Determination of A^+ in Open Clusters
Khushboo K. Rao, Kaushar Vaidya, Manan Agarwal, Souradeep Bhattacharya
MMNRAS , 1–9 (2021) Preprint 16 February 2021 Compiled using MNRAS L A TEX style file v3.0
Determination of A + in Open Clusters Khushboo K. Rao , ★ Kaushar Vaidya , Manan Agarwal , Souradeep Bhattacharya Department of physics, Birla Institute of Technology and Science-Pilani, 333031 Rajasthan, India Inter University Centre for Astronomy and Astrophysics, Ganeshkhind, Post Bag 4, Pune 411007, India
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The sedimentation level of blue straggler stars (BSS) has been shown to be a great tool toinvestigate the dynamical states of globular clusters (GCs). The area enclosed between thecumulative radial distributions of BSS and a reference population up to the half-mass radius ofthe clusters, 𝐴 + rh , is known to be a measure of the sedimentation of BSS in GCs. In this work,we calculate 𝐴 + rh for 12 open clusters (OCs) using main-sequence turn-off stars as a referencepopulation. The BSS as well as the main-sequence turn-off stars for these clusters are identifiedusing the proper motions and parallaxes from the Gaia DR2 data, with spectroscopicallyconfirmed BSS populations for some of these clusters available in the literature. Using thePearson and Spearman rank correlation coefficients, we find weak correlations between theestimated values of 𝐴 + rh and other markers of dynamical ages of the clusters, i.e., the number ofcentral relaxations a cluster has experienced since its formation, and the structural parametersof the clusters. Based on statistical tests, we find that these correlations are similar to thecorresponding correlations among the less evolved GCs, albeit within large errors. Key words: blue stragglers – open clusters: general – methods: statistical
A star cluster is a gravitationally bound system of stars having a widemass spectrum, in which various kinds of stellar interactions occurand lead to several dynamical processes in the cluster environment,such as two-body relaxation, mass segregation due to equiparti-tion of energy, stellar encounters, and binary system evolution, etc.(Meylan & Heggie 1997). These dynamical processes result in ex-otic populations like blue straggler stars (BSS; Stryker 1993; Bailyn1995), millisecond pulsars (Bhattacharya & van den Heuvel 1991),and cataclysmic variables (Ritter 2010). Among these stellar pop-ulations, BSS are particularly interesting because they form in themajority and are commonly found in diverse environments such asglobular clusters (hereafter GCs; Sandage 1953; Simunovic & Puzia2016), open clusters (hereafter OCs; Johnson & Sandage 1955; deMarchi et al. 2006), Galactic fields (Santucci et al. 2015), and dwarfgalaxies (Monelli et al. 2012), and are easy to be identified based ontheir location on color-magnitude diagrams (CMDs). The standardconcept of BSS formation is that their progenitor gains additionalmass and rejuvenates hydrogen burning in the core, though thetheories differ in how the extra mass is acquired. Currently, thereare three fundamental theories through which BSS can originate,a stellar collision in dynamical interaction of binaries with singlestars or with another binary (Leonard 1989; Leigh et al. 2019), themerger of an inner binary in a triple system through the Kozai mech-anism (Perets & Fabrycky 2009; Naoz & Fabrycky 2014), and masstransfer in a binary system (McCrea 1964; Gosnell et al. 2014). ★ E-mail: [email protected]
As a star cluster evolves, dynamical friction (hereafter DF) seg-regates sources in the cluster core based on the descending order oftheir masses (Chandrasekhar 1943). DF impacts massive stars firstand brings them from a more considerable distance to the clustercenter to place them in the cluster core, at the same time pushing thelighter stars further away from the cluster center. Since BSS are themost massive members among the cluster populations (Shara et al.1997), DF starts to influence those BSS first which are nearer to thecluster center, gradually affecting BSS from the peripheral regions,as the cluster evolves. One cannot directly measure the extent of thecluster up to which the DF is effective. Ferraro et al. (2012) plotteddouble normalized radial distributions of BSS against a referencepopulation (horizontal branch stars in their case) and classified GCsin three distinct families. Family I GCs have flat BSS radial distribu-tion in which DF has not been effective yet to segregate the clusterBSS in its core. These are dynamically young clusters. Family IIGCs show bimodal radial distributions, a central peak followed bya minima at a certain radial distance from the cluster center, anda rising trend in the outskirts of the cluster. The minima in the bi-modal radial distribution, 𝑟 min , is a fraction of the cluster radius upto which the DF is effective such that the BSS up to this radius aremass segregated. The Family II clusters are of intermediate dynam-ical age. Family III GCs show unimodal BSS radial distributionswith a central peak that monotonically decreases throughout thecluster extension. These are dynamically old clusters in which DFhas segregated all the BSS in the cluster core. Ferraro et al. (2012)found a strong anticorrelation between 𝑟 min and the central relax-ation time of GCs normalized to the Hubble time, 𝑡 rc / 𝑡 H . Severalauthors, e.g., Beccari et al. (2013), Dalessandro et al. (2013), Sanna © a r X i v : . [ a s t r o - ph . GA ] F e b Khushboo K. Rao et al. et al. (2014), Dalessandro et al. (2015), estimated 𝑟 min for individualclusters and compared them with other markers of the dynamicalage and proved that 𝑟 min is a powerful indicator of the dynamicalevolution for GCs.For the first time in the literature, such a study in OCs wasundertaken by Vaidya et al. (2020). They found that even amongthe OCs, a similar correlation between the two cluster parameters, 𝑁 relax , defined as the ratio of the cluster age to the central relaxationtime ( 𝐶 Age / 𝑡 rc ), and 𝑟 min , as previously reported in a large numberof GCs, is observed. For such an analysis, however, one needs to becareful in estimating the 𝑟 min . If the bin size is too large, the error in 𝑟 min increases up to the width of the chosen bin size, however, if itis too small, one gets noisy points near 𝑟 min because of diminishingnumbers of BSS near this region (zone of avoidance; Miocchi et al.2015). This fine-tuning of bin size for an accurate determination of 𝑟 min becomes particularly challenging for the radial distributionsof OCs since they typically contain much fewer stars compared toGCs.Alessandrini et al. (2016) performed N-body simulations ofGCs with different fractions of dark remnants (neutron stars andblack holes), and proposed a new parameter, 𝐴 + , to measure thesedimentation level of the BSS that indicates the dynamical stateof the cluster, where 𝐴 + is the area confined between the cumulativeradial distributions of the BSS and a reference population . Unlike 𝑟 min , 𝐴 + is a directly measurable quantity which measures dynam-ical evolution for all three families of GCs and not just Family II(Lanzoni et al. 2016; Raso et al. 2017; Ferraro et al. 2018, 2020).As a cluster evolves, BSS start to segregate in the cluster centermore rapidly than any reference population, leading to increasingseparation between the two cumulative radial distributions. Blackholes delay the mass segregation process, though do not preventBSS from segregation, which implies that 𝐴 + always increases withtime, whether it is a slow or a rapid increment (Alessandrini et al.2016).Lanzoni et al. (2016) performed the first observational estima-tion of 𝐴 + in 25 GCs. This work was extended to ∼
33% populationof GCs by Raso et al. (2017) and Ferraro et al. (2018), and to 5 LMCGCs by Ferraro et al. (2020). OCs are still unexplored systems in thisdomain, therefore, we pursue a similar study on OCs. OCs and GCsare vastly different stellar systems in terms of their shapes, ages,stellar densities and locations (Janes & Adler 1982; Lada 2010;Harris & Racine 1979; Freeman & Norris 1981). There are certainadvantages to study the BSS of OCs. Because of lower stellar densi-ties, the individual BSS can be studied in detail. With the Gaia DR2(Gaia Collaboration 2018) and now Gaia EDR3 (Collaboration et al.2020), the precise information of positions and proper motions forbillions of stars, secure cluster members and BSS populations canbe identified. In two OCs, NGC 188 and M67, the BSS populationand their binary origin, has been examined with great details thathas shed light on the formation channels of BSS. Also, Geller et al.(2008) have used N-body simulations to reproduce the observedBSS (binarity, radial distributions), which is only possible in OCs.More recently, the BSS populations of OCs have been studied to Due to the effect of DF in a star cluster, the most massive cluster membersstart to move towards the cluster center and settle in the cluster core. Depending on the available photometric data and clusters properties dif-ferent cluster populations such as horizontal branch stars (HBs), red-giantbranch stars (RGBs), sub-giant branch stars (SGBs), and main-sequenceturn-off stars (MS-TO) have been used as a reference population in the liter-ature to estimate the value of 𝐴 + (Lanzoni et al. 2016; Ferraro et al. 2018,2020). Table 1.
The list of clusters studied in this work, their ages, distances, andthe number of BSS.Cluster Age Dist No. of BSS(Gyr) (pc)Berkeley 17 𝑎
10 3138 23Berkeley 39 𝑏 𝑒 𝑏 𝑏 𝑏 𝑏 𝑐 𝑏 𝑏 𝑑 𝑓 𝐴 + rh are taken from: 𝑎 Bhattacharya et al. (2019); 𝑏 Vaidya et al. (2020); 𝑐 Geller et al. (2015); 𝑑 Nine et al. (2020); 𝑒 Rain et al. (2020); 𝑓 Rain et al. (2021) explore the link between the clusters’ dynamical status and the ob-served BSS radial distributions (Bhattacharya et al. 2019; Vaidyaet al. 2020; Rain et al. 2020, 2021).In this work, we present our measurements of 𝐴 + in twelveOCs whose BSS populations have been identified in the literature.These OCs contain 13 or more BSS. We use MS-TO stars of theseclusters as a reference population in this study. The rest of the paperis arranged as follows. The section 2 is devoted to the informationof the clusters presented in this work. We give the details of thecalculation of 𝐴 + and its error estimation for each OC in section 3.In section 4, we present the results of the paper and discuss thoseresults. In the end, section 5 concludes work of the present study. Gaia is a space mission to map the whole sky in three dimensions.The Gaia DR2 data (Gaia Collaboration 2018) provides five pa-rameters astrometric solution - positions ( 𝛼 , 𝛿 ), parallaxes, propermotions for more than 1.3 billion stars of the Galaxy and throughoutthe local group of galaxies. This dataset has unparalleled precisionin parallaxes and proper motions. In parallaxes, the uncertaintiesare of the order 0.4 milliarcsecond (hereafter mas) for G < 15 mag,0.1 mas for G = 17 mag, and 0.7 mas at the faint end, G = 20 mag.In proper motions, the corresponding uncertainties are up to 0.06mas yr − for G < 15 mag, 0.2 mas yr − for G = 17 mag, and 1.2mas yr − for G = 20 mag (Gaia Collaboration 2018).The present study focuses on studying the correlation of 𝐴 + with the clusters’ structural parameters and with the theoreticalestimate of the relaxation status of the clusters. For this purpose,it is imperative that we choose OCs which contain a reasonablylarge number of BSS that allow such an analysis. Since most OCscontain small numbers of BSS, finding such OCs is not an easy task.Vaidya et al. (2020) studied 7 OCs, Berkeley 39, Melotte 66, NGC188, NGC 2158, NGC 2506, NGC 6791, and NGC 6819 whichcontain a minimum of 14 BSS, using the Gaia DR2 data to identifycluster members and BSS candidates. Some of these clusters werepreviously part of WIYN Open Cluster Survey (WOCS) and hadspectroscopically confirmed BSS, i.e., NGC 188 (Geller et al. 2008), MNRAS , 1–9 (2021) etermination of A + in Open Clusters Table 2.
The dynamical and the structural parameters, and the estimated values of 𝐴 + rh and errors in 𝐴 + rh of the OCs.Cluster 𝑟 c 𝑟 t 𝑟 h 𝑟 min / 𝑟 c 𝑟 e c 𝑡 rc 𝑁 relax 𝐴 + rh Error(arcmin) (arcmin) (arcmin) (arcmin) (Myr) ( 𝜖 𝐴 + )Berkeley 17 𝑎 𝑏 𝑐 𝑏 𝑏 𝑏 𝑏 𝑐 𝑏 𝑏 𝑐 𝑐 𝑟 c , 𝑟 t , and 𝑟 min are taken from: 𝑎 Bhattacharya et al. (2019) 𝑏 Vaidya et al. (2020); for these clusters, the value of 𝑡 rc and 𝑁 relax are taken from Vaidya et al. (2020) 𝑐 This work
NGC 6791 (Tofflemire et al. 2014), and NGC 6819 (Milliman et al.2014). We choose all of these seven clusters in our present work.In addition, there are 5 clusters that we include in this work. Forthree of them, the BSS populations have been identified using theGaia DR2 in recent studies. Bhattacharya et al. (2019) studied BSSpopulations of Berkeley 17. Rain et al. (2020) studied the BSS ofCollinder 261, and Rain et al. (2021) studied the BSS of Trumpler 5.In both, Collinder 261 and Trumpler 5, Rain et al. (2020, 2021) alsohad a small number of BSS with spectra from the high-resolutionspectrograph FLAMES/GIRAFFE@VLT. In two additional OCs,NGC 2682 and NGC 7789, the complete BSS populations as well asa large number of MS-TO stars have been studied spectroscopicallyunder the WIYN Open Cluster Survey (Geller et al. 2015; Nineet al. 2020). We adopt the BSS samples of these twelve clustersfrom these studies, listed in Table 1. The reason why we prefer touse MS-TO stars of the clusters as a reference population instead ofRGBs, is because they are greater in number, and thus minimize thestatistical fluctuations in the 𝐴 + estimation. For the seven clustersfrom Vaidya et al. (2020), we already have the cluster members. ForBerkeley 17, we get the cluster members from Bhattacharya et al.(2019).For the remaining four clusters, i.e., Collinder 261, Trumpler 5,NGC 2682, and NGC 7789, we identify the cluster members usingthe Gaia DR2 in order to select the MS-TO stars as the referencepopulation, whereas the BSS lists are taken from the literature. Themembership identification procedure and the related figures for thefour clusters are presented in Appendix A. 𝐴 + In order to measure the BSS sedimentation level in the OCs, wecalculate 𝐴 + , i.e., the area enclosed between the cumulative radialdistribution of BSS and a reference population (hereafter REF; theyare MS-TO stars in this work). The sources above the main-sequenceturn-off point, extending up to the blue end of the SGB, and fainteras well as redder than the BSS are chosen as our MS-TO stars(see Figure A7). We avoid sources residing in the location of the binary stars and near the BSS region as our MS-TO stars. Sincethe Gaia DR2 data is 99% complete up to G ∼
18 mag (Boubert& Everall 2020), our results of the 𝐴 + estimation are not affectedby the photometric incompleteness. The area enclosed between thecumulative radial distributions of BSS ( 𝜙 BSS ) and REF ( 𝜙 REF ) isgiven as: 𝐴 + = ∫ 𝑥𝑥 𝑚𝑖𝑛 𝜙 BSS ( 𝑥 (cid:48) ) − 𝜙 REF ( 𝑥 (cid:48) ) 𝑑𝑥 (cid:48) (1)where 𝑥 = log ( 𝑟 / 𝑟 h ) and 𝑥 min are the outermost and the innermostradii from the cluster center, respectively, and 𝑟 h is the half-massradius of the cluster. Following Lanzoni et al. (2016), we calcu-late 𝐴 + only up to one 𝑟 h of the cluster (hereafter 𝐴 + rh ), mainly tocompare the parameters in different stellar systems and take intoaccount the cluster portion that is most sensitive to the phenomenonof BSS sedimentation that occurs due to DF. Moreover, to maximizethe sensitivity of the parameter 𝐴 + rh , we plot the cumulative radialdistributions as a function of the logarithm of the radius normal-ized to the cluster 𝑟 h , estimated using the equation A3 of Freireet al. (2005), and listed in Table 2. Figure 1 shows the cumulativeradial distributions of the BSS and the REF population of the 12OCs. The grey shaded portion enclosed between the BSS and theREF cumulative radial distributions corresponds to the estimatedvalue of 𝐴 + rh for the respective cluster and is noted in Table 2. Wehave estimated the errors in our calculated values of 𝐴 + rh using theBootstrap method (Lupton 1993), listed in Table 2. Lanzoni et al. (2016), Ferraro et al. (2018, 2020) ascertained that 𝐴 + rh is a powerful indicator of the dynamical evolution of GCs. Toinvestigate whether 𝐴 + rh values of OCs truly indicate the dynamical 𝑀 ( 𝑟 ) = ∫ 𝑟 𝜌 ( 𝑟 (cid:48)) 𝜋𝑟 (cid:48) 𝑑𝑟 (cid:48) = 𝜋𝜌 𝑟 𝑐 sinh − (cid:18) 𝑟𝑟𝑐 (cid:19) − 𝑟𝑟𝑐 √︃ + ( 𝑟 / 𝑟𝑐 ) MNRAS000
18 mag (Boubert& Everall 2020), our results of the 𝐴 + estimation are not affectedby the photometric incompleteness. The area enclosed between thecumulative radial distributions of BSS ( 𝜙 BSS ) and REF ( 𝜙 REF ) isgiven as: 𝐴 + = ∫ 𝑥𝑥 𝑚𝑖𝑛 𝜙 BSS ( 𝑥 (cid:48) ) − 𝜙 REF ( 𝑥 (cid:48) ) 𝑑𝑥 (cid:48) (1)where 𝑥 = log ( 𝑟 / 𝑟 h ) and 𝑥 min are the outermost and the innermostradii from the cluster center, respectively, and 𝑟 h is the half-massradius of the cluster. Following Lanzoni et al. (2016), we calcu-late 𝐴 + only up to one 𝑟 h of the cluster (hereafter 𝐴 + rh ), mainly tocompare the parameters in different stellar systems and take intoaccount the cluster portion that is most sensitive to the phenomenonof BSS sedimentation that occurs due to DF. Moreover, to maximizethe sensitivity of the parameter 𝐴 + rh , we plot the cumulative radialdistributions as a function of the logarithm of the radius normal-ized to the cluster 𝑟 h , estimated using the equation A3 of Freireet al. (2005), and listed in Table 2. Figure 1 shows the cumulativeradial distributions of the BSS and the REF population of the 12OCs. The grey shaded portion enclosed between the BSS and theREF cumulative radial distributions corresponds to the estimatedvalue of 𝐴 + rh for the respective cluster and is noted in Table 2. Wehave estimated the errors in our calculated values of 𝐴 + rh using theBootstrap method (Lupton 1993), listed in Table 2. Lanzoni et al. (2016), Ferraro et al. (2018, 2020) ascertained that 𝐴 + rh is a powerful indicator of the dynamical evolution of GCs. Toinvestigate whether 𝐴 + rh values of OCs truly indicate the dynamical 𝑀 ( 𝑟 ) = ∫ 𝑟 𝜌 ( 𝑟 (cid:48)) 𝜋𝑟 (cid:48) 𝑑𝑟 (cid:48) = 𝜋𝜌 𝑟 𝑐 sinh − (cid:18) 𝑟𝑟𝑐 (cid:19) − 𝑟𝑟𝑐 √︃ + ( 𝑟 / 𝑟𝑐 ) MNRAS000 , 1–9 (2021)
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Figure 1.
The cumulative radial distributions of the BSS (blue curve) and the REF population (MS-TO stars, black dashed curve), plotted against logarithmof the radial distance from the cluster center in the units of 𝑟 h , of the 12 OCs. The values of 𝐴 + rh shown on each plot correspond to the grey shaded portionbetween the cumulative radial distributions of the BSS and the REF population. status of the clusters, we plot 𝐴 + rh against 𝑁 relax . For the sevenclusters for which we adopt the BSS and MS-TO samples fromVaidya et al. (2020), we also adopt the structural parameters, coreradius ( 𝑟 c ), central relaxation times ( 𝑡 rc ), and 𝑁 relax , from their work.For the remaining 5 clusters, we used the same method as used byFerraro et al. (2012) and Lanzoni et al. (2016), to calculate the 𝑁 relax . For that, we first determine the central relaxation time, 𝑡 rc , using theEquation 10 of Djorgovski (1993), 𝑡 rc = 1 . × 𝑦𝑟 × 𝑘𝑙𝑛 ( . 𝑁 ∗ ) <𝑚 ∗ > − 𝜌 / 𝑀,𝑂 𝑟 𝑐 , where 𝑘 ∼ . 𝑁 ∗ is the total number of starsidentified in the cluster, < 𝑚 ∗ > is the average mass of the cluster, 𝜌 𝑀,𝑂 is the central density of the cluster, and 𝑟 c is the core radius ofthe cluster. The estimated values of 𝑁 relax and 𝑡 rc of these clusters, MNRAS , 1–9 (2021) etermination of A + in Open Clusters Figure 2.
The correlation between the values of 𝐴 + rh and the number ofcurrent central relaxation a cluster has undergone since its formation, 𝑁 relax ,for 12 OCs (blue filled circles) and 48 GCs (red filled circles) of Ferraroet al. (2018). The blue dashed line represents the best fitted line for the OCswhereas the red dashed line shows the best-fit correlation (Ferraro et al.2018) for the GCs. as well as the adopted values of 𝑁 relax and 𝑡 rc of the other clustersare listed in Table 2. We plot 𝐴 + rh against the 𝑁 relax for our 12OCs in Figure 2. For comparison, we also show the two identicalquantities of GCs from Ferraro et al. (2018) on the same figure. Thebest-fit relation for OCs plotted in Figure 2 islog ( 𝑁 relax ) = . (± . ) × 𝐴 + rh + . (± . ) (2)Whereas the best-fit correlation for GCs islog ( 𝑁 relax ) = . (± . ) × 𝐴 + rh + . (± . ) (3)The OC datapoints exhibit a positive correlation between 𝐴 + rh and 𝑁 relax with a smaller slope and a higher intercept than the correla-tion among the GC datapoints (Ferraro et al. 2018). Moreover, theOC correlation has a low statistical significance with the Spearmanrank correlation coefficient being 0.301, and the Pearson correlationcoefficient being 0.329. The OCs are seen to lie near the middle tothe lower end of the plot, implying, as expectantly, that they areamong the least dynamically evolved GCs. Moreover, physicallyit is not possible for OCs to be as evolved as GCs. Therefore wecompare OCs with the less evolved GCs. In particular, we calculatethe Spearman rank correlation coefficient, and the Pearson corre-lation coefficient for GCs with log ( 𝑁 relax ) ≤ tool, which covers a broad range of tests including the comparisonsof independent and dependent correlations with either overlappingor nonoverlapping variables (Diedenhofen & Musch 2015). TheCOCOR tool, however, is limited to the comparison between twocorrelations only. In our case, the correlation coefficients are inde-pendent to each other, so it compares the correlation coefficientsthrough two tests: the Fisher test (Fisher 1992), and the Zou test(Zou 2007). We chose the values of 𝛼 , p-value threshold, and theZou’s confidence level as 0.05 and 0.95, respectively. We then con-duct a two-tailed test that gives the results in the form of whether the http://comparingcorrelations.org Figure 3.
The correlation between 𝐴 + rh and structural parameters for OCs(blue filled circles) and GCs (red filled circles). We utilized the values of 𝐴 + rh and structural parameters of GCs from Lanzoni et al. (2016), Ferraroet al. (2018), and the references therein. two correlations are equal or not. We get the p-value, i.e., the signif-icance level of the comparison done by the Fisher test as 0.1389 and0.1092 for the Pearson correlation coefficient and the Spearmanrank correlation coefficient, respectively. We get the Zou’s confi-dence interval as − + − + 𝛼 value and the estimated Zou’s confidenceintervals also contain zero which indicate that the null hypothesisthat the two distributions are similar is not rejected. Hence, thesetests suggest that the correlation coefficients estimated for the lessevolved GCs and OCs are not different. However, given the smallsample size of OCs, we need a larger sample size to see if they aresimilar.Ferraro et al. (2018, 2020) demonstrated that the long-termdynamical evolution tends to produce compact stellar systems. Toshow this effect in OCs, we plot 𝐴 + rh against three structural pa-rameters, 𝑟 c , 𝑐 (defined as logarithm of the ratio of tidal radius tocore radius), and 𝑟 c / 𝑟 e (the ratio of core radius to effective radius ofthe cluster) as shown in Figure 3. For this, we derived the effectiveradius, defined as the radius of the circle in projection including half MNRAS000
The correlation between 𝐴 + rh and structural parameters for OCs(blue filled circles) and GCs (red filled circles). We utilized the values of 𝐴 + rh and structural parameters of GCs from Lanzoni et al. (2016), Ferraroet al. (2018), and the references therein. two correlations are equal or not. We get the p-value, i.e., the signif-icance level of the comparison done by the Fisher test as 0.1389 and0.1092 for the Pearson correlation coefficient and the Spearmanrank correlation coefficient, respectively. We get the Zou’s confi-dence interval as − + − + 𝛼 value and the estimated Zou’s confidenceintervals also contain zero which indicate that the null hypothesisthat the two distributions are similar is not rejected. Hence, thesetests suggest that the correlation coefficients estimated for the lessevolved GCs and OCs are not different. However, given the smallsample size of OCs, we need a larger sample size to see if they aresimilar.Ferraro et al. (2018, 2020) demonstrated that the long-termdynamical evolution tends to produce compact stellar systems. Toshow this effect in OCs, we plot 𝐴 + rh against three structural pa-rameters, 𝑟 c , 𝑐 (defined as logarithm of the ratio of tidal radius tocore radius), and 𝑟 c / 𝑟 e (the ratio of core radius to effective radius ofthe cluster) as shown in Figure 3. For this, we derived the effectiveradius, defined as the radius of the circle in projection including half MNRAS000 , 1–9 (2021)
Khushboo K. Rao et al.
Table 3.
The COCOR tool results of OCs and the less evolved GCs for the correlation between 𝐴 + rh and structural and dynamical parameters of the clusters.Column 1 gives the fitted correlation, column 2 gives the name of the correlation coefficient where P refers to the Pearson correlation coefficient and S refersto the Spearman rank correlation coefficient, columns 3 and 4 list the calculated correlation coefficients for the OCs and the less evolved GCs, column 5 and 6give the results obtained by employing the statistical tests to compare the correlation coefficients of the OCs and the less evolved GCs using the COCOR tool,the last column denotes whether the null hypothesis is or is not rejected, with a (cid:88) sign impling that the null hypothesis is not rejected.Correlation Correlationcoefficient OCs GCs p-value(Fisher test) 95% CI ★ (Zou test) COCOR tool result 𝐴 + rh vs 𝑁 relax PS + + + + − + − + (cid:88)(cid:88) 𝐴 + rh vs 𝑟 c PS − − − − − + − + (cid:88)(cid:88) 𝐴 + rh vs 𝑟 c / 𝑟 e PS − − − − − + − + (cid:88)(cid:88) 𝐴 + rh vs 𝑐 PS + + + + − + − + (cid:88)(cid:88) ★ Confidence interval of the total counted stars of a cluster for all the 12 OCs included inthe present work. The values of 𝑟 c , 𝑟 t , distances, and the ages of 7clusters are utilized from (Vaidya et al. 2020), while for Berkeley17, we use these parameters derived by Bhattacharya et al. (2019).Following Vaidya et al. (2020), we fit the King’s profile to our iden-tified members of the remaining clusters and derive the values of 𝑟 c and 𝑟 t (see Figure A5), and we determine the ages and the distancesby fitting the PARSEC isochrones to the cluster members. The esti-mated structural parameters of each cluster are listed in Table 2, andthe values of the ages and distances of the clusters are listed in Ta-ble 1. In Figure 3, we plot our estimated values of 𝐴 + rh against thesestructural parameters, log ( 𝑟 c ) , log ( 𝑟 c / 𝑟 e ) , and 𝑐, in the top, middle,and the bottom panel, respectively. As can be seen in Figure 3, OCsfollow the same trend as GCs, and occupy the same parameter spacewith smaller values of 𝐴 + rh . We calculated the Pearson correlationcoefficients and the Spearman rank correlation coefficients for OCsof all the three plots of Figure 3. In order to compare OCs with GCs,we also calculate the correlation coefficients for less evolved GCs,i.e., GCs which are having values log ( 𝑟 c ) , log ( 𝑟 c / 𝑟 e ) , and 𝑐 similaras OCs. The estimated correlation coefficients of OCs and GCs forall the three cases and the COCOR tool results are listed in the Table3. Again, we see that they are not different within the errors, but weneed a larger sample size of OCs since errors are large due to oursmall sample size.Since 𝐴 + rh and 𝑟 min both have been used as indicators of thedynamical ages of the clusters, it is useful to see how these two ob-served parameters of clusters themselves correlate with one another.In fact, Lanzoni et al. (2016) plotted 𝐴 + rh against the logarithm of 𝑟 min (in the units of 𝑟 c ) for their GC sample and found that thesetwo quantities are in a linear correlation with each other. We plotthese two parameters against one another for our 12 OCs (blue filledcircles) and show them in Figure 4. We use the values of log ( 𝑟 min ) of 5 OCs with bimodal BSS radial distributions from Vaidya et al.(2020) whereas for Berkeley 17 we use the value derived by Bhat-tacharya et al. (2019). Collinder 261 (Rain et al. 2020) and Trumpler5 (Rain et al. 2021) and two OCs in Vaidya et al. (2020) have flatBSS radial distributions. For NGC 7789 and NGC 2682, we plottedBSS radial distributions using the method described by Vaidya et al. Figure 4.
The correlation between 𝐴 + rh and the location of minima in BSSradial distributions, log ( 𝑟 min / 𝑟 c ) , for OCs (blue filled circles) and GCs (redfilled circles) of Lanzoni et al. (2016). Following Lanzoni et al. (2016), weconsider the value of 𝑟 min / 𝑟 c as 0.1 for those OCs which have flat BSS radialdistributions. (2020). As shown in Figure A6, these two clusters also have flat BSSradial distributions. Thus we have 6 OCs with bimodal distributionsand 6 OCs with flat BSS radial distributions for which we consider 𝑟 min / 𝑟 c as 0.1. For a comparison, we also plot all the GCs datapoints (red filled circles) from Lanzoni et al. (2016, their Figure 4).The OCs parameters, 𝐴 + rh and log ( 𝑟 min / 𝑟 c ) , appear to occupy thesame parameter space of GCs. Not surprisingly, the OCs are amongthe least dynamically evolved GCs.Among the 12 OCs studied in the current work, NGC 7789 andTrumpler 5 are the least dynamically evolved. Their values of 𝐴 + rh are also consistent with the values of 𝑁 relax (see Figure 2). Rain et al.(2021) presented the BSS radial distribution of Trumpler 5 usingRGBs as a reference population and found it to have flat BSS radialdistribution, which is in agreement with our finding that the clusteris dynamically young. In contrast to our conclusion of the dynamicalstatus of NGC 7789, Wu et al. (2007) reported that the cluster ismass-segregated by estimating concentration parameter for different MNRAS , 1–9 (2021) etermination of A + in Open Clusters mass range sources and by fitting mass-function in different spatialranges. However, neither our estimate of the number of relaxationsundergone by the cluster since its formation, nor our estimated 𝐴 + rh suggest that NGC 7789 is a dynamically evolved cluster. Berkeley17 seems to be less dynamically evolved as per its 𝐴 + rh value thoughits 𝑁 relax is very large. Its small 𝐴 + rh is consistent with the value of 𝑟 min reported by (Bhattacharya et al. 2019). It is worth mentioningthat Berkeley 17 shows a core-tail morphology (Bhattacharya et al.2017) which indicates that the cluster evolution is not only governedby its internal dynamics, but also by its external environments thatare also affecting its evolution. Therefore, the sedimentation level ofBSS is not commensurate with the estimated 𝑁 relax . We notice thatNGC 2682 is another cluster which is expected to be a dynamicallyevolved cluster according to previous works in the literature inwhich signatures of extra-tidal sources and mass segregation havebeen found (Fan et al. 1996; Bonatto & Bica 2003; Geller et al. 2015;Carrera et al. 2019). However, its 𝐴 + rh and 𝑁 relax suggest that it is ofintermediate dynamical age. This is the second example of a clusterin addition to Berkeley 17, where we can speculate the presence ofexternal environments possibly affecting the dynamical evolutionof the cluster. Vaidya et al. (2020) had shown the evidence of thedynamical ages of 7 OCs using BSS radial distributions. Amongthose, 5 OCs, Melotte 66, NGC 188, NGC 2158, NGC 2506, andNGC 6791 were classified as intermediate dynamical age clustersand are consistent with the present finding. Berkeley 39 was foundto show a flat radial distribution (Vaidya et al. 2020). With a 𝐴 + rh ∼ 𝐴 + rh as per our current work. We believe thatthis discrepancy is due to the small numbers of BSS in the clusterwhich did not allow us to detect the bimodality and the associatedminima in its radial distribution. The present study is the extended version of the work done byVaidya et al. (2020), in which they identified BSS and RGBs of 7OCs and performed the analysis of dynamical evolution of the clus-ters using the BSS radial distributions. This work presents the firstever attempt at estimating 𝐴 + rh in 12 OCs, including the previous7 clusters studied by Vaidya et al. (2020). 𝐴 + rh can be calculatedfor more OCs than 𝑟 min which is limited only to Family II clusters.While the correlation between 𝐴 + rh and 𝑟 min is not clear for OC dat-apoints alone, the use of 𝐴 + rh as a tracer of dynamical evolution isclear from its relation to 𝑁 relax (Figure 2). Our study shows that the 𝐴 + rh when plotted against the theoretical estimation of the relaxationstatus of the clusters, 𝑁 relax , or against the structural parametersof the clusters, the OC datapoints are seen to fall in the categoryof the less evolved GCs. OCs are young stellar systems of 10 − Gyr age (Lada 2010), in contrast with GCs formed during theearly stages of the Milky Way (Vandenberg et al. 1996). Also, GCshave a dense star distribution with ∼ member stars while OCsare sparser, containing only ∼ member stars in general. Becauseof the high density, GCs have enough gravitational force to resistthe tidal force and remain in a spherical shape and gravitationallybounded (Harris & Racine 1979; Freeman & Norris 1981) whilesparser OCs are more easily stretched by the external forces and donot remain gravitationally bound over time and spread out (Chenet al. 2004; Zhai et al. 2017). Therefore, we compared the correlationcoefficients of OCs with the less evolved GCs using the statisticaltests, the Fisher test and the Zou test, implemented by the COCOR tool. The employed statistical tests suggest that the compared corre-lation coefficients are consistent within the large errors. In order todetermine a more sound correlation between the 𝐴 + and the othermarkers of the dynamical ages of the OCs, we will extend this workusing the recently released Gaia EDR3 data through an applicationof an automated membership determination algorithm (ML-MOC;Agarwal et al. 2020) to a large number of OCs having greater than10 – 12 BSS. ACKNOWLEDGEMENTS
This work has made use of second data release from the EuropeanSpace Agency (ESA) mission
Gaia ( ), Gaia-DR2 (Gaia Collaboration 2018), processed by the Gaia
Data Processing and Analysis Consortium (DPAC, ). Fundingfor the DPAC has been provided by national institutions, in partic-ular the institutions participating in the
Gaia
Multilateral Agree-ment. This research has made use of the VizieR catalog accesstool, CDS, Strasbourg, France. This research made use of
AS-TROPY , a
PYTHON package for astronomy (Astropy Collabora-tion 2013),
NUMPY (Harris et al. 2020),
MATPLOTLIB (Hunter2007), and
SCIPY (Virtanen et al. 2020). This research alsomade use Astrophysics Data System (ADS) governed by NASA( https://ui.adsabs.harvard.edu ). DATA AVAILABILITY
The data underlying this article are publicly available at https://gea.esac.esa.int/archive . The derived data generated in thisresearch will be shared on reasonable request to the correspondingauthor.
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APPENDIX A: MEMBERSHIP DETERMINATION ANDMS-TO STARS IDENTIFICATION IN COLLINDER 261,NGC 2681, NGC 7789, AND TRUMPLER 5
We follow the method developed by Vaidya et al. (2020) to deter-mine the membership of 4 OCs, Collinder 261, NGC 2682, NGC7789, and Trumpler 5 using the Gaia DR2 data. The method mainlyinvolves three steps: (a) Estimation of proper motion selection range,(b) Estimation of parallax selection range, and (c) Estimation ofcluster radius from the sample of sources following the proper mo-tion and parallax selection criteria. For (a), we used two methods, aGaussian fitting and an application of the mean-shift algorithm onthe sources pre-identified as probable cluster members due to theirseparate proper motions from the field stars, in order to determinethe mean values and the spread in the cluster proper motions. Ta-ble A1 lists the mean and the standard deviation of proper motionsestimated using the mean-shift method and by fitting the Gaussianfunction (see Figure A1). For (b), we use the parallax values andthe corresponding errors of the previously known, spectroscopically
MNRAS , 1–9 (2021) etermination of A + in Open Clusters confirmed members to fix the parallax selection range for NGC 2682and NGC 7789, which have many sources studied spectroscopically(Geller et al. 2015; Overbeek et al. 2014). For Collinder 261 andTrumpler 5, there are very few sources that have been studied spec-troscopically (Mitschang et al. 2013; Donati et al. 2014), thereforeto fix the parallax ranges in these two clusters, we use those propermotion selected sources which are bright (G <= 15 mag). FigureA2 shows the over plotted histograms of the parallax and the propermotions selected members and only proper motions selected mem-bers. For (c), we plot the radial distribution of sources whose propermotions and parallaxes fall in our selected ranges of the two param-eters, for a field of radius that ranges from 40 (cid:48) to 120 (cid:48) around thecluster center. The radius at which cluster members merge with fieldstars is considered as the cluster radius (see Figure A3). The propermotions and parallax selected sources up to the cluster radius areour bonafide cluster members.We then estimate the cluster centers using the mean-shift al-gorithm and by fitting a Gaussian function to the cluster members(see Figure A4). Table A3 lists the estimated cluster centers. FigureA5 shows the fitted King’s profiles to the cluster members. Thederived values of 𝑟 c and 𝑟 t are listed in column 2 and column 3 ofTable 2, respectively. Table A2 includes the derived fundamentalcluster parameters, e.g. age, distance, extinction, color-excess, andmetallicity estimated by fitting the Parsec isochrone to the identi-fied cluster members. Figure A7 shows the CMDs of the 12 OCsincluded in the present work. The sources which are above the main-sequence turn-off but fainter and redder than BSS, and bluer thanSGB stars, are selected as MS-TO stars. We avoid sources residingin the location of the binary stars and near the BSS region as ourMS-TO stars (see Figure A7). MNRAS000
MNRAS , 1–9 (2021) etermination of A + in Open Clusters confirmed members to fix the parallax selection range for NGC 2682and NGC 7789, which have many sources studied spectroscopically(Geller et al. 2015; Overbeek et al. 2014). For Collinder 261 andTrumpler 5, there are very few sources that have been studied spec-troscopically (Mitschang et al. 2013; Donati et al. 2014), thereforeto fix the parallax ranges in these two clusters, we use those propermotion selected sources which are bright (G <= 15 mag). FigureA2 shows the over plotted histograms of the parallax and the propermotions selected members and only proper motions selected mem-bers. For (c), we plot the radial distribution of sources whose propermotions and parallaxes fall in our selected ranges of the two param-eters, for a field of radius that ranges from 40 (cid:48) to 120 (cid:48) around thecluster center. The radius at which cluster members merge with fieldstars is considered as the cluster radius (see Figure A3). The propermotions and parallax selected sources up to the cluster radius areour bonafide cluster members.We then estimate the cluster centers using the mean-shift al-gorithm and by fitting a Gaussian function to the cluster members(see Figure A4). Table A3 lists the estimated cluster centers. FigureA5 shows the fitted King’s profiles to the cluster members. Thederived values of 𝑟 c and 𝑟 t are listed in column 2 and column 3 ofTable 2, respectively. Table A2 includes the derived fundamentalcluster parameters, e.g. age, distance, extinction, color-excess, andmetallicity estimated by fitting the Parsec isochrone to the identi-fied cluster members. Figure A7 shows the CMDs of the 12 OCsincluded in the present work. The sources which are above the main-sequence turn-off but fainter and redder than BSS, and bluer thanSGB stars, are selected as MS-TO stars. We avoid sources residingin the location of the binary stars and near the BSS region as ourMS-TO stars (see Figure A7). MNRAS000 , 1–9 (2021) Khushboo K. Rao et al.
Figure A1.
The proper motion selection criteria of the cluster members. The left panels show the proper motion scatter diagram of all sources within 10 (cid:48) radiusof the cluster centers. The red rectangular region shows our initial range of proper motion selected by visual examination of the proper motion scatter diagrams.The two middle panels show the frequency distribution of proper motions in RA and DEC, respectively, with orange curve showing the fitted Gaussian functionto the distributions. The proper motion scatter-diagram of the cluster members selected within 𝐺 𝑚𝑒𝑎𝑛 ± 𝜎 range are shown in the rightmost panels,where the black plus sign shows the mean of those sources computed using the mean-shift algorithm, whereas the red open circles and the filled circle arespectroscopically confirmed cluster members from the literature and their mean, respectively. MNRAS , 1–9 (2021) etermination of A + in Open Clusters Table A1.
The mean value and the standard deviation of the proper motions in RA and DEC determined from the Gaussian fit, and the peak of the propermotions in RA and DEC estimated by the mean-shift algorithm.cluster G mean (RA) 𝜎 (RA) Peak ms (RA) G mean (DEC) 𝜎 (DEC) Peak ms (DEC)mas yr − mas yr − mas yr − mas yr − mas yr − mas yr − Collinder 261 − − − − − − − − − − − − − − + + Figure A2.
The parallax distribution of the proper motion selected sources (blue histogram) and the proper motion as well as parallax selected sources (orangehistogram) of the clusters. The parallax range is selected as ¯ 𝜔 ± Δ ¯ 𝜔 , where ¯ 𝜔 and Δ ¯ 𝜔 are the mean of the parallaxes of the proper motion selected sourceswhich are having G <= 15 mag and mean of their parallax errors, respectively, for Collinder 261 and Trumpler 5, and the mean of the parallaxes of thespectroscopically confirmed members and the mean of their parallax errors, respectively, for NGC 2682 and NGC 7789.MNRAS000
The parallax distribution of the proper motion selected sources (blue histogram) and the proper motion as well as parallax selected sources (orangehistogram) of the clusters. The parallax range is selected as ¯ 𝜔 ± Δ ¯ 𝜔 , where ¯ 𝜔 and Δ ¯ 𝜔 are the mean of the parallaxes of the proper motion selected sourceswhich are having G <= 15 mag and mean of their parallax errors, respectively, for Collinder 261 and Trumpler 5, and the mean of the parallaxes of thespectroscopically confirmed members and the mean of their parallax errors, respectively, for NGC 2682 and NGC 7789.MNRAS000 , 1–9 (2021) Khushboo K. Rao et al.
Figure A3.
The radial distribution of the proper motion and parallax selected sources. The radius at which the cluster members merge with the field stars ischosen as the cluster radius.
Table A2.
The fundamental cluster parameters from the isochrone fitting of the cluster members and a comparison with the literature values.This Work LiteratureCluster Age d Radius Metallicity 𝐴 G E ( Bp − Rp ) Age d Metallicity(Gyr) (parsec) ( (cid:48) ) (Z) (mag) (mag) (Gyr) (parsec) ( [ Fe / H ] )Collinder 261 6.0 3053 20 0.0127 0.6 0.4 6 – 11 2.7 – 2.9 − − − − − + − − , 1–9 (2021) etermination of A + in Open Clusters Figure A4.
The estimation of cluster centers using the mean-shift algorithm (left panels), and by the fitting of the Gaussian functions to the frequencydistributions of RA and DEC (middle and right panels).MNRAS000
The estimation of cluster centers using the mean-shift algorithm (left panels), and by the fitting of the Gaussian functions to the frequencydistributions of RA and DEC (middle and right panels).MNRAS000 , 1–9 (2021) Khushboo K. Rao et al.
Table A3.
The cluster center coordinates determined by the mean shift algorithm and the fitting of the Gaussian function to frequency distributions of RA andDEC of the cluster members. cluster RA (deg) DEC (deg) RA (deg) DEC (deg)Mean Shift Mean Shift Gaussian GaussianCollinder 261 + . − . + . ± . − . ± . + . + . + . ± . + . ± . + . + . + . ± . + . ± . + . + . + . ± . + . ± . Figure A5.
The King’s profile fitted to the surface density profile of the cluster members. The error bars are the 1 𝜎 poisson errors. The estimated values ofthe central surface density (A), the core radius ( 𝑟 c ), and the tidal radius ( 𝑟 t ) of each cluster are marked on the respective plots. MNRAS , 1–9 (2021) etermination of A + in Open Clusters Figure A6.
The ratio 𝑁 BSS / 𝑁 RGB is plotted against the radial distance in the units of 𝑟 c , for NGC 2682 (left panel) and NGC 7789 (right panel). To plot this,BSS and RGBs of the same magnitude range have been used. The error bars are estimated using propagation of errors. The dip statistic, D, and the p-valueestimated from the dip test for bimodality are marked on the plots.MNRAS000
The ratio 𝑁 BSS / 𝑁 RGB is plotted against the radial distance in the units of 𝑟 c , for NGC 2682 (left panel) and NGC 7789 (right panel). To plot this,BSS and RGBs of the same magnitude range have been used. The error bars are estimated using propagation of errors. The dip statistic, D, and the p-valueestimated from the dip test for bimodality are marked on the plots.MNRAS000 , 1–9 (2021) Khushboo K. Rao et al.
MNRAS , 1–9 (2021) etermination of A + in Open Clusters Figure A7.
The CMDs of 12 OCs with fitted PARSEC isochrones. BSS are shown as the blue filled squares and MS-TO stars are shown as black filled squares.The remaining sources are shown as grey dots.MNRAS000