002] SMC-6908 and 46241,show candidate SMC AGB and RGB stars respectively(Boyer et al. 2011), which are therefore likely field starsin the line of sight. These stars are much less luminousin B and V than our targets. But in general, we cautionthat occasional evolved supergiants may be present nearother target stars in our sample.With the estimates from our non-runaway data in Ta-ble 4, we can set an upper limit on the frequency ofTIB clusters in our sample. For the non-runaways, theaverage excess over the random fields is 11% ± . j = 8 −
12, corrected for the false positive rate; while forthe excess based on the median calculations, the averageTIB cluster frequency is 8 . ± . ∼ −
5% since non-runaways represent abouthalf of the full sample.Although Figure 13 and Table 4 show few clear ex-amples of TIBs, our results suggest that our estimated4 −
5% fraction of TIB clusters might be real. On theone hand, statistical tests indicate a lack of positive de-tections for TIB clusters. There are mixed results forNN, with positive results from Wilcoxon but at leastone negative result from the Rosenbaum test for allthree datasets. These results seem to indicate a lackof evidence for TIBs within our sample. But on theother hand, there is a contrast between our runaway andnon-runaway populations that is consistent with expec-tations if TIB clusters are present. The positive skewslead to estimated TIB cluster fractions for non-runawaysfrom all the algorithms that are roughly double the valuefor the full data set, consistent with TIB clusters beingassociated with non-runaways, as expected. We also seethat the non-runaways show a statistically significant0
Vargas-Salazar et al. distinction from the runaway NN density distributionsusing the KS test, and p -values below 0.1 for the ADtest. Furthermore, the stacked fields show clear, cen-trally concentrated densities only for the non-runaways.Thus, although the statistical results are quantitatively inconclusive, the evidence does support a TIB clusterfrequency of up to 4 or 5%. Table 5 . Top 10 Candidates from NN for Non-Runaway Targets a Target b NN Median NN Average FOF N ∗ FOF M -valueValue Rank Value Rank Value Rank Value Rank47459 4.0 (cid:70) (cid:70) (cid:70) (cid:70) (cid:70) (cid:70) • • (cid:70) (cid:70) · · · · · · (cid:70) (cid:70) (cid:70) • (cid:70) (cid:70) · · · · · · • • (cid:70) (cid:70) • • • • • • · · · · · · • • • · · · • ◦ · · · ◦ ◦ • • (cid:70) a Open circles, filled circles, and stars correspond to objects identified among the top20, 10, and 5 TIB candidates, respectively. There are 11 candidates since the medianand average results share the same top 10 with one exception. b ID from Massey (2002).5.
DISCUSSIONOur results show that in situ star formation is rare atbest, with at most 4 −
5% of our target field OB stars be-ing in small, TIB clusters. This result is consistent withthe work of Dorigo Jones et al. (2020) who use stellarkinematics to determine that runaways and walkawayscomprise the overwhelming majority of our sample.Furthermore, the fact that 5 out of the 14 candidatesfor in situ field OB stars found by Oey et al. (2013) turnout to be runaways (Section 1.1) suggests that their cri-teria for identifying in situ field OB stars are surprisinglyineffective, therefore casting some doubt on the remain-ing 9 candidates in their sample. This result is consis-tent with Gvaramadze et al. (2012) who determined thatmany isolated in situ candidates are actually runaways.Moreover, for the other 5 of the in situ candidates thatare in our sample, 3 are not among our top cluster can-didates, suggesting that they are not TIB stars. Thishowever does not rule out the possibility that they areactually rare cases of isolated field OB-star formation and that the 5 runaways may be a product of the selec-tion bias within the sample.There are two possible explanations for the apparentlack of TIB clusters: either they could be evaporatingon very short timescales, or they simply do not form.Oey et al. (2004) found that the cluster mass function isfully consistent with the existence of TIBs, which wouldrepresent the lowest-mass clusters containing single Ostars. Indeed, TIBs could comprise up to ∼
50% of fieldOB stars for the observed cluster MF (Lamb et al. 2016).Our results may be consistent with the smallest clustersundergoing infant mortality, and therefore causing theirOB stars to appear isolated. de Grijs & Goodwin (2008)show that a large fraction of clusters in the SMC evap-orate on 3 −
10 Myr timescales. Therefore, the presenceof small, unbound associations with OB stars would notbe unlikely, as also suggested by Ward et al. (2020).On the other hand, the smallest clusters that form OBstars may have masses larger than those probed by oursample selection criteria. Lamb et al. (2010) estimateda lower limit of ∼ M (cid:12) for the cluster MF, based onobservations and Monte Carlo simulations that assumed n-Situ OB Star Formation in SMC Figure 13.
The top 10 cluster candidates from NN, for the non-runaways in our study. There are 11 candidates shown sincethe median and average j = 8 −
12 results share the same top 10 targets, except for one. North is to the right and East is up.100 px corresponds to 26 (cid:48)(cid:48) in angular scale. Each target is showed in magenta with its 20 j th nearest neighbors in green. Theirtop-20 rank in each of our criteria are shown in Table 5 . Vargas-Salazar et al.
Figure 13. - continued. n-Situ OB Star Formation in SMC m max ∝ M / relation from Bonnell et al. (2004). In this scenario,the smallest clusters do exist, but never form OB stars.Stellar mergers have been proposed to explain observedexceptions (Oh & Kroupa 2018), perhaps including theestimated fractions in Section 4.Since the core collapse model for massive star forma-tion does allow the occasional formation of OB stars asTIBs, the observed lack of TIB clusters may favor thecompetitive accretion model. However, it may be thatour non-runaway field OB sample is not large enough todistinguish between these models, and, as discussed inSection 4, relatively isolated star formation could stilloccur in very rare situations. Our results remain con-sistent with the estimate of de Wit et al. (2004), whofound that 4% ±
2% of their sample cannot be tracedto a formation to a cluster/OB association, suggest-ing that these could be either TIBs or candidates forisolated in-situ star formation. Additionally, our re-sults are also consistent with OB stars forming in small,unbound associations, which would support the forma-tion of massive stars by monolithic cloud collapse (Wardet al. 2020). The small fraction of TIB clusters in ourobserved dataset would occur if these associations dis-perse quickly, thereby leaving apparently isolated OBstars. In any case, our new limits on the existence ofTIBs set much more stringent constraints on the forma-tion of massive stars in relative isolation. CONCLUSIONIn summary, we use two cluster finding algorithms,friends-of-friends and nearest neighbors, to determinewhether our field OB stars are the “tips of icebergs” ontiny clusters based on stars having
I < .
0. Our 210target stars are a subset of the statistically complete, RI-OTS4 survey of field OB stars in the SMC (Lamb et al.2016), that are also included in the I -band imaging fromthe OGLE-III survey (Udalski et al. 2008). We com-pare our observed data to three realizations of random-field datasets for each field. We also measure the stellardensity as a function of radius from the targets in thestacked fields to search for a signal of TIB clusters inour sample.Our results show that there are very few TIBs in oursample, but that a small number likely do exist. Results for both cluster-finding algorithms show strong statisti-cal similarities in the spatial distribution of our observeddata and random-field datasets. Indeed, the FOF algo-rithm, which we confirm to be less effective than NN(Schmeja 2011), is unable to statistically identify a dif-ference between runaway and non-runaway subsamples,highlighting the low occurrence of TIB clusters.However, the NN algorithm and the stacked fieldsanalysis do show significant differences between the run-aways and non-runaways, suggesting the presence of asmall number of TIB clusters. The 101 non-runawaystars show higher stellar-density environments, consis-tent with the expectation that any TIB OB stars can-not be runaways. The stacked fields also show an excessdensity relative to the random fields at radii <
60 px (4.6pc). In general, the estimated fraction of TIB clustersfor non-runaway fields is 2 – 3 times the estimated fre-quency in the full sample, which is again consistent withreal clusters being present, since non-runaways make uproughly half of our full sample. Overall, our results showthat ∼ −
5% of the field OB stars in the SMC are mem-bers of small clusters, and thus almost all are runawaysand walkaways.The low detection rate of TIB clusters implies that ei-ther such clusters evaporate on very short timescales, orthey form rarely or not at all. This may imply that thecluster lower-mass limit is higher than that probed inour sample selection criteria. If so, these results wouldbe consistent with the m max ∝ M / relation (Bonnellet al. 2004), which would support the competitive ac-cretion model of massive star formation. However, oursample may not be large enough to rule out the alter-native, core collapse model.On the other hand, we note that our findings do sup-port a frequency of ∼ −
5% for the presence of TIB clus-ters, and moreover, we cannot definitively rule out thepossibility that some OB stars may form in highly iso-lated conditions, which would not be identified as TIBs.Although our findings cast doubt on many such candi-dates identified in our earlier work (Oey et al. 2013), afew still remain as compelling possible candidates of iso-lated OB star formation. However, it may be expectedthat these occur with even lower frequencies than TIBclusters. Thus, our results set strong constraints on theformation of massive stars in relative isolation.We are grateful to Johnny Dorigo Jones and MattDallas for providing data related to the RIOTS4 sur-vey. We thank our anonymous referee for insightful com-ments and suggestions. This research was supported bythe National Science Foundation, grant AST-1514838 toM.S.O., and by the University of Michigan. M.S.O. also4
Vargas-Salazar et al. gratefully acknowledges hospitality from the Universityof Arizona that enhanced this work.
Facilities: