A Census of Blue Stragglers in Gaia DR2 Open Clusters as a Test of Population Synthesis and Mass Transfer Physics
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A CENSUS OF BLUE STRAGGLERS IN GAIA DR2 OPEN CLUSTERS AS A TEST OF POPULATIONSYNTHESIS AND MASS TRANSFER PHYSICS
Emily M. Leiner and Aaron Geller Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, NorthwesternUniversity, 1800 Sherman Ave., Evanston, IL 60201, USA Adler Planetarium, Department of Astronomy, 1300 S. Lake Shore Drive, Chicago, IL 60605, USA
ABSTRACTWe use photometry and proper motions from Gaia DR2 to determine the blue straggler star (BSS)populations of 16 old (1-10 Gyr), nearby ( d < ∼ .
35 by 4 Gyr. Fitting stellar evolutionary tracks to these BSS, we find that their mass distributionpeaks at a few tenths of a solar mass above the main-sequence turnoff. BSS more than 0.5 M (cid:12) abovethe turnoff make up only ∼
25% of the sample, and BSS more than 1.0 M (cid:12) above the turnoff arerare. We compare this to Compact Object Synthesis and Monte Carlo Investigation Code (
COSMIC )population synthesis models of BSS formed via mass transfer. We find that standard populationsynthesis assumptions dramatically under-produce the number of BSS in old open clusters. We alsofind that these models overproduce high mass BSS relative to lower mass BSS. Expected numbers ofBSS formed through dynamics do not fully account for this discrepancy. We conclude that in order toexplain the observed BSS populations from Roche lobe oveflow, mass-transfer from giant donors mustbe more stable than assumed in canonical mass-transfer prescriptions, and including non-conservativemass transfer is important in producing realistic BSS masses. Even with these modifications, it isdifficult to achieve the large number of BSS observed in the oldest open clusters. We discuss someadditional physics that may explain the large number of observed blue stragglers among old stellarpopulations. INTRODUCTIONOne discovery of the first cluster color-magnitude di-agrams was that members stars often could be foundbrighter and bluer than the main-sequence turnoff ofthe cluster. These blue straggler stars (BSS) shouldhave evolved into giants millions or billions of years ago.Several explanations have been proposed to explain howblue stragglers may form. They may result from stellarcollisions during dynamical encounters (Leonard 1989);mergers of close binary systems, perhaps the inner bina-ries in triple systems driven to merger by Kozai cycles(Perets & Fabrycky 2009); or the result of mass-transferfrom a binary companion (McCrea 1964). It is likelythat all these mechanisms contribute to the blue strag-gler population to varying degrees.Blue stragglers are found in both open and globularclusters. Perhaps the most detailed study of a bluestraggler population to-date has been in the old opencluster NGC 188 (6 Gyr). 80% of the blue stragglers in [email protected] the cluster are spectroscopic binaries (Mathieu & Geller2009, 2015; Gosnell et al. 2015). Geller & Mathieu(2011) found that the statistical mass distribution of thecompanions to these blue stragglers peaks near 0.5 M (cid:12) ,suggestive of white dwarf secondary stars. Hubble SpaceTelescope UV photometry later confirmed that 7 of the20 blue stragglers in this cluster host hot white dwarfcompanions that are less than 400 Myr old, pointingto recent mass transfer from a red giant branch (RGB)or asymptotic giant branch (AGB) companion (Gosnellet al. 2014, 2015). The authors conclude that 2/3 of theblue straggler population in the cluster likely formedthrough this channel, since some of the blue stragglerswould likely have older, fainter white dwarfs below thedetection limit of the study.Such detailed observations are not available in otherclusters, but other studies also point to mass-transfer asthe likely origin for most blue stragglers in old clusters.In globular clusters, the lack of correlation between clus-ter density and blue straggler formation suggests that in-ternal binary evolution, including mass transfer, (ratherthan dynamical collisions) may be the dominant forma- a r X i v : . [ a s t r o - ph . S R ] J a n Leiner et al. tion channel there (Knigge et al. 2009). The significantpopulation of blue stragglers and other AFG-type post-mass-transfer binaries in the field also indicates masstransfer is likely a common formation channel (Mur-phy et al. 2018; Escorza et al. 2019; Carney et al. 2001,2005). These populations therefore offer a snapshot ofpost-mass-transfer outcomes in solar-type binaries. Yetdetailed studies of blue straggler populations, especiallyacross large populations including different cluster ages,densities, and metallicities, are still scarce. The mostcomprehensive catalog of blue stragglers in open clustersis given in Ahumada & Lapasset (1995) and updated inAhumada & Lapasset (2007), although the data qualityis inhomogenous and many of the clusters included havelimited astrometric membership information.The release of Gaia DR2, proper-motions and paral-laxes to 1.3 billions stars allows for kinematic member-ships for any nearby star cluster, enabling a much im-proved census of their blue straggler populations. Herewe use Gaia DR2 to select members of a sample of 16 old( > . ≤ ≤ (cid:12) ) stars. Thismass transfer physics is important not only for bluestraggler stars, but for many other astrophysical objectsthat result from binary evolution in low-mass stellar sys-tems (e.g. double white dwarfs, Type Ia supernovae andother transients resulting from white dwarf mergers, X-ray binaries, etc.)In this paper we first present our technique for select-ing clusters and determining membership (Section 2),then present analysis of the number of blue stragglersin each cluster and search for trends in these results(Section 3). We compare these observations to COSMIC (Breivik et al. 2020) population synthesis models todetermine how well these models reproduce observedcluster blue straggler populations (Section 4). Finally,we discuss discrepancies between our observations andmodels, and consider some of the mass-transfer physicsthat may need to be improved in population synthesis inorder to obtain more realistic results (Sections 5 and 6).In Section 7 we provide our summary and conclusions. CLUSTER SAMPLE2.1.
Initial Selection
We start by compiling a list of open clusters that isas complete as possible by combining multiple catalogsin the literature. Specifically, we use the Milky WayStar Clusters Catalog (MWSC Kharchenko et al. 2012, 2013; Schmeja et al. 2014; Scholz et al. 2015), Lynga(1995), Piskunov et al. (2008), Salaris et al. (2004), vanden Bergh (2006), Cantat-Gaudin et al. (2018), and wealso download and include all the open clusters fromWEBDA . From this catalog we select the clusters thathave masses > M (cid:12) , distances < . > Gaia Memberships
We perform a membership selection for all 39 clustersin our sample using using Gaia astrometric measure-ments as follows.First, we use literature values of cluster RA and Dec.We select from the Gaia catalog all stars with g < . <
5% from the next smaller aperture size,and use this as the aperture for the rest of our analysis.At this 5% level we are adding 0-3 RGB stars to mostclusters, depending on cluster size. At this level we areno longer adding many cluster stars, but will continuingto add field contamination as we increase the aperture.This threshold therefore allows us to select an aperturethat will contain most cluster members, while ensur-ing we don’t select an unnecessarily large aperture that https://webda.physics.muni.cz/ SS Gaia DR2 ≈ − hm ) from our catalog compilation inTable 1.The Gaia magnitude limit is G= 21.0, but incom-pleteness, astrometric errors, and photometric errors be-come larger at the faint end. We therefore select allstars in the Gaia DR2 database brighter than G= 18.This magnitude limit is inclusive of solar-type main se-quence stars in old clusters out to 3.5 kpc, but elim-inates faint stars where the astrometry may be poor.We flag sources with non-zero values for the parameter astrometric excess noise . These sources have largerthan expected errors in the Gaia astrometric solution,possibly due to the presence of a binary companion thatperturbs their motion. The errors on the parallax andproper motion of these stars may be larger than typi-cal, and therefore we exclude these sources from the fitsdescribed below. We do include these sources in ourmembership determinations, though their membershipsstatus is less certain.We then fit two 2D Gaussians to the proper motiondistribution. One Gaussian fits the field proper motiondistribution, and the other fits the cluster proper mo-tion distribution. For each source, we then determine acluster proper motion membership probability by divid-ing the value of the cluster Gaussian by the sum of thecluster and field Gaussians at the proper motion of thatsource.For each proper-motion member, we find the distancedetermined by Bailer-Jones et al. (2018). We fit a Gaus-sian function to the distribution of these distances. Wedetermine the cluster distance by taking the mean ofthis Gaussian. We note that there may be small offsetsin these distances due to complex Gaia systematics thathave not been well quantified. These errors, however,are expected to be small and do not significantly impactour membership analysis.For the color-magnitude diagrams presented below, weuse membership cutoffs of P PM >
50% to qualify as amember. We also require that the Bailer-Jones et al.(2018) distance to the star is < σ from the cluster meanto exclude any clear field interlopers. We do not use anyradial-velocity (RV) membership information, as GaiaRVs are only available for a small subset of bright starsin our sample.We also experiment with using proper motion mem-bership cuts of 20% and 80%. In most cases these dif-ferent cuts do not significantly change the blue stragglercounts in a cluster. In two cases where the field and clus-ter proper-motion distributions are not well separated,different cuts result in blue straggler populations thatvary by more than 1 σ from the value using a 50% cut. Figure 1 . A color-magnitude diagram of IC4651 showingGaia members of the cluster (gray points). Lines separatethe RGB stars (red), blue straggler stars (blue) and yellowstraggler stars (between blue and red, above the yellow line).In black we show a 1.4 Gyr MIST isochrone. We show thisfigure as an example; CMDs for the rest of the clusters inour sample are in the Appendix(Figure A.1).
In these two cases the blue straggler memberships aremore uncertain, and so we exclude these clusters fromour analysis. 2.3.
Reddening Corrections
In clusters with low reddening (E[bp-rp] < > -30 degrees). Thismap is an improvement over typical 2D dustmaps, whichprovide integrated values of extinction along a givenline-of-sight, but do not provide distance-dependent cor-rections. Green et al. (2019) uses far infrared emissionto map the 2D dust distribution across the sky, and thenincorporates stellar photometry and parallax measure-ments from Gaia and other surveys, using the observedreddening of these stars to infer the distance distribu-tion of the absorbing dust. We find reddening valuesfrom this dustmap to be in reasonable agreement withprevious literature estimates.In three clusters with considerable reddening, we com-pute star-by-star reddening corrections using this dustmap (Green et al. 2019). Using these corrections yieldsnarrower features in the cluster CMDs by correcting forthe effects of substantial differential reddening. We elect Leiner et al. to use these corrections for our analysis of NGC 7789,NGC 6939, and Berkeley 17. For these clusters we usestar-by-star reddening corrections in our CMDs, and wecompute the mean reddening measurement of all clustermembers and report this mean reddening value in Table1. Despite its high reddening, we find that the star-by-star reddening corrections in Collinder 110 broaden theCMD features considerably. Therefore, we adopt themean cluster value as the global cluster reddening forCollinder 110.In some cases, particularly for distant ( > > . < −
30 degrees) or does notdo an adequate job in correcting for differential extinc-tion. A large spread in color of the main sequence andgiant branch remains, which creates uncertainty in se-lecting blue straggler candidates near the main sequenceturnoff. We exclude 9 clusters with very high reddeningfrom our blue straggler analysis for this reason.2.4.
Sparse Clusters
Finally, we remove 2 clusters from our initial listthat are sparsely populated with fewer than 10 RGBstars. These clusters likely have over-estimated massesin the literature, and thus are not rich enough to relyon isochrone-derived ages or confidently identify bluestraggler populations.After this membership and reddening analysis, we areleft with 16 clusters from our sample with high-qualitymemberships and color-magnitude diagrams. These 16clusters are the basis of our analysis below. Propertiesfor these clusters can be found in Table 1.2.5.
Isochrones
We determine the appropriate MIST isochrone (Dot-ter 2016) for each cluster as follows. First, we determinethe metallicity for the cluster from literature sources(provided in Table 1). In most cases the metallicityis near solar, and we adopt solar metallicity isochrones.We adopt reddening and extinction parameters listedin Table 1 (see Section 2.3). We use a distance to thecluster of the mean of all distances to cluster members(Section 2.2). Based on these parameters, we pick theisochrone by eye from a grid of MIST models with aspacing of log( δt = 0 . BLUE STRAGGLER POPULATIONS IN GAIACLUSTERS3.1.
Cluster CMDs
We show an example color-magnitude diagram (CMD)for IC 4651 in Figure 1. CMDs for our entire sample canbe found in the Appendix ( Figure A.1). The CMDs arecorrected assuming the distance and reddening valuesgiven in Table 1. For each cluster, we also show a MISTisochrone using the age and metallicity listed in Table 1.We plot all Gaia members with gray points, and high-light blue stragglers (blue points), yellow stragglers (yel-low points) and red giants (red points). We discuss howwe select these stars below.3.2.
Defining the Blue Straggler Domain
There is substantial variation in the definition of bluestraggler stars between different observational works(Leigh et al. 2011).For this work, we define blue stragglers as being bluerthan the bluest point on the cluster isochrone. If thecluster has a blue hook, we set this limit at the bluestpoint in the isochrone. This excludes the region slightlyblue of the main sequence just below the blue hook; sometrue blue stragglers may reside in this area, but becausenormal photometric binaries may fall in this region andbecause the exact shape of the hook is sensitive to de-tailed properties of stellar models such as overshooting,we cannot confidently classify these stars as blue strag-glers. If the cluster does not have a blue hook, we re-quire that the blue straggler is at least 0.03 bluer thanthe bluest point on the isochrone, which allows for somephotometric scatter among normal main sequence stars.For all clusters, we also require that the blue straggleris no more than 1 magnitude below the cluster turnoff.This magnitude cut excludes white dwarf-main sequencebinaries, which sometimes are present at high tempera-tures and low luminosities well below the turnoff. It alsoallows us to consider similar CMD regions regardless ofcluster distance, as our magnitude limit of g= 18 is onlyabout 1 magnitude below the turnoff in our oldest andmost distant clusters. Visual examination shows thismagnitude cut rarely excludes possible blue stragglersfrom our analysis. We mark this blue straggler regionwith the blue dashed lines in Figure 1.Some stars, called “yellow stragglers” or “evolved bluestragglers” have been found in clusters (Leiner et al.2016; Landsman et al. 1997). These stars are signifi-cantly brighter than the main sequence, but redder thanblue stragglers, falling in between the blue straggler re-gion and red giant branch. These stars are thought to beblue stragglers that have evolved into giant or subgiantstars. We define the yellow straggler region to be morethan 0.75 magnitudes above the brightest point on the
SS Gaia DR2 Cluster Distance [Fe/H] Age E ( bp − rp ) r hm r N BSS N YSS N RGB (pc) (pc) (pc)NGC 752 440 ±
20 0.0 1.4 Gyr 0.04 4.2 16 1 0 15NGC 7789 2080 ±
190 0.0 1.4 Gyr 0.38 6.4 20 10 0 165Collinder 110 2200 ±
300 0.0 1.6 Gyr 0.50 5.4 20 8 0 74NGC 6939 1850 ±
140 0.0 1.6 Gyr 0.44 2.3 16 4 0 44IC 4651 920 ±
50 0.0 1.8 Gyr 0.18 1.7 8 4 1 24NGC 2506 2900 ±
500 –0.25 2.0 Gyr 0.08 5.0 16 9 0 72NGC 6819 2600 ±
400 +0.25 2.5 Gyr 0.10 2.4 16 17 0 76Ruprecht 147 300 ±
10 0.0 2.8 Gyr 0.08 2.2 8 3 0 10Ruprecht 171 1530 ±
150 +0.4 2.8 Gyr 0.26 1.8 8 5 0 33NGC 6253 1670 ±
160 +0.4 3.2 Gyr 0.28 1.6 8 16 1 55Berkeley 98 3400 ±
160 0.0 3.5 Gyr 0.22 2.3 8 5 1 21M67 840 ±
60 0.0 4.0 Gyr 0.05 2.5 12 16 0 43NGC 2243 3500 ±
700 –0.5 4.5 Gyr 0.06 2.6 12 14 0 34NGC 188 1830 ±
170 0.0 6.3 Gyr 0.12 4.6 16 16 0 60Berkeley 39 3200 ±
800 –0.25 8.0 Gyr 0.10 2.1 12 24 4 70Berkeley 17 2800 ±
750 –0.25 10 Gyr 0.62 2.7 12 20 2 59
Table 1 . Parameters of open clusters in our sample adopted for this study, including the mean Gaia distance of all members(Bailer-Jones et al. 2018), isochrone age and metallicity (Dotter 2016), average cluster reddening, cluster half-mass radius (r hm ),and radius for membership selection (r) . We also give blue straggler, yellow straggler and RGB counts. subgiant branch of the isochrone (the brightest magni-tude expected for the combined light of an equal massbinary at the cluster turnoff), with a color redder thanthe defined blue straggler region and bluer than the baseof the RGB (see Figure A.1). These stars are rare, andin the remainder of our analysis we combine the bluestraggler and yellow straggler populations to determine N BSS .We classify stars that are brighter and redder thanthe base of the RGB on our isochrone as standard RGBstars.Using this approach we count the total number of redgiants, blue stragglers and yellow stragglers within thecolor-magnitude diagrams of our 16 open clusters. Wereport the number of such stars in Table 1, along withthe aperture we used to determine these star counts.We plot the observed ratio of N BSS N RGB in the left panelof Figure 5. We note that the results of these starcounts are not very sensitive to the aperture adopted,and adjusting the aperture size up or down one stepresults in minimal changes to our results (i.e. < N BSS N RGB < Combined CMDs
We show in Figure 2 combined color-magnitude dia-grams made by plotting the CMDs of every cluster inour sample, shifting the photometry by a small amount δG and δ ( bp − rp ) such that the main sequence turnoffsof each cluster align. This allows us to compare the po-sitions of the blue straggler stars relative to the main sequence turnoff in a much larger sample than any onecluster alone. For these plots, we split our sample into3 bins by age: 1-2 Gyr, 2-4 Gyr, and more than 4 Gyr.This allows us to compare clusters with similar turnoffmasses, and splits our sample into 3 groups containing4-5 clusters each. For clarity, we exclude clusters thathave 1 or fewer blue stragglers.We see in Figure 2 that for intermediate age clustersthere appear to be two clumps of blue stragglers, onebrighter clump about 1.5 magnitudes above the turnoff,and one fainter clump stretching from the MSTO toabout 1 magnitude above. A less prominent, narrowergap may also be present in the old clusters ( > Blue Straggler Masses
We determine masses for each blue straggler by fittingMIST evolutionary tracks (Dotter 2016) to the CMDpositions of the blue stragglers in our clusters using themetallicity and cluster parameters given in Table 1. Weshow histograms of the derived blue straggler massesin Figure 3, expressing them in terms of the differencebetween the turnoff mass of the cluster and the bluestraggler mass ( δM = M BSS − M MSTO ).Clusters of all ages show a distribution skewed towardslower mass blue stragglers, with a peak 0.3 M (cid:12) above
Leiner et al.
Figure 2 . We show a combined CMDs of the youngest open clusters ( < > the turnoff among the entire sample of 171 blue strag-glers. Only 5 stars in the whole sample have masses (cid:38) . (cid:12) above the turnoff, and no blue stragglers aremore than 1.5 M (cid:12) above the turnoff. For context, wenote that the maximum δM expected for conservativeAGB mass transfer onto an MSTO star would lead to aBSS with δM (cid:104) . δM (cid:104) . δM (cid:104) . Blue Straggler Fractions
Given that more blue stragglers should be expectedin larger clusters, we scale the number of blue stragglersobserved in the cluster by the number of RGB stars inthe cluster. We use this scaling because RGB stars areeasy to count in the cluster, and because the RGB popu-lation traces the number of stars near the main-sequenceturnoff. This is likely to be the progenitor population ofthe blue straggler stars formed through mass-transfer.We show this ratio as a function of cluster age in Fig-ure 5 (black points). We find that the blue stragglerfraction increases with cluster age from 1-4 Gyr, andthen plateaus at a near constant N BSS N RGB ≈ .
35 at laterages. We find no trends with cluster distance or redden-ing. Most of our sample consists of clusters of near-solarmetallicity, and we do not have enough high and lowmetallicity clusters to determine whether there are dif- ferences in blue straggler fraction in clusters of differentcompositions.Ahumada & Lapasset (1995) also found in their cat-alog that old clusters have more blue stragglers thanyoung clusters. With our result, we confirm that thistrend holds with the more accurate photometry, astro-metric memberships, and cluster parameters enabled byGaia, and provide a more well-sampled picture of theevolution with age for clusters older than 1 Gyr.In the next section, we investigate whether populationsynthesis models can produce the blue straggler popu-lations observed or the trend seen with increasing bluestragglers in older clusters. POPULATION SYNTHESIS WITH
COSMIC
We use the population synthesis code
COSMIC (Breiviket al. 2020) to model cluster blue straggler popula-tions and determine if these models reproduce the trendwe see between blue straggler population and clusterage. COSMIC implements the population synthesis codeBinary Stellar Evolution (
BSE
Hurley et al. 2002) in-side an easy-to-use python wrapper, and includes someupdates to the binary evolution physics.4.1.
Prescriptions for q crit We focus our investigation on the impact of the choiceof q crit , the mass ratio required for mass-transfer stabil-ity. Population synthesis models do not compute fullstellar structure models to determine the outcome ofmass transfer in a binary system. Instead, these mod-els use fitting formulae to predict whether mass transferwill proceed stably– resulting in significant mass trans-fer and possibly producing a blue straggler– or unstably–resulting in a common envelope (CE), little mass accre- https://github.com/COSMIC-PopSynth/COSMIC SS Gaia DR2 Figure 3 . A histogram of the number of blue stragglers in our cluster sample binned by the difference between the blue stragglermass and the turnoff mass ( δM ). We show from left to right the sample of young clusters (1-2 Gyr), intermediate aged clusters(2-4 Gyr), and old clusters ( > (cid:12) (cid:12) inthe 2-4 Gyr clusters, and 1.3-1.0 M (cid:12) in a 4-10 Gyr cluster. Figure 4 . A plot of the critical mass ratio for mass-transfer stability ( q crit ) as a function of the donor core mass. We assumea donor mass of 1.2 M (cid:12) on the left and 2.0 M (cid:12) on the right. The colored lines represent different q crit prescriptions. The grayshaded region shows the region where the donor is more massive than the accretor, and the core mass of the donor is consistentwith expectations for a low mass RGB or AGB star. This is the region where stable mass transfer would have to occur in orderto produce blue stragglers. tion by the companion, orbital shrinkage, and possiblya merger of the binary system. For giant donor stars,this prescription typically depends on the mass fractionof the donor star’s core, M c1 /M , and the mass ratio ofthe binary system ( q = M M ).Because many binaries will interact as the donor starexpands into an RGB or AGB star, the critical massratio for mass transfer from a giant donor is essential tounderstanding the outcomes of mass transfer. This massratio for stability has been the subject of much debate in the literature, and varying prescriptions have beenimplemented across different population synthesis codes.There are a large range of prescriptions in use that canyield dramatically different mass-transfer outcomes.We show in Figure 4 four examples of stability crite-ria, all of which we test in our COSMIC models. In thisexample, we show these criteria for both a 1.2 M (cid:12) giantdonor star on the left, and a 2.0 M (cid:12) giant donor on theright (though in our population synthesis models, weapply these criteria at the modeled mass of the star).
Leiner et al.
These masses are at the upper and lower range of donorstars we would expect in our cluster sample given ourage range of ∼ −
10 Gyr. We implement these criteriafor giant star donors only (in
BSE , kstar types 3, 4, 5,6), and use the default Hurley et al. (2002) prescriptionfor all other stellar types. We describe each of these fourmodels briefly below.4.1.1. Hurley et al. (2002) In BSE , the default option for q crit for mass-transferring binaries with a giant donor star is given bythe equation: q crit = (cid:34) . − x + 2 (cid:18) M c1 M (cid:19) (cid:35) / .
13 (1)Here M C1 is the core mass of the donor star, M is thetotal mass of the donor star, and x is the helium massfraction of the star. For mass transfer to proceed stablyin population synthesis models that use this presciption, M donor M accretor > q crit . This is Equation 57 in Hurley et al.(2002) and derived by fitting results from detailed stellarevolution calculations. Note that this model assumesmass transfer is conservative (i.e., all the material lost bythe donor is accreted by the companion and no angularmomentum is lost from the system). This equation isshown in Figure 4 with the pink line.4.1.2. Hjellming & Webbink (1987)
An alternative stability criterion is given by (Hjellm-ing & Webbink 1987). This equation is determined an-alytically using condensed polytropes as stellar models.Like the Hurley et al. (2002) expression, this assumesfully conservative mass transfer. This model is shownwith the red line in Figure 4: q crit = 0 .
362 + (cid:20) (cid:18) − M c1 M (cid:19)(cid:21) − (2)with variables defined as in Equation 1.4.1.3. Non-conservative Mass Transfer
Woods & Ivanova (2011) demonstrate that stabilitycriteria like the Hurley et al. (2002) and Hjellming &Webbink (1987) prescriptions underestimate the stabil-ity of mass transfer because of the assumption thatmass transfer will be conservative. Woods & Ivanova(2011) re-derive the stability predictions including amass transfer efficiency parameter, β , that representsthe fraction of material lost by the donor that is ac-creted by the companion. When β is small (i.e. masstransfer is highly non-conservative), mass transfer canproceed stably for higher mass ratio systems.Here we use a Hjellming-like model that approxi-mates the non-conservative stability criteria of Woods & Ivanova (2011) assuming β = 20% over the relevantrange of core masses for low-mass giants (M core < . q crit = 0 . (cid:20) . (cid:18) − M c1 M (cid:19)(cid:21) − (3)This stability criterion is shown with the cyan linein Figure 4. We note that while we use this relationthat assumes non-conservative mass transfer to deter-mine whether a binary will evolve to common envelopeor stable mass transfer, BSE determines mass transferefficiency independently from stability calculations bycomparing the Kelvin-Helmholtz timescale of the accre-tor to the mass-transfer time scale (Hurley et al. 2002).That is, though we assume 20% efficiency in this rela-tion,
BSE will still assume conservative mass transferunless the calculated mass loss rate exceeds a thermaltimescale of the accretor. A fully consistent treatment ofmass transfer with this model would couple the calcula-tion of the mass-transfer stability with a determinationof mass-transfer efficiency, but this is beyond the scopeof this paper. For now, we adopt this criterion simply toprovide an upper limit on how many blue stragglers maybe created if allowing for non-conservative mass transferwith a Hjellming-like law.4.1.4.
L2/L3 Overflow
Pavlovskii & Ivanova (2015) argue that mass transferfrom giant donors may be much more stable than typi-cally thought because a common envelope will not com-mence unless the donor exceeds its Roche lobe to such anextent that mass-loss occurs through the outer Lagrangepoints. They perform hydrodynamic modeling to deter-mine the q crit values relevant for this L2/L3 overflowscenario. They find that for evolved giants with deepconvective envelopes, q crit ranges from 1.5-2.2, twice thecritical mass ratio predicted by polytrope models likeHjellming & Webbink (1987). Based on these results,we run population synthesis models using q crit = 1 . (cid:12) in these de-tailed simulations. This model is shown with the orangeline in Figure 4.4.1.5. No Common Envelope
We also consider a scenario in which mass transferfrom an RGB or AGB donor always results in stablemass transfer. In this case, we set q crit = 100 for allred giant, red clump, AGB, and TP-AGB donor starsin COSMIC . While this is not a realistic scenario, it setsan upper limit on how many blue stragglers may be cre-ated through mass transfer in
COSMIC if all binaries thatinteract during their giant evolution evolve through sta-ble mass transfer. This model is not shown in Figure 4
SS Gaia DR2
Model Parameters
We implement each of the mass-transfer stability con-ditions discussed above in a
COSMIC population synthe-sis model. For all of our calculations, we implement
COSMIC ’s multidim sampling feature to select an ini-tial population of binaries. We seed our models with aninitial population of 100,000 binaries, all solar metallic-ity, and use the delta burst star formation model inwhich all stars are born at the same time. Binary massratios and orbital parameters are selected from the dis-tributions reported in Moe & Di Stefano (2017). Thisbinary population is based on observations of large pop-ulations of binaries in the field, but investigations ofopen cluster binary populations have found similar re-sults (Geller et al. 2021, 2013; Geller & Mathieu 2012;Duchˆene & Kraus 2013)). Very wide binaries that existin the field will be dynamically disrupted in a clusterenvironment, but as these binaries are much wider thanwould be expected to interact and transfer mass, thisdifference should not impact blue straggler production.For all models, we lower the Reimer’s wind massloss coefficient from the default value of η = 0 . η = 0 .
1, consistent with asteroseismic measurementsof RGB mass loss in older open clusters (Miglio et al.2012). Wind mass transfer occurs via standard Bondi-Hoyle wind accretion in our models using the defaultBondi-Hoyle accretion factor of a w = 1 .
5. For all otherparameters, we use the default
COSMIC values.We note that this model population of 100,000 bina-ries is far larger than a typical open cluster in our sam-ple, but by scaling the number of blue stragglers to thenumber of RGB stars in our models, the ratio N BSS N RGB is di-rectly comparable to the observations. We discuss belowhow we define the blue straggler and RGB populationsin our models.4.3.
Model Blue Straggler Selection
From our simulations, we select stars as blue strag-glers that have temperatures >
5% larger than the mainsequence turnoff, and no more than a factor of 10 in lu-minosity below the main sequence turnoff. We use thisluminosity cut to remove stars such as MS+hot WD bi-naries, and to be consistent with photometric cuts wemake in our observational sample, and find that it ex-cludes only a few possible blue stragglers in the models.We also count yellow stragglers in our model, selectingthem to be at least twice the luminosity of the mainsequence turnoff, with T eff <
5% larger than the main-sequence turnoff and T eff > the maximum temperaturefound among subgiants in our model ( kstar =2) thathave not been through any prior interactions. 4.4. Model RGB Selection
Since our simulations contain only binaries, not singlestars, and includes very wide binaries that would be pastthe hard-soft boundary in an open cluster, we cannotsimply scale our BSS count to the number of RGB starsin our simulations as we do for the observations. Instead,we select all binaries from our simulation containing anRGB or red clump star ( kstar = 3 or 4) with orbitalperiods less than 10 ,
000 days. The spectroscopic binaryfraction in both the field and in open clusters withinthis period domain is found to be ∼
25% (Geller et al.2009; Milliman et al. 2014; Leiner et al. 2015; Moe & DiStefano 2017). We multiply our selection of red giantbinaries by 4 to calculate the total number of both RGBsingle and binary stars that would be in this population.We use this binary-fraction-corrected total to scale theblue straggler counts and compare to observations. RESULTS5.1.
Blue Straggler Fractions Using Different MassTransfer Stability Prescriptions
In the left panel of Figure 5 we show the ratio of thenumber of blue straggler to the number of RGB stars inour 5 models compared to the observations. All modelsunder-produce blue stragglers in the older clusters. Allmodels also show a similar shape, with a large increasein the ratio of blue stragglers to RGB stars between 1Gyr and 2 Gyr, followed by a more gradual decrease andflattening. This shape is set by a few things: the for-mation frequency of blue stragglers, the lifetime of bluestragglers, and number of RGB stars at a given age. Inthe models, the steep rise from 1-2 Gyr is set primarilyby a decreasing number of RGB stars, as at young agesRGB stars have longer lifetimes. After 2 Gyr in the mod-els, the number of RGB stars flattens and the shape ofthis curve is dominated by the changing number of bluestragglers. The number of blue stragglers in the mod-els declines slowly with increasing age. The lifetimes ofblue stragglers in the older models is generally longer,since the blue stragglers are lower mass and have longermain-sequence lifetimes. Therefore, this decline reflectsa decreasing production rate of blue stragglers. Thisdecreasing production rate is because 1) more massivestars reach larger radii on the AGB, and therefore widerbinaries evolve through mass transfer. 2) Wind accre-tion contributes more to younger models, and 3) masstransfer on the subgiant branch or Hertzprung gap isquite stable in BSE ( q crit = 4 . q crit prescriptionmore specifically below.We find that the two most commonly used mass trans-fer prescriptions of Hjellming & Webbink (1987) and0 Leiner et al.
Figure 5 . On the left, we show the observed ratio of the number of blue stragglers to the number of RGB stars in open clusters(black points) compared to
COSMIC population synthesis models. We show models with varying critical mass ratios for stabilitywith colors as in Figure 4. For these models, we use the default Bondi-Hoyle wind prescription. On the right, we show thesame set of models, but increase the Bondi-Hoyle wind accretion by a factor of 10. We discuss this enhanced wind scenario inSection 6.2.1.
Hurley et al. (2002) (red and pink lines in Figure 5)produce very few, if any, blue stragglers in our simula-tions. This is consistent with the youngest clusters inour study ( < < > ∼ .
05 compared to the observationed ratio of 0.3-0.4.Our model using a constant q crit = 1 .
8, approxi-mately the value found in the L2/L3 overflow modelsof Pavlovskii & Ivanova (2015), produces a higher BSSto RGB ratio of ∼ .
15 in clusters older than 2 Gyr.While much improved, this is only half the rate observedin older clusters. In clusters younger than 2 Gyr, thismodel produces approximately the observed number ofblue stragglers.Finally, our model with q crit = 100, effectively sendingall mass transferring binaries with RGB or AGB donorsthrough stable mass transfer, produces a blue stragglerto RGB ratio of ≈
25% in the oldest clusters. This isstill slightly below the observed fraction for clusters withages > < HR Diagram Distribution of Blue Stragglers
We produce some synthetic Hertzprung-Russell (HR)diagrams from these
COSMIC runs for comparison toreal clusters. Figure 6 shows an example HR diagramfor the L2/L3 overflow model. For readability, the HRdiagrams for the rest of the q crit prescriptions are givenin the Appendix (Figure A.2– Figure A.5). These plotsshow the results of COSMIC population synthesis runsfor each of our stability criteria using 100,000 binariesat three different age snapshots: 1.5 Gyr, 3 Gyr, and7 Gyr. In all plots we show two versions of these HRdiagrams; one with points colored by the stellar type ofthe primary and one colored by the orbital period of thebinary. We note that this sample size is much largerthan the typical open cluster in our sample, which havetypical stellar masses of a few thousand M (cid:12) . The totalnumber of blue stragglers in these models is thus notdirectly comparable to a cluster, but provides a clearerpicture of the distribution of blue straggler propertiesproduced in each model.In nearly all cases, these HR diagrams show that theblue stragglers in our models have companions that are SS Gaia DR2 Figure 6 . HR diagrams showing the results of COSMIC population synthesis simulations of 100,000 binaries using the L2/L3overflow mass transfer stability prescription ( q crit = 1.8). We show three snapshots in age: 1500 Myr (left), 3000 Myr (middle),and 7000 Myr (right). In the top panels we plot stars colored by the stellar type of the primary (i.e., the originally more massivestar in our simulation). In the bottom panels we show the same simulations, but we color the stars by orbital period. Graypoints indicate systems that are now single stars (i.e., due to a binary merger) and thus do not have orbital periods. Thedashed lines indicate the boundaries of the blue straggler domain. We show the same plots for the other q crit prescriptions inthe Appendix. either Helium white dwarfs (green points) or carbon-oxygen white dwarfs (orange points). In the youngestmodels (1.5 Gyr plots), a few blue stragglers are still ac-tively accreting and contain various types of giant pri-maries (blue or cyan points) or have been stripped oftheir hydrogen envelopes and are now Helium stars.These HR diagrams also show that the Hurley et al.(2002) and Hjellming & Webbink (1987) prescriptionsproduce few blue stragglers, particularly at older ages.Nearly all blue stragglers produced have low-mass He-lium white dwarf companions in short-period orbits onof tens of days or less (see Figure A.2 and A.3). Thisstands in contrast with the orbits determined for binaryblue stragglers in old ( > q crit = 1.8; Figure 6) produce similar distribu-2 Leiner et al. tions to the non-conservative prescription, but producemany more blue stragglers. Again, these models produceblue stragglers with orbits ranging from a few to thou-sands of days, most with C/O white dwarfs or He whitedwarf companions. A similar gradient in orbital periodcan be observed, with longer period blue stragglers ap-pearing fainter than shorter-period blue stragglers.The HR diagram produced by eliminating commonenvelope evolution ( q crit = 100; Figure A.5) again pro-duces lots of blue stragglers with orbits ranging froma few to thousands of days, most with Helium or C/OWD companions. The relationship between brightnessand orbital period is less apparent, however, and bluestragglers near the MSTO have a mix of Helium whitedwarf and C/O white dwarf companions and possess arange of orbital periods.5.3. Model Blue Straggler Mass Distributions
In Figure 7, we show the distribution of masses in eachof the snapshots discussed above: 1.5 Gyr, 3 Gyr, and 7Gyr for each of our mass transfer prescriptions.We see that the Hurley et al. (2002) and Hjellm-ing & Webbink (1987) produce similar mass distri-butions, with more blue stragglers produced near theMSTO and more massive blue stragglers produced morerarely. However, these models produce so few blue strag-glers the comparison to the observed mass distributionsdoes not yield much new insight. These models under-produce blue stragglers of every mass.The non-conservative prescription under-produceslow-mass blue stragglers near the MSTO and over pro-duces the more massive blue stragglers compared to theobserved populations across all ages. This limitation isparticularly noticeable in the youngest snapshot, wherethe models produce many massive blue stragglers morethan 1.0 M (cid:12) above the turnoff. Blue stragglers thisbright are rarely observed in the real clusters. Notably,the model mass distribution at 3.0 Gyr does show apaucity of blue stragglers near δ M= 0 .
5, a feature alsoseen in the mass distribution of the observations.The q crit = 1.8 prescription produces a double-peakedmass distribution at 1.5 Gyr, which is not observedin the true 1-2 Gyr population. This model againover-produces very massive blue stragglers, and under-produces intermediate-mass blue stragglers with massesof a few tenths M (cid:12) above the turnoff. At 3 Gyr, themodel under-produces blue stragglers near the MSTOand does not produce the double peaked mass distribu-tion observed in the real clusters. At 7 Gyr, the massdistribution reproduces the observed distribution fairlywell. If there is a small mass gap around 0.5 M (cid:12) (0.1M (cid:12) in width) in the old clusters, it is not reproducedby the simulations.The no common-envelope prescription produces mass distributions very similar to the q = 1 . q crit models of q = 1 . q = 100 do the best job producing the observedmass distributions in old clusters, but both modelsproduce too many high-mass blue stragglers and toofew low-mass blue stragglers at young and intermediateages. The non-conservative model is the only one whichmay produce a mass-gap like the one observed in theintermediate-age cluster, though the peak at lower BSSmasses is smaller than observed. This model, too, un-derproduces low mass blue stragglers and overproduceshigh mass blue stragglers relative to the observations.Taken together, the HR diagrams and mass distribu-tions of our models reveal two things about the BSSpopulation models. 1) Default assumptions about mass-transfer stability far under-produce observed blue strag-glers in old clusters. Even setting q crit =100 is unable toproduce the observed numbers of blue stragglers in realold open clusters. 2) Models produce relatively morehigh mass blue stragglers than observed, but don’t pro-duce enough low-mass blue straggler near the turnoff.We discuss possible reasons for these discrepancies be-low. DISCUSSION6.1.
The Contribution of Stellar Dynamics, Collisionsand Mergers
We have focused thus far on mass-transfer from RGBand AGB donors as the dominant blue straggler forma-tion channel, but several other formation mechanismshave been proposed that likely contribute to the ob-served blue straggler population including stellar col-lisions, binary mergers, and interactions in triple starsystems. 6.1.1.
Stellar Dynamics
Dynamics have long been thought to produce bluestragglers in dense cluster environments during stel-lar collisions (Leonard 1989). N -body simulations ofopen clusters do produce blue stragglers via this channel(Geller et al. 2013; Hurley et al. 2005). A particularlyrelevant N -body model was done of NGC 188 (Gelleret al. 2013), an old (6 Gyr) cluster in our sample witha large blue straggler population. This N -body simula-tion uses a BSE framework to perform stellar and binaryevolution calculations, and thus uses much of the sameinput stellar physics as
COSMIC . This model can thereforegive us some insight into how the addition of dynamicsmight modify our results.
SS Gaia DR2 Figure 7 . We show the mass distributions of blue stragglers produced in our models. On the x-axis, we plot the differencebetween the cluster turnoff mass and the blue straggler mass. On the y-axis we plot probability density distribution (PDF) ofthe blue straggler masses. Each row is a different model, colored as in Figure 4 (pink for Hurley et al. (2002); red for Hjellming& Webbink (1987); cyan for non-conservative; orange for q crit = 1.8; teal for q crit = 100). For comparison, we also plot thenormalized number of blue stragglers from our Gaia cluster sample in black. Leiner et al.
Geller et al. (2013) produce a mean of 6 blue stragglersin their cluster simulation, about 1/3 of their observednumber of blue stragglers (20; Mathieu & Geller 2015)in NGC 188. At old ( > ∼ ∼ ∼ (cid:46)
15% (3/20) ofthe blue straggler population in our older clusters, andperhaps even fewer in the youngest clusters.6.1.2.
Main Sequence Mergers
Nearly all the blue stragglers in Geller et al. (2013) N -body models of NGC 188 not produced via collisions areproduced via mergers of two main sequence stars. Thisamounts to 2-4 blue stragglers across all ages in theirmodel. Thus Geller et al. (2013) predict that mergerscan account for ∼ ≥ orb ≤ . COSMIC ), other prescriptions forangular momentum loss, or altering the stability criteriaadopted for mass transfer from main-sequence or sub-giant donors (for which we use the default Hurley et al.2002 settings).We conclude that our models under-estimate the con- tribution of main-sequence mergers compared to someinvestigations in the literature. Considering the resultsof Geller et al. (2013) and Andronov et al. (2006), merg-ers might make up an additional 10-30% of the observedblue straggler populations.6.1.3.
Triple Systems
Triple systems, which are not included in our
COSMIC models, may also contribute to the blue straggler pop-ulations. Kozai cycles plus tidal friction (“KCTF”) inhierarchical triples may lead to the merger of the innerbinary system, thus forming a blue straggler (Ivanovaet al. 2008; Perets & Fabrycky 2009), and dynamicallyinduced collisions may also be more likely in triple sys-tems than in binaries (adding a wider tertiary star in-creases the cross section for interaction of the system).Geller et al. (2013) allow for the dynamical formationof triples in their NGC 188 simulation, and the modelattempts to account for Kozai cycles and tidal friction.Geller et al. (2013) did not seed their canonical NGC 188model with primordial triples, and find that insufficienttriples form dynamically, as compared to observations.In this canonical model, essentially no blue stragglers areformed from triples. Geller et al. (2013) also producea few simulations with varying numbers of primordialtriples, whose masses and orbital periods were specif-ically chosen to facilitate blue straggler production atthe age of NGC 188 and with the appropriate period dis-tribution. Even with the maximum number of possibletriples (assigning every available binary in the appropri-ate mass range to a triple), Geller et al. (2013) find only1 - 2 additional blue stragglers created above the canon-ical model. However, they also note that the efficiencyof the KCTF mechanism in the nbody6 code may beunderestimated, and/or the implementation may needimprovements.Most short-period binaries (those that would lead tomain sequence – main sequence mergers) are found tohave hierarchical triple companions, with the frequencyof tertiary companions dropping off for inner binarieswith wider orbits (Raghavan et al. 2010; Moe & DiStefano 2017). Therefore, there is considerable overlapbetween those systems that may merge due to Kozaioscillations in triples, and those that may merge viaa standard main-sequence mergers in close binaries asdiscussed above. The Perets & Fabrycky (2009) KCTFmechanism, therefore, does not boost blue straggler pro-duction substantially above the estimates of Andronovet al. (2006). The important difference is that whenthese inner binaries merge, they will have a wide binarycompanion (most likely a low-mass main sequence star).We conclude that triples may enhance the numbers ofstars produced through mergers or dynamics. However,this effect is already largely included in the estimates of
SS Gaia DR2
Correcting Straggler Production for StellarDynamics, Collisions and Mergers
In the analysis above, we estimate that ∼
15% of bluestragglers may be produced by stellar collisions, and an-other ∼ −
30% may arise from mergers. Overall, thismeans 25-45% of all blue stragglers in our open clus-ter sample may arise from mergers and dynamics. Thisleaves 55-75% of the blue straggler population to haveformed via mass transfer from a giant donor.As we explain in the introduction, significant observa-tional evidence points to mass-transfer as an importantformation mechanism for blue stragglers in clusters andin the field. To revisit one important finding, Gosnellet al. (2014, 2015) use Hubble Space Telescope UV pho-tometry to search for hot white dwarf companions to theblue stragglers in NGC 188. They detect 7 white dwarfcompanions, all hot enough to have formed in the last400 Myr. Given that older, cooler white dwarfs are un-detectable with their method, they estimate that withan incompleteness correction, two-thirds of the NGC 188blue stragglers have white dwarf companions. This im-plies that 2/3 of the blue straggler population of thiscluster formed via mass transfer from a giant donor.This study only looks at the population of one opencluster, but it is the most direct measurement availablefor the fraction of blue stragglers formed through masstransfer. This result is in line with our estimate fromtheoretical models that 55-75% of blue stragglers formedfrom mass transfer.If we assume that the blue stragglers formed in ourCOSMIC models account for only 55% of the real bluestraggler population, then the L2/L3 overflow modeland the no CE model are consistent with the observedblue straggler fraction found in old clusters, but signifi-cantly over-predict the number of blue stragglers foundin 1-2 Gyr clusters (Figure 5). The non-conservativemodel would still underestimate the number of bluestragglers in old clusters by a factor of 3, but wouldbe approximately correct for the 1-2 Gyr clusters. TheHurley et al. (2002) and Hjellming & Webbink (1987)prescriptions still produce far too few blue stragglers atall ages.The underproduction of blue stragglers through masstransfer was also noted by Geller et al. (2013) in theirNGC 188 model, that used the Hjellming & Webbink(1987) q crit equation. Geller et al. (2013) also investigatethe effects of using various q crit approaches, though not directly in their canonical NGC 188 N -body model, topotentially explain both the model’s lack of blue strag-glers and also the model’s over-abundance of long-period( ∼ N -body model would pro-duce roughly the correct number and orbital period dis-tribution of the observed NGC 188 blue stragglers. Ourno CE model investigates this possibility as well, and in-deed does produce the predicted ∼
50% of blue stragglersthrough mass transfer for old open clusters.We conclude that correcting for other productionmechanisms may bring the L2/L3 model and no CEmodel into agreement with the total number of bluestragglers produced in older clusters, but using the samecorrection in young clusters results in too many bluestragglers. Perhaps the relative number of blue strag-glers produced via collisions and mergers varies with age.(Indeed the Geller et al. 2013 NGC 188 model suggeststhat collisions may only be relevant for clusters at (cid:38)
Additional Mechanisms that can Contribute to theNumber of Blue Straggler
Wind Mass Transfer
Recent modeling has showed that mass transfer mayoccur at wider separations than expected through typi-cal Roche lobe overflow (e.g. Abate et al. 2013; Chenet al. 2017; Mohamed & Podsiadlowski 2007). Thismechanism has been suggested to be particularly im-portant in creating barium stars and carbon enhancedmetal poor stars with s-process enrichment (CEMP-s).These stars are relatives of blue stragglers and are iden-tified as products of mass-transfer because of observedenhancements in s-process elements like barium that arelikely the result of accretion of s-processed enriched ma-terial from an AGB donor star (Boffin & Jorissen 1988).In the context of s-process enhanced stars like these,hydrodynamic models show that the slow wind drivenfrom the star during the final phases of AGB evolutioncan, in some circumstances, be accreted by the compan-ion with efficiencies above typical Bondi-Hoyle accretion(e.g. Abate et al. 2013; Saladino et al. 2019), in some6
Leiner et al.
Figure 8 . Same as Figure 7, but showing model blue straggler mass distributions when increasing the Bondi-Hoyle windaccretion by a factor of 10.
SS Gaia DR2
COSMIC/BSE does not include a Wind Roche Lobeoverflow mechanism. Our models use only standardBondi-Hoyle accretion, and thus may underestimate thenumber of wide blue straggler binaries forming as a re-sult of wind accretion from an AGB companion.To test the impact of our wind assumption on theresulting blue straggler populations, we re-ran all ourmodels using an enhanced Bondi Hoyle wind accretionin which we raise the constant by a factor of 10 ( a w = 15,raised from a w = 1 . ∼ − q crit still overproduce high-mass blue stragglers (Figure 8). The non-conservativeprescription with wind matches the observed mass dis-tribution particularly well, showing a double peaked dis-tribution in intermediate and old populations like theobservations (see Figure 8, cyan plots).We suggest that wind mass transfer may be a viablemechanism to produce more blue stragglers close to theturnoff. However, in our COSMIC models wind masstransfer seems to increase the population of blue strag-glers as much or more in the young clusters as in the oldclusters. A larger increase in blue straggler productionin the old clusters is needed to resolve the discrepancywith observations. A more careful treatment of windmass transfer in population synthesis models is neededto examine this possibility more closely.6.2.2.
Binary “Twins” and Blue Straggler Lifetimes
In our 1 −
10 Gyr models, the binary systems thatwill interact and transfer mass will have primaries withmasses of 1 − (cid:12) and orbital periods (cid:46) . < M M < . P orb < . < M < .
0, 20%of these systems are twins. In the L2/L3 overflow model( q crit =1.8), twins represent 42% of the initial binaries inour sample that are likely to go through stable Rochelobe overflow because they have mass ratios M M < . < M < (cid:12) and P orb < M M = 0 . .
95, we wouldproduce more blue stragglers from twin binaries in oldclusters. Similarly, if the population favors mass ratioscloser to 0.95 rather than being uniformly distributed,we would produce more blue stragglers. These granulardetails of the twin population are beyond the precisionof current studies, but relevant to the formation of bluestraggler stars.Furthermore,
BSE simply assumes that the fractionallifetime remaining for the accretor star will be preservedonce it gains mass and becomes a blue straggler. Ifthe blue straggler is rejuvenated at formation becausemore hydrogen is mixed into the core, the lifetime ofthe blue stragglers could be significantly longer. Thisrejuvenation is particularly important to understand inblue stragglers forming from twin binaries, where the re-maining fractional lifetime of the blue straggler is smallwithout including some rejuvenation (Figure 9). Theamount of rejuvenation in blue stragglers is uncertain,and depends on the initial and final mass of the ac-cretor, the rotational evolution of the system, and theamount of internal mixing, but it could be substantial.For example, Sun et al. (2020) create a detailed stellarevolution model of a blue straggler in NGC 188 (6 Gyr).The system has an initial mass ratio of q = 0 .
85, goesthrough non-conservative mass transfer, and experiencesmodest core hydrogen rejuvenation. The resulting bluestraggler has a remaining main-sequence lifetime of ∼ Leiner et al.
Figure 9 . We show the blue straggler lifetime as a function of the initial binary mass ratio for three different time snapshots inour L2/L3 overflow model ( q crit = 1.8 model): 1500 Myr (left), 3000 Myr (middle) and 7000 Myr (right). The colors indicate theinitial orbital period of the binary that formed the blue straggler. These give an indication of the type of Roche lobe overflowthat formed the blue straggler: ∼ −
10 day periods binaries (purple) go through Roche lobe overflow beginning on the mainsequence or subgiant branch, periods of a few hundred days (blue) begin RLO on the RGB, periods of ∼ ∼ would be (cid:46) q crit = 1.8 and q crit = 100models, and less in other stability models.6.2.3. Mass Transfer Efficiency
As we state in Section 4.1.3,
COSMIC/BSE generally as-sumes mass transfer will be conservative (i.e. all of themass lost by the donor star during Roche lobe overflowis accreted by the companion). Nearly all the blue strag-glers created in our models are produced via nearly con-servative mass transfer, except in the case where the bluestraggler is formed entirely via Bondi-Hoyle wind accre-tion (see Figure 10). This is, of course, inconsistent withour non-conservative and L2/L3 overflow prescriptions,as these stability predictions assume significant materialcan be lost during the mass transfer process. Changing q crit can change the total number of blue stragglers pro-duced by our models, but if nearly all these blue strag-gler form via conservative mass transfer, changing q crit does not change the mass distribution of the blue strag-glers. Incorporating non-conservative mass transfer intopopulation synthesis models, then, is likely required to produce realistic blue straggler mass distributions.A straight-forward population synthesis prescriptionthat self-consistently calculates q crit and mass transferefficiency has not been developed. Nevertheless, non-conservative mass transfer is probably common whenforming blue straggler stars. Gosnell et al. (2019) showthat 2 post-mass-transfer blue stragglers in NGC 188both likely formed via non-conservative mass transfer.Their formation history for one of the brightest bluestragglers in the cluster requires an overall mass trans-fer efficiency of just 60%, suggesting even the most mas-sive blue stragglers in a cluster are not forming via fullyconservative mass transfer.Allowing for lower mass transfer efficiencies duringRoche lobe overflow could reduce the number of highmass (bright) blue stragglers and increase the number oflower-mass blue straggler (closer to the turnoff), bring-ing the mass distributions of the model blue stragglersin line with the observed distributions (Figure 3).We also note that lower mass blue stragglers wouldalso have longer lifetimes than higher mass blue strag-glers, so this modification could also increase the num-ber of blue stragglers stars in our models by increasingblue straggler lifetimes (Figure 9). This would be par-ticularly relevant for the blue stragglers forming fromtwins we discuss in the previous section, since twin bi-naries form the most massive blue stragglers with themost depleted hydrogen cores. We suggest allowing thatnon-conservative mass transfer is an essential step in im-proving population synthesis models. SUMMARYWe use Gaia DR2 to determine memberships for asample of 16 nearby open clusters with ages rangingfrom 1.4-10 Gyr. This age range corresponds to a range
SS Gaia DR2 Figure 10 . For our L2/L3 overflow model, we show the overall efficiency of the mass transfer (i.e. fraction of material lost by thedonor that is accreted by the companion) that formed the blue stragglers. We calculate this simply by dividing the combinedmass of the blue straggler binary by the combined mass of the progenitor binary. This mass may have been lost during RLO orvia a wind. We show these plots for blue stragglers in 3 different snapshots: 1.5 Gyr (left), 3 Gyr (middle) and 7 Gyr (right). in main-sequence turnoff masses of 2.0-1.0 M (cid:12) .We compare the observed blue straggler populationsbetween clusters, finding that there is a steep rise inthe ratio of blue stragglers to RGB stars in our clus-ters up to an age of ≈ N BSS N RGB ≈ .
35. This result indicates that oldclusters produce more blue stragglers than young clus-ters, and/or that their blue straggler populations arelonger lived. We compare this observed relationship to
COSMIC population synthesis models using five differentprescriptions for the critical mass ratio for mass-transferstability ( q crit ). We find that all options under-produceblue stragglers in old ( > q crit =100.0, in effect as-suming all mass-transfer is stable and avoids commonenvelope evolution.We discuss other factors that may contribute to thenumber of observed blue stragglers, including blue strag-glers produced via stellar collisions or mergers in triplesystems. These other production mechanism could ac-count for 25-45% of blue straggler productions in clus-ters, making up for some of the discrepancy betweenmodels and observations in number blue stragglers.We also find that in aggregate, the observed bluestraggler populations in clusters older than 2 Gyr aresplit into two groups: a bright group of blue stragglers0.6-1.0 M (cid:12) above the MSTO and a fainter group 0.0-0.4 M (cid:12) above the MSTO. In these clusters, there is anobserved paucity of blue stragglers at ∼ . (cid:12) abovethe turnoff. The faint blue stragglers are the majoritygroup, accounting for 70% of the overall blue stragglerpopulation summed over all clusters.This mass distribution is at odds with the results ofmost of our population synthesis models, which produce more bright blue straggler than observed, regardless ofstability condition, and show no mass gap. We findmore promising agreement in blue straggler mass dis-tributions when adding in enhanced Bondi-Hoyle accre-tion that produces fainter blue stragglers via highly non-conservative wind mass transfer, particularly when us-ing our non-conservative stability prescription (i.e., thecyan histograms in Figure 8).We therefore suggest one explanation for the twogroups of blue stragglers we observe in old clusterscould be that stable, nearly conservative mass trans-fer forms the brighter group of blue stragglers, whereasnon-conservative mass transfer (e.g., via an AGB wind)forms the fainter group of blue stragglers. Alternatively,perhaps the two groups form via different channels (e.g.,the brighter group forms from RGB mass transfer, thefainter group from AGB mass transfer). More detailedstudies of blue straggler orbital properties to comparewith model predictions could offer more insight into thispuzzle.We conclude that current population synthesis mod-els have difficulty producing the observed blue strag-gler production rates and mass distributions in openclusters. It is clear that the standard prescriptions ofHjellming & Webbink (1987) or Hurley et al. (2002) failto produce reasonable numbers of blue stragglers and ahigher q crit should be used in population synthesis. Us-ing a substantially higher q crit value such as we use forour L2/L3 overflow model produces more realistic bluestraggler counts, but over-produces bright blue strag-glers relative to near-turnoff blue stragglers. These q crit prescriptions are not exhaustive, and implementing new q crit prescriptions based on model grids of detailed stel-lar structure calculations may reveal prescriptions that0 Leiner et al. perform better (e.g. Ge et al. 2020).However, our results indicate that q crit is not the onlyparameter that needs adjustment in population synthe-sis in order to match the observed blue straggler popu-lations. We suggest in particular that the mass distri-butions of blue straggler stars indicate non-conservativemass transfer is important to their formation, and al-lowing for non-conservative mass transfer in populationsynthesis is a necessary ingredient to produce more re-alistic blue straggler populations. Coupling one of themore stable q crit prescriptions (e.g. the L2/L3 model)with a self-consistent calculation of the mass transferefficiency may yield results more consistent with obser-vations. Formation of blue stragglers via a highly non-conservative mass-transfer channel such as wind accre-tion may also be important.In addition, core hydrogen rejuvenation in blue strag-glers would increase blue straggler lifetimes, and a morecareful treatment of this effect may be needed to pro-duce observed quantities of blue stragglers in open clus-ters. We leave a detailed investigation of these factorsto future modeling work.Blue stragglers are one window into the outcome ofmass transfer processes in low mass binaries. They formfrom the first mass transfer event in a binary’s evolution,leaving a white dwarf – blue straggler binary. These systems may later go on to form many types of inter-esting astrophysical binaries– e.g. double white dwarfs,low-mass white dwarf mergers and exotic transients, X-ray binaries, and others. We demonstrate in this paperthe difficulty of reproducing the observed blue stragglerpopulations using common implementations of mass-transfer physics in population synthesis codes, and showthat the blue straggler populations can provide insightsinto the mass-transfer process. Improving these mass-transfer models is important not only in understandingblue straggler formation, but in understanding a widevariety of post-mass-transfer binaries.EML is supported by an NSF Astronomy andAstrophysics Postdoctoral Fellowship under awardAST-1801937.This work has made use of datafrom the European Space Agency (ESA) mission Gaia ( ), processedby the Gaia
Data Processing and Analysis Con-sortium (DPAC, ). Funding for the DPAC hasbeen provided by national institutions, in particularthe institutions participating in the
Gaia
MultilateralAgreement. The authors also thank the anonymous ref-eree for helpful comments on this work.REFERENCES
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Leiner et al.
APPENDIXIn this Appendix, we show CMDs for all the open clusters in our sample (Figure A.1). We also show HR diagrams forthe COSMIC population synthesis models assuming q crit prescriptions of Hurley et al. (2002) (Figure A.2), Hjellming& Webbink (1987) (Figure A.3), our non-conservative prescription (Figure A.4) and q crit = 100 (Figure A.5). For moredetails about these models and HR diagrams see see Sections 4.1 and 5.2 in the main text. (a) (b) (c)(d) (e) (f)(g) (h) (i) SS Gaia DR2 (j) (k) (l)(m) (n) (o)(p) Figure A.1 . Color-magnitude diagram showing Gaia members of the cluster (gray points). Colored points mark the RGB stars(red), blue stragglers (blue) and yellow straggler region (yellow). The blue straggler region is brighter and bluer than the bluedashed lines. The RGB region is redder and brighter than the red dashed lines. The yellow straggler region falls between thered and blue regions brighter than 0.75 magnitude above the turnoff (yellow line). We also show MIST (Dotter 2016) isochroneswith the age and metallicity given in Table 1. Leiner et al.
Figure A.2 . HR diagrams showing the results of COSMIC population synthesis simulations of 100,000 binaries using Hurleyet al. (2002) mass transfer stability prescription. We show three snapshots in age: 1500 Myr (left), 3000 Myr (middle), and 7000Myr (right). In the top panels we plot stars colored by the stellar type of the primary (i.e., the originally more massive starin our simulation). In the bottom panels we show the same simulations, but we color the stars by orbital period. Gray pointsindicate systems that are now single stars (i.e., due to a binary merger) and thus do not have orbital periods. The dashed linesindicate the boundaries of the blue straggler domain.
SS Gaia DR2 Figure A.3 . Same as Figure A.2, but showing results using the Hjellming & Webbink (1987) prescription for mass transferfrom RGB and AGB donors. Leiner et al.
Figure A.4 . Same as Figure A.2, but showing results using our non-conservative q crit prescription for mass transfer from RGBand AGB donors. SS Gaia DR2 Figure A.5 . Same as Figure A.2, but showing results using q crit = 100 ..