Draftversion January6,2016 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 INTERRUPTED BINARY MASS TRANSFER IN STAR CLUSTERS Nathan W. C. Leigha DepartmentofAstrophysics,AmericanMuseumofNaturalHistory,CentralParkWestand79thStreet, NewYork,NY10024and DepartmentofPhysics,UniversityofAlberta,CCIS4-183,Edmonton,ABT6G2E1,Canada Aaron M. Gellerb,c CenterforInterdisciplinaryExplorationandResearchinAstrophysics(CIERA)andDepartmentofPhysicsandAstronomy, 6 NorthwesternUniversity,2145SheridanRd,Evanston, IL60208,USAand 1 DepartmentofAstronomyandAstrophysics,UniversityofChicago,5640S.EllisAvenue,Chicago,IL60637, USA 0 2 Silvia Toonend n LeidenObservatory,LeidenUniversity,POBox9513, NL-2300RALeiden,TheNetherlands a Draft version January 6, 2016 J ABSTRACT 4 Binarymasstransferisattheforefrontofsomeofthemostexcitingpuzzlesofmodernastrophysics, ] including Type Ia supernovae, gamma-ray bursts, and the formation of most observed exotic stellar R populations. Typically,the evolutionis assumedtoproceedinisolation,evenindensestellarenviron- S ments such as star clusters. In this paper, we test the validity of this assumption via the analysis of . a large grid of binary evolution models simulated with the SeBa code. For every binary, we calculate h analytically the mean time until another single or binary star comes within the mean separation of p the mass-transferring binary, and compare this time-scale to the mean time for stable mass transfer - o to occur. We then derive the probability for each respective binary to experience a direct dynam- r ical interruption. The resulting probability distribution can be integrated to give an estimate for t s the fraction of binaries undergoing mass transfer that are expected to be disrupted as a function of a the host cluster properties. We find that for lower-mass clusters (. 104 M⊙), on the order of a few [ to a few tens of percent of binaries undergoing mass-transfer are expected to be interrupted by an 1 interloping single, or more often binary, star, over the course of the cluster lifetime, whereas in more v massiveglobularclustersweexpect 1%tobeinterrupted. Furthermore,usingnumericalscattering 1 experimentsperformedwiththe FEW≪BODYcode,we showthatthe probabilityofinterruptionincreases 5 if perturbative fly-bys are considered as well, by a factor 2. ∼ 6 Subject headings: binaries: general — galaxies: star clusters: general — globular clusters: general 0 — open clusters and associations: general — stars: kinematics and dynamics — 0 binaries: close . 1 0 1. INTRODUCTION 2005; Hurley et al. 2007; Marks et al. 2011; Geller et al. 6 2013a,b, 2015; Leigh & Geller 2012, 2013, 2015). 1 Binary mass transfer (MT) is thought to be respon- By necessity, full star cluster dynamical sim- : sible for the production of most observed exotic stellar v ulations, including both direct N-body (Aarseth populations, including blue straggler stars, low-mass X- i 2003, e.g., NBODY6) and Monte Carlo methods (e.g. X ray binaries, millisecond pulsars, cataclysmic variables, Spurzem & Giersz1996;Joshi et al.2000;Vasiliev2015), etc., and the process is thought to contribute signifi- r rely on a number of simplifying assumptions in order to a cantly to the rate of Type Ia supernovae, gravitational treat binary star evolution. In particular, binary MT waves, gamma-ray bursts, etc. A complete understand- is often parameterized, following a pre-calculated grid ing of binary evolution is a prerequisite for predicting of more detailed models that were evolved in isolation. the rates of these high-energy phenomena. Many are These models are based on a sophisticated treatment of thought to occur in star clusters, facilitated by their high densities that stimulate direct1 gravitational en- stellar evolution, with some basic assumptions for the actual MT process (e.g. energy conservation, mass con- counters leading to e.g, exchange interactions, fly-bys servation,etc.). The hydrodynamicsinvolvedin the MT and/or collisions. Such strong encounters can modify process is traditionally not modeled directly within full the binary orbital parameters, and hence directly af- star cluster simulations. Moreover, even if the parame- fect the course of binary evolution (e.g. Ivanova et al. terizations are able to approximate the binary evolution accurately, perturbations during MT are rarely treated [email protected] properly (and are often ignored). For example, a di- [email protected] rect encounter involving a binary undergoing MT could cNSFAstronomyandAstrophysicsPostdoctoral Fellow [email protected] disrupt the binary or exchange a different star into the 1Weusetheterm,direct,toindicatethatthedistanceofclosest system, thus halting the MT process. Scenarios such as approach between an interloping object and a binary undergoing these are not accountedfor in many star cluster dynam- MTisequaltoorlessthanthebinarysemi-majoraxis. 2 Leigh, Geller & Toonen ical models. We initially performed a set of simulations allowing The assumption that MT occurs in relative isolation binaries to occupy nearly the entire log-normal period (even in star clusters) is often justified by comparing distribution and determined that the longest-period bi- time-scales; in many clusters the duration of MT, par- narythatwillundergoMTinSeBawithinaHubbletime ticularly for high-mass stars, can be quite short when (over all primary masses) is log(P[days]) = 4.16. We compared to the single - binary (1+2) encounter time. then imposed this maximum orbital period, along with However, for lower-mass stars, the typical duration of a minimum orbital period chosen such that the binaries MT can greatly increase(e.g., due to the longer thermal are initially detached, and drew 106 random binaries for and nuclear time-scales within the stars). Furthermore, SeBa.3 Binaries in the contact stage are excluded in our the binary - binary (2+2) encounter time-scale can be analysis. much shorter than the 1+2 encounter time in clusters Weperformedtwosetsofexperiments,oneat[Fe/H]= with large binary frequencies.2 1.5, typical for an old globular cluster (GC), and the − In this paper, we begin to quantify the validity of the other at [Fe/H] = 0, appropriate for an OC. We use assumption that MT can be treated as an isolated pro- the SeBa code to evolve these binaries for a Hubble cess within star clusters. Due to mass segregation and timeandproducetime-averagedorbitalparameters(i.e., dynamicalfriction,mostbinariesliveinthedensecluster meanseparation,mean binary mass)calculatedoverthe core, where they may not be allowed to evolve fully be- course of stable MT, as well as the cluster age at the forebeinginterrupted. Weusethebinaryevolutioncode onsetofMT andthe totaldurationofthe MT phase,t , d SeBa (Portegies Zwart & Verbunt 1996; Nelemans et al. for each binary. 2001;Toonen, et al.2012)tosimulatethetimeevolution 3. ENCOUNTERTIME-SCALES of the binary orbital parameters of individual binaries undergoing stable MT, as described in Section 2. We WeusetheoutputfromSeBatocalculatetheexpected then compare the MT time-scales to analytic estimates time-scalesforanothersingleorbinarystarinthecluster for the time-scales over which other single and binary to (directly) encounter, or “interrupt”,each of the bina- stars in the cluster are expected to encounter, or “inter- ries while MT is ongoing,which we will call te. We then rupt”, the ongoing MT, as calculated in Section 3. We compare this time-scale directly to the MT duration, td, show that the fraction of interrupted binaries can reach which is calculated and outputted by SeBa. a few tens of percent, for certain cluster parameters. Fi- Tocalculatethe encountertime-scale,te,wemustfirst nally, we discuss in Sections 5 and 6 the implications of findthetimeforasingle(1)orbinary(2)startointerrupt our results and offer some concluding remarks. an ongoing episode of stable MT, which we will refer to as, respectively, τ and τ . As in, Geller & Leigh 2+1 2+2 2. BINARYEVOLUTION (2015), we define these timescales as follows (Leonard We present results from 106 individual binary evo- 1989; Leigh & Sills 2011): lution simulations performe∼d using the SeBa code. SeBa 103 pc−3 vrms,0 0.5 M⊙ 1 AU is a fast stellar and binary evolution code, that parame- β = , terizesstarsbytheirmass,radius,luminosity,coremass, (cid:18) n0 (cid:19)(cid:18)5 km s−1(cid:19)(cid:18) m (cid:19)(cid:18)Renc (cid:19) etc. as functions of time and initial mass. The binary τ =1.4 1011(1 f )−1β yr , 2+1 b evolution includes mass loss, MT, angular momentum × − loss, common envelope evolution, magnetic braking and τ2+2 =5.4 1010(fb)−1β yr , × gravitationalradiation. We consider only stable MT be- (1) tween hydrogen-rich non-degenerate stars. The stability wheren isthestarclustercentralnumberdensity,v of MT and the MT rate depend mainly onthe adiabatic 0 rms,0 is the central root-mean-squarevelocity (and we assume and thermal response of the donor star’s radius, and the response of the Roche lobe, to the re-arrangement vrms,0 =σ0 =√3σ0,1D), m is the mean mass of a star in of mass and angular momentum within the binary. For the cluster (for which we adopt m = 0.5 M⊙), fb is the an overview of the method and a comparisonwith other binaryfrequency,andπR2 isthe meangravitationally- enc methods, see Toonen, et al. (2014), Appendix B. focused cross section, where R is either the time- enc We follow a nearly identical method to define the averagedseparationofthe binary undergoingstable MT massesandorbitalparametersofourinitialbinarypopu- (for 1+2 encounters) or the orbital separation corre- lation as in Geller et al. (2015). Briefly, primary masses sponding to the cluster hard-soft boundary4 (for 2+2 arechosenfromaKroupa et al.(1993)initialmassfunc- encounters). Finally, the total time until a subsequent tion between 0.1 M⊙and 100 M⊙. We draw mass ra- encounter is given by, tios (q =m /m ) from a uniform distribution such that q 1 and m22 >1 0.1 M⊙. We choose binary orbital pe- te =(Γ2+1+Γ2+2)−1 (2) ≤ riods andeccentricities fromthe log-normalandflatdis- 3 This number of binaries is chosen to be sufficiently large to tributions, respectively, as observed by Raghavan et al. providereliablestatistics. (2010) for binaries with solar-type primary stars in the 4 The hard-soft boundary is a theoretical separation in either Galactic field, which are also consistent with observa- semi-majoraxisororbitalperiod. Binariesinsidethisboundaryare tions of solar-type binaries in open clusters (OCs) (e.g., dynamically “hard”, and encounters with other stars in the clus- Geller et al. 2010, 2013a; Geller & Mathieu 2012). ter tend to shrink (i.e., harden) the binary. Binaries beyond this boundary are dynamically “soft”, and encounters tend to widen and eventually ionize the binary. Note that this results in an ap- 2 The exact binary fraction at which the 2+2 encounter proximate lower limit for the 2+2 time-scale. We also calculate rate begins to exceed the 1+2 rate is around fb ∼ 10% (e.g. a rough upper limit below by adopting instead the mean orbital Sigurdsson&Phinney1993;Leigh&Sills2011). separationbelowthehard-softboundaryforRenc. Interrupted Binary Mass Transfer in Star Clusters 3 where Γ=1/τ. TocalculatetheparametersinEquation1,weconsider total cluster masses Mcl ranging from 103 to 106 M⊙ in steps of 1 in log10(Mcl [M⊙] ), and assume a Plummer model (Plummer 1911) with a half-mass radius5 of 3 pc. See Geller et al. (2015) for specific details. In generaln 0 increaseswithincreasingclustermass,whilef decreases b withincreasingcluster mass(as isconsistentwith obser- vations, e.g., Leigh et al. 2015). 4. FRACTIONOFINTERRUPTEDBINARIES UNDERGOINGSTABLEMASSTRANSFER We find from these binary evolution simulations that, the probability that a given binary will be interrupted during MT in a star cluster core ranges from 1% to ≪ well over unity. The main results are plotted in Fig- ures 1 and 2. AsshowninFigure1,bothτ andτ aresensitive 1+2 2+2 to the total cluster mass, whereas the MT durations t d arenot. Thesensitivityofτ andτ toM isdueto 1+2 2+2 cl the dependence of the semi-major axis at the hard-soft Fig. 1.— Distributions of the total duration of stable MT (td; boundary on M , which results (on average) in tighter solidblackhistograms)andtimesuntilthenextdirect1+2(τ1+2; cl dotted blue histograms) and 2+2 (τ2+2; dashed red histograms) binariesandlowerbinaryfractionsinmoremassiveclus- encounter, drawn from clusters with half-mass radii of 3 pc and ters. In low-mass clusters, the mean binary separation the Mcl values indicated in each panel. The left and right panels and binary fraction are both large, causing τ to be distinguish between assumed metallicities of Z = 0.0006 and Z = 2+2 0.02,respectively. short. As M increases, both the mean binary separa- cl tion and binary fraction decrease along with the separa- tionatthe hard-softboundary,causingτ andτ to 1+2 2+2 decrease and increase, respectively. Importantly, most binariesundergoingstable MT havetime-averagedsepa- rations that are on the order of only a few stellar radii, causing τ to exceed a Hubble Time nearly indepen- 1+2 dent of M . On the other hand, τ is always much cl 2+2 shorterthanaHubbleTime,duetothelargermeansep- aration of the entire cluster binary population. InFigure1,abimodalityisapparentinthedistribution for τ at low metallicity (i.e. Z = 0.0006) that is not 1+2 apparentineitherthe τ distributionathighmetallic- 1+2 ity (i.e. Z = 0.02) or in either of the τ distributions. 2+2 This is due to a bimodality thatappearsin the meanbi- nary separation during mass transfer, which dominates the1+2cross-sectionbutnotthe2+2cross-section,since the mean binary separationin the cluster is much larger than the mean binary separation during mass transfer. The first peak in this bimodal distribution (at small or- bitalseparations)isduetosystemsforwhichstablemass transfer is initiated in a relatively tight binary, involv- Fig. 2.— Distributions of the probability for binaryMT inter- ing relatively unevolved stars. The second peak is due ruption,asdefinedinEquation3. Thebinarypopulationsusedto to systems with asymptotic giant branch (AGB) donors calculatetheseprobabilitiesarethesameasinFigure1. that have lost so much mass that they are less massive probabilityortheprobabilitythatagivenbinarywillun- than their companion. Consequently, the resulting mass dergoat least one encounter with another star or binary transferisstable. Forlowmetallicities,thissecondgroup within a time interval t : isbigger,andasecondpeakbecomesclearlyvisible(rela- d tivetohighmetallicities). Thisisbecausethelifetimesof P =1 e−td/te (3) low metallicity stars are shorter relative to more metal- − rich stars. Therefore, low mass stars are able to evolve Equation 3 approximates the probability that a given off the main sequence, become an AGB star, and hence binary undergoes a direct encounter with another star fill their Roche lobe all within a Hubble time. or binary in the cluster before MT stops (shown on the In Figure 2, we compare t and t for each individual x-axis in Figure 2). d e encounterdirectly,bycalculatingthecumulativePoisson Importantly, fly-by encounters will also perturb bina- riesundergoingMT,andthisisnotaccountedforinFig- 5 Note that for these parameters, the mean orbital separation ures 1 and 2. To quantify this effect, we ran simulations duringMTismuchsmallerthanthehard-softboundary,indepen- of1+2encountersusingtheFEWBODYcode(Fregeau et al. dentofMcl. 2004). We fix the binary orbitalparametersand relative 4 Leigh, Geller & Toonen velocity at infinity for the encounters, but sample the TABLE 1 impact parameter in multiples of 0.5b0, where b0 is the Parameters adopted in Equation 5 tocalculate the impact parameter corresponding to a distance of clos- numbersof interruptedMTbinaries. est approach equal to the binary semi-major axis a, or Mcl Nbin fMT fi Nbin,i (Leonard 1989; Leigh & Sills 2011): (inM⊙) 103 665 0.05 0.18 6 b = 6Gma 1/2 (4) 104 3837 0.07 0.10 27 0 v2 105 16431 0.12 0.03 59 (cid:16) inf (cid:17) 106 50338 0.23 0.01 116 Here, m is the mean single star mass in the cluster and v is the relative velocity at infinity. We perform 104 inf of the cluster lifetime is excluded from our calculation numerical scattering experiments for every choice of the for the fraction of binaries expected to be interrupted impact parameter. All stars are assumed to be point during MT, these estimates increase by a factor 2.6 particles with masses of 1M⊙. The initial binary semi- These fractions correspond approximately to upper∼lim- majoraxisandeccentricityare1AUand0,respectively. its, since we adopt the mean orbital separation at the The incoming velocity at infinity is set to v =5 km/s. inf hard-soft boundary when calculating the 2+2 encounter The results of these scattering experiments are illus- time. Ifweadoptinsteadthemeanorbitalseparationbe- trated in Figure 3, which shows the mean (absolute) low the hard-soft boundary (which does not include the changeinthebinarysemi-majoraxisandeccentricityfor presence of soft binaries), these fractions decrease by a all simulations, as a function of the impact parameter. factor of a few (of the same order as the increase from We assume here that a fly-by encounter “interrupts” a including fly-bys). givenbinaryundergoingMTifeitherofthetwofollowing Fromthesefractions,wecanalsoestimatethe number criteriaaremet: 1)thechangeinsemi-majoraxis ∆a is of binaries that are expected to be directly interrupted | | greater than or equal to the typical stellar radius at the during ongoing stable MT: onsetofRochelobeoverflow( 1R⊙ formostbinaries), and/or 2) the change in eccen∼tricity ∆e correspondsto Nbin,i =NbinfMTfi, (5) | | a change in the quantity r r that is greater than or equal to the typical stellarp−radaius at the onset of MT, where Nbin is the total number of initial binaries in the cluster (estimated as f N ), f is the fraction of where rp and ra are the pericenter and apocenter dis- binaries expected to unbdesrtgaorsMTM(Tcalculated by SeBa tances, respectively, from the binary center of mass. We for our assumed binary orbital parameter distributions) assumethatthemagnitudeofsuchaperturbationissuf- and f is the fraction of binaries undergoing MT that ficientlylargetoatleasttemporarilyhalttheMTprocess i are expected to be interrupted over a Hubble time of (provided ∆a>0 and/or ∆e<0). clusterevolution(ascalculatedaboveforourfourcluster As shown in Figure 3, the perturbative effects of fly- masses). These parameters are shown in Table 1, for all bysremainsignificantforb.1.5b . Moreover,both ∆a 0 | | four assumed values of the total cluster mass. and ∆e undergoasteepdrop-offbetweenb=1b andb | | 0 The resulting total numbers of binaries expected to =2b ,suchthatneitherofourcriteriafora“significant” 0 be interrupted over the lifetime of the clusters is then perturbationaresatisfiedforb&1.5b . Ouranalytices- 0 N 6, 27, 59 and 116 for our model clusters with timatesforthe1+2and2+2encounterratesaredirectly bin,i ∼ proportional to the gravitationally-focused cross-section Mcl[M⊙] = 103, 104, 105 and 106, respectively. If per- σ πb2,forsomeimpactparameterb. Hence,iffly-bys turbativefly-bysare included, these values eachincrease gf ∝ by a factor 2. The calculated rate of MT interruption areincluded,thetime-scalesfor1+2and2+2encounters for clusters∼is 1-10 binary per Gyr, with only a weak should each decrease by a factor .1.52 2.1, as should ∼ ∼ tendency for this rate to increase with increasing cluster t . Working from Equation 3, we find that the proba- e mass. bilities typically increase by a corresponding factor of ∼ 2. Thatis,the ratiobetweentheprobabilitiescalculated 5. DISCUSSION with and without this additional factor of 2.1 in our ∼ We find that a relatively large fraction f of binaries estimate for t peak at a value of 2 with very little i e ∼ undergoing stable MT may encounter another single or scatter. binary star in the core of a star cluster before the MT The fraction of binaries expected to be interrupted proceeds to completion. We calculate f 0.18, 0.10, during MT can be estimated from the probability dis- i ∼ tributions presented in Figure 2. To do so, we integrate 0.03 and 0.01 for our model clusters with Mcl[M⊙] = the probability distributions in Figure 2 and divide by 103,104,105and106,respectively,nearlyindependentof the total number of binaries. The resulting fractions are metallicity. These fractions are only slightly lower than f 0.18,0.10,0.03and0.01forourmodelclusterswith we calculated in Geller & Leigh (2015) for the fraction i Mmec∼tl[aMlli⊙ci]ty=Z10=3,01.0024., 1T0h5esaendfra1c0t6io,nrsesrpeemcatiivneltyh,easnadmae 6Notethatthefinalincreaseinthemeandurationofmasstrans- fer at ∼ 13 Gyr seen for the Z = 0.0006 case should be regarded to within 1% for a metallicity Z = 0.0006. As shown with caution. This is due in part to small number statistics, and in Figure 4±, the mean mass transfer duration is 105.5 thepresenceofahandfulofbinarieswithunusuallylongMTdura- years during the first 1 Gyr of cluster evolution∼, then tionsatlatetimes. Thisisnotvisibleintheerrorbars,sinceweare quickly increases by roughly an order of magnitude. Af- showing the standard error of the mean here, and the number of binariesincludedinthiscalculationislarge. Perhapsmoreimpor- ter this, the mean MT duration slightly and slowly de- tantly,theseerrorbarsdonotreflectanyinherentuncertaintiesin creasesoverthe next 12Gyr,butremainsmuchlarger theSeBacodeandbinaryevolutioningeneral,whicharecertainly than during the initia∼l 1 Gyr. Hence, if the initial 1 Gyr non-negligible. Interrupted Binary Mass Transfer in Star Clusters 5 small geometric cross section, the interloper in these cases are most often binary stars themselves, resulting in 4-body encounters that may dramatically alter the course of the binary evolution. Interrupted MT is ex- pectedtooccurmostofteninmassiveOCsandlow-mass GCs ( 104 M⊙), where in general the binary frequency ∼ is larger,and (detached) binaries can have larger orbital separationsandthereforelargergeometriccrosssections. These larger binary frequencies and geometric cross sec- tions present in the low-mass cluster regime also result in a larger fraction of interrupted stellar encounters. No publiclyavailablecodeforstarclusterdynamicsaccounts directly for the hydrodynamics of binary MT needed to accurately model such interrupted/perturbed MT bina- ries. As discussed in Geller & Leigh (2015) for interrupted stellarencounters,the dependenceoff onM isencour- i c aging for numerical models of star clusters, since low- mass clusters such as OCs are most often modeled with direct N-body simulations. Direct N-body codes, such Fig.3.— The (logarithm of the) mean change in eccentricity as NBODY6,can in principle apply perturbations to bina- —∆e— (red triangles) and semi-major axis —∆a— (black open riesundergoing(parameterized)MT,ifthetimestepsare circles; in AU) for binaries involved in single-binary interactions, chosenappropriately. Given our results, a more detailed as a function of the impact parameter (x-axis). The impact pa- rameterissampledinunitsof0.5b0,definedasinEquation4. The study of how often such cases of interrupted MT occur uncertainties correspond to the standard deviation of the mean. and how accurately they are treated in direct N-body (The distributions for —∆a— and —∆e—are not strictly Gaus- codes is warranted. Massiveclusters, on the other hand, sian,sotheuncertaintiesshouldformallybeasymmetric. However are more often modeled with Monte Carlo codes (e.g., the differences are negligible for our purposes here.) The dashed linescorrespondtothecriticalvaluesof|∆a|(black)and|∆e|(red; Chatterjee et al. 2010; Hypki & Giersz 2013). Here, bi- inAU)defined byour criteriafora”significant” perturbation de- nary evolution is assumed to run to completion in iso- scribedinthetext. lation. In massive GCs, we find that this assumption is much more valid than in OCs (albeit not perfect, see Figure 2), similar to what was found in Geller & Leigh (2015) for interrupted stellar encounters. The time-averaged separations of binaries undergoing stable MT are typically onthe orderof a few solarradii. However,stableandconservativeMTleadstoanincrease inthe binaryorbitalseparation(ifthe donorisless mas- sive than the accretor). Generally toward the end of the MT phase, after a mass-ratio inversion, the orbits expand significantly, increasing the binary cross-section anddecreasingthe1+2and2+2encountertimes. Hence, thetime-scaleforinterruptiondecreasesmarkedlyatthe end of the MT phase and beyond, which is not prop- erly accounted for in our models (since we only use the mean binary separation during MT). Interestingly, the final separation is on the order of & 1 AU in our mod- els, which is comparable to the hard-soft boundary in old massive ( 106 M⊙) GCs. Hence, these binaries be- ∼ come subject to ionization due to dynamical encounters withothersingleandbinarystarsafterMT.Assuminga MT origin, this could be of particular relevance for blue Fig. 4.—The(logarithmofthe)meandurationofmasstransfer straggler formation in GCs. In particular, this naively (shown on the y-axis; in years) as a function of the (logarithm of predictsaroughcorrelationwherebythe fractionofblue the)hostclusterage(x-axis;inGyr),forallclustermassesadopted inFigures1and2. Themeaniscalculatedin1Gyrintervals,using stragglers in binaries increases with decreasing Mcl. all binaries that initiate mass transfer during each 1 Gyr window oftheclusterlifetime. Errorbarscorrespondtothestandarderror 6. CONCLUSIONS ofthemean. Blackandredsolidcirclescorrespondtometallicities Even the very intimate dance of binary MT can be of,respectively,Z=0.0006andZ=0.02. interrupted by an interloping single or, more often, bi- of interrupted stellar encounters in star clusters, which nary star in a star cluster. Up to about a quarter of rangesfrom< 0.01inmassiveGCs to > 0.4in low-mass all binaries undergoing stable MT in the cores of mas- OCs. The origin of this analogous dependence between sive OCs and low-mass GCs ( 104 M⊙) may encounter ∼ the fractionofinterrupted”events”andthe totalcluster another star (or stars) before completing this phase of mass is ultimately the same (see below). evolution(seeFigure2). Thisfractiondecreasestowards As a binary undergoing MT generally has a relatively <<1% for more massive GCs, primarily because the bi- 6 Leigh, Geller & Toonen nary frequency in these clusters is much lower. Cases of evolution in star cluster simulations. interrupted MT may lead to significantly different out- comes than are assumed for isolated evolution, which in The authors would like to thank an anonymous re- turn may impact the formation rates of MT products in viewerforusefulsuggestionsforimprovement. N.W.C.L. starclusters,fromexoticstarslikebluestragglers,toSN is grateful for the generous support of an NSERC Post- Ia and gamma-ray burst progenitors. Our results high- doctoral Fellowship. A.M.G. is funded by a National light the need for a more detailed treatment of binary Science Foundation Astronomy and Astrophysics Post- doctoral Fellowship under Award No. AST-1302765. 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