3D Morphology of Open Clusters in the Solar Neighborhood with Gaia EDR3: its Relation to Cluster Dynamics
Xiaoying Pang, Yuqian Li, Zeqiu Yu, Shih-Yun Tang, František Dinnbier, Pavel Kroupa, Mario Pasquato, M.B.N. Kouwenhoven
DDraft version February 23, 2021
Typeset using L A TEX twocolumn style in AASTeX63
3D Morphology of Open Clusters in the Solar Neighborhood with Gaia EDR3:its Relation to Cluster Dynamics
Xiaoying Pang ,
1, 2
Yuqian Li, Zeqiu Yu , Shih-Yun Tang ,
3, 4
Frantiˇsek Dinnbier , Pavel Kroupa ,
5, 6
Mario Pasquato ,
7, 8 and M.B.N. Kouwenhoven Department of Physics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Dushu Lake Science and Education Innovation District,Suzhou 215123, Jiangsu Province, P. R. China Shanghai Key Laboratory for Astrophysics, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, P. R. China Lowell Observatory, 1400 W. Mars Hill Road, Flagstaff, AZ 86001, USA Department of Astronomy and Planetary Sciences, Northern Arizona University, Flagstaff, AZ 86011, USA Astronomical Institute, Faculty of Mathematics and Physics, Charles University in Prague, V Holeˇsoviˇck´ach 2, 180 00 Praha 8, CzechRepublic Helmholtz-Institut f¨ur Strahlen- und Kernphysik (HISKP), Universit¨at Bonn, Nussallee 14–16, 53115 Bonn, Germany Center for Astro, Particle and Planetary Physics (CAP ), New York University Abu Dhabi INFN- Sezione di Padova, Via Marzolo 8, I–35131 Padova, Italy (Accepted February 18, 2021)
Submitted to ApJABSTRACTWe analyze the 3D morphology and kinematics of 13 open clusters (OCs) located within 500 pc ofthe Sun, using
Gaia
EDR 3 and kinematic data from literature. Members of OCs are identified usingthe unsupervised machine learning method
StarGO , using 5D parameters (
X, Y, Z , µ α cos δ, µ δ ). TheOC sample covers an age range of 25 Myr–2.65 Gyr. We correct the asymmetric distance distributiondue to the parallax error using Bayesian inversion. The uncertainty in the corrected distance for acluster at 500 pc is 3.0–6.3 pc, depending on the intrinsic spatial distribution of its members. Wedetermine the 3D morphology of the OCs in our sample and fit the spatial distribution of stars withinthe tidal radius in each cluster with an ellipsoid model. The shapes of the OCs are well-describedwith oblate spheroids (NGC 2547, NGC 2516, NGC 2451A, NGC 2451B, NGC 2232), prolate spheroids(IC 2602, IC 4665, NGC 2422, Blanco 1, Coma Berenices), or triaxial ellipsoids (IC 2391, NGC 6633,NGC 6774). The semi-major axis of the fitted ellipsoid is parallel to the Galactic plane for most clusters.Elongated filament-like substructures are detected in three young clusters (NGC 2232, NGC 2547,NGC 2451B), while tidal-tail-like substructures (tidal tails) are found in older clusters (NGC 2516,NGC 6633, NGC 6774, Blanco 1, Coma Berenices). Most clusters may be super-virial and expanding. N -body models of rapid gas expulsion with an SFE of ≈ / (cid:12) , while clusters less massive than 250 M (cid:12) tend to agree with adiabatic gas expulsionmodels. Only six OCs (NGC 2422, NGC 6633, and NGC 6774, NGC 2232, Blanco 1, Coma Berenices)show clear signs of mass segregation. Keywords: stars: evolution — open clusters and associations: individual – stars: kinematics anddynamics – methods: statistical – methods: numerical INTRODUCTIONOpen star clusters (OCs) are stellar systems formedin giant molecular clouds (GMCs) located in the disk
[email protected] of the Milky Way (e.g., Lada & Lada 2003). Unliketheir compact halo counterparts (globular clusters), thestellar members of OCs have a looser spatial distribu-tion, hence the name “open”. The formation and evolu-tion of OCs is closely related to Galactic star formation.Studying the spatial distribution of stars in OCs there- a r X i v : . [ a s t r o - ph . GA ] F e b Pang et al. fore provides an opportunity to uncover the mechanismsand conditions of star cluster formation in the Galaxy.The earliest morphological study of OCs dates back acentury (Jeans 1916). In the decades that followed, fur-ther systematic studies were carried out; notable studiesinclude those of Oort (1979) and Bergond et al. (2001).They investigated the spatial distribution of a handfulof nearby OCs, and found that the flattening of the pro-jected shape of OCs tends to be parallel to the Galacticplane. The pioneering work of Chen et al. (2004) de-termined the 2D morphology of nearby 31 OCs using2MASS infrared photometry; they took an importantfirst step in the statistical investigation of OC morphol-ogy. However, they did not reach a firm quantitativeconclusion due to the challenges arising from member-ship determination.Differences between the morphologies of young and oldOCs were identified by (Jones & Basu 2002). Young OCstend to have a higher degree of substructure. S´anchez &Alfaro (2009) found that clusters with fractal-like struc-tures are generally younger than clusters with smoothradial density profiles. Kounkel & Covey (2019) identi-fied several hundreds of filamentary structures youngerthan 100 Myr, most of which were associated withnearby OCs. One string-like structure in Kounkel &Covey (2019), for example, hosts two coeval open clus-ters, NGC 2232 and LP 2439 (Pang et al. 2020). The ex-tended substructures of young OCs are thought to havebeen inherited from the primordial shape of the parentalGMCs (Ballone et al. 2020), in which star formationtakes place along the densest filamentary substructures(Jerabkova et al. 2019).Most regions in GMCs are not self-gravitating, and aresupported by large-scale turbulence. Elongated shapes,such as triaxial and prolate shapes are therefore com-mon among GMCs (Jones & Basu 2002). The triaxialityis consistent with the non-equilibrium state of GMCs.The dense cores in GMCs, where OCs are formed, arepulled together by self-gravity, with an observed elon-gated shape (Curry 2002). After the first stars haveformed, the gas surrounding the OCs is rapidly removedby stellar radiation (Krumholz & Matzner 2009; Dinn-bier & Walch 2020), stellar winds (Weaver et al. 1977),and/or supernovae (McKee & Ostriker 1977). Stars thatescape from the cluster after gas expulsion reduce thegravitational potential of the cluster, and form a tidal“tail I” (following the definition and nomenclature ofDinnbier & Kroupa 2020b). Expansion has been ob-served in very young OCs with ages less than 5 Myr(Kuhn et al. 2019), as well as in young clusters that aretens of millions of years old (Brandner 2008; Bravi et al. 2018; Getman et al. 2018; Karnath et al. 2019; Pang etal. 2020).Simultaneously, the stellar members of an OC inter-act with each other through two-body relaxation, whichresults in the observed “mass segregation” in star clus-ters (Hillenbrand & Hartmann 1998; Pang et al. 2013;Tang et al. 2018), in which low-mass stars are dispersedto the outskirts of the cluster and massive stars tendto migrate to the central region of the cluster. Con-sequently, a dense core will form, while low-mass starscontinue to escape from the cluster, mainly at low speedsthrough Lagrange points (K¨upper et al. 2008), and forman S-shaped tidal “tail II” (following the nomenclatureof Dinnbier & Kroupa 2020b). The reduction of clus-ter members further lowers the gravitational potential,which results in expansion of OCs and consequently alower stellar number density. Chen et al. (2004) pro-posed that the internal relaxation process causes the in-ner part of a cluster to evolve into a spherical spatialdistribution.As the Galactic disk is abundant in stars, spiral arms,and GMCs, OCs are subjected to external tidal pertur-bations, such as disk shocks, spiral arm passages, andencounters with molecular clouds (Spitzer 1958; Lamerset al. 2005; Kruijssen et al. 2012). Stars escape the clus-ter as a consequence of gas expulsion, close encountersor evaporation, and due to their exposure to the Cori-olis force produced by the Galactic tidal field, and mi-grate to more tangential orbits. As a consequence, thestar cluster stretches. Furthermore, when OCs cross theGalactic plane, the disk tidal field compresses them andflattens their shapes. The projected major axis of theelongated shapes of OCs are known to be aligned withthe Galactic plane in most cases (Oort 1979; Bergond etal. 2001; Chen et al. 2004). As OCs evolve, their shapescontinue to distort and members disperse, leading to theinevitable dissolution of the entire cluster. Expansionhas been identified in old open clusters as a sign of an on-going disruption process (Pang et al. 2018). Giant tidaltails extending from OCs have been directly observedin recent years (R¨oser et al. 2019; Meingast & Alves2019; Tang et al. 2019; F¨urnkranz et al. 2019; Zhang etal. 2020). These observed tidal tails are thought to becomposed of both a “tail I”, driven by gas expulsion,and by a “tail II”, driven by evaporation (Dinnbier &Kroupa 2020a,b).The
Gaia
Early Data Release 3 (EDR 3; Gaia Collab-oration et al. 2020) has revolutionized the study of OCmorphology by providing parallaxes with a 30% higherprecision and proper motions with double accuracy, ascompared to those in the
Gaia
Data Release 2 (DR 2;Gaia Collaboration et al. 2018a). It is desirable to rep-
D Morphology of Open Clusters in the Solar Neighborhood
Gaia
EDR 3 data. The distances to the target clustersrange between ≈
86 pc (Coma Berenices) and ≈
476 pc(NGC 2422). The target OCs span a representativerange in ages, from ≈
25 Myr (NGC 2232) to ≈ Gaia
EDR 3 data. At the same time, it is also a reminiscentanalogy to the studies that quantified the morphologyof elliptical galaxies (Benacchio & Galletta 1980; Padilla& Strauss 2008).The paper is organized as follows. In Section 2 we dis-cuss the quality and limitations of the
Gaia
EDR 3 data,and describe our input data-set for member star identifi-cation. We then present the algorithm,
StarGO , whichis used to determine cluster members. The properties ofthe identified member candidates of the 13 target OCsare discussed in Section 3. The 3D morphology of tar-get OCs and the parameterization of the cluster shapesare presented in Section 4, in which we reconstruct thedistances with a Bayesian method (Section 4.1). The dy-namical state of the OCs are quantified using kinematicdata in Section 5. In Section 5.2 we compare our obser-vational findings with N -body simulations. Finally, weprovide a brief summary in Section 6. DATA ANALYSIS AND MEMBERIDENTIFICATION2.1.
Gaia EDR 3 Data Processing and Analysis
The
Gaia
EDR 3 (Gaia Collaboration et al. 2020)has provided parallaxes ( (cid:36) ) and proper motions (PMs; µ α cos δ, µ δ ) with unprecedented precision and sensitiv-ity for more than 1.8 billion sources with magnitudebrighter than 21 mag in the G band (330 − G band photometry is in therange of 0.2–6 mmag for stars brighter than 20 mag. The median error of (cid:36) ranges from ∼ G <
15 mag) to ∼ G ≈
21 mag). The corresponding uncertaintiesin the PMs for these sources are 0.02–0.03 mas yr − and 1.4 mas yr − , respectively (Gaia Collaboration et al.2020). Beside PMs, about 7.2 million stars have radialvelocity (RV) measurements in the Gaia
DR 2, whichare transferred to EDR 3 (Torra et al. 2020; Seabroke etal. 2020). These RV measurements have a typical uncer-tainty of 2 km s − (Lindegren et al. 2018). Unreliable orerroneous RVs in the data release have been discarded(see Boubert et al. 2019).The following analysis is carried out for these 13 OCsin the the sample. The spatial and kinematic structuresof the ten target clusters are investigated using Gaia
EDR 3 data within 100 pc from the cluster center takenfrom the member catalogs of Liu & Pang (2019) andGaia Collaboration et al. (2018b) in Cartesian Galac-tocentric coordinates (see definition in Appendix A). Inorder to remove possible artifacts in the
Gaia
EDR 3from our sample, we apply a general astrometric qualitycut, as described in Lindegren et al. (2018, in their Ap-pendix C), which selects stars with parallaxes and pho-tometric measurements within 10 percent uncertainty.Hereafter, we refer to this set as “Sample I”. Generally,the number of stars in Sample I ranges from 122 154to 456 527 for the clusters in our study. The G -bandmagnitude of the sources in Sample I ranges between ∼ ∼ G (cid:38)
19 mag.We construct a 2D density map of PMs to select starsaround the over-density location of the 13 target clus-ters. Figure 1 (a) shows a 2D density map of PMs forthe cluster NGC 2516 as an example. This map onlyshows bins with over-densities > σ in Sample I. Severalover-densities stand out. There is an over-density nearthe average PMs of the cluster (indicated with a bluecross) provided by Liu & Pang (2019). An over-densityof nearby clusters can also be seen in Figure 1 (PM plotsfor other twelve OCs are provided in Appendix B; seeFigures 14, 15 and 16). In this work we focus on the tar-get clusters and we do not investigate their neighbors.We apply a circular cut (the black circle in Figure 1 (a))to only include the target cluster for further analysis.Note that the radius of the circle is chosen to include asmany potential members as possible, while simultane-ously excluding most unrelated nearby structures. Theradius is therefore different for each cluster in the sam-ple. Application of this circular cut reduces the numberof candidate members for each cluster. Hereafter, werefer to this set of stars as “Sample II”. The number of Pang et al. stars in Sample II drops to below 10 000 for most clus-ters. The stars in this sample have magnitudes rangingbetween G ∼ . G ∼ . G (cid:46) − . (cid:36) , µ α cos δ , and µ δ ) from Gaia
EDR 3. Since only a small fraction of the stars in eachcluster have RV measurements, we adopt the higher-accuracy radial velocities from Jackson et al. (2020) andBailey et al. (2018) as supplementary data. The RVs ofstars in the ten target clusters are obtained from Jack-son et al. (2020); these are part of the Gaia-ESO Sur-vey (GES, Gilmore et al. 2012) with an uncertainty of0.4 km s − , obtained using FLAMES (the Fiber LargeArray Multi Element Spectrograph) combined with theGIRAFFE and UVES (Ultraviolet and Visual EchelleSpectrograph) spectrographs mounted on the 8-m UT2-Kueyen telescope of the ESO Very Large Telescope fa-cility. Bailey et al. (2018) obtained RVs of stars inNGC 2422 with M2FS (the Michigan/Magellan FiberSystem), a multi-object fibre-fed spectrograph on theMagellan/Clay 6.5-m telescope, with a median uncer-tainty of 0.08 km s − . We use the RVs from Gaia DR 2for the clusters Coma Berenice and NGC 6774, neither ofwhich is included in the above-mentioned spectroscopicsurveys.The distance to each individual star is computed as1 /(cid:36) , from which we compute for each source the Galac-tocentric Cartesian coordinates ( X, Y, Z ). The transfor-mation is performed by using the Python
Astropy pack-age (Astropy Collaboration et al. 2013, 2018). There isan asymmetric error in the distance that arises from thedirect inversion of (cid:36) (Zhang et al. 2020). We adopt aBayesian method to correct individual distances of stars,as outlined in Section 4.2.2.
Membership determination
The unsupervised machine learning method,
StarGO (Yuan et al. 2018) has proven to be successful inmembership determination of OCs, e.g., for the ComaBerenices cluster (Tang et al. 2019), Blanco 1 (Zhang etal. 2020), NGC 2232 and LP 2439 (Pang et al. 2020).The algorithm is based on the Self-Organizing-Map(SOM) method that maps high-dimensional data ontoa two-dimension neural network, while preserving thetopological structures of the data.We apply StarGO to map a 5D data set (
X, Y, Z , µ α cos δ, µ δ ) of ten target clusters (Sample II) onto a2D neural network in order to determine member can- https://github.com/salamander14/StarGO didates. Stars are fed to the neural network sequen-tially. We therefore scale the number of neurons to thenumber of stars in Sample II. We adopt a network with100 × ×
150 neurons (depending on the numberof stars in Sample II of each cluster) represented bythe 100 ×
100 (150 × X, Y, Z , µ α cos δ, µ δ ) that are provided to thealgorithm. During each iteration, the weight vector ofeach neuron is updated so that it is closer to the inputvector of an observed star. The learning process is iter-ated 400 times (600 times for 150 ×
150 grids) until theweight vectors converge. When stars associated withneurons are spatially and kinematically coherent (e.g.,when they are cluster members), the 5D weight vectorsof the adjacent neurons are similar. Therefore, the valueof the difference of weight vectors between these adja-cent neurons, u , is small. Neurons with similar smallvalues of u group together in the 2D neural network aspatches (see Figure 1 (c)). Different groups of stars formdifferent patches. The value of u is smaller for neuronslocated inside the patch, and larger for neurons out-side the patch. The u values of neurons inside patchesgenerate an extended tail towards small values in the u -histogram (see panel (b) in Figure 1).The selection of u is made by applying a cut to the tailof the u -distribution. This cut is made to ensure a sim-ilar contamination rate of ∼
5% among members, whichhas been applied to NGC 2232 in Pang et al. (2020). Weadopt this 5% field star contamination u -cut as a mem-ber selection criteria for the ten target clusters, whichcorresponds to the blue patch in Figure 1 (c). We eval-uate the contamination rate from the smooth Galacticdisk population using the Gaia
DR 2 mock catalog (Ry-bizki et al. 2018). An identical PM cut as describedin Section 2.1 is also applied to the mock catalog inthe same volume of the sky. Each of these mock starsis attached to the trained 2D neural network. We thenconsider the mock stars associated with selected patchesas contamination. The numbers of identified membersof each target cluster are listed in Table 1. We pro-vide a detailed member list of all 13 target clusters inTable 2, with parameters obtained in this study. Themembers lists of these 13 target clusters therefore formhomogeneous data sets. GENERAL PROPERTIES OF TARGET OPENCLUSTERSTo evaluate the validity our membership identifica-tion, we cross-match the members in target clusters
D Morphology of Open Clusters in the Solar Neighborhood −20−10010 μ α cos δ [mas yr −1 ] μ δ [ m a s y r − ] (a)NGC 2516 −4 −3 −2 u (b) (c) N u m be r Figure 1. (a) 2D density map of the proper motion vectors for the regions around NGC 2516 in sample I. The blue crossindicates the mean over-density generated by NGC 2516 obtained from Liu & Pang (2019). Each bin is smoothed by neighboringeight bins. Only bins with a number count > σ are shown, where σ is the standard deviation of all bins. The color indicatesthe number count in each bin. (b) Histogram of the distribution of u . The orange line denotes the selections of u that producesa 5% contamination rate among the identified candidates, for the orange patch in the 2D neural network (panel (c)). (c) 2Dneural networkresulting from SOM, the neurons with a u -selection of 5% contamination rate (orange line in panel (b)) are shaded as orange.Among these, the neurons corresponding to the member candidates of the target cluster NGC 2516 are highlighted in blue. with two independently published catalogs that bothidentify star clusters using all-sky Gaia DR 2 data: Liu& Pang (2019) and Cantat-Gaudin et al. (2020). Liu& Pang (2019) used a friend-of-friend (FoF) clusterfinder to identify star clusters in Gaia
DR 2 in the five-dimensional parameter space ( l, b, (cid:36), µ α cos δ , and µ δ ).Members in the catalog of Cantat-Gaudin et al. (2020)are compiled from Cantat-Gaudin & Anders (2020);Castro-Ginard et al. (2018, 2019, 2020) and are iden-tified using the unsupervised membership assignmentcode UPMASK (Cantat-Gaudin et al. 2018).All of the target clusters presented in this work aregenerally in good agreement with both catalogs, andhave a comparable number of identified members (seethe last two columns in Table 1). Coma Berenices,Blanco 1, and NGC 6774 are absent in Liu & Pang(2019)’s catalog.We display the positions of all identified members ofthe 13 target clusters in the Galactic coordinates in Fig-ure 2. Coma Berenices (grey triangles) and Blanco 1(grey diamonds) occupy the regions of the Northern andSouthern Galactic poles, respectively. The other OCsare within ∼
15 degrees from the Galactic plane. Al-though NGC 2451A and NGC 2451B appear to overlapin the 2D projection, they are separated by a distance of ≈
200 pc along the line-of-sight (see Table 1). Extendedtidal tails are clearly visible in Coma Berenices andBlanco 1. An elongated shape is observed in the otherclusters, notably in NGC 2547, NGC 2516, NGC 2232and NGC 2451B. Note that a 2D elongated projectedmorphology that we see in projection, must have an even more prominent elongation in its 3D morphology.We will carry out an detailed investigation of the 3Dmorphology of the clusters in our sample in Section 4.We show the members of each cluster in the color-magnitude diagram (CMD; Figure 3). The memberstars of each cluster track a clear locus of a main se-quence, which is consistent with the PARSEC isochrone(black solid curves in Figure 3) for which the sensitivitycurves are provided by Ma´ız Apell´aniz & Weiler (2018).The distribution of the stars in the CMD shows thatthe field stars’ main sequence (which is bluer than thatof the clusters) is largely filtered out, therefore furtherconfirming the reliability of our identified members ofeach cluster. We adopt ages for the target clusters fromprevious studies (except for NGC 2451B) when they arein good agreement with the locations of the membersin the CMD. We fit the values of E ( B − V ) and of themetallicity, which are not available from literature. Welist the cluster ages and related parameters in Table 1.The ages of the clusters span a wide range, from 25 Myrfor the youngest cluster (NGC 2232) to 2.65 Gyr for theoldest cluster (NGC 6774). Such a wide age range in thesample of target clusters allows us to probe the influ-ence of the secular dynamical evolution of the star clus-ters and the interaction with their environments on theirmorphology. The majority of the clusters in the sampleare relatively young, with ages younger than 100 Myr.Four clusters are of intermediate-age, with ages between100 Myr and 800 Myr.An extended main sequence turn-off (eMSTO) of ∼ G BP − G RP is observed in Pang et al. -150° -120° -90° -60° -30° 0° 30° 60° 90° 120° 150°-75°-60°-45°-30°-15°0°15° 30° 45° 60° 75°
IC 2391IC 2602IC 4665NGC 2422NGC 2516NGC 2547Coma Berenices NGC 6633NGC 6774NGC 2451ANGC 2451BNGC 2232Blanco 1
Figure 2.
2D projection of identified member stars of each target cluster in Galactic coordinates ( l, b ). Each of the 13 clusterfor which the members are obtained via
Gaia
EDR 3 in this study are denoted with different colours and symbols. Amongthese, three clusters are colored indicated in grey. The members of these clusters were also identified using
Gaia
DR 2 in earlierstudies. Member stars of NGC 2232 are indicated with grey stars, those of Coma Berenices with grey triangles, and those ofBlanco 1 with grey diamonds. two intermediate-age clusters, NGC 2516 (123 Myr) andNGC 6633 (426 Myr). The eMSTO region has been ob-served in many other star clusters (Li et al. 2014, 2017;Milone et al. 2018; Li et al. 2019), which is a result ofstars with a wide distribution of rotation rates (Bastian& de Mink 2009; D’Antona et al. 2017). At the sametime, the binary sequence locus of equal-mass systemsis clearly seen for most clusters. In the oldest clusterNGC 6774, we observe blue straggler candidates.Sixteen white dwarf members are found in five ofthe target clusters (IC 2391, Blanco 1, NGC 2516, ComaBerenices, and NGC 6774). The majority of these havebeen cataloged in Gentile Fusillo et al. (2019). The whitedwarfs in NGC 2516 and NGC 6774 gather at very sim-ilar locations in the CMD. A detailed study has beencarried out for three of the white dwarfs in NGC 2516 byKoester & Reimers (1996, IDs: NGC 2516-1,2,5). Thesewhite dwarfs were estimated to have ages of 120–160 Myr(based on the cooling age, the main sequence lifetime,and the lifetime of red giants), which is consistent withthe age of the cluster determined in our study.
3D MORPHOLOGY OF OPEN CLUSTERS4.1.
Distances through Bayesian parallax inversion
It is known that the morphology of star clusters ap-pears to be stretched along the line-of-sight, when dis-tances are obtained by simple parallax inversion (see,e.g., Carrera et al. 2019). Such artificial elongation isa consequence of computing the distance to each starby directly inverting the Gaia EDR 3 or DR 2 parallax,1 /(cid:36) . Even when the errors in the parallax measure-ments ∆ (cid:36) have a symmetric distribution, taking thereciprocal introduces a skewed distribution of errors onthe distances, which results in a systematic bias in thedistance to each cluster. We perform Monte Carlo sim-ulations to estimate the contribution of the parallax er-ror ∆ (cid:36) on the uncertainty in the cluster distance (seealso Zhang et al. 2020). The mean value of ∆ (cid:36) of eachcluster is adopted to estimate the parallax induced un-certainty in the distance, which is typically 0.4–12.6 pcfor the clusters in our sample.To mitigate this issue, we follow the method intro-duced by Bailer-Jones (2015) and treat the inversionproblem within a Bayesian framework. Our approachclosely follows the distance correction procedure de-scribed in Carrera et al. (2019). In this approach a priordistribution is assumed for each star. The Bayesian the-orem is adopted to estimate the prior through the likeli-hood function computed from the observed parallax and
D Morphology of Open Clusters in the Solar Neighborhood Table 1.
General parameters of target clusters
Cluster Age
Dist cor erDist cor r h r t M cl M dyn Z E ( B − V ) memb. CG20 LP19(Myr) (pc) (M (cid:12) ) (dex) (mag) (number)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)IC 2391 50 a,C . . . . W . b,C
219 190 (86%) 135 (94%)IC 2602 45 a,C . . . . W . W
318 267 (86%) 135 (94%)IC 4665 36 c,C . . . . W . W
197 142 (85%) 74 (91%)NGC 2422 73 d,C . . . . d,C . W
466 312 (75%) 335 (68%)NGC 2516 123 e,C . . . . e,C . e,C e,C . . . . e,C . e,C
452 192 (89%) 214 (88%)NGC 6633 426 f,C . . . . W . W
300 133 (89%) 164 (88%)NGC 6774 2650 g,C . . . . g,C . g,C
154 136 (80%) · · ·
NGC 2451A 58 h,C . . . . W . h,C
311 266 (80%) 204 (93%)NGC 2451B 50 W . . . . W . h,C
359 207 (73%) 109 (85%)NGC 2232 25 i,C . . . . i,C . i,C
281 169 (90%) 93 (89%)Blanco 1 100 j,C . . . . j,C . j,C
703 369 (97%) · · ·
Coma Berenices 700 k,C . . . . k,C . k,C
158 129 (84%) · · ·
Note — Dist cor is the mean corrected distance of members in each cluster. erDist cor is the error in corrected distance following the Bayesian modeldescribed in Section 4.1. r h and r t are half-mass and tidal radii of each cluster. The metallicity, Z , and reddening, E ( B − V ), of several clusters aretaken from literature (indicated with a capital C ), some are fitted in this work indicated with a capital W . When the referenced age fits the members,we adopt the age from previous works (indicated with a capital C ). The quantity M cl is the mass of each star cluster. The last two columns showthe number of matched members in Cantat-Gaudin et al. (2018) (CG18) and Liu & Pang (2019) (LP19),and the corresponding percentages. Theage, Z and E ( B − V ) for some clusters are adopted from a:Marsden et al. (2009); b:Postnikova et al. (2020); c: Miret-Roig et al. (2019), d: Baileyet al. (2018), e:Gaia Collaboration et al. (2018a), f:Williams & Bolte (2007), g: Olivares et al. (2019), h: Balog et al. (2009), i:Pang et al. (2020), j:Zhang et al. (2020), k: Tang et al. (2019). its nominal error. The prior is composed of two compo-nents, one representing the star cluster density and theother representing the field. The former follows a nor-mal distribution, the latter an exponentially decreasingdensity as in Bailer-Jones (2015). The standard devia-tion of the cluster component coincides with the stan-dard deviation of cluster-centric distance of the memberstars (i.e., the distances of the stars from the center ofeach star cluster). We combine these two componentswith weights proportional to the membership probabil-ity. We apply a value of 95% for the cluster term, and5% for the field star term (see Section 2.2). The mean ofthe posterior is considered as the corrected distance foreach star. Further details about this parallax inversionapproach can be found in Carrera et al. (2019) and inPang et al. (2020).Furthermore, we carry out Monte-Carlo simulations toestimate the uncertainty in the corrected distances thatresult from our procedure. Three types of clusters aresimulated to test our Bayesian procedure: (i) sphericalstar clusters with a uniform spatial distribution of stars;(ii) elongated star clusters with an elongation perpendic-ular to the line-of-sight; and (iii) elongated star clusterswith an elongation along the line-of-sight. For the uni-form model, the uncertainty in our corrected distanceincreases monotonically with distance (solid black curvein Figure 4). At a distance of 500 pc, the error in themean corrected distance of all stars becomes as large as 3.0 pc. When the elongation of the cluster is perpendic-ular to the line-of-sight, errors are very similar to thoseof a uniform cluster, and reach a slightly larger error of3.4 pc at a distance of 500 pc (dotted black curve in Fig-ure 4). The situation is different for the model in whichthe elongation is along the line-of-sight; in this case theuncertainty is as large as 6.3 pc at a distance of 500 pc(dashed black curve in Figure 4). Details of the proce-dure of the simulations are described in Appendix C.These findings show that the quality of the Gaia par-allaxes play an important role in determining the recov-ered intrinsic cluster morphology from measurements.Artificial morphological elongation due to parallax er-rors becomes most severe when the intrinsic elongationhappens to align with the line-of-sight. Intrinsicallyelongated clusters will suffer from larger uncertainty inthe corrected distance, especially when their elongationis aligned with the line-of-sight.4.2.
Presentation of 3D morphology
In Figures 5, 6 and 7 we show the 3D spatial distribu-tions after correcting distances for member stars in the13 target clusters. The corrected 3D positions of themembers in all 13 target clusters are presented in Ta-ble 2, together with other parameters from
Gaia
EDR 3.We also present the 3D positions of the 13 target clus-ters in Table 3. The Bayesian method has provided areasonable correction of the stretched shapes along the
Pang et al. G BP − G RP −2024681012 M G IC 2391
Age = 50 MyrZ = 0.030 E ( B − V ) = 0.01 G BP − G RP −2024681012 IC 2602
Age = 45 MyrZ = 0.020 E ( B − V ) = 0.01 G BP − G RP IC 4665
Age = 36 MyrZ = 0.015 E ( B − V ) = 0.23 G BP − G RP −20246810 M G NGC 2422
Age = 73 MyrZ = 0.017 E ( B − V ) = 0.13 G BP − G RP −4−2024681012 NGC 2516
Age = 123 MyrZ = 0.020 E ( B − V ) = 0.05 G BP − G RP NGC 2547
Age = 40 MyrZ = 0.015 E ( B − V ) = 0.04 G BP − G RP −2024681012 M G NGC 6633
Age = 426 MyrZ = 0.022 E ( B − V ) = 0.18 G BP − G RP NGC 6774
Age = 2650 MyrZ = 0.020 E ( B − V ) = 0.11 G BP − G RP −2024681012 NGC 2451A
Age = 58 MyrZ = 0.015 E ( B − V ) = 0.01 G BP − G RP −4−2024681012 M G NGC 2451B
Age = 50 MyrZ = 0.020 E ( B − V ) = 0.05 G BP − G RP NGC 2232
Age = 25 MyrZ = 0.015 E ( B − V ) = 0.07 G BP − G RP Blanco 1
Age = 100 MyrZ = 0.015 E ( B − V ) = 0.00 G BP − G RP M G Coma Berenices
Age = 700 MyrZ = 0.015 E ( B − V ) = 0.00 Figure 3.
The color-magnitude diagrams obtained from the
Gaia
EDR 3 absolute magnitude M G (adopting the distance afterthe correction described in Section 4.1) for member stars (blue dots) of 13 target OCs identified by StarGO . The PARSECisochrones of the adopted/fitted age are indicated with the black solid curves, with E ( B − V ) and metallicities provided byliterature or estimated in work (Table 1). D Morphology of Open Clusters in the Solar Neighborhood C o rr e c t e d D i s t a n c e E rr o r [ p c ] UniformElongated (along line of sight)Elongated (perpendicular to line of sight)
Figure 4.
Dependence of the uncertainty in the corrected distance on cluster distances based on simulations described inAppendix C. The black solid curve represents a star cluster in which the members have a uniform spatial distribution. Thedotted and dashed curves are clusters with elongated shape perpendicular to and parallel to the line-of-sight, respectively. Thecolored solar symbols and grey symbols indicate errors in the corrected distances when adopting a star cluster with a uniformstellar density located at the distance of each of the clusters in our study. The color coding of each cluster is identical to thatin Figure 2. line-of-sight of each cluster (grey dots in Figures 5, 6and 7).To estimate the uncertainty in the corrected distanceto each cluster, we carry out additional simulations foreach individual target cluster, with a uniform model (fordetails, see Appendix C). We apply the mean paral-lax error to the members of each cluster and move thesimulated cluster to the same distance as each targetcluster. The corresponding uncertainty in the distancecorrection of each cluster (Table 1) is represented witha colored symbol in Figure 4, that follows the curveof the uniform model. The most distant star cluster,NGC 2422 (476 pc), has an uncertainty of 3.2 pc in thecorrected distance. The uncertainty in the distance ob-tained through the Bayesian distance correction is muchsmaller than the error that arises from directly inverting
Gaia parallax (see Section 4.1).To quantify the size of each star cluster, we computetheir tidal radii as r t = (cid:18) GM cl A − B ) (cid:19) , (1)(Pinfield et al. 1998). Here, G is the gravitational con-stant, M cl is the total mass of the star cluster (i.e.,the sum of the masses of the individual member stars),and the parameters A and B are the Oort constants ( A = 15 . ± . − kpc − and B = − . ± . − kpc − ; see Bovy 2017).In the analysis below, we assume that candidatemembers located within tidal radius are gravitationallybound to the star cluster, while members outside areunbound. The mass of each individual member star isobtained from the nearest point in the fitted isochronethat is searched for using the k -D tree method (Millmanet al. 2011). The tidal radius of each cluster is indicatedwith a black circle in each panel of Figures 5, 6 and 7.The global morphology of an OC can generally be de-scribed with a dense central core (or nucleus) and anouter halo (or corona). The halo is much more ex-tended and has a low stellar number density (Nilakshiet al. 2002). However, the number of members in thehalo can be substantial (Meingast et al. 2020). BothBlanco 1 and Coma Berenices show two grand tidal tailsspanning up to 50–60 pc from the cluster center, whichbelong to the halo region, accounting for more than 36%and 50% of their members, respectively. The directionof the tidal tails in Coma Berenices and Blanco 1 arefound to be parallel to the Galactic plane, in agreementwith previous studies (Bergond et al. 2001; Chen et al.2004). No apparent elongation is present in the youngclusters IC 2391, IC 2602. IC 2391 is more centrally com-0 Pang et al.
Table 2.
Columns for the table of corrected 3D positions of members in all target clusters.
Column Unit DescriptionCluster Name Name of the target cluster
Gaia
ID Object ID in
Gaia
EDR 3ra degree R.A. at J2016.0 from
Gaia
EDR 3er RA mas Positional uncertainty in R.A. at J2016.0 from
Gaia
EDR 3dec degree Decl. at J2016.0 from
Gaia
EDR 3er DEC mas Positional uncertainty in decl. at J2016.0 from
Gaia
EDR 3parallax mas Parallax from
Gaia
EDR 3er parallax mas Uncertainty in the parallaxpmra mas yr − Proper motion with robust fit in α cos δ from Gaia
EDR 3er pmra mas yr − Error of the proper motion with robust fit in α cos δ pmdec mas yr − Proper motion with robust fit in δ from Gaia
EDR 3er pmdec mas yr − Error of the proper motion with robust fit in δ Gmag mag Magnitude in G band from Gaia
EDR 3BR mag Magnitude in BR band from Gaia
EDR 3RP mag Magnitude in RP band from Gaia
EDR 3Gaia radial velocity km s − Radial velocity from
Gaia
DR 2er Gaia radial velocity km s − Error of radial velocity from
Gaia
EDR 3Jackson radial velocity km s − Radial velocity from Gaia/ESO survey (Jackson et al. 2020)er Jackson radial velocity km s − Error of radial velocity from Gaia/ESO survey (Jackson et al. 2020)Bailey radial velocity km s − Radial velocity from Bailey et al. (2018)er Bailey radial velocity km s − Error of radial velocity from Bailey et al. (2018)Mass M (cid:12)
Stellar mass obtained in this studyX obs pc Heliocentric Cartesian X coordinate computed via direct inverting
Gaia
EDR 3 parallax (cid:36)
Y obs pc Heliocentric Cartesian Y coordinate computed via direct inverting
Gaia
EDR 3 parallax (cid:36)
Z obs pc Heliocentric Cartesian Z coordinate computed via direct inverting
Gaia
EDR 3 parallax (cid:36)
X cor pc Heliocentric Cartesian X coordinate after distance correction in this studyY cor pc Heliocentric Cartesian Y coordinate after distance correction in this studyZ cor pc Heliocentric Cartesian Z coordinate after distance correction in this studyDist cor pc The corrected distance of individual member
Note —A machine readable full version of this table is available online.
Table 3.
3D positions and velocities of 13 target clusters
Cluster X m Y m Z m U V W (pc) (km s − )(1) (2) (3) (4) (5) (6) (7)IC 2391 1 . − . − . − . − . − . . − . − . − . − . − . .
76 167 .
91 101 . − . − . − . − . − .
47 26 . − . − . − . . − . − . − . − . − . − . − . − . − . − . − . .
25 228 .
84 56 . − . − . − . .
57 106 . − .
20 49 . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . .
91 11 . − . − . − . − . − . − .
07 85 . − . − . − . Note — X m , Y m , Z m is 3D position of 13 target clusters in the heliocentric Cartesiancoordinates, taken as the median value of all members. U , V , W are mean 3Dvelocities of each cluster in the heliocentric Cartesian coordinates. D Morphology of Open Clusters in the Solar Neighborhood −20 0 20−180−170−160−150−140−130 Y [ p c ] IC 2391
40 60 80−170−160−150−140−130−120 Y [ p c ] IC 2602
260 280 300140150160170180190 Y [ p c ] IC 4665 −320 −300 −280 X [pc] −390−380−370−360−350−340 Y [ p c ] NGC 2422 −20 0 20−40−30−20−10010 Z [ p c ]
40 60 80−40−30−20−10010 Z [ p c ]
260 280 3008090100110120130 Z [ p c ] −320 −300 −280 X [pc] Z [ p c ] −180 −160 −140−160 −140 −120140 160 180−380 −360 −340 Y [pc] Figure 5.
3D spatial position of member stars in four target clusters: IC 2391, IC 2602, IC 4665, NGC 2422, in heliocentricCartesian coordinates (
X, Y, Z ; see definition in Appendix A) after distance correction via a Bayesian approach (see Section 4.1).The blue dots represent member stars in each cluster. The tidal radius of each cluster is indicated with a black circle. Thedashed line indicates the direction of the line-of-sight. The grey dots in the background show the spatial distribution of memberswithout distance correction. Pang et al. pact showing a clear core, while IC 2602 is more popu-lous. Despite the age of 36 Myr, IC 4665 has a sparsedistribution without a clear central concentration, whichmay be a consequence of rapid gas expulsion (Pang et al.2020; Dinnbier & Kroupa 2020a, see more discussion inSection 5.2). An elongated shape along the line-of-sightis apparent for the region containing the stars that aregravitationally bound to the cluster (i.e., inside tidal ra-dius) for NGC 2422, NGC 2547, NGC 6633, and Blanco 1(the angle between elongation and the line-of-sight, φ ,is presented in Table 4). The errors in the corrected dis-tances to these clusters ( ≈ ≈ Gaia
DR 2. Approxi-mately 60–90% of our members cross-match with mem-bers determined by Meingast et al. (2020). The ma-jority of the matched members is located within tidalradius of the cluster. Our current member identificationmethod is unable to confirm the membership of the starsin the vast extended stellar corona that were identifiedby Meingast et al. (2020).As a result of the higher accuracy of the proper mo-tion measurements in
Gaia
EDR 3, the extended fila-mentary structures of NGC 2232 that were once identi-fied as two separate groups (purple and green) in Panget al. (2020) (using
Gaia
DR 2 data and with the sameselection technique) are now identified as members ofNGC 2232. This confirms the conclusion of Pang et al.(2020) that the coeval filamentary structures are closelyrelated to NGC 2232, which are formed at the same timein the parental molecular clouds (Jerabkova et al. 2019;Beccari et al. 2020; Tian 2020). Similar filament-likesubstructures are also found in another two young clus-ters, NGC 2547 and NGC 2451B. On the other hand,tidal-tail-like structures extending up to 10–20 pc aredetected in three older clusters: NGC 2516, NGC 6633and NGC 6774. The diffuse spatial distribution of theoldest cluster (NGC 6774) implies its advanced disso-lution state, after having experienced substantial secu-lar dynamical evolution. The 3D morphology of OCsagain confirms the presence of the 2D elongation thatwe observed in NGC 2547 and NGC 2516, NGC 2232 andNGC 2451B in Figure 2.4.3.
Parameterization of 3D morphologies
From the 3D distribution of member stars in each clus-ter (Figures 5, 6, and 7), the general shape of memberdistribution within the tidal radius can be approximated with an ellipsoid. We perform ellipsoid fitting to the3D morphology of each cluster in order to quantify theshape of the distribution of bound stars in the targetclusters (we do not include the members located outsidetidal radius, since their number is small). NGC 2516 isshown as an example to illustrate the ellipsoid fittingfor the bound member stars (see Figure 8). The fit-ted ellipsoid (green surface) is centered at the medianposition of bound members, which we consider as thecluster center. The three semi-axes of the ellipsoid a , b , c are the free parameters in this fit, where a is thesemi-major axis (red line), b the semi-intermediate axis(pink line), and c the semi-minor axis (orange line). Weuse the lengths of the semi-axes a , b , c , and axis ratios b/a and c/a to describe the morphology of the clusters,and the direction of the semi-major axis a of the fittedellipsoid as the direction of elongation of the star cluster.Smaller values of the axis ratios b/a and c/a indicate amore elongated structure. The fitted values of the mor-phological parameters ( a , b , c , b/a and c/a ) are listed inTable 4. We also compute for each cluster the angle θ between the direction of a and the Galactic plane (theprojection of a on the Galactic plane), and the angle φ between the direction of a and the line-of-sight. The val-ues of these two angles are listed in Table 4. The fittedellipsoids for stars inside the tidal radius of other twelvetarget OCs are presented in Appendix D (Figure 17).The clusters NGC 2516, NGC 2547, NGC 2451A,NGC 2451B and NGC 2232 have axes ratios of approxi-mately b/a = 0 . − .
95, while c/a = 0 . − .
7. The mor-phologies of these five clusters resemble oblate spheroids.The other clusters, IC 2602, IC 4665, and NGC 2422,have shapes that are well described by prolate spheroids,with a difference between b/a and c/a of less than ∼ https://github.com/marksemple/pyEllipsoid Fit D Morphology of Open Clusters in the Solar Neighborhood Table 4.
Morphological and kinematic parameters of target clusters
Cluster Name a b c b/a c/a θ φ σ RV σ pmra σ pmdec (pc) (axis ratio) (degrees) (km s − )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)IC 2391 5 . ± .
16 3 . ± .
06 2 . ± .
05 0 . ± .
02 0 . ± .
02 16 .
76 46 .
94 0 . +0 . − . . +0 . − . . ± . . ± .
11 4 . ± .
04 3 . ± .
05 0 . ± .
02 0 . ± .
02 5 .
44 41 .
35 0 . +0 . − . . +0 . − . . +0 . − . IC 4665 6 . ± .
03 4 . ± .
40 3 . ± .
48 0 . ± .
37 0 . ± .
31 38 .
78 22 .
50 0 . ± .
16 0 . +0 . − . . ± . . ± .
04 5 . ± .
49 4 . ± .
56 0 . ± .
57 0 . ± .
47 19 .
01 79 .
28 0 . +0 . − . . +0 . − . . ± . . ± .
75 9 . ± .
48 6 . ± .
53 0 . ± .
35 0 . ± .
27 11 .
93 31 .
29 0 . ± .
07 0 . ± .
04 0 . ± . . ± .
78 6 . ± .
39 2 . ± .
44 0 . ± .
31 0 . ± .
16 4 .
20 10 .
81 0 . +0 . − . . ± .
04 0 . ± . . ± .
21 5 . ± .
31 3 . ± .
38 0 . ± .
19 0 . ± .
12 1 .
22 23 .
90 0 . +0 . − . . +0 . − . . ± . . ± .
45 4 . ± .
22 2 . ± .
27 0 . ± .
16 0 . ± .
12 18 .
30 17 .
53 0 . +0 . − . . ± .
04 0 . +0 . − . NGC 2451A 5 . ± .
56 4 . ± .
11 3 . ± .
12 0 . ± .
10 0 . ± .
07 14 .
33 81 .
06 0 . +0 . − . . +0 . − . . +0 . − . NGC 2451B 5 . ± .
62 5 . ± .
37 4 . ± .
42 0 . ± .
42 0 . ± .
31 22 .
12 48 .
56 0 . +0 . − . . +0 . − . . +0 . − . NGC 2232 6 . ± .
01 5 . ± .
28 3 . ± .
36 0 . ± .
28 0 . ± .
19 4 .
55 26 .
05 0 . +0 . − . . +0 . − . . +0 . − . Blanco 1 8 . ± .
66 4 . ± .
25 4 . ± .
31 0 . ± .
12 0 . ± .
10 78 .
35 12 .
53 0 . ± .
08 0 . ± .
03 0 . ± . . ± .
02 4 . ± .
02 3 . ± .
01 0 . ± .
01 0 . ± .
01 14 . .
38 0 . ± .
18 0 . +0 . − . . +0 . − . Note — a , b , c are the semi-major, semi-intermediate and semi-minor axes of the fitted ellipsoid for each star cluster in the sample. θ is the angle between thedirection of a and the Galactic plane. The quantity φ is the angle between the direction of a and the line-of-sight. σ RV is the RV dispersion (within thetidal radius); σ pmra and σ pmdec are the dispersions of the R.A. and Decl. components of the PMs (within the tidal radius). The values in columns 9–11are obtained using the MCMC method; each best-fit value is the median of the posterior distribution, and the uncertainties are the corresponding 16- and84-percentiles of the posterior. the cluster orbit (Tang et al. 2019). That is the rea-son why the tidal tails of Coma Berenices and Blanco 1are parallel to the Galactic plane (see Figure 7). Thebound region (within the tidal radius) of most clustershas an elongation direction that is more or less alignedwith the Galactic plane (see Figure 17), with an angle(between a and the disk) of | θ | < ◦ (see Table 4).This result is thus in agreement with earlier findings(Oort 1979; Bergond et al. 2001; Chen et al. 2004), andalso indicates that despite their young age, most clus-ters have already been affected by the external Galactictides. The direction of the semi-major axis a does notalign with the direction of the line-of-sight (see the val-ues of φ in Table 4), confirming the reliability of ourdistance correction (Section 4.1).Although Blanco 1 appears to show evidence of hav-ing been affected by the tidal force, its bound regionhas an elongation that is closely aligned with the verti-cal ( Z ) direction of perpendicular to the Galactic plane(with an angle of ≈ . ◦ ). While stars escape mainlythrough the two Lagrange points, the evaporation pro-cess stretches all the way between the two Lagrangepoints (K¨upper et al. 2008) and generates the elongatedshape in the distribution of the bound stars. The un-bound stars are subjected to Galactic tides so that theirorbits become more tangential and form the tidal tailsaround Blanco 1, which probably constitutes both “tailI” and “tail II” (Dinnbier & Kroupa 2020a). The distribution of the bound population of starsin the oldest cluster NGC 6774 can be describe withan triaxial ellipsoid. The secular relaxation process inNGC 6774 results in significant mass loss, that resultsin the formation of tidal-tail-like structures beyond thetidal radius (see Figure 6 and Yeh et al. 2019), and theescape velocity is consequently greatly reduced. A phaseof global evaporation must have taken place.Internal stellar dynamics, such as two-body relax-ation, tend to produce an isotropic velocity distributionin the radial direction. Therefore, the core of the OCbecomes more spherical as it evolves. Chen et al. (2004)indeed noted that the projected flattening of OCs de-creases as cluster grow older. The axis ratios b/a and c/a provide appropriate tools to investigate this phe-nomenon, since they probe the bound region of OCswhere internal dynamical processes dominate. However,among the 13 target clusters, no correlation appears toexist between b/a and age. One may expect a decreasingtrend between elongation and age if star clusters inherittheir elongated shape from their parent GMC. As OCsevolve toward older age, their initial shape is “forgotten”as internal relaxation processes increase the sphericity ofthe clusters, especially the shape of the bound region.This process continues until the time when evaporationbecomes the dominant process in the evolution of thecluster.A larger sample of OCs is required to further quantifythe relation between morphology and cluster dynamics.4 Pang et al. −20 0 20 40 60−440−420−400−380−360 Y [ p c ] NGC 2516 −60 −40 −20−410−400−390−380−370−360 Y [ p c ] NGC 2547
300 320 340210220230240250260 Y [ p c ] NGC 6633
260 280 300 X [pc] Y [ p c ] NGC 6774 −20 0 20 40 60−160−140−120−100−80 Z [ p c ] −60 −40 −20−80−70−60−50−40−30 Z [ p c ]
300 320 340304050607080 Z [ p c ]
260 280 300 X [pc] −90−80−70−60−50−40 Z [ p c ] −440 −420 −400 −380 −360−400 −380 −360220 240 26080 100 120 Y [pc] Figure 6.
3D spatial position of members in four target clusters: NGC 2516 and NGC 2547, NGC 6633, NGC 6774, in heliocentricCartesian coordinates (
X, Y, Z ; see definition in Appendix A), after distance correction via a Bayesian approach (see Section 4.1).Colors and symbols are the same as in Figure 5. Filament-like substructures present in the young cluster NGC 2547, and tidal-tail-like substructures in the older clusters NGC 2516, NGC 6633, and NGC 6774.
D Morphology of Open Clusters in the Solar Neighborhood DYNAMICAL STATES OF OPEN CLUSTERS5.1.
3D velocity and velocity dispersion
The observed 3D morphology of OCs is thought to bedriven by cluster dynamics. However, few studies havebeen carried out to investigate the relationship betweencluster morphology and stellar dynamics. In this study,we connect the 3D morphology and the dynamical stateof open clusters for the first time. We use PMs andRVs from
Gaia
EDR 3, and RVs from the literature, asdiscussed in Section 2.Considering extended structures in target clusters, weadopt the median position of the members in each clus-ter as the cluster centers, and we use the average veloc-ity in each cluster as the origin of the reference frame,values of which are listed in Table 3. We present the3D velocity vectors of the member stars superposed onthe 3D spatial positions (relative to that of the clustercenter) in Figures 9 and 10. The tidal radii and the pro-jections of the a , b and c axes of the fitted ellipsoids ofeach cluster are overplotted. The majority of the mem-bers with 3D velocity measurements are located withinthe tidal radius. Members with 3D velocities that differmore than 2 σ from the mean value are excluded from thevelocity vector plots. These high-velocity stars all havelarge RVs. They are most likely binary candidates (see,e.g., Kouwenhoven & de Grijs 2008, for details), and arealso located on the binary sequence in the CMD. Onestar with extraordinary RV is a blue straggler candidatein NGC 6774. Its peculiar velocity may have originatedfrom a close encounter or from a merger event.Figures 9 and 10 show that the directions of the ve-locity vectors of a large number of members align withthe major axis of the fitted ellipsoid that coincides withthe direction of elongation. All clusters from young toold, are expanding, as the majority of members are seento move away from the cluster center. Expansion inyoung clusters is thought to be driven by gas expul-sion (Baumgardt & Kroupa 2007; Dinnbier & Kroupa2020a,b; Pang et al. 2020). After the gas expulsion,member stars expand radially and therefore reduce thedepth of the gravitational potential wells of the OCs.In order to quantitatively analyse the dynamical statesof our target OCs, we compute the RVs and PMs’ dis-persion of the bound members in each cluster. The like-lihood function for the RV distribution is a combinationof two Gaussian components, one for cluster members, and the other one for field stars (equations 1 and 8 inCottaar et al. 2012). The Gaussian distribution of clus-ter members is broadened by the orbital motions of un-resolved binary systems, and also by the uncertainties inthe RV measurement. To model the broadening intro-duced by binary stars, we adopt distributions of orbitalparameters that are characteristic for solar-type stars inthe Galactic field: (i) a log-normal orbital period dis-tribution for the binaries (Raghavan et al. 2010); (ii)a flat mass ratio distribution between q = 0 and q = 1(Duchˆene & Kraus 2013); and (iii) a flat eccentricity dis-tribution between e = 0 and the maximum value (Parker& Goodwin 2009). The adopted parameters of binarystars are more computationally efficient but still com-parable to the more realistic models (e.g., Marks et al.2011; Marks & Kroupa 2011). However, as pointed outby Bravi et al. (2018), the selected binary properties donot significantly affect the final fitted results. The like-lihood function of the PM distribution ( µ α cos δ and µ δ )is described by two Gaussian profiles (Equations 1–3 inPang et al. 2018): the component of the cluster mem-bers, and the field component (the latter accounts for5%). We use the Markov Chain Monte Carlo (MCMC)method to obtain the best-fit values and the correspond-ing uncertainties for the RV and PM velocity dispersions(columns 9–11 in Table 4).The derived velocity dispersions can be used to quan-tify the rate of expansion of each cluster. Clusters with ahigher velocity dispersion tend to expand faster. Basedon the half-mass radius, r h (Table 1), and the 3D ve-locity dispersion of each cluster (Table 4), we estimatetheir dynamical masses using Equation 1 in Fleck et al.(2006). The resulting dynamical masses (M dyn ) of eachcluster ranges from 263 M (cid:12) to 3368 M (cid:12) (Table 1 col-umn 8), higher than the estimated photometric masses,101 M (cid:12) to 1973 M (cid:12) (column 7 in Table 1). This discrep-ancy is not resolved, even when we correct the mass offaint stars below Gaia
EDR 3’s detection limit by extrap-olating the mass function (see demonstration in Tang etal. 2019). Therefore, this suggest the majority of clus-ters might be supervirial and may end up expanding asthe kinetic energy overtakes the gravitational potential.We display the dependence of ratio between the dy-namical mass and the photometric mass on the clusterage in Figure 11. The ratio M dyn /M cl increases as clus-ters grow older, especially after 300 Myr. The oldestcluster in the sample, NGC 6774, has the highest ratioof M dyn /M cl , further confirms its state of disruption.On the contrary, the youngest cluster in the sample,NGC 2232, has the lowest ratio (M dyn /M cl ∼ . Pang et al. −80 −70 −60 −50 −40−200−190−180−170−160 Y [ p c ] NGC 2451A −140 −120 −100 −80−370−360−350−340−330−320−310 Y [ p c ] NGC 2451B −300 −280 −260 −240 −220−220−200−180−160−140 Y [ p c ] NGC 2232 Y [ p c ] Blanco 1 −40 −20 0 20 X [pc] −40−20020 Y [ p c ] Coma Berenices −80 −70 −60 −50 −40−40−30−20−100 Z [ p c ] −140 −120 −100 −80−70−60−50−40−30−20−10 Z [ p c ] −300 −280 −260 −240 −220−80−60−40−200 Z [ p c ] Z [ p c ] −40 −20 0 20 X [pc] Z [ p c ] −200 −190 −180 −170 −160−360 −340 −320−220 −200 −180 −160 −140−50 0 50−40 −20 0 20 Y [pc] Figure 7.
3D spatial position of members in five target clusters: NGC 2451A, NGC 2451B, NGC 2232, Blanco 1 and ComaBerenices, in heliocentric Cartesian coordinates (
X, Y, Z ; see definition in Appendix A), after distance correction via a Bayesianapproach (see Section 4.1). Colors and symbols are the same as in Figure 5. Filament-like substructures present in the youngclusters NGC 2451B and NGC 2232, and tidal tails in the older clusters Blanco 1 and Coma Berenices.
D Morphology of Open Clusters in the Solar Neighborhood X ( p c ) Y ( p c ) Z ( p c ) NGC 2516 a axisb axisc axis
Figure 8.
Ellipsoid fitting for the 3D spatial positions in heliocentric Cartesian cooridinates, (
X, Y, Z ), for the cluster memberswithin tidal radius of NGC 2516, after distance correction through a Bayesian approach (see Section 4.1). The green surfacerepresents the fitted ellipsoid. Blue dots are members within tidal radius. The three axes of the ellipsoid ( a , b , and c ) areindicated in red, pink, and orange, respectively. revirialization. The cluster probably reaches its maxi-mal expansion before its re-collapse to form a virialisedcluster (e.g., Kroupa et al. 2001).As suggested from simulations (Baumgardt &Kroupa 2007), stellar members will acquire highly-anisotropic velocity dispersions after rapid gas expul-sion, and isotropic velocity dispersion after slow gasremoval. Some degree of velocity anisotropy is ob-served in target clusters with detected elongated struc-tures (NGC 2547, NGC 2451B, NGC 2516, NGC 6633,NGC 6774, Blanco 1, Coma Berenices) but not inNGC 2232 (Table 4, columns 9–11). Velocity anisotropymay also originate from global rotation in star clusters.Global rotation is not uncommon; it was recently dis-covered in the open cluster Tr 15 (Kuhn et al. 2019).Although OCs may inherit angular momentum fromparental GMCs, the merging substructures, or merg- ing clusters (e.g., Priyatikanto et al. 2016; Zhong et al.2019; Darma et al. 2019), few attempts have been madeto measure rotation in OCs. Unlike OCs, rotation hasbeen measured in globular clusters (e.g., Bianchini et al.2018; Kamann et al. 2018). According to N -body sim-ulations by Einsel & Spurzem (1999) and Hong et al.(2013), rotation enhances mass loss and therefore speedsup the disruption process of clusters. The global rota-tion speeds are typically much lower in OCs than globu-lar clusters, down to sub km s − level. Higher-resolutionof spectroscopy is required in order to quantify the ro-tational properties of our target OCs.5.2. Comparison with numerical models
In order to determine the properties of the gas expul-sion process of the 13 target clusters in this study, wecarry out N -body simulations of OCs with four differ-8 Pang et al. −10 0 10−160−150−140 Y [ p c ] IC 2391 (a.1) s −1 −30 −20 −10 (a.2) s −1 abc
40 50 60−150−140−130
IC 2602 (b.1) s −1 −20 −10 0 (b.2) s −1
280 290160170180 Y [ p c ] IC 4665 (c.1) s −1
90 100 110 (c.2) s −1 −310 −300 −290−380−370−360 NGC 2422 (d.1) s −1
10 20 30 40 (d.2) s −1
10 20 30 40−410−400−390−380 Y [ p c ] NGC2516 (e.1) s −1 −130 −120 −110 −100 −90 (e.2) s −1 −50 −40 −30−390−380−370 NGC 2547 (f.1) s −1 −70 −60 −50 (f.2) s −1
310 320 330 X [pc] Y [ p c ] NGC 6633 (g.1) s −1
50 60 70 Z [pc] (g.2) s −1
270 280 290 X [pc] NGC 6774 (h.1) s −1 −80 −70 −60 Z [pc] (h.2) s −1 Figure 9.
The relative 3D velocity vectors for members of eight target clusters, projected onto X - Y and Y - Z planes. Theblue vectors represent the velocities of member stars, relative to the mean motion of each cluster. The center of each clusteris indicated with the (+) symbol. Black circles denote the tidal radii. The scale of the velocity vectors is indicated in thebottom-left corner of each panel. D Morphology of Open Clusters in the Solar Neighborhood −70 −60 −50−190−180−170 Y [ p c ] NGC 2451A (a.1) s −1 −30 −20 −10 (a.2) s −1 abc −120 −110 −100−350−340−330 NGC 2451B (b.1) s −1 −50 −40 −30 (b.2) s −1 −270 −260 −250−190−180−170 Y [ p c ] NGC 2232 (c.1) s −1 −50 −40 −30 (c.2) s −1
30 40 50 X [pc] Blanco 1 (d.1) s −1 −240 −230 −220 Z [pc] (d.2) s −1 −10 0 X [pc] −100 Y [ p c ] Coma Berenices (e.1) s −1
80 90 Z [pc] (e.2) s −1 Figure 10.
The relative 3D velocity vectors for members of five target clusters, projected onto X - Y and Y - Z planes. Colorsand symbols are identical to those in Figure 9 Pang et al. Age [year]012345678 M d y n / M c l IC2391IC2602IC4665NGC2422NGC2516 NGC2547NGC6633NGC6774NGC2451A NGC2451BNGC2232Blanco1Coma Berenices
Figure 11.
The relation between the ratio of dynamicalmass over photometric mass, M dyn /M cl , and cluster age. Thelarge values of the ratio suggests that most of the clustersmay be super-virial. ent sets of initial conditions, and compare our numericalfindings with the observations.5.2.1. Initial conditions
The initial mass M cl (0) of the models is chosen suchthat the cluster mass M cl at an evolved stage at theage t is comparable to the mass of the observed clus-ters. Accordingly, we adopt M cl (0) = 250 M (cid:12) , 500 M (cid:12) ,1000 M (cid:12) , 2000 M (cid:12) , and 4000 M (cid:12) , which is consistentwith the simulations of cluster-formation in molecularclouds (Bate 2012). All the cluster models are initializedwith a Plummer model in virial equilibrium (Aarsethet al. 1974), that is characterised by the initial clus-ter mass M cl (0) and the half-mass radius r h . We notethat a much larger variety of initial conditions, includ-ing non-spherically symmetric substructures, are possi-ble (e.g. Moeckel & Bate 2010; Fujii & Portegies Zwart2015). However, such the substructure typically disap-pears quickly (e.g., through feedback from photoioniza-tion; Gonz´alez-Samaniego & Vazquez-Semadeni 2020),as the cluster relaxes and obtains a spherically symmet-ric configuration (Kroupa et al. 2001; Goodwin & Whit-worth 2004; Sills et al. 2018; Banerjee & Kroupa 2018).This process occurs prior to the onset of gas expulsion inour models. Since the uncertainty of the mechanism ex-pelling the gas has likely a much more prominent impacton the cluster dynamics than the initial substructure, wefocus on the gas expulsion mechanism in spherical sys-tems in the present work.Stellar masses are sampled from the Kroupa (2001)initial mass function (IMF), with a minimum mass of m min = 0 .
08 M (cid:12) , and a maximum mass that is ob-tained following the m max − M cl relation of Weidneret al. (2013), where m max is the maximum mass of a starformed in a cluster of mass M cl . We assume a binaryfraction of 100% among member stars (see, e.g., Good-win & Kroupa 2005) and initial distributions for the or-bital elements (periods or binding energies, mass-ratios,and eccentricities) derived from unifying the observedpopulations in very young populations that are in dif-ferent stages of dynamical processing with the Galacticfield population (Kroupa 1995a,b; Kroupa et al. 2001;Marks & Kroupa 2011; Belloni et al. 2017).The clusters move on circular orbits through theGalaxy at galactocentric radius d GC = 8 kpc, and withan orbital speed of 220 km s − . The simulated clustersare evolved until t = 100 Myr; dynamical evolution ismost prominent during this time span. This age rangecovers two thirds of the age of our target clusters. All thepresent models are initialized as embedded star clusters,i.e. containing both the stellar and gaseous components.Over time, the gas is removed from the cluster due tofeedback from massive stars. Technical details of the N -body simulations are described in Appendix E.In the first model S0, the initial half-mass radius r h ofthe cluster is related to the initial cluster mass M cl (0),following the relation of Marks & Kroupa (2012). Theseinitial conditions generate star clusters of rather com-pact sizes ( r h ≈ . M cl (0) = 4000 M (cid:12) . Model S0 has a star formationefficiency (SFE) of 1 / τ M = 0 .
03 Myr, which removes the gas on a time-scaleshorter than the stellar crossing time. In other words,the gas expulsion is impulsive. No primordial mass seg-regation is present in this model.The clusters in the second model S5 are identical tomodel S0, apart from its primordial mass segregation.Mass segregation is generated using the method of ˇSubret al. (2008), with an initial mass segregation index of S = 0 .
5. The clusters in the third scenario (model AD)have a longer gas expulsion time-scale of τ M = 1 Myr.Thus, the gas is removed on a time-scale longer thanstellar crossing time; i.e., the gas expulsion is adiabatic.Adiabatic gas expulsion typically impacts the clusterless than impulsive gas expulsion of the same SFE (e.g.,Baumgardt & Kroupa 2007; Dinnbier & Kroupa 2020b).Clusters in the fourth scenario (model WG) contain nogas, i.e. M gas (0) = 0 and SFE = 0, and have a largeinitial r h of 1 pc.5.2.2. Comparison with target clusters
Figure 12 shows the relationship between the cluster’shalf-mass radius r h , its total mass M cl and its age t D Morphology of Open Clusters in the Solar Neighborhood r h substantially. They reviri-alize, at which stage both r h and M cl decrease. Primor-dially mass-segregated clusters (model S5) expand some-what less. The evolution of cluster mass and half-massradius for both models S0 and S5 is in agreement withthe majority of the target clusters (triangles), whereonly two clusters (IC 2391 and IC 2602) have their radiitoo compact for their age and mass. In contrast, theproperties of models AD and WG are inconsistent withmany of the target clusters.The agreement between the models S0 and S5 with thetarget clusters more massive than log ( M cl ( t )) (cid:38) . ( M cl ( t )) (cid:38) ≈
250 M (cid:12) . Another piece of evidencefor rapid gas expulsion in these clusters is the highervalue of dynamical mass as compared to the photomet-ric mass.As mentioned above, the state of the two most com-pact clusters (IC 2391 and IC 2602) appears to be indisagreement with models S0 and S5. However, thesetwo clusters are consistent with models AD. It is im-possible to draw a firm conclusion from this based onlyon two star clusters, but the data may suggest that thegas expulsion time-scale transitions from adiabatic toimpulsive at cluster mass of ≈
250 M (cid:12) , while the SFEdoes not change substantially. A decrease of the gasexpulsion time-scale with cluster mass is expected theo-retically because the maximum stellar mass in a clusterincreases with the mass of the cluster (see Weidner et al.2013, and references therein), so that the total energy ofphotoionising feedback of the cluster increases with clus-ter mass. The reduction of the gas expulsion time-scalewith increasing cluster mass is also reported in hydrody-namic simulations of Dinnbier & Walch (2020) and theobservational analysis carried out by Pfalzner (2020).5.3.
Mass segregation
Mass segregation is commonly found in embeddedclusters and young star clusters, and can be a conse-quence of internal dynamical relaxation, violent relax-ation and/or primordial mass segregation (Hillenbrand& Hartmann 1998; Allison et al. 2009; Pang et al. 2013;Pavl´ık et al. 2019). The youngest cluster in our targets,NGC 2232 does not manifest any evidence of mass seg-regation, based on measurements of the mean mass indifferent annuli (Pang et al. 2020). Two mediate-age clusters in our sample, however, do show mass segrega-tion. Coma Berenices shows evidence of mass segrega-tion that was quantified by comparing the mass distribu-tions in different annuli (Tang et al. 2018), and Blanco 1shows evidence of mass segregation obtained using theΛ-method (Zhang et al. 2020).The Λ-method, developed by Allison et al. (2009), is atool to analyse the degree of mass segregation a star clus-ter without the necessity of determining of cluster cen-ter. The Λ-method compares the minimum path lengthamong the N massive most massive members ( l massive )of the cluster, to that of the minimum path length of N normal random members ( l normal ).This average minimum path length l is calculatedfrom the minimum spanning tree (MST) of the sam-ple of stars, which is obtained using the Python pack-age MiSTree (Naidoo 2019). When the N massive massivestars are segregated, the average path length for thisset of stars, l massive , is smaller than that for the set ofrandomly selected stars ( l normal ).Previous studies have applied the Λ-method to starclusters using the observed 2D positions of stars in theclusters. Examples include the studies of NGC 3603 inPang et al. (2013) and of Blanco 1 in Zhang et al. (2020).However, the 2D projection can overestimate the degreeof segregation by projecting background stars that arelocated behind the cluster center into the inner region.With the distance-corrected 3D spatial positions of thetarget cluster members, we are able to improve the de-termination of the degree of mass segregation, in 3Dspace. The significance of the mass segregation is mea-sured using the “mass segregation ratio” (Λ MSR , Allisonet al. 2009), which defined asΛ
MSR = (cid:104) l normal (cid:105) l massive ± σ normal l massive , (2)where σ normal is the standard deviation of the 100 dif-ferent sets of l normal , and (cid:104) l normal (cid:105) is the average lengthof a hundred random sets.Figure 13 presents the Λ MSR for the clusters, basedon the 3D and 2D positions of members in each clus-ter. As can be seen from the figure, robust evidenceof mass segregation is found in six clusters, NGC 2422(segregated down to 3.6 M (cid:12) ), NGC 6633 (2.2 M (cid:12) ),NGC 6774 (1.6 M (cid:12) ), NGC 2232 (2.1 M (cid:12) ), Blanco 1 (1.5M (cid:12) ) and Coma Berenices (1.1 M (cid:12) ), consistent with pre-vious works (Prisinzano et al. 2003; Kraus & Hillenbrand2007; Moraux et al. 2007; Tang et al. 2018; Yeh et al.2019). In Coma Berenices, the most massive stars arenot the most concentrated; they have likely been ex-pelled of the cluster center via close encounter with bi-nary stars (e.g., Oh et al. 2015; Oh & Kroupa 2018).2
Pang et al. . . . . . . . . log ( M cl ( t )[ M (cid:12) ]) r h [ p c ] model S0 M cl (0) = 4000 M (cid:12) M cl (0) = 2000 M (cid:12) M cl (0) = 1000 M (cid:12) M cl (0) = 500 M (cid:12) M cl (0) = 250 M (cid:12) observations M cl (0) = 4000 M (cid:12) M cl (0) = 2000 M (cid:12) M cl (0) = 1000 M (cid:12) M cl (0) = 500 M (cid:12) M cl (0) = 250 M (cid:12) observations . . . . . . . . log ( M cl ( t )[ M (cid:12) ]) r h [ p c ] model S5 . . . . . . . . log ( M cl ( t )[ M (cid:12) ]) r h [ p c ] model AD . . . . . . . . log ( M cl ( t )[ M (cid:12) ]) r h [ p c ] model WG t [Myr] Figure 12.
Evolution of the cluster half-mass radius r h versus the cluster mass M cl in N -body simulations. Each panelfeatures a different set of cluster initial conditions as indicated by the model name at the upper-right corner of each panel. Theevolution of clusters is shown with the curves, where the thickness of the curve represents the initial cluster mass, and its colourrepresents the age (shown in the colourbar). The observed 13 target clusters are represented with triangles, again with colorsindicating their ages. Note that models S0 and S5 are consistent with most of the observational data (particularly with clustersof M cl (cid:38)
250 M (cid:12) ), while models AD are consistent only with clusters of M cl (cid:46)
250 M (cid:12) . Models WG are largely inconsistentwith most observed clusters.
D Morphology of Open Clusters in the Solar Neighborhood (cid:104) l normal (cid:105) in Equation 2 by projecting stars that are lo-cated further away from cluster center onto the innerregion. This will result in a decrease in Λ MSR . There-fore, the 2D MST will most likely under-estimate thedegree of mass segregation in a star cluster (Figure 13). SUMMARYUtilizing high-precision
Gaia
EDR 3 astrometry andphotometry, we apply the cluster finding method
StarGO to identify member stars in 13 target clus-ters: IC 2391, IC 2602, IC 4665, NGC 2422, NGC 2516,NGC 2547, NGC 6633, NGC 6774, NGC 2451A, andNGC 2451B, NGC 2232, Blanco 1, and Coma Berenicesin the 5D phase space of stars (
X, Y, Z , µ α cos δ, µ δ ).The selected members are cross-matched with membersin catalogs of Cantat-Gaudin et al. (2020) and Liu &Pang (2019). The ages obtained from isochrone fittingfor each cluster agree with those of previous studies.Altogether we have 13 target clusters with members de-termined via the same method, covering an age rangefrom 25 Myr to 2.65 Gyr, and located in the solar neigh-bourhood up to a distance of 500 pc. We analyze the 3Dmorphology and cluster dynamics of these 13 clusters,and quantify their morphology and dynamical state. N -body simulations are carried out to determine which gasexpulsion scenario best describes the history of thesestar clusters. Our findings can be summarized as fol-lows.1. We recovered the individual distance of each can-didate member from the parallax by means of aBayesian method. The uncertainties in the cor-rected distances are estimated by simulations ofspherical clusters with a uniform spatial distribu-tion of members, and of clusters with elongatedshapes. The estimated distance for a uniform-density, spherical model has an uncertainty of3.0 pc in the distance when the cluster is located at500 pc. Elongated models suffer from larger uncer-tainty. Notably, when the elongation is along theline-of-sight, uncertainties in the distance reach6.3 pc at the distance of 500 pc.2. We have determined the 3D morphology of 13 tar-get OCs, with corrected position in the Cartesianheliocentric coordinates ( X , Y , and Z ). An ellip-soid model is chosen to fit the spatial distributionof the stars within the tidal radius in all clusters.The semi-major axis a , semi-intermediate axis b ,and semi-minor axis c of the ellipsoid are obtained from fitting. We use the axes lengths a , b , c , andaxis ratios b/a and c/a as morphological param-eters to quantify the 3D distribution of the stel-lar population within the tidal radii of the OCs.We consider the direction of the major axis as thedirection of the morphological elongation of eachcluster. Most clusters have semi-major axes a par-allel to the Galactic plane or slightly inclined withrespect to the Galactic plane. A notable exceptionis Blanco 1, for which a is closer to the vertical( Z ) direction. The shapes of the distribution ofthe stellar population within the tidal radius forfive clusters (NGC 2547, NGC 2516, NGC 2451A,NGC 2451B, and NGC 2232) resemble that of anoblate spheroid, while those of other five clusters(IC 2602, IC 4665, NGC 2422, Blanco 1 and ComaBerenices) resemble prolate spheroids. The shapeof the stellar population within the tidal radiiof the other three clusters (IC 2391, NGC 6633,NGC 6774) are well-described by triaxial ellip-soids.3. A significant elongation is observed for the boundregions of NGC 2422, NGC 2457, NGC 6633 andBlanco 1. Considering that the uncertainty inthe corrected distance is much smaller than thesize of elongated structures, the elongations mea-sured for these clusters are robust. Among these,Blanco 1 is notable in the sense that its elongatedshape is significantly inclined (by 78 ◦ ) with re-spect to the Galactic plane. The elongation of thebound region might be driven by evaporation ofstars via the two Lagrange points. The 3D mor-phology of Blanco 1 might be a result of expan-sion due to fast gas expulsion and virialisation.Elongated filament-like substructures are foundin three young clusters, NGC 2232, NGC 2547and NGC 2451B, while tidal-tail-like substructuresare found in older clusters NGC 2516, NGC 6633,NGC 6774. Giant tidal tails are again confimed inBlanco 1 and Coma Berenices with Gaia
EDR 3.4. We combine
Gaia
EDR 3 PMs and RVs, togetherwith RVs from Jackson et al. (2020) and Baileyet al. (2018) to measure the 3D velocity of stel-lar members in the 13 target clusters. All clustersshow evidence of expansion in their 3D velocitydistributions. There is an anisotropy in the ve-locity dispersion for stars inside the tidal radius,which may be driven by gas expulsion.5. Four models of N -body simulations are carried outto determine the properties of the the gas expul-sion process that have occurred in the target clus-4 Pang et al. M S R IC2391 M IC2602
IC4665
NGC2422
NGC2516 M S R NGC2547 M NGC6633
NGC6774
NGC2451A
NGC2451B N M S R NGC2232 M N Blanco1 N Coma Berenices
Figure 13.
The “mass segregation ratio” (Λ
MST ) for the 60 most massive members, with a bin size of 12 stars in eachtarget cluster. The dashed line (Λ
MST = 1) indicates an absence of mass segregation. Increasing value of Λ
MST indicate a moresignificant degree of mass segregation. The error bars indicate the uncertainties obtained from a hundred realizations of l normal .The bin size is selected to avoid large stochastic errors in (cid:104) l normal (cid:105) for small N MST . ters: (i) a non mass-segregated model with impul-sive gas expulsion; (ii) a mass-segregated modelwith impulsive gas expulsion; (iii) a model withadiabatic gas expulsion; and (iv) a model with-out gas. All the target clusters with a mass largerthan 250 M (cid:12) are consistent with models of rapid(impulsive) gas expulsion with a rather low SFEof ≈ /
3, for models both with and without pri-mordial mass segregation. The target clusterswith mass smaller than 250 M (cid:12) are consistent withmodels of slow (adiabatic) gas expulsion with anSFE of ≈ /
3. Although the results for clusterswith masses above 250 M (cid:12) appear to be robust,the results for lower mass clusters are only ten-tative as they are based only on a sample of twoclusters. If the decrease of gas expulsion time-scalewith increasing cluster mass is confirmed for moreclusters in the future, this may point towards aprominent role of feedback from massive stars onthe early evolution of star clusters. Models with-out gas expulsion, i.e., models assuming a star for- mation efficiency of 100 percent, are not compati-ble with the data.6. In order to quantify the degree of mass segregationin each cluster, we apply both 3D and 2D MSTmethods to the OCs in the sample. Six of the OCsin our sample are found to have mass segregation:NGC 2422, NGC 6633, and NGC 6774, NGC 2232,Blanco 1, and Coma Berenices.Our study of these 13 open clusters is a pioneering at-tempt in quantitative study of cluster morphology andits relation to the formation and early evolution of starclusters. The methods developed in this work can beapplied to study a much larger sample of OCs coveringdifferent locations with data from the
Gaia
EDR 3 andDR 3, with the aim of achieving a better explanation ofthe dependence of 3D morphology of open clusters onthe location of the star clusters in the Galaxy.
D Morphology of Open Clusters in the Solar Neighborhood .This work made use of data from the European SpaceAgency (ESA) mission Gaia
Gaia
Software:
Astropy (Astropy Collaboration et al.2013, 2018),
SciPy (Millman et al. 2011),
TOPCAT (Taylor2005), and
StarGO (Yuan et al. 2018)6
Pang et al.
REFERENCES
Aarseth, S. J. 2003, Gravitational N-Body Simulations(Cambridge: Cambridge University Press)Aarseth, S. J., Henon, M., & Wielen, R. 1974, A&A, 37, 183Ahmad, A. & Cohen, L. 1973, Journal of ComputationalPhysics, 12, 389Allen, C. & Santillan, A. 1991, RMxAA, 22, 255Allison, R. J., Goodwin, S. P., Parker, R. J., et al. 2009,ApJL, 700, L99. doi:10.1088/0004-637X/700/2/L99Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., etal. 2013, A&A, 558, A33Astropy Collaboration, Price-Whelan, A. M., Sip˝ocz, B. M.,et al. 2018, AJ, 156, 123Bailer-Jones, C. A. L. 2015, PASP, 127, 994Bailey, J. I., Mateo, M., White, R. J., et al. 2018, MNRAS,475, 1609. doi:10.1093/mnras/stx3266Ballone, A., Mapelli, M., Di Carlo, U. N., et al. 2020,MNRAS, 496, 49. doi:10.1093/mnras/staa1383Balog, Z., Kiss, L. L., Vink´o, J., et al. 2009, ApJ, 698, 1989.doi:10.1088/0004-637X/698/2/1989Bate, M. R. 2012, MNRAS, 419, 3115.doi:10.1111/j.1365-2966.2011.19955.xBanerjee S., Kroupa P., 2017, A&A, 597, A28Banerjee, S. & Kroupa, P. 2018, Formation of Very YoungMassive Clusters and Implications for Globular Clusters,ed. S. Stahler, Vol. 424, 143Bastian, N. & de Mink, S. E. 2009, MNRAS, 398, L11.doi:10.1111/j.1745-3933.2009.00696.xBaumgardt, H., & Kroupa, P. 2007, MNRAS, 380, 1589Beccari, G., Boffin, H. M. J., & Jerabkova, T. 2020,MNRAS, 491, 2205Belloni, D., Askar, A., Giersz, M., et al. 2017, MNRAS,471, 2812. doi:10.1093/mnras/stx1763Benacchio, L. & Galletta, G. 1980, MNRAS, 193, 885.doi:10.1093/mnras/193.4.885Bergond, G., Leon, S., & Guibert, J. 2001, A&A, 377, 462.doi:10.1051/0004-6361:20011043Bianchini, P., van der Marel, R. P., del Pino, A., et al.2018, MNRAS, 481, 2125. doi:10.1093/mnras/sty2365Boubert, D., Strader, J., Aguado, D., et al. 2019, MNRAS,486, 2618. doi:10.1093/mnras/stz253Bovy, J. 2017, MNRAS, 468, L63Brandner, W. 2008, arXiv:0803.1974Bravi, L., Zari, E., Sacco, G. G., et al. 2018, A&A, 615, A37Cantat-Gaudin, T., Jordi, C., Vallenari, A., et al. 2018,A&A, 618, A93.Cantat-Gaudin, T., Anders, F., Castro-Ginard, A., et al.2020, A&A, 640, A1. doi:10.1051/0004-6361/202038192Cantat-Gaudin, T. & Anders, F. 2020, A&A, 633, A99.doi:10.1051/0004-6361/201936691 Castro-Ginard, A., Jordi, C., Luri, X., et al. 2018, A&A,618, A59. doi:10.1051/0004-6361/201833390Castro-Ginard, A., Jordi, C., Luri, X., et al. 2019, A&A,627, A35. doi:10.1051/0004-6361/201935531Castro-Ginard, A., Jordi, C., Luri, X., et al. 2020, A&A,635, A45. doi:10.1051/0004-6361/201937386Carrera, R., Pasquato, M., Vallenari, A., et al. 2019, A&A,627, A119Chen, W. P., Chen, C. W., & Shu, C. G. 2004, AJ, 128,2306. doi:10.1086/424855Chen, B., Stoughton, C., Smith, J. A., et al. 2001, ApJ,553, 184.Cottaar, M., Meyer, M. R., & Parker, R. J. 2012, A&A,547, A35. doi:10.1051/0004-6361/201219673Curry, C. L. 2002, ApJ, 576, 849. doi:10.1086/341811D’Antona, F., Milone, A. P., Tailo, M., et al. 2017, NatureAstronomy, 1, 0186. doi:10.1038/s41550-017-0186Darma, R., Arifyanto, M. I., & Kouwenhoven, M. B. N.2019, Journal of Physics Conference Series, 1231, 012028.doi:10.1088/1742-6596/1231/1/012028Dinnbier, F. & Kroupa, P. 2020, A&A, 640, A85.doi:10.1051/0004-6361/201936572Dinnbier, F. & Kroupa, P. 2020, A&A, 640, A84.doi:10.1051/0004-6361/201936570Dinnbier, F. & Walch, S. 2020, MNRAS, 499, 748.doi:10.1093/mnras/staa2560Duchˆene, G. & Kraus, A. 2013, ARA&A, 51, 269.doi:10.1146/annurev-astro-081710-102602leck, J.-J., Boily, C. M., Lan¸con, A., et al. 2006, MNRAS,369, 1392. doi:10.1111/j.1365-2966.2006.10390.xFujii, M. S. & Portegies Zwart, S. 2015, MNRAS, 449, 726Goodwin, S. P. & Whitworth, A. P. 2004, A&A, 413, 929Einsel, C. & Spurzem, R. 1999, MNRAS, 302, 81.doi:10.1046/j.1365-8711.1999.02083.xF¨urnkranz, V., Meingast, S., & Alves, J. 2019, A&A, 624,L11. doi:10.1051/0004-6361/201935293Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al.2020, arXiv:2012.01533Gaia Collaboration, Babusiaux, C., van Leeuwen, F., et al.2018, A&A, 616, A10Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al.2018, A&A, 616, A1Gentile Fusillo, N. P., Tremblay, P.-E., G¨ansicke, B. T., etal. 2019, MNRAS, 482, 4570. doi:10.1093/mnras/sty3016Getman, K. V., Kuhn, M. A., Feigelson, E. D., et al. 2018,MNRAS, 477, 298. doi:10.1093/mnras/sty473Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ,692, 1075.
D Morphology of Open Clusters in the Solar Neighborhood Gilmore, G., Randich, S., Asplund, M., et al. 2012, TheMessenger, 147, 25Goodwin, S. P. & Kroupa, P. 2005, A&A, 439, 565.doi:10.1051/0004-6361:20052654Gonz´alez-Samaniego, A. & Vazquez-Semadeni, E. 2020,MNRAS, 499, 668. doi:10.1093/mnras/staa2921Hillenbrand, L. A. & Hartmann, L. W. 1998, ApJ, 492, 540.doi:10.1086/305076Hong, J., Kim, E., Lee, H. M., et al. 2013, MNRAS, 430,2960. doi:10.1093/mnras/stt099Hurley, J. R., Pols, O. R., & Tout, C. A. 2000, MNRAS,315, 543Hurley, J. R., Tout, C. A., & Pols, O. R. 2002, MNRAS,329, 897Jackson, R. J., Jeffries, R. D., Wright, N. J., et al. 2020,MNRAS, doi:10.1093/mnras/staa1749Jeans, J. H. 1916, MNRAS, 76, 567.doi:10.1093/mnras/76.7.567Jerabkova, T., Boffin, H. M. J., Beccari, G., et al. 2019,MNRAS, 489, 4418Jones, C. E. & Basu, S. 2002, ApJ, 569, 280.doi:10.1086/339230Kamann, S., Husser, T.-O., Dreizler, S., et al. 2018,MNRAS, 473, 5591. doi:10.1093/mnras/stx2719Karnath, N., Prchlik, J. J., Gutermuth, R. A., et al. 2019,ApJ, 871, 46. doi:10.3847/1538-4357/aaf4c1Koester, D. & Reimers, D. 1996, A&A, 313, 810Kounkel, M., & Covey, K. 2019, AJ, 158, 122Kouwenhoven, M. B. N. & de Grijs, R. 2008, A&A, 480,103. doi:10.1051/0004-6361:20078897Kraus, A. L. & Hillenbrand, L. A. 2007, AJ, 134, 2340.doi:10.1086/522831Kroupa, P. 1995, MNRAS, 277, 1491.doi:10.1093/mnras/277.4.1491Kroupa, P. 1995, MNRAS, 277, 1507.doi:10.1093/mnras/277.4.1507Kroupa, P., Aarseth, S., & Hurley, J. 2001, MNRAS, 321,699. doi:10.1046/j.1365-8711.2001.04050.xKroupa, P. 2001, MNRAS, 322, 231Kruijssen, J. M. D., Maschberger, T., Moeckel, N., et al.2012, MNRAS, 419, 841.doi:10.1111/j.1365-2966.2011.19748.xKrumholz, M. R. & Matzner, C. D. 2009, ApJ, 703, 1352.doi:10.1088/0004-637X/703/2/1352Kuhn, M. A., Hillenbrand, L. A., Sills, A., et al. 2019, ApJ,870, 32. doi:10.3847/1538-4357/aaef8cKustaanheimo, P. & Stiefel, E. 1965, Reine Angew. Math.,218, 204K¨upper, A. H. W., Maschberger, T., Kroupa, P., &Baumgardt, H. 2011, MNRAS, 417, 2300 K¨upper, A. H. W., MacLeod, A., & Heggie, D. C. 2008,MNRAS, 387, 1248. doi:10.1111/j.1365-2966.2008.13323.xLada, C. J., & Lada, E. A. 2003, ARA&A, 41, 57Lamers, H. J. G. L. M., Gieles, M., Bastian, N., et al. 2005,A&A, 441, 117. doi:10.1051/0004-6361:20042241Li, C., de Grijs, R., & Deng, L. 2014, Nature, 516, 367.doi:10.1038/nature13969Li, C., de Grijs, R., Deng, L., et al. 2017, ApJ, 844, 119.doi:10.3847/1538-4357/aa7b36Li, C., Sun, W., de Grijs, R., et al. 2019, ApJ, 876, 65.doi:10.3847/1538-4357/ab15d2Lindegren, L., Hern´andez, J., Bombrun, A., et al. 2018,A&A, 616, A2Liu, L., & Pang, X. 2019, ApJS, 245, 32Makino, J. 1991, ApJ, 369, 200Makino, J. & Aarseth, S. J. 1992, PASJ, 44, 141Marsden, S. C., Carter, B. D., & Donati, J.-F. 2009,MNRAS, 399, 888. doi:10.1111/j.1365-2966.2009.15319.xMa´ız Apell´aniz, J. & Weiler, M. 2018, A&A, 619, A180.doi:10.1051/0004-6361/201834051Marks, M. & Kroupa, P. 2011, MNRAS, 417, 1702.doi:10.1111/j.1365-2966.2011.19519.xMarks, M., Kroupa, P., & Oh, S. 2011, MNRAS, 417, 1684.doi:10.1111/j.1365-2966.2011.19257.xMarks, M. & Kroupa, P. 2012, A&A, 543, A8Meingast, S. & Alves, J. 2019, A&A, 621, L3.doi:10.1051/0004-6361/201834622Mikkola, S. & Aarseth, S. J. 1990, Celestial Mechanics andDynamical Astronomy, 47, 375Milone, A. P., Marino, A. F., Di Criscienzo, M., et al. 2018,MNRAS, 477, 2640. doi:10.1093/mnras/sty661Miret-Roig, N., Bouy, H., Olivares, J., et al. 2019, A&A,631, A57. doi:10.1051/0004-6361/201935518Millman, K. J., Aivazis, M.. 2011, Computing in Science &Engineering, 13, 2, 9Moe, M. & Di Stefano, R. 2017, ApJS, 230, 15Moeckel, N. & Bate, M. R. 2010, MNRAS, 404, 721Moraux, E., Bouvier, J., Stauffer, J. R., et al. 2007, A&A,471, 499. doi:10.1051/0004-6361:20066308Naidoo, K. 2019, The Journal of Open Source Software, 4,1721. doi:10.21105/joss.01721Nilakshi, Sagar, R., Pandey, A. K., et al. 2002, A&A, 383,153. doi:10.1051/0004-6361:20011719Oh, S., Kroupa, P., & Pflamm-Altenburg, J. 2015, ApJ,805, 92. doi:10.1088/0004-637X/805/2/92Oh, S. & Kroupa, P. 2018, MNRAS, 481, 153.doi:10.1093/mnras/sty2245Olivares, J., Bouy, H., Sarro, L. M., et al. 2019, A&A, 625,A115. doi:10.1051/0004-6361/201834924Oort, J. H. 1979, A&A, 78, 312 Pang et al.
McKee, C. F. & Ostriker, J. P. 1977, ApJ, 218, 148.doi:10.1086/155667Meingast, S., Alves, J., & Rottensteiner, A. 2020,arXiv:2010.06591Padilla, N. D. & Strauss, M. A. 2008, MNRAS, 388, 1321.doi:10.1111/j.1365-2966.2008.13480.xPang, X., Grebel, E. K., Allison, R. J., et al. 2013, ApJ,764, 73Pang, X., Shen, S., & Shao, Z. 2018, ApJL, 868, L9.doi:10.3847/2041-8213/aaedaaPang, X., Li, Y., Tang, S.-Y., et al. 2020, ApJL, 900, L4.doi:10.3847/2041-8213/abad28Parker, R. J. & Goodwin, S. P. 2009, MNRAS, 397, 1041.doi:10.1111/j.1365-2966.2009.15037.xPavl´ık, V., Kroupa, P., & ˇSubr, L. 2019, A&A, 626, A79.doi:10.1051/0004-6361/201834265Pfalzner, S. 2020, in Star Clusters: From the Milky Way tothe Early Universe, ed. A. Bragaglia, M. Davies, A. Sills,& E. Vesperini, Vol. 351, 208–211Pinfield, D. J., Jameson, R. F., & Hodgkin, S. T. 1998,MNRAS, 299, 955R¨oser, S., Schilbach, E., & Goldman, B. 2019, A&A, 621,L2Postnikova, E. S., Elsanhoury, W. H., Sariya, D. P., et al.2020, Research in Astronomy and Astrophysics, 20, 016.doi:10.1088/1674-4527/20/2/16Prisinzano, L., Micela, G., Sciortino, S., et al. 2003, A&A,404, 927. doi:10.1051/0004-6361:20030524Priyatikanto, R., Kouwenhoven, M. B. N., Arifyanto, M. I.,et al. 2016, MNRAS, 457, 1339.doi:10.1093/mnras/stw060Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010,ApJS, 190, 1. doi:10.1088/0067-0049/190/1/1Rybizki, J., Demleitner, M., Fouesneau, M., et al. 2018,PASP, 130, 74101.S´anchez, N. & Alfaro, E. J. 2009, ApJ, 696, 2086.doi:10.1088/0004-637X/696/2/2086 Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science,337, 444 sSeabroke, G., Cropper, M., Baker, S., et al. 2020,arXiv:2010.16337Sills, A., Rieder, S., Scora, J., McCloskey, J., & Jaffa, S.2018, MNRAS, 477, 1903Spitzer, L. 1958, ApJ, 127, 17. doi:10.1086/146435Spurzem, R. 1999, Journal of Computational and AppliedMathematics, 109, 407ˇSubr, L., Kroupa, P., & Baumgardt, H. 2008, MNRAS, 385,1673Tang, S.-Y., Chen, W. P., Chiang, P. S., et al. 2018, ApJ,862, 106. doi:10.3847/1538-4357/aacb7aTang, S.-Y., Pang, X., Yuan, Z., et al. 2019, ApJ, 877, 12Taylor, M. B. 2005, Astronomical Data Analysis Softwareand Systems XIV, 29Tian, H.-J. 2020, ApJ, 904, 196.doi:10.3847/1538-4357/abbf4bTorra, F., Casta˜neda, J., Fabricius, C., et al. 2020,arXiv:2012.06420Tout, C. A., Pols, O. R., Eggleton, P. P., & Han, Z. 1996,MNRAS, 281, 257Wang, L., Spurzem, R., Aarseth, S., et al. 2015, MNRAS,450, 4070. doi:10.1093/mnras/stv817Wang, L., Spurzem, R., Aarseth, S., et al. 2016, MNRAS,458, 1450. doi:10.1093/mnras/stw274Weaver, R., McCray, R., Castor, J., et al. 1977, ApJ, 218,377. doi:10.1086/155692Weidner, C., Kroupa, P., & Pflamm-Altenburg, J. 2013,MNRAS, 434, 84Williams, K. A. & Bolte, M. 2007, AJ, 133, 1490.doi:10.1086/511675Yeh, F. C., Carraro, G., Montalto, M., et al. 2019, AJ, 157,115. doi:10.3847/1538-3881/aaff6cYuan, Z., Chang, J., Banerjee, P., et al. 2018, ApJ, 863, 26Zhang, Y., Tang, S.-Y., Chen, W. P., et al. 2020, ApJ, 889,99Zhong, J., Chen, L., Kouwenhoven, M. B. N., et al. 2019,A&A, 624, A34. doi:10.1051/0004-6361/201834334
D Morphology of Open Clusters in the Solar Neighborhood A. CARTESIAN GALACTOCENTRIC/HELIOCENTRIC COORDINATES USED IN THIS STUDYThe Galactic center ( l = 0 ◦ and b = 0 ◦ ) is located at the origin of the Cartesian Galactocentric coordinate system.The Sun is located 27 pc above the Galactic midplane, and 8.3 kpc from the Galactic center (Chen et al. 2001; Gillessenet al. 2009). The positive X -axis points from the projection of the Sun’s position onto the Galactic mid-plane towardsthe Galactic center. The positive Y -axis points towards l = 90 ◦ , and the positive Z –axis points towards b = 90 ◦ .The origin of the Cartesian heliocentric coordinate system is the solar system barycenter, while the direction of axesremains unchanged.0 Pang et al. B. FIGURES FOR SECTION 2 −40−30−20−1010203040 μ δ [ m a s y r − ] (1.a)IC 2391 −30−20−1000102030 μ δ [ m a s y r − ] (2.a)IC 2602 −10010−20−15−10−505 μ δ [ m a s y r − ] (3.a)IC 4665 −10−5 μ α cos δ [mas yr −1 ] −4−20246 μ δ [ m a s y r − ] (4.a)NGC 2422 −4 −3 −20200400600800 (1.b) −4 −3 −20200400600800 (2.b) −4 −3 −202004006008001000 (3.b) −4 −3 −2 u (4.b) (1.c) (2.c) (3.c) (4.c) N u m be r N u m be r N u m be r N u m be r Figure 14. (a) 2D density map of the proper motion vectors for the regions around four target clusters in sample I. The bluecrosses indicate the mean over-densities generated by the target clusters taken from Liu & Pang (2019). Each bin is smoothed byneighboring 8 bins, and here only bins with a number count > σ are shown, where σ is the standard deviation of all bins. Thegreyscale indicates the number count in each bin. (b) Histogram of the distribution of u . The orange line denotes the selectionsof u that produces a 5% contamination rate among the identified candidates, for the orange patch in the 2D neural network(panel (c)). (c) 2D neural network resulting from SOM, the neurons with a u -selection of 5% contamination rate (orange linein panel (b)) are shaded in orange. Among these, the neurons corresponding to the member candidates of the target cluster arehighlighted in blue. D Morphology of Open Clusters in the Solar Neighborhood −15−10−50510 μ δ [ m a s y r − ] (5.a)NGC 2547 −10010−10−50510 μ δ [ m a s y r − ] (6.a)NGC 6633 −10−505−35−30−25−20 μ δ [ m a s y r − ] (7.a)NGC 6774 −30−25−20−15 μ α cos δ [mas yr −1 ] μ δ [ m a s y r − ] (8.a)NGC 2451A −4 −3 −20200400600800 (5.b) −4 −3 −20100020003000 (6.b) −4 −3 −202004006008001000 (7.b) −4 −3 −2 u (8.b) (5.c) (6.c) (7.c) (8.c) N u m be r N u m be r N u m be r N u m be r Figure 15. (a) the 2D density map of the proper motion vectors for the regions around five target clusters in sample I.(b) Histogram of the distribution of u . (c) 2D neural network resulting from SOM. The symbols and color coding are identicalto those in Figure 14. Pang et al. −20−100−10−5051015 μ δ [ m a s y r − ] (9.a)NGC 2451B −10−50−505 μ δ [ m a s y r − ] (10.a)NGC 2232 μ δ [ m a s y r − ] (11.a)Blanco 1 −30−20−100 μ α cos δ [mas yr −1 ] −20−10010 μ δ [ m a s y r − ] (12.a)Coma Berenices −4 −3 −2050010001500 (9.b) −4 −3 −202004006008001000 (10.b) −4 −3 −202004006008001000 (11.b) −4 −3 −2 u (12.b) (9.c) (10.c) (11.c) (12.c) N u m be r N u m be r N u m be r N u m be r Figure 16. (a) the 2D density map of the proper motion vectors for the regions around five target clusters in sample I.(b) Histogram of the distribution of u . (c) 2D neural network resulting from SOM. The symbols and color coding are identicalto those in Figure 14. D Morphology of Open Clusters in the Solar Neighborhood C. ESTIMATION OF UNCERTAINTY IN THE CORRECTED DISTANCEIn this section we describe Monte-Carlo simulations that were carried out to quantify the uncertainty in the correcteddistances to the individual stars using Bayesian method. In step I, we generate simulated clusters to model observations.A thousand stars are uniformly distributed within a radius of 10 pc. The cluster is first placed at a distance of 50 pc.An initial parallax (hereafter parallax (I), in step I) is assigned to each star through inverting its original distance.To simulate the observed parallax errors, we resample the observed parallax (I) from a Gaussian distribution, withthe initial parallax as the mean and the mean parallax error among members of all clusters (0.046 mas yr − ) as thestandard deviation. The observed parallax (I) is converted into observed distance (I) by reciprocation. An artificialelongated cluster is generated by stretching the stellar population along the line-of-sight, similar to observations. Weapply the Bayesian method to correct the observed distances of the individual stars. To maintain consistency withthe membership determination applied in Section 2.2, we adopt a membership probability of 95% for each star. Thedifference between the corrected distance (I) and the original distance is adopted as the uncertainty of the Bayesianmethod. We increase the distance of the simulated cluster with a steps of 50 pc until a distance of 500 pc. We repeatthis procedure for an ensemble of 100 simulations in order to obtain a statistically reliable result.OCs can be intrinsically elongated. To further investigate the dependence of the distance correction on the intrinsicmorphology of star clusters, we also generate OCs that are intrinsically elongated. To simplify the procedure, weconsider the artificially elongated cluster resulting from step I as a starting point in step II. In this case, the elongatedshape is considered as the original morphology of cluster. Two types of intrinsically elongated clusters are simulated:(i) clusters with an elongation along the line-of-sight and (ii) clusters with an elongation perpendicular to the line-of-sight. We obtain the initial parallax (II), observed parallax (II), observed distance (II), and corrected distance (II)following the same procedure as in step I. The elongated simulated clusters are located at distances ranging from 50 pcto 500 pc from the Sun. The uncertainty in the corrected distances for the elongated clusters (dotted and dashedcurves in Figure 4) follow similar trend as the those of the uniform cluster.In general, intrinsically elongated clusters have larger uncertainties in their corrected distances than clusters with aspherical uniform stellar distribution. When the cluster is elongated perpendicular to the line-of-sight, the uncertaintyin the corrected distance is close to that of a uniform model for distances smaller than approximately 300 pc. Atdistances larger than 300 pc, errors in elongated models are larger than those of uniform cluster 3.0 pc, and reach3.4 pc at 500 pc. The situation is different in the model with elongation along the line-of-sight. For distances largerthan 200 pc, such clusters show significant deviations in the uncertainty of the corrected distance when compared tothe uniform model, and reach an uncertainty of 6.3 pc at a distance of 500 pc.4 Pang et al. D. ELLIPSOID FITTING X ( p c ) Y ( p c ) Z ( p c ) IC 2391 a axisb axisc axis X ( p c ) Y ( p c ) Z ( p c ) IC 2602 X ( p c ) Y ( p c ) Z ( p c ) IC 4665 X ( p c ) Y ( p c ) Z ( p c ) NGC 2422 X ( p c ) Y ( p c ) Z ( p c ) NGC 2547 X ( p c ) Y ( p c ) Z ( p c ) NGC 6633 X ( p c ) Y ( p c ) Z ( p c ) NGC 6774 X ( p c ) Y ( p c ) Z ( p c ) NGC 2451A X ( p c ) Y ( p c ) Z ( p c ) NGC 2451B X ( p c ) Y ( p c ) Z ( p c ) NGC 2232 X ( p c ) Y ( p c ) Z ( p c ) Blanco1 X ( p c ) Y ( p c ) Z ( p c ) Coma Berenices
Figure 17.
Ellipsoid fitting for the 3D spatial positions (in heliocentric Cartesian coordinates
X, Y, Z ) of cluster memberswithin the tidal radii of twelve target clusters after distance correction via a Bayesian approach (see Section 4.1). The fittedellipsoid is shown with the green surface. Blue dots are members within tidal radii. The a , b , and c axes of the ellipsoid areindicated in red, pink, and orange, respectively. D Morphology of Open Clusters in the Solar Neighborhood E. N -BODY SIMULATION SETUPE.0.1. Numerical method
The N -body simulations are carried out by the code NBODY6. The code uses state-of-the-art numerical techniques(Kustaanheimo & Stiefel 1965; Ahmad & Cohen 1973; Aarseth et al. 1974; Mikkola & Aarseth 1990; Makino 1991;Makino & Aarseth 1992) to deal with the large dynamical range of time-steps of the stars under integration. Stellarevolution and binary evolution algorithms are adopted from Tout et al. (1996); Hurley et al. (2000, 2002). The modelledclusters are subjected to the external gravitational potential of the Galaxy, which is approximated by the model ofAllen & Santillan (1991). A detailed description of NBODY6 and further applications can be found in Spurzem (1999),Aarseth (2003) and Wang et al. (2015, 2016).E.0.2. Initial binary conditions
Binaries of lower mass primary ( m break < m , where m break = 5M (cid:12) ) have orbital parameters (i.e. the semi-majoraxis and eccentricity) and mass ratios generated from the initial binary distribution of Kroupa (1995a), while binariesof the primary more massive than that are generated according to the distribution of Sana et al. (2012) and Moe &Di Stefano (2017). The initial conditions for the star clusters are generated using the software package mcluster (K¨upper et al. 2011). E.0.3. Gas expulsion
The gaseous component of the cluster is approximated with an analytical gravitational potential, which also followsthe Plummer profile with the same half-mass radius as the stellar component. The initial mass M gas (0) of the gaseouscomponent is given by the definition of the SFE, which we adopt for simplicity as SFE = M cl (0) / ( M cl (0) + M gas (0)).The gaseous potential does not evolve up to t d = 0 . M gas ( t ) = M gas (0) exp { ( t − t d ) /τ M } , (E1)where τ M is the gas expulsion time-scale. This follows the procedure of Kroupa et al. (2001) and the conditions forthe ultra compact HII region phase.For each set of model, we obtain a realization of the most massive cluster twice with a different random numberseed, and cluster of initial mass M cl (0) = 2 − i × (cid:12) are realised 2 i +1+1