The incidence of stellar mergers and mass gainers among massive stars
S.E. de Mink, H. Sana, N. Langer, R.G. Izzard, F.R.N. Schneider
aa r X i v : . [ a s t r o - ph . S R ] D ec Draft version June 29, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE INCIDENCE OF STELLAR MERGERS AND MASS GAINERS AMONG MASSIVE STARS
S.E. de Mink , H. Sana , N. Langer , R.G. Izzard & F.R.N. Schneider Draft version June 29, 2018
ABSTRACTBecause the majority of massive stars are born as members of close binary systems, populationsof massive main-sequence stars contain stellar mergers and products of binary mass transfer. Wesimulate populations of massive stars accounting for all major binary evolution effects based on themost recent binary parameter statistics and extensively evaluate the effect of model uncertainties.Assuming constant star formation, we find that 8 +9 − % of a sample of early type stars to be theproduct of a merger resulting from a close binary system. In total we find that 30 +10 − % of massivemain-sequence stars are the product of binary interaction.We show that the commonly adapted approach to minimize the effects of binaries on an observedsample by excluding systems detected as binaries through radial velocity campaigns can be coun-terproductive. Systems with significant radial velocity variations are mostly pre-interaction systems.Excluding them substantially enhances the relative incidence of mergers and binary products in thenon radial velocity variable sample.This poses a challenge for testing single stellar evolutionary models. It also raises the question ofwhether certain peculiar classes of stars, such as magnetic O-stars, are the result of binary interactionand it emphasizes the need to further study the effect of binarity on the diagnostics that are used toderive the fundamental properties (star-formation history, initial mass function, mass to light ratio)of stellar populations nearby and at high redshift. Subject headings:
Stars: early-type — Stars: massive — Binaries: close — Binaries: spectroscopic —blue stragglers — Galaxy: stellar content INTRODUCTIONYoung massive stars are predominantly found in closebinary systems (e.g. Mason et al. 2009; Sana & Evans2011; Chini et al. 2012). This implies that the majorityof massive stars interacts with a companion before end-ing their lives as supernovae (Sana et al. 2012). Such in-teraction has drastic consequences for the further evolu-tion and final fate of both stars (e.g. Podsiadlowski et al.1992; Wellstein & Langer 1999; Eldridge et al. 2008) andgives rise to a variety of exotic phenomena includingblue stragglers, X-ray binaries, mili-second pulsars andgamma-ray bursts.The strong preference for close systems, with orbitalperiods of less than a few days, implies that a third of thesystems interacts while both stars are still on the mainsequence (Sana et al. 2012). Binary evolutionary modelspredict that a large fraction of these systems evolve intoa contact configuration (e.g Pols 1994; Wellstein et al.2001; Nelson & Eggleton 2001; de Mink et al. 2007).The evolution of massive contact binaries is uncertain,but it is anticipated that the stars are driven deeper intocontact due to the continuing expansion of the stars andshrinkage of the orbit due to angular momentum loss.In particular, the loss of mass with high specific angular Observatories of the Carnegie Institution for Science, 813Santa Barbara St, Pasadena, CA 91101, USA Cahill Center for Astrophysics, California Institute of Tech-nology, Pasadena, CA 91125, USA Space Telescope Science Institute, Baltimore, MD 21218,USA Argelander Institut f¨ur Astronomie der Universit¨at Bonn,Germany * Einstein Fellow. momentum through the outer Lagrangian point, possibletorques resulting from circum-binary material and dissi-pative processes occurring in the common envelope arebelieved to eventually result in a merger of the two stars(e.g. Podsiadlowski et al. 1992; Wellstein et al. 2001).Recently, the merger of a close binary system, althoughat lower mass, was caught in the act as the transientV1309 Sco (Tylenda et al. 2011). It has also been sug-gested that the light echo of V838 Mon resulted froma massive merger (Munari et al. 2002; Tylenda & Soker2006), but this case remains more controversial.For massive stars the fraction of stars that will mergeas a result of contact in a close binary is predicted to beas high as 20-30% according to the most recent binarystatistics (Sana et al. 2012). In addition, mergers maybe triggered by dynamical interactions between threebodies systems (e.g. Kozai 1962; Perets & Fabrycky2009; Hamers et al. 2013) or as a result of (multiple)collisions in dense star clusters (Portegies Zwart et al.2004; Glebbeek et al. 2009) or near the galactic center(Antonini et al. 2010, 2011).The chance of witnessing the merger event of two mas-sive stars is small, because of the scarcity of massivestars. The rate of such events in our galaxy is ex-pected to be about 20–30% of the Galactic supernovarate, about one every two hundred years (e.g. Langer2012). In contrast, the products of such mergers may berather common, in particular for mergers between twomain sequence stars. These objects are expected to berejuvenated as fresh fresh is mixed into the central burn-ing regions (e.g. Glebbeek et al. 2013). These mergerproducts are expected to be among the brightest stars inyoung clusters forming a massive analogue of blue strag-glers (e.g. Mermilliod 1982; Chen & Han 2009; Lu et al.2010, Schneider et al. 2013a, subm.).For the population of early-type stars in a typ-ical galaxy, which is not characterized by a singleburst of star formation, mergers and mass gainerscannot easily be identified. However, if such starsare abundant, they can in principle affect various di-agnostics that are used to derive the fundamentalproperties such as the star-formation history and ini-tial mass function (e.g. van Bever & Vanbeveren 1998;Eldridge 2012; Zhang et al. 2012; Li et al. 2012). Suchproperties are generally derived using population syn-thesis models such as GALAXEV (Bruzual & Charlot2003), STARBURST99 (Leitherer et al. 2010) and FSPS(Conroy et al. 2009), in which all stars are in principleassumed to evolve in isolation. Estimating the incidenceof binary products is therefore ultimately motivated bythe need to improve our understanding of the validity ofthe properties derived for stellar populations nearby aswell as those at high redshift.A more direct motivation is the need to test state-of-the-art stellar evolution models, which contain uncertaineffects of convection, rotation and magnetic fields (e.g.Brott et al. 2011a; Ekstr¨om et al. 2012; Potter et al.2012). Lacking prescriptions from first principles makescalibration against observed populations indispensable.These models generally assume the stars to evolve in iso-lation. However, interaction with a binary companioncan lead to drastic changes in the observable properties.To evaluate the validity of tests and calibrations of themodels against observed samples, it is necessary to esti-mate the contamination of such samples with stars thatare the products of binary interaction.In this work we take a first step towards quantify-ing the implications of the newly derived binary fractionand distribution of orbital properties of massive binarystars (Sana et al. 2012). For this purpose we employ arapid synthetic binary evolution code that has been up-dated to adequately describe the main relevant processesin Sect. 2. We simulate a population of young massivestars in a typical galaxy assuming continuous star for-mation to compute the incidence of mergers and otherproducts of binary evolution. In particular, we examinewhich binary products can be detected as binary sys-tems through radial velocity variations in Sect. 3.1. Weassess the significance of our results by varying uncertaininput distributions and the adopted treatment of uncer-tain physical processes in Sect. 3.2. In Sect. 4 we discussthe presence of binary products in observed samples andhow they may be recognized and we conclude in Sect. 5. METHODTo estimate the effects of binary interaction on a pop-ulation of early-type stars we adopt the binary frac-tion, and distribution of orbital periods and mass ra-tios based on observations of O-stars in nearby ( . p , f p (log p ) ∼ (log p ) π for log p (days) ∈ [0 . , . π = − . ± . q defined as the mass of theless massive star over the mass of the more massive star, f q ( q ) ∼ q κ for q ∈ [0 . ,
1] where κ = − . ± .
6. The binary fraction,i.e. the number of binary system with periods and massratios in the range specified with respect to the totalnumber of single and binary systems, is f bin = 0 . ± . f m ( m ) ∼ m − α with α = 2 .
35 (Salpeter 1955). Given the young agesof the clusters in this sample, we consider these in oursimulations as initial conditions at the onset of hydrogenburning.In our standard simulation we adopt f bin = 0 . π = − . κ = 0 and α = 2 .
35. We we consider variationson these parameters (see Tab. 1) that generously includethe confidence interval quoted by Sana et al. (2012). Theremaining fraction 1 − f bin is included as single stars,even though these may in reality have a nearby low masscompanion or a companion in a wide orbit.To model the effect of stellar evolution and binary in-teraction we employ a synthetic binary evolution codethat is described in detail in de Mink et al. (2013) andreferences therein (hereafter Paper I). This code wasoriginally developed by Hurley et al. (2000, 2002) andIzzard et al. (2004, 2006, 2009) based on stellar modelsby Pols et al. (1998). It has been updated and extendedto include various processes relevant for this study (Pa-per I). A brief summary of the main aspects is givenhere.We account for mass and angular momentumloss through winds (Nieuwenhuijzen & de Jager 1990;Vink et al. 2001), effects of rotation on the stellar winds(Maeder & Meynet 2000) and deformation due to rota-tion in the Roche approximation (Paper I). We adopta metallicity of Z = 0 .
02. We account for interactionthrough tides (Zahn 1977; Hurley et al. 2002) and Rochelobe overflow (Hurley et al. 2002) assuming circular or-bits. In our standard simulation, we limit the accretionrate by the thermal rate of the accreting star (Tout et al.1997; Hurley et al. 2002) and we assume that the re-mainder of the mass is lost from the system taking awaythe specific orbital angular momentum of the accretingstar. Because the efficiency of mass transfer, β , definedas the fraction of the mass transferred from the donorstar to the companion star that is actually accreted bythe companion, and the specific angular momentum, γ ,of material lost from the system are highly uncertain, weconsider the extreme cases of conservative mass transfer( β = 1) and the highly non-conservative case where astar can no longer accrete after it has been spun up toits Keplerian limit (c.f. Paper I). This latter case is indi-cated as β = β K and is equal to the assumptions madein Petrovic et al. (2005) and de Mink et al. (2009). Forthe angular momentum loss we consider the extreme casethat all mass is lost through the outer Lagrangian point(indicated as γ = γ L ) and the extreme limit where thespecific angular momentum of the mass lost from thesystem is negligible ( γ = 0).We assume that binary systems comein contact when M acc /M don < q crit (e.g.Kippenhahn & Meyer-Hofmeister 1977), where M acc and M don refer to the mass of the accreting star andthe donor star. For systems with a main sequencedonor star we adopt q crit , MS = 0 .
65, based on a cali-bration against a grid of detailed binary evolutionarymodels (de Mink et al. 2007). Given the considerableuncertainties concerning the formation of contact (c.f.Sect 5.2.1 in Paper I) we explore the extreme cases of q crit , MS ∈ { . , . } . For systems with a Hertzsprung-gap donor star we adopt q crit , HG = 0 . q crit , HG ∈ { . , q W01 } ,where q W01 = 1 .
0, for p >
30 d and 0.65 for p ≤
30 dto mimic the detailed models by Wellstein et al. (2001).We follow Hurley et al. (2002) for the treatment ofevolved donors.We assume that contact binaries merge. During thecoalescence we assume in our standard simulations thata fraction µ loss = 0 . µ loss = 0, and µ loss = 0 .
25. The merger product is assumed to settleto its thermal equilibrium structure while rotating nearits Keplerian rotational velocity. It is assumed that thecore of the most evolved star, which has the lowest en-tropy, sinks to the center of the merged star. In ourstandard simulation only a small fraction µ mix = 0 . µ mix = 1, as is assumed in theoriginal version of the code, and the case of no additionalmixing µ mix = 0. For accreting stars we account for reju-venation by assuming that the star adapts it structure toits new mass, mixing in fresh hydrogen as the convectivecore expands (Paper I and references therein).In our simulations of a population of early-type starswe select stars that are undergoing central hydrogenburning for which the combined brightness of the main-sequence components exceeds 10 L ⊙ (see Paper I). Wechose this approach instead of estimating the spectraltypes directly from the effective temperatures, becauseour predictions of the luminosity and evolutionary phaseare more reliable than effective temperatures. Theadopted limit corresponds in our models to the bright-ness of a single main-sequence star with initial mass & M ⊙ , where the range reflects the fact that starsbecome brighter as they evolve over the main sequence.This roughly corresponds to stars of spectral types earlyB and O. We consider continuous star formation, whichis a good approximation for a large system with multiplebursts of star formation, such as the complete early-typestar population in a typical Galaxy. The case of differentstar formation histories (starbursts) computed with thesame code and very similar assumptions are discussed inSchneider et al. (2013, subm.). semi-detached systems:3% single: 22% pre mass transferbinaries: 50%companions after mass transfer: 17% mergers: 8% p r o d u c t s o f b i n a r y i n t e r a c t i o n : % Fig. 1.—
The incidence of stellar mergers and (post) mass trans-fer systems in our standard simulation of a population of massivemain sequence stars assuming continuous star formation and aninitial binary fraction of 70%. Percentages are expressed in termsof the number of systems containing at least one main sequencestar. See Sec. 3 for a discussion and Table 1 for the impact ofmodel uncertainties on these predictions. THE INCIDENCE OF BINARY PRODUCTS IN APOPULATION OF MASSIVE EARLY-TYPESTARSThe properties of our simulated stellar population aresummarized in Fig. 1, which indicates the relative contri-bution of single stars, binary stars and products of binaryinteraction. The percentages are expressed in terms ofthe number of systems, either single or binary, that con-tain at least one main sequence star and for which thecombined brightness of the main-sequence componentsexceeds 10 L ⊙ . A binary system containing two main-sequence stars is counted only once.The simulations started with 70% binaries and 30%single stars at birth. For the case of continuous star for-mation we find that the contribution of pre interactionbinaries is reduced from 70% at birth to 50% in the cur-rent population as a result of stellar evolution and binaryinteraction. About a fifth consists of stars that we referto as single stars, even though some may have a compan-ion in a very wide orbit, or very low mass companion,which are not accounted for in our simulation.More than a quarter of the systems have been severelyaffected by interaction with their companion. This groupmainly consists of mergers (8%) and stars that have pre-viously gained mass from a (former) companion (17%).The fraction of systems that are in a semi-detached con-figuration, i.e. that are currently undergoing mass trans-fer through Roche-lobe overflow, is small (3%). Thisis because the mass transfer phase typically lasts for afew thermal timescales at most, which is short comparedto the stellar lifetime. An exception are close systemsthat experience case A mass transfer (Nelson & Eggleton2001), i.e. mass transfer from a main-sequence donor,which can last for up to about a third of the main se-quence lifetime (e.g. Fig. 3 in de Mink et al. 2007).Practically all semi-detached systems, accounting for afew percent of the population, are undergoing slow CaseA mass transfer. These systems form a subset of the * [km s -1 ] semi-amplitude of the radial velocity curve: * [km s -1 ] apparantly single stars detectable as binaries pre mass transferbinaries singlemergers t o t a l c u m u l a t i v e d i s t r i b u t i o n ` s e m i - d e t a c h e d s y s t e m s c o m p a n i o n s a f t e r m a s s t r a n s f e r F ( * ) f ( * ) detectable as binaries p r e m ass tr ans f e r b i na ri es c o m pan i on s a ft e r m a ss t r an s f e r se m i- de t ached sys t e m s s i ng l e m e r g e r s t o t a l d e n s i t y d i s t r i b u t i o n Fig. 2.—
The density and cumulative distribution of the semi-amplitudes, i.e. the maximum line-of-sight velocities due to the orbitalmotion, for a population of massive early-type stars including a fraction of binaries characteristic for the Milky Way. The distribution isnormalized after adding single stars and mergers. Note that we adopted logarithmic-linear and linear-logarithmic scales to enhance differentfeatures. observed algol type systems.The large fraction of mergers and mass gainers canbe understood as the combined effect of different mech-anisms. First, their production is favored by the largeclose binary fraction at birth. Second, binary productshave gained mass through accretion or coalescence, re-sulting in an increase in brightness. Stars that were ini-tially not massive and luminous enough to be included inour brightness limited sample can become bright enoughafter mass accretion. In other words, the binary prod-ucts in our sample come from a wider range of ini-tial masses than the single stars and stars that havenot yet interacted. In addition, these binary productsoriginate from lower mass systems, which are favoredby the slope of the initial mass function. Third, whena main sequence star accretes mass it typically adaptsits internal structure leading to an increase of the sizeof the convective core. As a result the hydrogen richlayers above the original core are mixed to the centralburning regions providing fresh fuel, which effectivelyrejuvenates the star (Kippenhahn & Meyer-Hofmeister1977; Podsiadlowski et al. 1992; Braun & Langer 1995;Dray & Tout 2007; Claeys et al. 2011). The prolongedlifetime increases the fraction of stars that is expected tobe observed after mass transfer.3.1.
The counter-productive effect of selecting againstbinaries.
The contamination of a sample of early-type stars withbinary products poses a challenge for their usefulness totest stellar models. A commonly adopted approach to tryto reduce the effects of binaries on an observed sampleis to exclude every object that is a known binary. Here,we demonstrate that removing detected binaries from asample is counter-productive (cf. de Mink et al. 2011).Spectroscopic binaries are detected through variationsin the radial velocity resulting from the orbital motion.In a single-lined circular spectroscopic binary, the maxi-mum radial velocity variation, ∆ v , that can be measuredif the orbital phase is well sampled near quadrature, isequal to 2 K ∗ , where K ∗ is the semi-amplitude of the ra-dial velocity curve for the brightest star, and i is the inclination angle. In a typical early-type system with aprimary mass M ∗ , a mass ratio q and an orbital period p , K ∗ ≈
80 km s − (cid:18) sin iπ/ (cid:19)(cid:18) M ∗ M ⊙ (cid:19) α m (cid:18) q . (cid:19) α q (cid:18) p
10 d (cid:19) α p where α m = 1 / α p = − / α q ≈ .
86 for q ∈ [0 . , − depending on the signal to noise ratio,the spectral type and the rotation rate. Pulsations andstellar winds can further induce apparent variations oflow amplitude. Therefore, radial velocity variations of∆ v ≥ ∆ v lim ≡
20 km s − (or K ∗ ≥
10 km s − ) is typicallyconsidered as an unambiguous sign of orbital motion dueto the presence of a companion (e.g. Sana et al. 2013).In Fig. 2 we show the probability density function f and the cumulative distribution F of semi-amplitudes ofthe binaries in our simulated population assuming ran-dom inclination angles. The distributions are normalized f after adding single stars and mergers, for which we set K ∗ = 0 km s − . The detection limit is indicated as a ver-tical dashed line. As can been seen in the right panel,about 45% of the objects do not show any radial velocityvariations that are large enough to be un ambiguouslydetected as caused by a companion star. This groupconsists of stars that appear to be single.The remaining 55% show radial velocity variationslarger than the detection limit. In principle these aredetectable as binaries. This requires multiple spectrato be taken that cover enough different phases of theorbit such that the full radial velocity curve can be re-constructed. For semi-amplitudes larger than the detec-tion limit, the distribution is dominated by the primarystars of binary systems that have not yet interacted bymass transfer. Their semi-amplitudes extend beyond 250km s − , although semi-amplitudes of around 20 km s − ,approaching the detection limit, are most common. Sys-tems that are currently undergoing Roche-lobe overflowtypically have semi-amplitudes around 100 km s − , wellabove the detection limit. This group is dominated bymass transfer systems in which the Roche lobe filling staris a main-sequence star. These systems have compact or-bits leading to large radial velocity variations.The post interaction systems indicated as “companionsafter Roche-lobe overflow” consist of a main-sequencestar accompanied by a helium star, a neutron star, blackhole or white dwarf. These systems typically have smallsemi-amplitudes that are below the unambiguous detec-tion limit. A significant fraction will be unbound as aresult of the birth kick of the neutron star during thesupernova explosion of the primary. Because the depen-dence of the birth kick distribution on the pre-explosionproperties is highly uncertain, we assume a zero kick ve-locity in all cases, such that the quoted number of post-interaction systems with detectable radial velocity vari-ations provides an upper limit.When removing systems from a sample, that are de-tected as radial velocity variables, one preferably removesbinary systems from the sample that have not yet inter-acted. The products of binary interaction, which typi-cally do not show measurable radial velocity variations,are left in the sample. Fig. 3 summarizes the conse-quences for an idealized observing campaign, in whichsufficient observing time is allocated to acquire full phasecoverage of each system. In this case each binary witha semi-amplitudes K ∗ > ∆ v lim ≡
10 km s − would beidentified.The left pie chart shows the population of stars withno significant radial velocity variations, i.e stars that ap-pear to be single. About half of this sample are indeedstars were born as single stars (48%). A small fraction(10%) corresponds to pre-interaction binary systems dis-playing radial velocity variations that are too small to bedetected. This group consists of the widest binaries andbinaries with orbits that are aligned nearly face. Thesetwo groups together, accounting for 58% of the sample,are stars that have lived their lives so far without expe-riencing any significant interaction with a companion.The remaining 42% consists of stars whose evolution-ary history deviates strongly from that of an isolatedstar. These consist of mergers, (19%) and stars thathave gained mass through Roche lobe overflow in thepast (27%). In reality systems with radial velocity vari-ations approaching the detection limit are hard to de-tect without extensive monitoring. The pie chart on therighthand side shows that those binaries that in principlecan be detected through radial velocity variations. Thissample consists preferentially of pre-interaction systems.We conclude that excluding detected binaries from asample to reduce the contamination of binary productsis counter-productive. The binary products are typicallynot detected and will therefore represent a larger fractionof the sample than single stars. Those objects that areidentifiable as binaries are dominated by stars that havenot yet interacted. Apart from the closest systems inwhich tides play a role these systems have lived theirlives effectively as single stars. 3.2. Effect of model uncertainties
In Table 1 we summarize the effect of varying the ini-tial conditions and the main uncertain physical assump-tions on the incidence of binary products and mergers inparticular.Concerning the adopted initial distributions of binaryparameters, it is the power law exponent in the distri-bution of orbital periods, which is the main cause foruncertainty. We find that the fraction of stars that arebinary products varies from 19 to 40% when consideringan initial distribution that is flat in log p (i.e. π = 0or “Opik’s law”) and a distribution that strongly favorsshort period systems ( π = − κ = −
1) to a distribution that favors systems with equalmasses ( κ = 1) increases the fraction of binary productsfrom 22 to 33%. This results from the fact that binarieswith comparable masses produce more massive binaryproducts that are luminous enough to meet our bright-ness limit. The fraction of mergers varies only from 10 to7%. This is because of two effects that partially compen-sate each other. Systems with unequal masses are morelikely to come in contact and merge. However, if systemswith comparable masses merge they produce brighter ob-jects. We refer to the discussion section in Paper I forfurther details.The effect of changing the initial mass function moremodest. We find a variation of 27 to 32% when adopt-ing flatter ( α = 1 .
65) or a steeper ( α = 3 .
05) steeperslope for the initial stellar mass function, respectively.When changing the adopted initial binary fraction downto f bin = 0 . f bin = 1 . to 10 L ⊙ we find a increase of the incidence of binaryproducts (to 34%) and mergers (to 12%).Investigating the effects of the uncertain physical as-sumptions requires recomputing the grid of models. Tospeed up the computations we recomputed them at lowerresolution. In Table 1 we provide the results for varia-tions of the physical assumptions as well as a referencesimulation at the same resolution adopting our standardassumptions.The dominant uncertainty affecting the predicted num-ber of binary products is the accretion efficiency. Thisstill remains one of the major uncertainties in binaryevolutionary models. Attempts to constrain the effi-ciency using large samples of binary systems with accu-rately determined parameters remained inconclusive (e.g.de Mink et al. 2007, and references therein).If we assume that all the mass lost by the primary starthrough Roche-lobe overflow is accreted by the compan-ion we find that the fraction of binary products increasesto 34%, because of the larger masses and thus lumi-nosities of the binary products. Adopting highly non-conservative mass transfer by assuming that stars do notaccrete after they have been spun up to their Keplerianrotation rate, which is typically after they have gained a a) Apparently single ( * < 10 km s -1 ) b) Detectable as binary ( * > 10 km s -1 ) single:48% companions after mass transfer:23% mergers:19% pre mass transfer binaries:10% p r o d u c t s o f b i n a r y i n t e r a c t i o n : % companions after mass transfer: 11% semi-detached systems: 7%pre mass transfer binaries: 82% Fig. 3.—
Selecting against stars with evidence for a companion in order to reduce the contamination of binaries in an observed samplecan be counter productive. A sample of apparently single stars ( K ∗ <
10 km s − ) contains a large fraction of mergers and other binaryproducts, cf Fig. 1 (left pie chart). The systems that are removed are preferentially binary systems that have not yet experienced interaction(right pie chart). few percent of their mass (Packet 1981), reduces the inci-dence of binary products to 18%. Changing the adoptedspecific angular momentum of material that is lost fromthe system during interaction affects the number of merg-ers and whether the systems display detectable radialvelocity variations after interaction. However, we findthat this effect is minor compared to the other uncertainassumptions.The number of mergers is mainly affected by theamount of mixing in the merger product. When themerger product is assumed to be fully mixed, the re-maining lifetime is enhanced as a result of the fresh hy-drogen that has become available in the central burningregions. The fraction of mergers is also sensitive to theadopted critical mass ratio for contact in systems wherethe donor star is on the main-sequence. These two uncer-tainties are responsible for variations of 7-12%. We findthat our predictions are not very sensitive to the adoptedamount of mass loss during the merger event, nor to theadopted critical mass ratio for contact in systems wherethe donor star is a post main-sequence object. BINARY PRODUCTS IN OBSERVED SAMPLESThe number of binary products present in observedstellar samples depends on the selection effects and thedetails of the underlying star-formation history. In pop-ulations younger than 2 Myr one might expect the overallcontamination by binary products to be small, since onlythe most massive stars evolve fast enough to expand andinteract. However, depending on the selection criteria,the contamination by binary products may very signif-icant. Even after only 1-2 Myr, the brightest star of awell populated star cluster is expected to be the productof binary evolution (Schneider et al. 2013). Therefore, ina sample consisting of the brightest stars of young starclusters, the contamination by mergers and other binary products may be significantly larger than predicted byour simulations for continuous star formation.As discussed in Section 3.1, the inclusion or selectionagainst known binaries can significantly change the rel-ative number of binary products. Modeling this effectis very challenging since the selection against binariesis often not done in a systematic way. For example, ina typical sample serendipitously discovered binaries de-scribed in the literature are excluded. Given the limitedsize of most samples these unsystematic selection effectscan not be ignored. Because of these sample dependentdifficulties, a quantitative comparison of our predictionswith observations beyond the scope of this paper. How-ever, we will make several general remarks.An example of a large homogenous survey of massivestars is the VLT-FLAMES Tarantula Survey (VFTS)of massive stars (Evans et al. 2011). This ESO largeprogram is a multi-epoch spectroscopic survey of 800randomly selected early-type stars, among which 360 Ostars, in the Tarantula Nebula or 30 Doradus region ofthe Large Magellanic Cloud (Walborn 1984). The sam-ple contains a mix of stellar populations with differentages including one that is at least 20 Myr old.The fraction of O stars in this survey that are detectedto be spectroscopic binaries 0 . ± .
03 (Sana et al. 2013).Even though this multi-epoch spectroscopic survey wasdesigned to systematically search for binaries, it is farfrom the idealized case demonstrated in Fig. 3b. Af-ter carefully modeling the specific biases fro this survey,Sana et al. (2013) derive an intrinsic binary fraction of0 . ± .
04. Based on our simulations we expect the con-tribution of post interaction systems among the detectedbinaries to to be very small. One may therefore ten-tatively compare this to the fraction of pre-interactionsystems in our simulation for continuous star formation,e.g. Fig. 1, where we find a remarkably similar fraction
TABLE 1Impact of uncertainties in the initial conditions and physical assumptions on the incidence of mergers and the totalnumber of binary products among a population of massive early-type stars. Symbols are explained in Sect. 2. Theunderlined values are the largest deviation in each category. standard extreme all binary products ∗ mergers onlyvalues values (%) (%) Initial conditions
Standard simulation for reference 28.7 8.4- primary mass distribution α = 2 .
35 1 . , .
05 26.8 – 32.1 7.8 – 9.4- mass ratio distribution κ = 0 − , π = − . , − f bin = 0 . . , . Z = 0 . . , .
03 26.5 – 36.8 4.5 – 9.0- brightness limit L lim ( L ⊙ ) = 10 Main physical assumptions
Standard simulation for reference 27.3 8.0- accretion efficiency β = β th β K , γ = γ acc , γ L q crit , MS = 0 .
65 0 . , . q crit , HG = 0 . , q W µ loss = 0 . , .
25 27.1 – 27.3 7.8 – 8.0- treatment mergers: mixing µ mix = 0 . , ∗ All binary products, i.e. mergers, systems undergoing Roche-lobe overflow and companions after Roche-lobe overflow of 50%.The intrinsic binary fraction derived for the TarantulaSurvey is lower than that derived for young galactic clus-ters, which is 0 . ± . Characteristics of binary products
Apart from statistical statements about the frequencyof binary product, we can consider ways to identify indi-vidual binary products. There are several characteristicsthat give hints for a binary origin. Not all binary prod-ucts are expected to show each of these characteristicsand none of these characteristics uniquely signifies bi-nary interaction.(a) The surface abundances of binary products mayshow signatures that indicate mixing or accretion of en-richer gas. In particular, a depletion of fragile elementsis expected, because these can only survive in the coolerouter most layers of the star. This concerns lithium,beryllium, boron, and fluorine (e.g. Langer et al. 2010).In more sever cases the burning products of hydrogenfusion can appear at the surface, i.e. an enhancementhelium, nitrogen and sodium and a depletion of carbonand oxygen.(b) Binary products are likely to have peculiar ro-tation rates. In principle, rapid rotation is ex-pected as a result of spin-up during mass accretionor a merger (Dufton et al. 2011; de Mink et al. 2013),of which Be/X-ray binaries are direct evidence (e.g.Rappaport & van den Heuvel 1982). However, if binaryproducts experience strong angular momentum loss, forexample through magnetic braking, this may result invery slow rotators.(c) If binary interaction occurred recently there may besigns of shedded material in the circum-stellar medium,either a (bipolar) ejection nebula (as in the case of thepromising candidate for a merger product, the mag-netic O6.5f?p star HD 148937, Leitherer & Chavarria-K.1987; Smith et al. 2004; Naz´e et al. 2008, 2010) or acircum-binary disk (as seen for example around the inter-acting system RY Scuti, Grundstrom et al. 2007). Thetypical lifetime of such a nebula is expected to be around10 yrs. Therefore one can expect several stars that havemerged recently and thus still display a circum-stellarremnant (Langer 2012).(d) It has been suggested that a merger process maylead to the generation of a magnetic field (e.g. Tout et al.2008; Ferrario et al. 2009). Strong, large-scale fields canbe detected through spectropolarimetry, as for examplefor the companion of Plaskett star that has recently beenspun up by mass accretion (e.g. Grunhut et al. 2013).(e) The supernova explosion of the primary star canbreak up the binary system, depending on the amountof mass lost from the system and the magnitude and ori-entation of the birth kick of the compact object (Blaauw1961; Hoogerwerf et al. 2001). When a tight system isdisrupted the companion will acquire a high velocity.Such “runaway stars” can be detected by measuring theradial velocity, the proper motion or by indirect evidencesuch as the presence of a bow shock or its remote locationaway from regions of star formation. In many cases theacquired velocities will be moderate, i.e. a few to tensof km s − (e.g. Eldridge et al. 2011). These stars wouldnot be classified as runaway stars; “walk-away star” maybe a more appropriate term. However, they can traveltens to hundreds of parsecs from their birth location, be-cause 1 km s − ≈ − and these binary productsmay live for tens of Myr before they explode.(f) Binary products are not expected to have a nearbyunaffected main-sequence companion.(g) If the former companion is still present it may bedetectable in some cases through an UV excess (e.g.Gies et al. 1998; Peters et al. 2013), in the case of astripped helium star or through its X-ray properties whenthe former companion is an accreting compact object.(h) Within coeval stellar populations such as star clus-ters, the binary products may stand out as the mostluminous objects, possibly appearing younger than theage of the cluster. In this case they are the massive ana-logue of blue stragglers (van Bever & Vanbeveren 1998;Chen & Han 2009). For young star clusters, where theturn-off is not well defined, binary products may stillstand out as the upper mass tail of the stellar mass func-tion (Schneider et al. 2013). CONCLUSION AND DISCUSSIONBased on our simulations, we predict that a populationof early-type stars characterized by continuous star for-mation is contaminated by stars that have experiencedinteraction with a companion. We estimate the frac-tion of binary products in such a sample to be 30 +10 − % and the fraction of mergers to be 8 +9 − %. The errorbars quoted here refer to the largest variation we obtainwhen varying the input distributions and the treatmentof the physics of binary interaction with respect to ourstandard model. Even though larger variations cannotbe excluded if multiple assumptions need to be adjustedin a way that systematically favors or disfavors binaryproducts, we conclude — given our current understand-ing — that the contamination of a sample of early-typestars with binary products is considerable. This poses apotential challenge when using these samples for variousapplications. .In particular, our findings raise questions about thevalidity of tests and calibration of single stellar modelsagainst observational samples. We have shown that thecommonly adopted strategy of excluding detected bina-ries from a sample is counter-productive. Removing sys-tems with detectable radial velocity variations preferen-tially removes binary systems that have not yet inter-acted from the sample. Post mass-transfer systems andmergers are left in the sample, accounting for 40 +25 − % intotal, mergers in particular account for 15 +20 − %.Our findings also shed new light on the interpreta-tion of classes of peculiar stars by raising the questionwhether binary interaction or mergers are responsible forthe peculiarity. While objects such as binary productsand stellar mergers in particular may sound exotic, wepredict that they are quite common. This also raisesconcerns about our understanding of resolved and un-resolved stellar populations, in particular the accuracywith which we can derive quantities such as the star for-mation rate, mass-to-light ratio and initial mass functionusing population synthesis models which do not accountfor the effects of binarity.Our results emphasize the need further constrain thedistribution of initial parameters, in particular the distri-bution of initial orbital periods. Concerning the physicalassumptions the certainty of our predictions is mainlylimited by our poor understanding of the mass trans-fer efficiency and the treatment of contact systems andmergers. Concerning the design of observational surveysour findings call for prioritizing efforts to devote greatcare to a careful and systematic inclusion or selectionagainst known binary systems. Acknowledgements: