Formation and evolution of compact binaries in globular clusters: II. Binaries with neutron stars
N. Ivanova, C. Heinke, F.A. Rasio, K. Belczynski, J. Fregeau
aa r X i v : . [ a s t r o - ph ] F e b Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 28 October 2018 (MN L A TEX style file v2.2)
Formation and evolution of compact binaries in globular clusters:II. Binaries with neutron stars.
N. Ivanova ⋆ , C. 0. Heinke , † , F. A. Rasio , K. Belczynski , ‡ , & J. M. Fregeau § Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George, Toronto, ON M5S 3H8, Canada Northwestern University, Dept. of Physics & Astronomy, 2145 Sheridan Rd, Evanston, IL 60208, USA University of Virginia, Dept. of Astronomy, PO Box 400325, Charlottesville, VA 22904, USA Los Alamos National Lab, P.O. Box 1663, MS 466, Los Alamos, NM 87545 New Mexico State University, Department of Astronomy, 1320 Frenger Mall, Las Cruces, New Mexico 88003-8001, USA
28 October 2018
ABSTRACT
In this paper, the second of a series, we study the stellar dynamical and evolutionary processesleading to the formation of compact binaries containing neutron stars (NSs) in dense globularclusters. For this study, 70 dense clusters were simulated independently, with a total stellarmass ∼ × M ⊙ , exceeding the total mass of all dense globular clusters in our Galaxy.We find that, in order to reproduce the empirically derived formation rate of low-massX-ray binaries (LMXBs), we must assume that NSs can be formed via electron-capture super-novae with typical natal kicks smaller than in core-collapse supernovae. Our results explainthe observed dependence of the number of LMXBs on “collision number” as well as the largescatter observed between different globular clusters. We predict that the number of quiescentLMXBs in different clusters should not have a strong metallicity dependence.We compare the results obtained from our simulations with the observed population ofmillisecond pulsars (MSPs). We find that in our cluster model the following mass-gainingevents create populations of MSPs that do not match the observations (either they are incon-sistent with the observed LMXB production rates, or the inferred binary periods or companionmasses are not observed among radio bMSPs): (i) accretion during a common envelope eventwith a NS formed through electron capture supernovae, and (ii) mass transfer (MT) from aWD donor. Some processes lead only to a mild recycling – physical collisions or MT in apost-accretion induced collapse system. In addition, for MSPs, we distinguish low-magnetic-field (long-lived) and high-magnetic-field (short-lived) populations, where in the latter NSsare formed as a result of accretion induced collapse or merger induced collapse. With thisdistinction and by considering only those mass-gaining events that appear to lead to NS re-cycling, we obtain good agreement of our models with the numbers and characteristics ofobserved MSPs in 47 Tuc and Terzan 5, as well as with the cumulative statistics for MSPsdetected in globular clusters of different dynamical properties.We find that significant production of merging double NSs potentially detectable as short γ -ray bursts occurs only in very dense, most likely core-collapsed clusters. Key words: stellar dynamics – binaries: close – binaries: general – stars: neutron – pulsars:general – globular clusters: general – X-rays: binaries.
Neutron stars (NSs) are seen in globular clusters (GCs) via their(current or past) participation in interacting binary systems. Fromthe earliest observations of X-ray binaries in GCs it has been notedthat their abundance per unit mass is ∼
100 times greater in GCs ⋆ E-mail:[email protected], Tremaine Fellow † Lindheimer Fellow ‡ Oppenheimer Fellow § Chandra Fellow than in the Galaxy as a whole (Katz 1975). This was understoodto be a consequence of the high stellar density of GCs, which maylead to the creation of compact NS binaries in close stellar encoun-ters (Clark 1975). NS binaries are often relatively easy to identifyin GCs (compared to other binaries), and they allow us to constrainthe dynamical history of clusters and the evolution of NS propertiesin binary systems. They have been observed so far in three guises:(i) as bright low-mass X-ray binaries (LMXBs); (ii) as quiescentLMXBs (qLMXBs); and (iii) as binary or single millisecond pul- c (cid:13) N Ivanova et al. sars (MSPs). In the future, they could also be detected as sourcesof gravitational radiation.Bright LMXBs have X-ray luminosities of typically – ergs s − . In bright LMXBs, the NS is actively accretingfrom a companion star via Roche-lobe overflow and an accretiondisk. Thirteen sources are known in Galactic GCs, of which 12are known to contain NSs through the detection of thermonuclearbursts. Six are transient systems (two have remained in outburstfor more than a decade; the others have outburst time-scales ofweeks), while the other seven seem to be persistent. Orbital peri-ods have now been identified for seven systems (Verbunt & Lewin2006; Dieball et al. 2005; Altamirano et al. 2007), of which threeare below one hour, indicating ultracompact systems composed ofNSs with white dwarf companions are common in globular clus-ters (Deutsch et al. 2000). Efficient surveys of the X-ray sky giveus confidence that we have detected all bright LMXBs that haveexisted in Galactic GCs during the past 10 years.Quiescent LMXBs have lower X-ray luminosities ( We adopt the binary evolution model from the population syn-thesis code StarTrack (Belczynski et al. 2002, 2008). In ourfirst study of binary populations in GC cores (Ivanova et al.2005a) we used a previous version of StarTrack describedin Belczynski et al. (2002). In our study of close binaries withwhite dwarfs (Ivanova et al. 2006, paper I) we employed a morerecent version of StarTrack available at the time (as inBelczynski et al. 2008, but including updates only for WD evo-lution compared to the previous version). In this paper we in-corporated all the latest updates of StarTrack , as described inBelczynski et al. (2008). Below we outline several of the most c (cid:13) , 000–000 inaries with NS in globular clusters important changes in StarTrack that affect the formation andevolution of NSs compared to our previous studies of NSs inGCs (Ivanova & Rasio 2004; Ivanova et al. 2005b, 2006). Wealso provide detailed descriptions of further modifications to the StarTrack model, beyond those in Belczynski et al. (2008),which we developed for the treatments of electron-capture super-novae and common-envelope events. At the moment of formation, both NSs and BHs receive addi-tional speed (a natal kick), most likely due to asymmetric su-pernova ejecta (Fryer 2004). Although most recent simulationsare in relatively good agreement with the measured distributionof pulsar velocities, the agreement is not yet firmly established(Scheck et al. 2006). In this paper we adopt the most recently de-rived pulsar kick velocity distribution from Hobbs et al. (2005),which is a Maxwellian distribution with one-dimensional RMSvelocity σ = 265 km s − (the mean three-dimensional velocityis ∼ km s − ). In contrast to some earlier studies (e.g., seeArzoumanian et al. 2002), no evidence for a bimodal velocity dis-tribution is present.At the moment of the explosion, the supernova progenitor canbe a binary companion. Because of the kick, the binary can becomeunbound (and the binary components will separate) or the entirebinary can be affected by the natal kick. In our previous studies, inorder to find the velocities of the components of an unbound binary,we used the derivation by Tauris & Takens (1998), where the pre-supernova orbit is assumed to be circular. The approach used inthe current version of StarTrack allows us to properly calculatecases with an arbitrary pre-explosion orbital eccentricity (for fulldetails, see § Through our studies of accreting WDs (Ivanova et al. 2006) wefound that electron-capture supernovae (ECSe) could be the dom-inant source of retained NSs in clusters. Similar results were alsorecently obtained in Kuranov & Postnov (2006). We have thereforeconsidered in great detail all possible types of ECSe for this work.It has been argued that when a degenerate ONeMg corereaches M ecs = 1 . M ⊙ , its collapse is triggered by electron cap-ture on Mg and Ne before neon and subsequent burnings startand, therefore, before the iron core formation (Miyaji et al. 1980;Nomoto 1984, 1987; Timmes & Woosley 1992; Timmes et al.1994). The explosion energy of such an event is significantly lowerthan that inferred for core-collapse supernovae (Dessart et al. 2006;Kitaura et al. 2006). There are several possible situations when astar can develop a degenerate ONeMg core that will eventuallyreach M ecs .First, this can occur during the normal evolution of singlestars. It was initially proposed by Barkat et al. (1974) that a starof 7–10 M ⊙ , after non-explosive carbon burning, develops anONeMg core. If the initial core mass is less than required forthe neon ignition, . M ⊙ , the core becomes strongly degener-ate. Through the continuing He shell burning, this core grows to M ecs . In more massive stars, & M ⊙ , carbon, oxygen, neon andsilicon burnings progress under non-degenerate conditions, and, inless massive stars, ONeMg cores never form. The formation of de-generate or non-degenerate ONeMg cores depends on the He coremass at the start of the asymptotic giant branch, where ∼ . M ⊙ is a rough boundary between the cases (Nomoto 1984). If the ini-tial He core mass is below . M ⊙ , no off-centre ignition willhappen—the carbon core burning will occur when the degeneratecore reaches the Chandrasekhar mass, resulting in a thermonuclearexplosion (Hurley et al. 2000, also J. Eldridge 2006, priv. comm.).The conditions for ECS were shown to occur in single stars ofinitial mass 8–10 M ⊙ by Nomoto (1984). This critical mass rangedepends on the properties of the He and CO cores, which, in turn,are highly dependent on the mixing prescription (semiconvection,overshooting, rotational mixing, etc.) as well as on the adoptedopacities. As a result, the range varies between different evolu-tionary codes (see discussion in Podsiadlowski et al. 2004; Siess2006), and is generally reduced compared to that proposed initiallyby Nomoto (1984, 1987). In the code that we use for our clustersimulations, a non-degenerate ONeMg core is formed when the ini-tial He core mass is about . M ⊙ (Pols et al. 1998; Hurley et al.2000) and the range of initial masses for single stars of solar metal-licity that leads to the formation of such a core is 7.66 to 8.26 M ⊙ . The equivalent mass ranges are from 6.85 to 7.57 M ⊙ andfrom 6.17 to 6.76 M ⊙ for single stars with GC-like metallicities, Z = 0 . and Z = 0 . , respectively.The initial stellar mass is not the only parameter that defineswhether a star will form a degenerate ONeMg core. It was recentlypointed out by Podsiadlowski et al. (2004) that the range of progen-itor masses for which an ECS can occur depends also on the masstransfer history of the star, and therefore may be different for binarystars, making it possible for more massive progenitors to collapsevia ECS. We will refer to the single and binary scenarios for ECSin non-degenerate stars, as described above, as evolution-inducedcollapse (EIC).The second possibility for an ECS to occur is by accretion onto a degenerate ONeMg WD in a binary: accretion-induced col-lapse (AIC). In this case, a massive ONeMg WD steadily accumu-lates mass until it reaches the critical mass M ecs (for more detail onthe adopted model of the accretion on the WD see Ivanova & Taam2004; Belczynski et al. 2008). In addition, ONeMg WDs can beformed from CO WDs via off-centre carbon ignition if the massof CO WDs is above 1.07 M ⊙ and the accretion rate is high, ˙ M > . × − M ⊙ yr − (Kawai et al. 1987) so that eventuallythis also leads to AIC.The third case that we consider is coalescing double WDs witha total mass exceeding M ecs . To distinguish it from AIC, we willrefer to this case as merger-induced collapse (MIC). The nature ofNS formation is the same as in the case of AIC (accumulation ofmass by a massive ONeMG WD). In the case of coalescing COWDs, then, as in the case of single stars, off-centre carbon ignitionoccurs first, which quiescently converts the star into an ONeMgcore, and this proceeds with an ECS (Saio & Nomoto 1985, 2004).Although the approach above was proposed for coalescing WDsin binaries only, we assume that in the case of collisions betweentwo WDs it is applicable as well: except for the rare cases of nearhead-on collisions, a less massive WD will be tidally disrupted andaccreted at high rate on to the more massive WD.We therefore assume in our simulations that a NS will beformed via ECS in the following cases: • if at the start of the AGB the He core mass is . M ⊙ . M c , BAGB . . M ⊙ (EIC) • if an accreting ONeMg WD reaches M ecs (AIC) • if the total mass of two merging or colliding WDs exceeds M ecs (MIC).For NSs formed via ECS we assume that the accompanying c (cid:13) , 000–000 N Ivanova et al. natal kick is 10 times smaller than in the CC case. This assump-tion follows from the results of numerical simulations which findthat the SASI instability, required by current understanding for thelarge explosion asymmetry in the case of core-collapse supernovae,fails to develop and resulting kicks are significantly smaller and onoverall do not exceed 100 km/s (Buras et al. 2006; Kitaura et al.2006). We will provide the separate statistics for NSs created bydifferent channels (core-collapse, EIC, AIC or MIC), so that if oneof the channels later turns out to have been overestimated it will beeasy to recalibrate.The gravitational mass of the newly formed NS is . M ⊙ .This is ∼ . of its baryonic mass (which cannot exceed the pre-collapse core mass) after subtracting the binding energy and is cal-culated as in Timmes et al. (1996): M baryon − M grav = ∆ M = 0 . M , (1)where the masses are in solar masses. As the angular momentum ofthe binary is conserved, in the case of no kick, the binary widens.The condition for ECS to occur depends essentially on thecentral density of the object. As a consequence, a rapidly rotatingWD can reach a much higher mass than the Chandrasekhar limitbefore the central density becomes high enough for electron cap-tures on Mg and Ne to occur. For example, its mass can beas high as . M ⊙ if the ratio of the WD rotational energy tothe gravitational binding energy is 0.0833 and the star is highlyoblate (Dessart et al. 2006). Such rapidly rotating heavy WDs canbe formed, e.g., during the coalescence of two WDs with total massexceeding the Chandrasekhar mass. During such a merger, less than0.5 per cent of mass will be lost from the system (Guerrero et al.2004). The collapse of a rapidly rotating WD can therefore lead tothe formation of a more massive and very fast spinning NS. The presence or absence of a deep outer convective envelope influ-ences a star’s behavior during various phases of close binary inter-actions (e.g., angular momentum loss via magnetic braking, tidalinteractions, stable or unstable mass transfer). In particular, it hasbeen pointed out in Ivanova (2006) that the absence of a deep outerconvective zone in low-metallicity GCs may explain the preferen-tial formation of LMXBs in metal-rich clusters (Grindlay 1993;Bellazzini et al. 1995; Zepf et al. 2006).In this study, for main-sequence stars we adopt the metallicity-dependent mass range to develop a convective envelope describedby (eq.(10) in Belczynski et al. 2008) In addition, in our previouswork, all giant-like stars were assumed to have outer convectiveenvelopes. Now stars crossing the Hertzsprung gap and stars in theblue loop are not assumed to have convective envelopes if theireffective temperature satisfies log T eff > . . For common-envelope (CE) events we use the standard energy for-malism (Livio & Soker 1988). In this approach, the outcome ofthe CE phase depends on the adopted efficiency for orbital energytransfer into envelope expansion energy α ce and on the donor en-velope central concentration parameter λ . The commonly used pre-scription is α ce × λ = 1 , and this is adopted in our models as well.However, for He stars with an outer convective envelope we deter-mine λ from the set of detailed He star evolution models calculatedby Ivanova et al. (2003). In this case, λ = 0 . R − . , where R is the radius of the He star in solar radii. As many He stars end their livesas NSs, the CE evolution of He stars is an important channel forthe formation of close binaries with NSs. Indeed, the prescriptionabove was recently used in a study of double neutron star formationas short γ -ray burst progenitors (Belczynski et al. 2006). We consider in our models the possiblity of (eccentric) binary for-mation via physical collisions involving a red giant (RG). For bina-ries formed through NS–RG collisions, the final binary separation a f and eccentricity e f depend on the closest approach distance p (Lombardi et al. 2006) and can be estimated using the results ofhydrodynamic calculations as e f = 0 . − p R RG (2) a f = p . − e ) (3)More details on our treatment of these collisions can be found inIvanova et al. (2006). In this paper, we refer to this event as to adynamical common envelope (DCE). We assume that the amount of matter accreted by a NS following amerger or collision depends on the evolutionary stage of the otherstar. If a NS collides or merges with a MS or a He MS star, weassume that it accretes no more than 0.2 M ⊙ (Rosswog 2006). Inthe case of a collision with a giant-like star, the maximum accretedmass is 0.01 M ⊙ (Lombardi et al. 2006). When the other star is aWD, all the WD mass is accreted. In essentially all mergers andcollisions, the accreted mass must have gone through an accretiondisk, implying that, if a significant amount of matter is accreted, arecycled pulsar will be formed. Stable hierarchical triples can be formed through binary–binary en-counters. Although these triples would be stable in isolation, it islikely that they will be destroyed during their next dynamical en-counter. As there are no developed population synthesis methodsfor handling triple star evolution, we cannot keep these triples inour simulations and instead we have to break them immediately af-ter formation into a binary and a single star. We do this while con-serving energy: the energy required to eject the outer companion isbalanced by shrinking the inner binary orbit. The outer companionis released unless the required energy is such that the inner binarymerges. In the latter case the inner system is allowed to merge andthe outer companion is kept in a new, wider orbit to form the finalbinary system.It is possible that in a triple the inner orbit’s eccentricity willbe increased via the Kozai mechanism (Kozai 1962). This secu-lar coupling causes large variations in the eccentricity and incli-nation of the orbits and could drive the inner binary of the triplesystem to merge before the next dynamical interaction. The maxi-mum eccentricity in systems with large initial inclinations i , suchthat sin i > (2 / / ( i & ◦ ) is (e.g., Innanen et al. 1997;Eggleton & Kiseleva-Eggleton 2001): c (cid:13) , 000–000 inaries with NS in globular clusters e max ≃ p − / ( i ) . (4)The period of the cycle to achieve e max is (Innanen et al. 1997;Miller & Hamilton 2002) τ Koz ≃ . 42 ln(1 /e i ) p sin ( i ) − . (cid:18) m + m m o b a (cid:19) / (cid:18) b Gm o (cid:19) / , (5)where e i is the initial eccentricity of the inner binary, m , m and m o are the masses of the inner binary companions and the mass ofthe outer star, a i and a o are initial orbital separations for the innerand outer orbits, and b o = a o (1 − e o ) / is the semiminor axis ofthe outer orbit.We compare the Kozai time-scale τ Koz with the collision time τ coll (computed as in Ivanova et al. 2005a): τ coll = 8 . × yr P − / M − / n − v − × (6) (cid:18) M tot + h M i ) kP / M / v (cid:19) − Here P td is the triple period in days, M tot is the total triple massin M ⊙ , h M i is the mass of an average single star in M ⊙ , v = v ∞ / (10 km / s) and n = n/ (10 pc − ) , where n is the stellarnumber density.If τ Koz > τ coll , the Kozai mechanism does not affect thetriple evolution before the next encounter occurs, and we breakthe triple as describe above. However, for triples with τ Koz <τ coll , we consider the circularization time-scale τ circ (see § τ circ / e (Mazeh & Shaham 1979). If τ circ / e < τ coll , the inner binarywill shrink until one of the components overfills its Roche lobeor its separation will small enough to cause the Darwin instabil-ity resulting in a merger (Eggleton & Kiseleva-Eggleton 2001). Inthe opposite case, when τ circ / e > τ coll , we assume that masstransfer starts if e max is such that at least one of the stars in theinner binary overfills its Roche lobe at pericentre. As a result, tripleformation may enhance the formation of mass-transferring binarieswith a NS.For sufficiently compact inner binaries, relativistic effects mayplay an important role. The relativistic precession period P pr of theinner binary, to first post-Newtonian order, is (see Weinberg 1972,p.197): P pr = 2 πc a / (1 − e )3 G / ( m + m ) / (7)If τ Koz > P pr , Kozai cycles are strongly suppressed (Holman et al.1997).We expect that some dynamically formed triples can also un-dergo significant secular eccentricity evolution even for orbital in-clinations smaller than the Kozai angle (see e.g. Ford et al. 2000),but here we neglect this possibility. The time over which a pulsar will spin down from its initial period P to a current period P is τ MSP = P − P P ˙ P , (8) where P ˙ P = B / [s]. With the assumption that the magneticfield B does not decay, P ˙ P remains constant with time.Several observed radio pulsars in GCs are slowly spinning( P > . s) and have high ˙ P s (higher than can be produced bygravitational acceleration in the cluster potential) implying rela-tively high B fields, > Gauss. These include NGC 6342A(van Kerkwijk et al. 2000), NGC 6624B (Biggs et al. 1994), andNGC 6440A (Lyne et al. 1996). Their inferred characteristic agesrange from yr to × yr, compared to − yr for“normal” MSPs. These pulsars cannot have been formed in theircurrent state through core-collapse SNe, which only occur withinthe first ten million years. The inferred birthrate for this class ofobjects is similar to that of “normal” MSPs, but observationallythey seem to be concentrated in the densest clusters, and 2 of 3 areisolated pulsars.A strong magnetic field of > -gauss can be producedby the collapse, assuming flux conservation, of a typical magneticWD with B ∼ -gauss. Observed magnetic WDs typically havehigher masses than nonmagnetic WDs(Wickramasinghe & Ferrario2005), indicating that they may dominate the population of WDsthat undergo AIC. Stellar dynamo theory supports the link betweenstrong magnetic fields and massive WDs (Thompson & Duncan1993). It is also possible that a strong magnetic field may be gen-erated during the merger of a WD pair, where one or more WDshad a strong magnetic field, leading to the production of -gaussmagnetic fields, or even magnetar B field strengths of ÷ -gauss (King et al. 2001; Levan et al. 2006; Chapman et al. 2006).MSPs formed with such high magnetic fields will have relativelyshort lifetimes, and thus will make up only a small portion of theobserved globular cluster MSP population. In our simulations, wedo not assign a specific magnetic field to formed neutron stars.However we keep a record of their formation and therefore cantest whether the population of AIC and MIC NSs is formed at therequired rates to explain the population of slow GC radio pulsars,assuming that their life-time is only years.We refer to young, recently formed pulsars that have not yetbeen recycled as “1a” and “1b” in Fig. 1. As a standard picture, weconsider that the origin of a NS does not affect its further evolu-tion if a MT event occurred in a dynamically formed binary (c.f.post-AIC evolution), but we note that it is possible that MIC NSsmight have a higher than usual magnetic field and their subsequentevolution might be different – they can be seen only for a shortertime after their formation as young slow pulsars and they might benot able to accrete and be spun up. Very little is known about whether a magnetic field is always re-duced during mass accretion, independently of its initial strength.E.g., the UCXB and 7.7 second X-ray pulsar 4U 1626-67 hasa strong ( × -gauss) magnetic field (Orlandini et al. 1998).On the other hand, known magnetic fields in accreting MSPsare usually - gauss (e.g., − × -gauss in XTEJ1751-305, Di Salvo & Burderi (2003); < × -gauss in IGRJ00291+5934 Torres et al. (2007); . -gauss in XTE J0929-314,Galloway et al. (2002)) and are consistent with the magnetic fieldsof MSPs. Even though 4U 1626-67 is evolving through the accre-tion phase and has passed the stage when the MT rate is maxi-mum, the NS has kept a high magnetic field and has not beenspun up to millisecond periods. To explain this, it has been sug-gested that the NS in 4U 1626-67 was recently formed via AIC(Yungelson et al. 2002). Several radio pulsars in the Galaxy possess c (cid:13) , 000–000 N Ivanova et al. relatively high B fields and long periods, with low-mass compan-ions in low-eccentricity orbits (Breton et al. 2007). They argue thatthese systems can be well-explained through AIC.It is unclear why post-AIC binaries may yet fail to produceMSPs. Reasons could include the high initial B field and mod-erately fast spin of newly-formed AIC NSs, which could disruptspin-up; or simply that after sufficient accretion onto the WD toreach the Chandrasekhar limit, there is usually not enough remain-ing mass in the companion to spin the NS up to millisecond periods.We will show below that producing MSPs in the same binaries inwhich AIC occurred contradicts observations. For our final results,we therefore adopt that NSs formed via AIC or MIC have shortlifetimes ( years), and that NSs recycled in their original AICsystems are only mildly recycled, also with lifetimes of years(case “2a” in 1). A common understanding of recycled pulsar formation is that theNS is recycled through disk accretion and has its magnetic fields re-duced during the accretion phase (van den Heuvel 1984). Since thecharacteristic lifetime of a MSP is − years and is signif-icantly larger than a GC life-time ( years), we do not considerin our simulations the evolution of MSPs towards a death zone afterthey are formed.The amount of material that needs to be accreted on to a NSto form a MSP is however not firmly established. In our simula-tions, we record for further analysis any event through which a NSacquires at least 0.01 M ⊙ . We also distinguish mass gain throughsteady accretion from that which can result from a physical colli-sion with a red giant (DCE); a CE event during binary evolution; ora merger or physical collision with a MS star or a WD.The connection between stable mass-transfer and the bi-nary MSP production in the field seems to be reasonably well-understood and is reviewed by Deloye (2007). We outline only themain branches: • If the donor is a RG or an asymptotic-giant branch companion,then the resulting binary MSP has a wide orbital separation (“4b”at the Fig. 1). • If the MT occurred in a NS-MS binary, then the evolution de-pends on the period at the start of the MT. Some systems will evolvetowards shorter periods(“3a” at the Fig. 1), and others to longer(“4d” at the Fig. 1) (for a review of the current understanding ofNS-MS LMXBs evolution see Deloye 2007). The progenitors tothe short-period binary MSPs evolve slowly enough that they maybe detected as (bright or quiescent) LMXBs; systems with theseorbital periods are indeed seen in globulars (Heinke et al. 2005a;Altamirano et al. 2007). • If a binary system with a NS has evolved through a CE re-sulting in a significant shrinkage of a binary orbit, MT with a WDdonor can start. This leads to the formation of an ultra-compactX-ray binary (UCXB) (“3b” at the Fig. 1).We do not discuss here the case of thermal-time scale MT thatoccurs with an intermediate-mass donor ( M d M ⊙ ) in aclose orbit at the start of the MT (0.3-2 days), as we find it un-likely for a NS in a GC environment to acquire an intermediatemass donor (c.f. Rasio et al. 2000). We will return to this point laterin § ∼ . M ⊙ (see Fig. 1 in Deloye 2007). There are discrepancies between the observed orbital periods ofshort-period radio binary MSPs (bMSPs) with very low-mass com-panions and the modeled periods. Using a standard mass-radiusrelation for WDs during the MT evolution of a NS-WD binarygives periods several times smaller than those of observed bM-SPs with companions of similar masses. It has been shown thateven the consideration of more realistic WD models does notchange the result strongly – hotter donors don’t, by themselves,produce the long periods needed to explain the short-period bM-SPs (Deloye & Bildsten 2003; Deloye 2007). If the mass-losingcompanion is a WD, some poorly understood process must be in-voked – e.g., strong tidal heating of the WD during MT (Rasio et al.2000). In that case, it is not possible to accurately predict the bMSPperiods during the MT or at its end.There is no clear termination point for MT from a degeneratedonor. As a result, the MT continues uninterrupted and within theHubble time the companion mass can reach below . M ⊙ . Noobserved radio bMSPs have been shown to have such low-masscompanions, in globular clusters or the field.Several accretion-powered MSPs with P orb < minuteshave been identified, consistent with (partly) degenerate He or C/Odonors (e.g. Krimm et al. 2007), and several are known in globularclusters (e.g. Dieball et al. 2005). However, no radio bMSPs havebeen observed with such short periods. These systems will spendGyrs as transient LMXBs, with continually decreasing mass trans-fer rates (“3b” and “6a” in Fig. 1). It is possible that the UCXBsin clusters remain in this configuration, never turning on as mil-lisecond pulsars, for a Hubble time. A sizable number of such sys-tems could remain hidden in globular clusters if their accretion out-burst intervals are sufficiently long. Alternatively, partly degener-ate (due to irradiation) donors will have a minimum companionmass for a given temperature, indicating that evaporation of thedonor star and production of isolated millisecond pulsars is a likelyoutcome (Bildsten 2002; Deloye & Bildsten 2003). Other possibil-ities include very strong tidal heating of the companion leading tostrong orbital expansion, producing the low-mass eclipsing pulsars(Rasio et al. 2000), or spin-down of the NS due to decreasing masstransfer rates (see discussion in Deloye 2007). Or, current pulsardetection algorithms may not yet be sufficiently sensitive to detectsuch short-period systems (e.g. Hessels et al. 2007). We do notknow which of these fates befalls ultracompact LMXBs, althoughwe know observationally that ultracompact radio bMSPs (“6a” inFig. 1) have not yet been seen. We therefore count the numbersof ultracompact systems that have evolved past 55 minutes (thelongest orbital period yet observed for an ultracompact LMXB,Krimm et al. 2007) and tabulate them as ultracompact MSPs, butdo not plot their final binary periods and masses; their true finalfate is uncertain.In the case of MT in a NS-MS binary, if the resulting bi-nary has a short period, it will have a (partly) degenerate hydro-gen companion. Such a companion is larger than a hydrogen-poorWD, and better matches the observed periods of known low-masseclipsing radio bMSPs than systems with WD donors. It is thoughtthat the MT becomes unstable and leads to a merger of the two c (cid:13) , 000–000 inaries with NS in globular clusters stars when the hydrogen-rich companion becomes less massivethan ∼ . M ⊙ (for analytic estimates see King et al. 2005),producing a single MSP (“4a” in fig. 1). However, current binaryevolution codes lack the equation of state appropriate for low tem-peratures. Therefore, as of today, there are no direct mass transfercalculations that can proceed below donor masses of 0.04 Msun(van der Sluys et al. 2005a,b), which is well above the masses ofobserved bMSPs with very low-mass companions. The main rea-son is that an equation of state that is appropriate for extremelylow-mass stars (e.g., Saumon et al. 1995) is not yet incorporatedin current binary evolutionary codes.Our code evolves a NS-MS binary down to very low massesassuming a simple mass-radius dependence for a hydrogen-rich de-generate donor. We find that this evolution ends as a merger whenthe donor mass is ∼ . M ⊙ , where this value varies a bit depend-ing on the initial donor mass. While evolving, the system appearsin a globular cluster as first a qLMXb and then as a bMSP for sub-stantial lengths of time. The values of the donor mass when theinstability occurs, and the orbital periods during the qLMXB stage,could use refinement when improved evolutionary simulations areavailable. We consider here mass gain which can result from a physical colli-sion with a red giant (DCE,case “4e”, see Fig. 1) ; a CE event duringbinary evolution (case “4c”); or a merger or physical collision witha MS star or WD (“2b” at the Fig. 1).During a physical collision with a giant, the NS retains . M ⊙ of the giant’s envelope (Lombardi et al. 2006). This mat-ter has high specific angular momentum and the resulting accretionmust proceed through an accretion disk, implying that a recycledpulsar can be formed.A CE event is also likely to lead to spin-up. As an example, itis believed that the double neutron star system PSR J0737-3039 isa post-common envelope remnant (Dewi & van den Heuvel 2004),where the first-formed NS has been spun up during a CE event tomillisecond periods.It is less clear whether possible accretion as a result of amerger leads to recycling. Such mergers have been considered toproduce only mild spin up (Lyne et al. 1996). For a standard sce-nario we assume that physical collision of NSs with other starsleads only to the formation of short-living slow (isolated) pulsars. Once recycled, a single or binary MSP can participate again in dy-namical encounters, and form new binary systems whose proper-ties will depend on the type of binary formation: binary exchange(“5a”), tidal capture (“5b”) or another DCE (“5c”). This type of bi-nary millisecond pulsars, along with post-DCE “4e”, should not beformed in the field population. Newly dynamically formed binarieswith MSPs may evolve again through stable MT. Our “standard” cluster model has an initial binary fraction of 100per cent. The distribution of initial binary periods is constant inthe logarithm between contact and d and the eccentricities aredistributed thermally. We emphasize that about 2/3 of these bina-ries are soft initially (the initial binary fraction for hard binaries is about per cent if the 1-D velocity dispersion is 10 km/s) andmost very tight binaries are destroyed through evolutionary merg-ers. Our initial binary fraction is therefore comparable to the initialbinary fractions that are usually adopted in N -body simulations,10-15 per cent in hard binaries (soft binaries are short-lived in denseclusters but slow down the calculations considerably). For a moredetailed discussion of the choice of primordial binary fraction, seeIvanova et al. (2005a).For single stars and primaries we adopt the broken power lawinitial mass function (IMF) of Kroupa (2002) and we assume a flatmass-ratio distribution for secondaries. The initial core mass is 5per cent of the total cluster mass and no initial mass segregation isassumed (an average object in the core is as massive as an averagecluster star). At the age of 11 Gyr, which we adopt as a typical ageof a Galactic GC, the mass of such a cluster in our simulations is ∼ . × M ⊙ for the metallicity of a typical metal-rich cluster( Z = 0 . ) and is comparable to the mass of typical GCs in ourGalaxy. In the case of a metal-poor cluster (Z=0.0005), the mass is ∼ . × M ⊙ . To make a comparison between cluster modelsof different metallicities we will show all results scaled to a clustermass × M ⊙ . As the metal-rich case observationally is moreimportant – most bright LMXBs and MSPs detected so far are lo-cated there – we take for our standard model the value Z = 0 . .For our standard model we fix the core density at n c =10 pc − . The link to an observed luminosity density can be foundusing the number density-to-mass density ratio, which in our sim-ulations is ∼ M − ⊙ at the ages of 7-14 Gyr, and mass-to-lightratio, which on average is ∼ . M ⊙ /L ⊙ and is M ⊙ /L ⊙ for47 Tuc (Pryor & Meylan 1993). We adopt a half-mass relaxationtime t rh = 1 Gyr (Harris 1996). The characteristic velocities arecomputed for a King model with dimensionless central potential W = 7 and the total model mass, giving a one-dimensional ve-locity dispersion σ = 10 km/s and a central escape velocity v esc = 40 km/s. If, after an interaction or SN explosion, an ob-ject in the core acquires a velocity higher than the recoil velocity v rec = 30 km/s, the object is moved from the core to the halo. Theejection velocity for objects in the halo is v ej , h = 28 km/s. Thisstandard model represents a typical dense globular cluster in ourGalaxy.In addition to this “standard” model we also considered clustermodels with the following modifications: • a metal-poor cluster with Z = 0 . – “metal-poor”; • different central densities: n c = 10 pc − , n c = 10 pc − and n c = 10 pc − – “high-dens”, “med-dens” and “low-dens”respectively; • a cluster with smaller velocity dispersion and smaller ejectionvelocity – “low- σ ”; • shorter half-mass relaxation time “long- t rh ”; • reduced initial binary fraction, f = 50 per cent – “BF05”; • magnetic braking prescription of Rappaport et al. (1983) –“fast MB”; • reduced efficiency of the common envelope, α CE λ = 0 . –“CE-reduced”; • natal kick distribution is doubled peaked Maxwellian as inArzoumanian et al. (2002) and NSs are produced only via core-collapse and double WD collisions (MIC), but not through EIC,AIC or evolutionary issued MIC – “oldkicks” • 47 Tuc-type cluster, characterized by a higher density n c =10 . pc − (with M/L=2 as in (Pryor & Meylan 1993) it corre-sponds to ρ c = 10 . L ⊙ pc − as in Harris (1996)) , metallic-ity Z = 0 . , t rh = 3 Gyr (Harris 1996) σ = 11 . km/s c (cid:13) , 000–000 N Ivanova et al. Figure 1. The scenario for the formation and evolution of MSPs in globular clusters. τ P is the life-time of a radio pulsar after formation. τ MT , τ mb , τ gw , τ coll indicate which time-scale determines the lifetime of the indicated binary: τ MT – while MT lasts, τ mb and τ gw – until magnetic braking or gravitationalradiation merges the binary, τ coll – before the next dynamical encounter. The final fate of radio-quiet X-ray transient bMSPs (“6a”) is not determined.c (cid:13) , 000–000 inaries with NS in globular clusters Table 1. Models. n c t rh Z σ v esc f α CE λ MB M cluster Runs (number of stars)standard × M ⊙ , 5 with × metal-poor · · · · · · · · high-dens · · · · · · · · med-dens · · · · · · · · low-dens · · · · · · · · low- σ · · · · · · · long- t rh · · · · · · · · BF05 · · · · · · · · fast-MB · · · · · · · RVJ · CE-reduced · · · · · · · · oldkicks · · · · · · · · · 47 Tuc . · · · · M ⊙ × Terzan 5 · · · · . × M ⊙ . × Notations: n c – core number density ( pc − ); t rh – half-mass relaxation time (Gyr); Z – metallicity; σ – one-dimensional velocity dispersion (km/s); v esc –central escape velocity (km/s); f – initial binary fraction (see text for corresponding binary fraction of hard binaries); α CE λ – CE efficiency; MB – adoptedprescription for magnetic braking (IT03 is from Ivanova & Taam (2003) and RVJ is from Rappaport et al. (1983)); M cluster – all the results for this modelare scaled to this adopted cluster mass. “Oldkicks” model uses the distribution of natal kick velocities from (Arzoumanian et al. 2002). Symbol “ · ” means thatthe value is the same as in the standard model. (Pryor & Meylan 1993), v esc = 57 (Webbink 1985), with the recoilvelocity of 51 km/s and v ej , h = 32 km/s (King model W = 12) )– “47 Tuc”; • Terzan 5 cluster, n c = 10 . pc − ( ρ c = 10 . L ⊙ pc − as in Heinke et al. (2003)), metallicity Z = 0 . (Origlia & Rich2004), t rh = 0 . Gyr (Harris 1996) σ = 11 . km/s, v esc = 49 . (Webbink 1985), with the recoil velocity of 39 km/s and v ej , h = 29 km/s (King model W = 8 . ) – “Terzan 5”. For the complete list of models see Table 1.We expect that only a few LMXBs can be formed in everycluster with a typical mass of 200,000 M ⊙ through 11 Gyr of dy-namical evolution, and therefore we perform 5 runs for each modelwith initial stars and for the standard models - five runs with × stars, to check that statistics for rare objects like LMXBsare not different when the resolution is increased (for most of ourresults presented below we did not find statistical differences be-tween high and low resolution models, and fluctuations for the rateevents are smaller in the case of high resolution). For Terzan 5 and47 Tuc models, though, we do all runs at high resolution. In partic-ular, in the case of Terzan 5 . × stars provide at 11 Gyr themass comparable to the estimated observed mass of 370,000 M ⊙ .In the case of 47 Tuc the reason for a higher resolution is that therelative core mass is smaller than in the case of a standard cluster,due to longer t rh . To estimate how many NSs can be formed and retained in GCs,we first studied stellar populations without dynamics, consideringseparately populations with only primordial single stars and only The recent study of Ortolani et al. (2007) suggests a shorter distance, andthus higher central density for Terzan 5 ( ρ c = 10 . L ⊙ /pc ). If this studyis correct, then Terzan 5’s characteristics may be more closely approximatedby our “high-density” cluster. Table 2. Production of NSs, field population.CC EIC AIC MIC N cc40 N tot40 N NS singleZ=0.0005 3354 594 - - 2.1 251 584848Z=0.001 3255 581 - - 1.9 245 565168Z=0.005 2833 570 - - 1.2 243 498513Z=0.02 2666 400 - - 1.2 172 424696oldkicks 3104 - - - 19.3 21.9 339669binaryZ=0.0005 3079 545 59 14 19 329 249775Z=0.001 3056 553 60 15 15 332 242601Z=0.005 2750 576 58 16 14 335 219236Z=0.02 2463 406 33 20 12 240 180582oldkicks 2929 - - - 58 58 237705Notations for channels – CC - core-collapse supernova; EIC - evolutioninduced collapse via electron-capture supernova; AIC - accretion inducedcollapse; MIC - merger induced collapse. Numbers are scaled per 200,000 M ⊙ stellar population mass at the age of 11 Gyr. N cc40 and N tot40 arethe retained numbers of NSs formed via core-collapse and through allchannels, if the escape velocity is 40 km/s. N NS are the total numbers offormed NSs in simulated populations. primordial binaries. We considered several metallicities – from so-lar (as in the Galactic field) to that of a typical metal-poor clus-ter: Z=0.02, 0.005, 0.001, 0.0005. In addition, for comparison withstudies done by different researchers in the past, we examined amodel (with Z=0.005) with the previously widely accepted natalkick distribution, 2 Maxwellians with a lower peak for kick veloc-ities at 90 km/s (Arzoumanian et al. 2002). In this model we adoptthat NSs are produced only via core-collapse or through collisionalWD mergers, no evolutionary ECS. The results for NS productionvia different channels are shown in Table 2.The production of NSs via core-collapse SNe (CC NSs), perunit of total stellar mass at 11 Gyr, decreases as metallicity in-creases. The difference between the Galactic field case and a metal-poor case is about 20 per cent, while the difference between metal-rich and metal-poor clusters (Z=0.005 and Z=0.0005) is about 10-15 per cent, for primordial single or binary populations. c (cid:13) , 000–000 N Ivanova et al. Figure 2. The retention fractions as a function of escape velocity (for stel-lar evolution unaffected by dynamics). Dotted and dash-dotted lines showthe retention fractions for single and binary populations, core-collapse NSsonly. Solid and short-dashed lines show the total retention fractions for sin-gle and binary populations, all NSs. Long-dashed line shows the total reten-tion fraction of a binary population with the old kick distribution. EIC in single stars for metal-rich populations occurs in starsof higher masses. The range of initial masses is, however, the samefor both populations and is about . M ⊙ . As a result, the numberof NSs produced via EIC (EIC NSs) from the population of singlestars is smaller in the case of a Galactic field population than inthe case of a GC population by 30 per cent, while the differencebetween metal-rich and metal-poor clusters is only 5 per cent. Thisis in complete agreement with the adopted power law for the IMF.In binaries, the mass interval where EIC could occur is expandedcompared to single stars of the same metallicity.NSs produced via AIC and MIC also have similar productionrates in metal-rich and metal-poor populations. We note that MICgives the smallest contribution to the production of NSs in popula-tions evolved without dynamics.It can be seen that once we adopt that a NS can be createdvia an ECS, the number of formed NSs via CC is depleted – somestars would be evolving via core-collapse, but have experienced thecondition for a EIC before that. In Fig. 2 we show the retention fraction of NSs at formation as afunction of fixed escape velocity the whole population, consider-ing populations of metal-poor primordial singles and binaries; CCNSs and the whole NS population are shown separately. It can beseen that, with the adopted natal kick distribution, CC NSs froma single star population are rarely retained (see also Table 2). In apopulation of primordial binaries, the fraction of retained CC NSsis higher than for single stars. None the less, the resulting numberof retained CC NSs is significantly smaller than we get from theother NS formation channels. We conclude that if indeed EIC, AICor MIC are not accompanied by a large natal kick, the total number of retained NSs is high and only a negligible number of CC NSsis present. As the EIC/AIC NS masses are only . M ⊙ , this maylead to a NS mass function skewed towards low values.With the old distribution for natal kick velocities, reasonablenumbers of NSs can be produced and retained in a typical GC,while a massive cluster like 47 Tuc could retain as many as 600NSs (see also Fig. 2). This result agrees with the retention fractionsobtained in Pfahl et al. (2002). The effects of dynamics on the production and retention of NSs areillustrated in Table 3. The number of formed CC and EIC NSs is inagreement with the retained fraction of the stellar population wherebinaries have been partially depleted and some of the SN progeni-tors were located in the halo, where the escape speed is lower thanin the core. We find that the production of NSs via AIC and MICis strongly enhanced by dynamical encounters compared to a fieldpopulation. As a result, those formation channels together becomecomparable to the EIC channel, and produce many more retainedNSs than the CC channel. A metal-poor cluster, following the trendof field populations, possesses a few per cent more NSs.As noted above ( § yr. In this case, AXPs could be found in GCsonly if their formation rate is above 1 per 200 (GCs) per 10 yr, orat least 500 per typical cluster, over the cluster lifetime. Our forma-tion rates are significantly smaller and therefore the hypothesis of aMIC connection to AXPs does not contradict our results. We consider the position of retained NSs in the cluster. The progen-itors of CC NSs are mainly massive stars with masses & M ⊙ .However a significant fraction of progenitors of EIC NSs, espe-cially those that evolved from a primordial binary, have intermedi-ate initial masses—as low as 4 M ⊙ . The evolutionary time-scalesfor NSs to be formed are ∼ yr for an CC NS and ∼ yrfor an EIC NS. These are only ∼ per cent and ∼ per cent,correspondingly, of the cluster half-mass relaxation time t rh , where t rh = 10 yr in a typical cluster. Even massive stars do not experi-ence strong mass segregation during ∼ . t rh , the same is true forstars of intermediate masses that produce EIC NSs (see, e.g., Fig.7in G¨urkan et al. 2004, , where the mass segregation with time in acluster with stars between 0.2 M ⊙ and 120 M ⊙ is shown). When aNS is formed, the mass of the star is significantly decreased. As aresult, a significant fraction of NSs is formed in the halo, and manywill remain there. As many as ∼ per cent of all EIC NSs in thecase of a typical cluster ( t rh = 10 yr) remain in the halo at 11Gyr. AIC and MIC NSs are also present in the halo, but in a lesserproportion, as they are mainly produced in a dense environment.Many of the halo AIC and MIC NSs have recoiled from the core asa result of dynamical encounters. A significant difference with thedistribution of NSs over the clusters can only be expected for ini-tial mass segregation, or if the initial t rh was much shorter than thecurrent t rh resulting in faster mass segregation at early stages. Ob-servations find (e.g. Grindlay et al. 2002) that the radial distributionof MSPs within globular clusters in 47 Tuc is consistent with the c (cid:13) , 000–000 inaries with NS in globular clusters Table 3. Retained NSs and their locations at 11 Gyr, for different globular cluster models.model Core HaloCC EIC AIC MIC CC EIC AIC MICstandard . ± . . ± . . ± . . ± . . ± . . ± 11 18 . ± . . ± . metal-poor . ± . . ± 12 30 . ± . . ± . . ± . . ± . . ± . . ± . high-den . ± . . ± . . ± . . ± . . ± . ± 12 29 . ± . . ± . med-den . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . low-den . ± . . ± . . ± . . ± . . ± . . ± 11 15 . ± . . ± . lowsig . ± . . . ± . . ± . . ± . . ± . . . ± . . ± . . ± . longtrh . ± . . ± . . ± . . ± . . ± . ± 15 22 . ± . . ± . BF05 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . fast-MB . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . CE-reduced . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . oldkicks . ± . . ± . . ± . . ± . . . ± . . ± . . ± . . ± . Tuc-47 . ± . ± 20 91 . ± 23 27 . ± . . ± . ± 41 102 ± . . ± Terzan-5 . ± . ± . . ± 11 47 . ± . . ± . ± 10 38 . ± . . ± . Notations for channels as in Table 2. For all GC models, except 47 Tuc and Terzan 5, the numbers are scaled per 200 000 M ⊙ stellar population mass at theage of 11 Gyr; for 47 Tuc the numbers are given per its total mass taken as M ⊙ , for Terzan 5 - per 370 000 M ⊙ . Numbers show the results averagedover all runs for each cluster model (for the number of runs see Table 1). production of all MSPs within the core (but cf. G¨urkan & Rasio2003). The probability for a NS to remain in a primordial binary dependsstrongly on its formation channel (see Table 4). By definition, MICleads to the formation of a single NS. AIC occurs in a very tightbinary. Accordingly, a NS formed this way will most likely remainin a binary. Among CC NSs or EIC NSs, only several per cent re-main in binaries immediately after NS formation. Binaries that sur-vive SN generally have relatively massive secondaries that are alsolikely to experience SN, or the binary may evolve through dynam-ically unstable mass transfer resulting in a merger, with strongerbinary depletion in the case of initially more massive binaries witha CC NS. As a result, very few NS binaries formed through thosechannels remain in binaries. Post-AIC binaries are the GC primor-dial binaries most likely to contain a NS. The low probability forCC NSs and EIC NSs to be members of a primordial binary doesnot change strongly with metallicity.The formation rate of AIC NSs in GCs shows that as manyas 40 NS-MS binaries can be formed in a typical cluster. Most ofthem will evolve through the LMXB stage before the age of 11 Gyr,producing as a result 20-30 recycled pulsars, although those MSPsmay have short lifetimes and not be observable today (see § Table 4. Binary fractions for NSs in primordial binaries.Z after NS formation at 11 GyrCC EIC AIC CC EIC AIC0.0005 2.9% 3.0% 92.3% 0.28% 1.38% 78.6%0.001 2.8% 3.4% 93.6% 0.25% 1.41% 81.8%0.005 2.6% 3.4% 96.0% 0.18% 1.54% 88.0%0.02 1.4% 4.2% 95.2% 0.12% 1.49% 85.9%The fractions of NSs that remain in primordial binaries immediately afterNS formation (columns ÷ ) and at 11 Gyr (columns ÷ ), for eachformation channel. immediately after NS formation strongly increases the chance for aNS to participate in an encounter and, accordingly, to get another,non-primordial, companion. The fraction of NS binaries with a NSformed via AIC is therefore expected to be higher than for otherNS channels.The number of double neutron star (DNS) systems formed inthe field is very low, typically just a few DNSs per 200,000 M ⊙ at11 Gyr. For a typical GC this number will be smaller since someof the primordial binaries will be destroyed dynamically and mostDNSs will be ejected. In fact, more than 99 per cent of the first-born NSs in primordial DNS systems are CC NSs, and, accord-ingly, have high natal kicks. The formation of primordial DNSsoccurs very early in the GC evolution. Most of them are very tightand merge quickly, leaving very little chance for one to be observedtoday. As this paper only uses the results of the NS formation andevolution in the field as a reference point, we refer the reader to thecomplete study of DNS formation channels from primordial bina-ries, rates, appearance with time, metallicity dependence and otherdetails provided in Belczynski et al. 2007 (in prep). We concludethat essentially no primordial DNSs can be present in Galactic GCsat the current time. A binary with a NS and a WD can be formed via physical colli-sion with a giant (see § c (cid:13) , 000–000 N Ivanova et al. Table 5. NS binaries formed in scattering experiments.Z σ DCE TCkm/s Tot MT Tot MTstandard 0.005 10 3.68% 1.75% 2.35% 1.20%metal-poor 0.0005 10 3.25% 1.30% 1.93% 0.95%low- σ Figure 3. Companions in binary systems formed via physical collision witha red giant during the scattering experiment (Z=0.001). Triangles are WDs,circles are He stars; solid symbols denote the systems that started the masstranfer before 11 Gyr. single-single star collisions, although it may occur during a binary-single encounter as well. As we showed above, the number of NSsthat can be retained in globular clusters is about a couple hundredfor a typical globular cluster. The cross-section for two stars thathave a relative velocity at infinity v ∞ to pass within a distance r max is σ = πr (1 + 2 G ( m + m ) r max v ∞ ) , (9)where the second term is gravitational focussing and in, stellar sys-tems with low velocity dispersion like GCs, it exceeds the first term.The time-scale for a NS to undergo an encounter with a red giantcan be estimated as τ coll = 1 /f RG nσv ∞ , where f RG is the relativefraction of red giants in the stellar population: τ coll = 1 . × yr v n − ( M NS + M RG ) − R − f − , (10)where M NS and M RG are the masses of the NS and red giant in M ⊙ , and R max = r max /R ⊙ is the maximum distance betweenthe stars during the encounter that leads to the binary formation.Lombardi et al. (2006) shown that r max ≈ . R RG , where R RG is the radius of the red giant. Averaged over time, R RG is a few solar radii (Ivanova et al. 2005d), and f RG ≈ per cent. Usingeq. (10) we estimate that during 11 Gyr about 5 per cent of NSs canparticipate in the formation of binaries through physical collisions.As the estimate shows, having about NSs in our mod-els of GCs (and about 400 for the models with the highest reso-lution, × stars), we can form a maximum of 10 binary sys-tems through the whole cluster simulation. To study statisticallythe characteristics of post-encounter binaries, we performed a scat-tering experiment; we considered a population of single stars,with an IMF from 0.2 M ⊙ to 3.0 M ⊙ and 10000 NSs of . N ⊙ . Allstars are initially placed in the core with n c = 10 pc − . We con-sidered three different models - standard, low metallicity, and withlow velocity dispersion (see also Table 5). In these simulations onlysingle-single encounters were allowed.The scattering experiment shows that 3.7 per cent of all NSssuccessfully formed a binary via physical collision with a RG oran AGB star. This fraction slightly decreases when the metal-poorpopulation is considered and increases by a factor of about 2 (inaccordance with eq. 10) in the case of lower velocity dispersion.The fraction of NS–WD systems that started MT before 11 Gyr ishigh, ∼ per cent (see Table 5).In Fig. 3 we show the masses of companions in post-collisionbinaries vs. orbital periods. We form about the same number of sys-tems with WD companions and stripped He star companions (fromcollisions with AGB stars). Systems with He star companions aremainly formed within the first 2 Gyr and start MT mostly within 3Gyr. After 3 Gyr both the formation rate and the appearance rateare slightly higher for systems with WD companions. The evolu-tion of mass-transferring systems with He star companions is a bitdifferent from those with WD companions. They are a few timeswider (for the same companion mass) and do not live as long in themass-transferring phase at high luminosities.In our numerical simulation of different models of GCs theactual rates are smaller. For instance, not all NSs are immediatelypresent in the core at the time of their formation, and about half ofthem are still in the halo at the age of 11 Gyr. As a result, our “stan-dard” model shows about 2 per cent formation rate per NS presentin the core (or 1 per cent per all NSs in the cluster), producing only2 systems. The scatter between several realizations is big. In somesimulations we formed only half as many binaries via this channel,and, in one simulation, 3 times more binary systems were formed.For our standard model, no simulations failed to produce binaries.We note that the low- σ cluster model, unlike the scattering results,does not produce twice as many NS-WD binaries as the standardmodel – as the low- σ cluster retains less NSs than the standard. In our previous study, devoted to the formation of cataclysmic vari-ables (CVs) in globular clusters (Ivanova et al. 2006), we foundthat tidal capture plays a relatively small role in the formation ofdynamical binaries, only ∼ − per cent of the total formationrate. Two main reasons were responsible: (a) a significant fractionof CVs have been formed directly from primordial binaries and (b)both physical collisions and exchange encounters occurring withWDs form tighter binaries, increasing their chance to start the MT.For the case of tidal captures with NSs, the results are different.In our scattering experiments, we find that only ∼ per centof NSs form a binary via TC (see Table 5); with a small decreasein the case of a metal-poor population and an increase (4 times)for the lower velocity dispersion case. The latter effect is due to c (cid:13) , 000–000 inaries with NS in globular clusters r max being a function of the relative velocity, and increasing as therelative velocity decreases (see Ivanova et al. (2006)).The rate of tidal captures is flat both with time and companionmass. In the case of the metal-poor population, most TC systemsthat start MT before 11 Gyr have formed before 5 Gyr. For our stan-dard metal-rich population, among systems formed in the past fewGyrs, only those with MS companions > . M ⊙ are able to startMT. We conclude that < per cent of NSs can form a LMXB viatidal captures, and that the number of LMXBs mainly depends noton the current cluster properties, but on the dynamical conditions inthe cluster core more than 5 Gyr ago. For metal-rich clusters, thereis an exception for initially more massive MS companions.In our numerical simulations the formation rate is evensmaller, due to the same reasons as in § § As can be seen from eq. 10, due to a larger r max , binary encountersare much more likely than single star collisions. However, com-pared to the case of physical collisions and TC, where very tightbinaries are formed, a smaller fraction of the successful exchangeencounters are expected to result in a binary with MT. The rea-son is that when the binary exchange occurs, the binary separationin the formed post-exchange binary roughly scales with the pre-exchange binary separation as the ratio of the new companion massto the mass of the replaced companion (Heggie et al. 1996). There-fore, when a single NS becomes a member of a binary system atthe age of several Gyrs, the post-exchange system expands, as themass of the replaced companion is always less than the mass ofthe NS. In addition, when a binary encounter involves a very tightbinary, which would be likely to start MT soon even in the case ofpost-exchange expansion, a physical collision during the binary en-counter is more likely than a simple exchange (Fregeau et al. 2004).Overall, in our standard model ∼ − per cent of NSsin a cluster (40-50 per cent if we consider only NSs in the core)will successfully form binaries through exchange encounters. ∼ per cent of post-exchange binaries will have a MS companion, and ∼ per cent a WD companion. Only 1.5 per cent of NSs willbegin MT with a MS companion, and only 0.2 per cent with aWD companion. The role of binary exchanges in the formation ofNS-WD LMXBs is therefore negligible compared to physical col-lisions. From the discussion above, one can see that the expected forma-tion rate of LMXBs is rather small. For all dynamical formationchannels that involve a NS directly, in a typical dense metal-richcluster, the combined rate is only ∼ per cent per NS. This is 6LMXBs per 11 Gyr of the cluster life, where 2 LMXBs are witha WD companion and 4 LMXBs are with a MS or a RG compan-ion. A small number of LMXBs may come from primordial, butdynamically modified binaries (where the encounters changed thebinary eccentricity or separation). In addition, however, some frac-tion of LMXBs is expected to be provided by dynamically formedbinaries with heavy WDs (see Ivanova et al. (2006)). Such binaries, Table 7. Average number of LMXBs with different donorsmodel MS RG WDstandard . ± . 56 0 . ± . . . ± . metal-poor . ± . 05 0 . ± . 07 0 . ± . high-den . ± . 21 0 . ± . . . ± . med-den . ± . 40 0 . ± . . . ± . . low-den . ± . . . ± . 01 0 . ± . . low- σ . ± . 25 0 . ± . 06 0 . ± . long- t rh . ± . 73 0 . ± . . . ± . . BF05 . ± . 32 0 . ± . . . ± . fast-MB . ± . 93 0 . ± . . . ± . CE-reduced . ± . 33 0 . ± . 06 0 . ± . oldkicks . ± . . . . 47 Tuc . ± . 10 0 . ± . 09 4 . ± . Terzan 5 . ± . 95 0 . ± . 11 7 . ± . Average number of LMXBs present at ages ± . Gyr, for differentdonors (RG includes also subgiants and Hertzsprung gap stars). if evolved through AIC, create binaries that reinstate MT on a NSsoon after AIC, with a MS, WD or RG donor. From our currentsimulations, we find that the channel of LMXB production fromAIC of dynamically formed WD binaries produces more LMXBsthan any of the other dynamical LMXB production channels dis-cussed above. Overall, in a typical cluster it is 2-3 times more ef-ficient during 9.5-12.5 Gyr than physical collisions, tidal capturesand binary exchanges (see Table 6). The exceptions are only veryhigh density clusters. We also note that if the majority of LMXBsare produced through AIC, but these binaries do not produce long-lived MSPs ( § τ LMXB of LMXBs with various donors.An average τ LMXB for NS-MS LMXBs is about 1 Gyr. Depend-ing on the metallicity and the initial donor mass, a system canbe persistent 5-40 percent of the MT time for metal-rich donors > . M ⊙ and transient during the rest; for donors of lower metal-licities or smaller masses, a NS-MS LMXB will be transient all thetime (Ivanova 2006), and therefore more likely seen as a qLMXBrather than as a bright LMXB. An LMXB with a red giant com-panion or a companion in the Hertzsprung gap is very short-lived, − yr, and in only very rare cases can they live as long as yr. In the case of NS-WD LMXBs – ultra-compact X-ray bi-naries (UCXBs) – the total τ LMXB is a few Gyr, however the timeduring which the system is persistent and has an X-ray luminosityabove erg/s is only ∼ yr. UCXBs at ∼ years willfall below L b ol = 1 . × ergs/s. According to the theory ofirradiated disks (Dubus et al. 1999) and the evolution of UCXBs(Deloye & Bildsten 2003), ergs/s is roughly the critical L X for maintaining persistent accretion. in’t Zand et al. (2007) presenta table of 40 persistent X-ray bursters, which shows a striking low- L X pileup and cutoff at L X = 1 . × ergs/s, or 0.5% of thehydrogen-poor L Edd (incorporating their relativistic correction).Incompleteness in Galactic surveys does not fully explain this cut-off, as the PCA bulge scans monitor many sources at half this flux. c (cid:13) , 000–000 N Ivanova et al. Table 6. Average appearance rate of LMXBs formed via different dynamical channels.model AIC TC DCE BE/MS BE/WDstandard . ± . 54 0 . ± . . . ± . . . ± . . . ± . . metal-poor . ± . 54 0 . ± . . . ± . . . ± . . . high-den . ± . 42 0 . ± . 76 1 . ± . 66 1 . ± . 71 0 . med-den . ± . 19 0 . ± . . . . ± . . . low-den . ± . 23 0 . . . . low- σ . ± . 44 0 . ± . . . . ± . 27 0 . ± . . long- t rh . ± . 68 0 . . . ± . . . ± . . BF05 . ± . 52 0 . ± . . . ± . . . ± . 14 0 . fast-MB . ± . 36 0 . ± . . . ± . 15 0 . ± . 36 0 . CE-reduced . ± . 53 0 . . . ± . 23 0 . oldkicks . . . . ± . . . 47 Tuc . ± . 16 0 . ± . . . ± . . . ± . 36 0 . ± . . Terzan 5 . ± . 20 1 . ± . 49 2 . ± . 42 2 . ± . 70 0 . ± . . Number of LMXBs that appear per Gyr at age ± . Gyr, for different formation channels: AIC in a dynamically formed WD-binary, TC – tidal capture,DCE – a dynamical common envelope event (physical collision with a red giant), BE/MS – binary exchanges when a MS companion is acquired, BE/WD –binary exchanges when a WD companion is acquired. Figure 4. Number of appearing LMXBs per Gyr with different donor types (with all donors, with MS donors, with WD donors and with donors that are RGs,subgiants or are in the Hertzsprung Gap). The left panel shows the results for the standard model (solid line) and the metal-poor model (dotted line). The rightpanel shows the results for 47 Tuc (solid line) and Terzan 5 (dotted line). Therefore we adopt L X = 1 . × ergs/s as the break betweenpersistent and transient behavior for UCXBs.Combining the number of appearing LMXBs with their life-times in the simulations, we find that a typical dense cluster at ± . Gyr age can contain up to 2 LMXBs with a MS com-panion and up to 1 LMXB with a WD companion (see also Ta-ble 7, note the very large scatter). We specify, that Table 7 showsthe average expected number of LMXBs – both bright LMXBs andqLMXBs. We choose a cutoff period of 55 minutes for identifyingUCXBs as LMXBs. We justify this by noting that the longest pe-riod UCXB yet identified has a period of 55 minutes (Krimm et al.2007), and that the evolution of UCXBs will produce very largenumbers of longer-period systems which have not yet been seen.As expected, the number of LMXBs shows a strong dependence on the core density; dependence on the velocity dispersion and thehalf-mass relaxation time is also observed.A strong dependence of LMXB production on cluster densityhas been inferred observationally, both in Galactic globular clusters(e.g., Verbunt & Hut 1987; Pooley et al. 2003; Heinke et al. 2003)and in extragalactic globular clusters (e.g., Jord´an et al. 2004;Sivakoff et al. 2007). The interpretation has been that N LMXB scales roughly with the encounter frequency Γ = ρ r /σ , where r c is the core radius (Verbunt & Hut 1987).The observations of LMXBs in our galaxy suffer from lownumber statistics–only 12 clusters have LMXBs–so attempts to un-derstand the LMXB formation rate from them encounter difficul-ties (e.g. Bregman et al. 2006). Observations of qLMXBs in ourgalaxy also suffer from low number statistics, as only ∼ two dozen c (cid:13) , 000–000 inaries with NS in globular clusters Figure 5. The specific collision numbers γ and numbers of LMXBs in sim-ulated clusters. The solid line corresponds to n LMXB = (1 +7 − ) + 0 . γ .The error bars correspond to the scatter in our simulations. clusters have been observed with Chandra sufficiently to iden-tify most qLMXBs. Pooley & Hut (2006) and Heinke et al. (2006)study overlapping samples of clusters using different assumptions.Their estimates of the density dependence of qLMXB productionfor the case of no metallicity dependence also overlap, suggestinga density dependence larger than suggested by Γ , but with large er-rors. Heinke et al. (2006) also investigate (inconclusively) the pos-sibility of metallicity dependence, which decreases the requireddensity dependence.Observations of LMXBs in globular clusters of other galax-ies do not suffer from low statistics, and thus have clearly demon-strated a strong dependence of LMXB production on metallicity(Kundu et al. 2002, 2007). Analyses by Jord´an et al. (2004) andSivakoff et al. (2007) both find that LMXB production has a den-sity dependence lower than Γ . However, it is extremely difficult todetermine the structural properties of extragalactic globular clus-ters with high accuracy, so uncertainty in the density dependenceremains.In Fig. 5 we demonstrate how the specific number of LMXBs n LMXB = N LMXB /M depends on the specific collision frequency γ = Γ /M , taken per units of M ⊙ , as in Pooley & Hut (2006).To calculate Γ in the same way as observers, we extracted valuesof ρ c and r c from our models at the cluster age of 11 Gyr, andused our input parameter σ . When only cluster density is varied, n LMXB depends linearly on Γ , however all the other parameterscreate significant scatter. We therefore urge not to overinterpret thedependence of the number of observed LMXBs on Γ , or the mean-ing of the power-law dependence of Γ on density; the scatter in Γ in the observed GCs could easily be explained by the variancein other dynamical properties. If one assumes that in the observedsample of GCs the density increase is likely to be accompanied byincreases of the cluster mass, and, accordingly, t rh and σ , then weexpect from our results that the power law dependence of n LMXB with γ is a bit less than one and close to the value of 0.8, as in Sivakoff et al. (2007), where metallicity effects are separated fromdynamical effects.Even though a typical cluster contains 1 or 2 NS-MS LMXBs,there is only ∼ − per cent chance that those LMXBs willbe bright and persistent, and then only for metal-rich clusters. Theprobability to contain a bright persistent UCXB is no more than10 per cent and does not depend strongly on the metallicity. Over-all, there is at best a 30 per cent chance to contain a bright LMXBin a typical metal-rich cluster and a 10 per cent chance in a metal-poor cluster. This may explain the observed ratio between the brightLMXBs in metal-poor and metal-rich clusters. However, to confirmthat, it is necessary to have observational statistics on qLMXBs inboth metal-poor and metal-rich clusters – if the explanation aboveis correct, the number of qLMXBs will not show a dependenceon the metallicity. Considering rich Galactic globular clusters, wefind that our formation rate of UCXBs at 11 Gyr in Terzan 5 isabout 10 per Gyr and in 47 Tuc, ∼ per Gyr (see Fig. 4). Since τ UCXB ∼ yr, this agrees with the presence of one bright (possi-bly ultracompact) LMXB in Terzan 5 and no bright UCXB detectedin 47 Tuc.To compare the entire population of UCXBs in GCs with theresults of our simulations, we integrated through the whole sam-ple of galactic globular clusters, using data for velocities fromGnedin et al. (2002) and for other quantities from Harris (1996).For this calculation, we used the formation rates shown in Table 6and assumed that an UCXB will be bright for only ∼ yr. Asa result, the integral value of bright UCXBs in our Galaxy is 7.5,which is pretty close to the observed number of 4 to 9 UCXBs inGCs. The clusters that are most likely to contain UCXBs overlapvery well with the clusters where those UCXBs are observed: NGC1851 and NGC 7078 each have a probability to contain an UCXB > 20 per cent. NGC 6712 is known to have been heavily tidallystripped, so its current cluster conditions do not reflect the con-ditions that produced its UCXB (see discussion in Ivanova et al.2005d). Core-collapsed clusters are not properly modeled by ourcode, so our failure to predict UCXBs in NGC 6624 and NGC 6652(a likely UCXB) is not meaningful. Other clusters that have a > 20 per cent chance to contain a UCXB include 47 Tuc, Terzan 5,NGC 6266, NGC 6388, NGC 6440, NGC 6441, NGC 6517, andNGC 6715 (M54). Unsurprisingly, this list consist of clusters witha high collision number. Those clusters have not been shown tocontain bright UCXBs, but they generally contain numerous quies-cent LMXBs (some of which may be UCXBs) and MSPs (some ofwhich may be products of UCXB evolution). Overall, it seems thatphysical collisions are able to produce UCXBs in numbers compa-rable to those observed in globular clusters in the Milky Way.It is more difficult to make a similar comparison for NS-MS LMXBs, as the lifetime at the bright stage is much less cer-tain. In the whole GC population we expect to have about 180qLMXBs formed through tidal captures and binary exchanges. Wenote that since our numbers for core-collapse clusters are lowerlimits, the real number of qLMXBs is expected to be higher. Weidentified those clusters that are predicted to contain 5 or moreNS-MS LMXBs, most of which will be in quiescence. Those clus-ters are the same 10 as noted above for UCXBs: 47 Tuc, Terzan5, NGC 1851, NGC 6266, NGC 6388, NGC 6440, NGC 6441,NGC 6517, M 54, and NGC 7078. 5 of them, 47 Tuc, Terzan 5,NGC 6266, NGC 6388 and NGC 6440 are known to contain 5 ormore qLMXBs (Heinke et al. 2003, Cohn et al. 2008 in prep, ). (cid:13) , 000–000 N Ivanova et al. NGC 6441 and NGC 6440 contain bright LMXBs with MS donors(Verbunt & Lewin 2006; Altamirano et al. 2007). We cannot makeclear predictions for the core-collapsed, heavily reddened clustersTerzan 1, Liller 1, and Terzan 6 because we do not model dynami-cal evolution of clusters experiencing core collapse. We note that ifAIC binaries contribute to LMXB formation, the numbers will goup, although not strongly as the characteristic MT time in this caseis shorter than in the case of an average dynamically formed NS-MS binary. According to our models, several clusters have proba-bility 0.3 or more to contain a RG-NS LMXB, one of which – NGC7078 – indeed holds one.For extragalactic globular clusters, (Bildsten & Deloye 2004)showed that the observed luminosity functions can be explained bya constant birthrate of UCXB binaries, 10 per Gyr per 200,000 M ⊙ ,for all GCs in a galaxy. We obtained almost constant birthrates forsuch binaries, but our rates are significantly lower (one per Gyr for200,000 M ⊙ in a typical “dense” cluster). In fact, a 10 times higherUCXB formation rate will produce as many as 10 bright persistentLMXBs in Terzan 5. It is possible that the typical density of GCs inellipticals is larger than that of GCs in our galaxy, which may ex-plain the variation in LMXB production. However, since we do notfind any difference in UCXB production between metal-poor andmetal-rich populations, our simulations do not support the idea thatmost X-ray binaries in extragalactic GCs are UCXBs. To explainmost X-ray binaries in extragalactic GCs with UCXBs, we wouldneed to assume that more NSs are retained (or formed) in metal-richthan metal-poor clusters (since the model of an irradiation-inducedwind from Maccarone et al. (2004) is not applicable for degeneratedonors). In Table 8 we show the number of plausible pulsars that are formedin our simulations. In this table, we count as pulsars (i) all NSs thatgained more than 0.01 M ⊙ after their formation, through all thepossible mechanisms of mass gain (mass transfer, common enve-lope accretion and during mergers or physical collisions) – includ-ing all recycled pulsars (see also § years – “slow” high- B fieldpulsars (see also § § § § § 3, not all the mechanisms for mass gain lead tothe formation of a radio MSP.Let us analyze the population of NSs which have gained massin more detail. In Tables 9 and 10 we show the results of simula-tions of 47 Tucanae and Terzan 5, distinguishing NSs with gainedmass by their formation mechanism and by the process throughwhich they acquired mass after the NS formation. Note that theplausible pulsars that were formed via AIC or MIC constitute morethan half of all NSs that gained mass after their formation. As wediscussed in § years (see Tables 9and 10). From this moment, we remove from our further consider-ation of the population of “present” radio pulsars those AIC NSsthat were recycled or formed more than yr ago (see also § § In Figs. 6 and 7 we compare the simulated populations of NSs thathave gained mass with the observations of 47 Tuc and Terzan 5.We use all simulated models for those clusters, and, as a result, thesimulated populations plotted represent stellar populations about 2times larger than 47 Tuc and 5 times larger than Terzan 5. There arethree panels for each cluster. Each panel shows the observed pop-ulation, compared to the simulated population of pulsars that werespun up via a different mechanism: (i) stable MT; (ii) CE or dynam-ical common envelope (DCE) during a physical collision with a gi-ant; and (iii) as a result of a merger. For most observed MSPs, thecompanion masses are calculated assuming a pulsar mass of 1.35solar masses and an inclination of 60 degrees , leading to some un-certainty in the direct comparison between observations and simu-lations. In the following we consider MSP populations, grouped bybinary formation mechanism. This population is present on the panels that show pulsars that gainmass via MT and via CE. The first population (MT) is character-ized by rather large periods, small companion masses, and low ec-centricities. Systems which may have been formed in this way existin Terzan 5 (Ter 5 E), M53 (B1310+18), and M4 (B1620-26; notethat a dynamical exchange is necessary to add the planet, Ford et al.(2000)).The second population, characterized by a companion massaround M ⊙ (see the second panels for each cluster), is howeverundetected in both Ter 5 and 47 Tuc. No promising candidates arefound among the entire GC bMSP population. Typical progenitors ∼ pfreire/GCpsr.htmlc (cid:13) , 000–000 inaries with NS in globular clusters Table 8. Plausible pulsars and their location at 11 Gyr.model Halo Core EjectedAll psrs Bin psrs NS-bin All psrs Bin psrs NS-bin All psrs Bin psrs NS-binstandard . ± . . ± . . ± . . ± . . ± . . ± . ± . . ± . . ± . metal-poor . ± . . ± . . ± . . ± . . ± . . ± . ± 11 37 . ± . . ± high-den . ± . . ± . . ± . . ± . . ± . . ± . ± . ± 12 159 ± med-den . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± low-den . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± low- σ . ± . . ± . . ± . . ± . . ± . . ± . ± 10 88 . ± . ± long- t rh . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± BF05 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± fast-MB . ± . . ± . . ± . . ± . . ± . . ± . ± 11 39 . ± . . ± . CE-reduced . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± oldkicks . ± . . ± . . . ± . . . ± . . ± . . ± . . ± . . ± . . ± . 47 Tuc ± 15 84 . ± . ± . ± 16 77 . ± 11 85 . ± 15 409 ± 18 111 ± 13 136 ± Terzan 5 . ± . . ± . . ± . ± 12 58 . ± . . ± . ± 12 142 ± 10 178 ± The number of “plausible” pulsars that are retained in the halo or core of the cluster, as well as the number of ejected pulsars. As a pulsar we define here a NSthat gained mass > . M ⊙ after its formation. “All psrs” - the number of all pulsars, “Bin psrs” - the number of binary pulsars and “NS-bin” - the numberof all NSs in binaries, provided for comparison. For all GC models, except 47 Tuc and Terzan 5, the numbers are scaled per 200 000 M ⊙ stellar populationmass at the age of 11 Gyr; for 47 Tuc the numbers are given per its total mass taken as M ⊙ , for Terzan 5 - per 370 000 M ⊙ . would be initially wide binaries containing two stars of rather sim-ilar masses from 6 to 10 M ⊙ . Most of the primordial bMSPs arelocated in the halo. There are several possible explanations for thesebMSPs to not be present in real clusters: • accretion during the CE either does not occur or does not leadto NS spinup and MSP formation; • as NSs in those bMSPs are mainly post-EIC (see Tables 9 and10) – perhaps the NS natal kicks are higher, causing such systemsto be destroyed rather than evolve via CE; • the CE prescription adopted in our code ( α CE λ = 1 ) does notproduce realistic results for these systems; • our mass segregation does not work properly for these systems– if the progenitors segregate into the cluster core faster, then theyare destroyed and do not produce bMSPs.If indeed the kicks for EIC are higher than adopted in our code,then we will be able to retain very few NSs, as their main supply isfrom EIC. This will have a rather dramatic effect on the formationrate of LMXBs, decreasing it strongly. If the CE efficiency is theproblem, then its reduction might help, as binaries will be formedwith smaller binary separations, and will shrink their orbits fasterand start MT. In our ”CE-reduced” GC model (with CE efficiencyis 10 times smaller) their number is a few times less, but still largecompared to observations. Perhaps CE in these systems must be de-scribed by a much smaller λ parameter, e.g., Podsiadlowski et al.(2003) showed that λ can be as low as 0.01. In this case all sys-tems evolved through a CE start MT or merge shortly after the CE.Alternatively, post-CE systems could be described by the angularmomentum loss prescription proposed in Nelemans & Tout (2005).Then post-CE systems will be much wider and thus more easilydestroyed through dynamical encounters. Whatever the true rea-son, from now on we assume that in our simulations a CE event inprimordial binaries with a NS formed via EIC does not lead to theformation of a MSP.Another population of systems which we do not show in Figs.6 and 7 are NSs produced in AIC, and then mildly recycled in thesame binary system (see § ∼ yrs) and highest B -field of any GC pulsar. Both characteristicssuggest that this system may have been produced from a differentmechanism than the other slow pulsars. The short life-time alsoimplies a fairly high formation rate (see Tables 9 and 10). DCE leads at first to the formation of an eccentric binary MSP withorbital periods of 0.1-10 days, where a NS is spun up during theDCE. Later this system may start a new MT and become an UCXB.As a result, the population of DCE bMSPs can be found on twopanels in Figs. 6 and 7 – among the NSs that gained mass via MTand via DCE.Binary MSP systems produced in NS-WD systems (fromUCXBs) are produced in large numbers by our simulations, but nosuch MSPs have been observed in any globular cluster (see Figs. 6and 7). The possibilities for the fate of these systems are summa-rized in § B , binary WDs with P¡0.1d), although we donot know how they truly manifest themselves.The population of simulated bMSPs produced via DCE nicelyexplains observations of eccentric bMSPs, such as Ter 5 I, J, Q,X, Y, and Z, and 47 Tuc H. Our DCE simulations produce somebinaries with low eccentricity, but cannot reach e < − . Thus,they cannot explain the low eccentricities of 47 Tuc E, Q, S, T, andU, as well as systems like NGC 6752 A. A stable mass-transfermechanism must be invoked here (see discussion below in § c (cid:13) , 000–000 N Ivanova et al. Figure 6. Binary MSPs in 47 Tucanae. Observed and modeled populations:circles – observed, stars – formed via TC, squares – formed via DCE, tri-angles – formed via binary encounters and diamonds – “primordial” binaryMSPs. Filled symbols represent binary MSPs with non-WD companions;in the case of observed systems filled symbols represent eclipsing bMSPs.Crosses mark eccentric bMSPs ( e > . ), adjacent small rotated dia-monds mark bMSPs in the halo. The modeled population represents accu-mulated results from 5 models and corresponds to a stellar population twiceas large as 47 Tucanae. Figure 7. Binary MSPs in Terzan 5. Observed (circles) and modeled pop-ulations, symbols as in Fig. 6. The modeled population represents accumu-lated results from 5 models and corresponds to a stellar population 5 timesbigger than Terzan 5. c (cid:13) , 000–000 inaries with NS in globular clusters Table 9. Types of pulsars, at 11 Gyr, Terzan 5.Halo CoreTotal Binary Total BinaryAll “plausible” pulsarsAll . ± . . ± . ± 12 58 . ± MT . ± . . ± . . ± . . ± . MT-WD . ± . . ± . . ± 12 37 . ± CE . ± . . ± . . . ± . . ± . DCE . ± . . ± . . ± . . ± . Merg . ± . . ± . . ± . . ± . Core CollapseAll . ± . . ± . . ± . . ± . MT . ± . . ± . . ± . . . ± . MT-WD . ± . . ± . . ± . . . ± . . CE . ± . . ± . . ± . . ± . DCE . ± . . ± . . ± . . . ± . . Merg . ± . . ± . . ± . . ± . . ECSAll . ± . . ± . . ± . . ± . MT . ± . . ± . . ± . . ± . MT-WD . ± . . . ± . . . ± . . ± . CE . ± . . ± . . . ± . . ± . DCE . ± . . ± . . ± . . ± . Merg . ± . . ± . . . ± . . ± . AICAll . ± . . ± . . ± 11 42 . ± . MT . ± . . ± . . ± . . ± . MT-WD . ± . . ± . . ± . . ± . CE . ± . . ± . . ± . . ± . DCE . ± . . ± . . ± . . . ± . . Merg . ± . . ± . . ± . . ± . MICAll . ± . . ± . . . ± . . ± . MT . ± . . . ± . . . ± . . ± . . MT-WD . ± . . . ± . . . ± . . ± . CE . ± . . ± . . ± . . ± . . DCE . ± . . ± . . ± . . . ± . . Merg . ± . . ± . . ± . . ± . . Recently formed or mildly recycled high B -fieldAll . ± . . ± . . ± . . ± . MT . ± . . . . ± . . ± . MT-WD . ± . . ± . . ± . . ± . CE . ± . . ± . . ± . . ± . DCE . ± . . ± . . ± . . ± . Merg . ± . . ± . . ± . . ± . The number of pulsars that are retained in the halo and the core of ourTerzan 5 model. We indicate pulsars formed via CC, EIC, AIC or MIC, andthat gained mass through different mechanisms (a mass transfer, a commonenvelope or a merger). This population is present on all panels, as the MSP’s spin-up gen-erally occurred before the MSP acquired its current partner (“5b”in Fig. 1). The eclipsing bMSPs Terzan 5 P ( P orb = 0 . d, M c = 0 . M ⊙ ) and 47 Tuc W ( P orb = 0 . d, M c = 0 . M ⊙ )may have entered their current configurations through this route. The top panels of Figs. 6 and 7 include those bMSPs which wereexchanged, and then underwent MT. While the companion mass re- Table 10. Types of pulsars at 11 Gyr, 47 Tuc.Halo CoreTotal Binary Total BinaryAll “plausible” pulsarsAll ± 15 84 . ± . ± 17 77 . ± MT . ± . . ± . . ± . . ± . MT-WD . ± . . ± . . ± 16 39 . ± CE . ± . . ± . . ± . . ± . DCE . ± . . ± . . ± . . ± . Merg . ± 11 0 . ± . . ± . . ± . Core CollapseAll . ± . . ± . . . ± . . ± . MT . ± . . . ± . . . ± . . . ± . MT-WD . ± . . . ± . . . ± . . . ± . . CE . ± . . . ± . . ± . . ± . DCE . ± . . ± . . ± . . . ± . . Merg . ± . . ± . . ± . . . ± . . ECSAll . ± . . ± . . ± 14 22 . ± MT . ± . . . ± . . . ± . . ± . MT-WD . ± . . ± . . ± . . ± . CE . ± . . ± . . ± . . ± . DCE . ± . . ± . . ± . . ± . Merg . ± . . ± . . ± . . ± . AICAll . ± . . ± . . ± 23 51 . ± MT . ± . . ± . . ± . . ± . MT-WD . ± . . ± . . ± 15 34 . ± CE . ± . . ± . . ± . . ± . . DCE . ± . . ± . . ± . . . ± . . Merg . ± . . ± . . ± 11 2 . ± . MICAll . ± . . ± . . ± . . ± . . MT . ± . . ± . . ± . . ± . MT-WD . ± . . ± . . ± . . . ± . . CE . ± . . ± . . ± . . ± . DCE . ± . . ± . . ± . . . ± . . Merg . ± . . ± . . ± . . ± . . Recently formed or mildly recycled high B -fieldAll . ± . . ± . . ± . . ± . MT . ± . . ± . . ± . . ± . . MT-WD . ± . . ± . . ± . . ± . CE . ± . . ± . . ± . . ± . DCE . ± . . ± . . ± . . ± . Merg . ± . . ± . . ± . . ± . Notations are as in Table 9, but for our model of 47 Tuc. mains above ∼ . M ⊙ , the MT rate in these systems is high, andmost of them will be seen as LMXBs or qLMXBs (see also § c (cid:13) , 000–000 N Ivanova et al. relatively frequently, but all of them have collision times well be-low 1 Gyr. After the MT, they experienced a dynamical encounterthat led either to an exchange of companions, or to an eccentric-ity increase. As a result, none of them are observed at 11 Gyr withlow eccentricities. Potentially, we should be able to form binarieswith a post-MT period of about a day. Such binaries have a col-lision time around 10 Gyr and thus should stay preserved in theirpost-MT shape for almost the entire cluster evolution. However,probably due to rather low number statistics, they are not formedin our simulations for 47 Tuc and Ter 5. Also, our post-exchangebinaries are very eccentric. A better treatment of the MT in eccen-tric binaries, like discussed in Sepinsky et al. (2007), may help tocreate low-eccentric binaries with low-mass WD companions.The bMSPS which were spun-up via mergers during a pre-vious dynamical interaction (bottom panels) generally have moremassive companions and longer periods than the observed popula-tions. The few good candidates are Ter 5 Q, NGC 1851 A, M28 D,NGC 6441 A, and possibly M30 B. Dynamically formed binarieswhich undergo stable MT with a giant, producing non-eccentriclong-period systems are rare in our simulations; these tend to comefrom the primordial population located outside the core. From § B MSP pro-duction: • recent NS formation via any ECS channel • AIC followed by continued accretion in the same system • mass gain via merger or physical collisionRadio MSPs are not produced via these channels, or can not bedetected yet: • accretion from a degenerate donor • mass transfer in an NS-MS systems with current MS mass & . M ⊙ , as they have not yet turned on as radio pulsars • accretion during a CE eventWith these restrictions (see Fig. 1), we find that our estimatesfor the numbers of produced MSPs are ± in 47 Tuc and ± MSPs in Terzan 5 (see Table 11). These estimates are close to theobservations for both 47 Tuc (22 detected MSPs) and Terzan 5 (33MSPs).Our simulations indicate that the fraction of single pulsars ishigher in Terzan 5 than in 47 Tuc, as observed. The origin of iso-lated MSPs depends on the cluster dynamical properties. For ex-ample, in our standard model, about half of all isolated MSPs wereproduced by an evolutionary merger at the end of the MT from aMS companion ( § v rec decreases. Roughly half of all observed GC pul-sars are found outside their cluster core radius (Camilo & Rasio2005), but the radial distribution of most pulsars is exactly as ex-pected for a population that is produced in or around the core (Grindlay et al. 2002; Heinke et al. 2005b), with the exception of afew, likely ejected, pulsars in M15 and NGC 6752 (e.g. Colpi et al.2002).The creation rate of high-magnetic field (AIC) MSPs, at theage of 11 Gyr, is about 20-40 per Gyr in Terzan 5 and 47 Tuc. Thisleads to the probability of forming high-magnetic field MSPs ( B . -gauss) with a period > ms of less than a few per cent,while for B . -gauss, it is about 50 per cent (see eq. 8). E.g.,one pulsar in Terzan 5 (J1748-2446J) has P=80 ms (Ransom et al.2005). The absence of slow-period MSPs in 47 Tucanae suggeststhat the magnetic field in newly formed AIC NSs is most likely tobe at the higher end, & -gauss. We note that the results of oursimulations are consistent with a higher probability of a high- B pulsar in Terzan 5 than in 47 Tucanae (see Table 11), though thestatistics are very poor.We compare the behavior of pulsar properties with increas-ing cluster density in our simulations, and in real clusters, in Table12. We divide the clusters containing MSPs with known proper-ties into categories roughly corresponding to the central densitiesused in the models: High-density ( ρ c > . , and core-collapsedwith ρ c > . ): NGC 1851, NGC 6397, NGC 6522, NGC 6544,NGC 6624, M15, and M30. Terzan-5-like ( . < ρ c < . ):NGC 6266, Terzan 5, NGC 6440, and NGC 6441. 47Tuc-like( . < ρ c < . ): 47 Tuc, NGC 6342, NGC 6626, and NGC6752. Standard ( . < ρ c < . ): M5, M4, and NGC 6760.Medium-density ( . < ρ c < . ): M53, M3, M13, NGC 6539,NGC 6749, and M71. No pulsars have been found to inhabit low-density ( . > ρ c ) clusters yet, nor do we produce any in our sim-ulations (Table 11). We exclude from consideration all MSPs withunknown binary properties, and those isolated pulsars in clusterswhich have been observed too rarely to measure binary properties(NGC 6624 D and E; NGC 6517 A and C) to reduce bias towardssingle MSPs.The column “All pulsars” indicates the total number of pul-sars used in that category; other columns refer to fractions of thatnumber. The total numbers of pulsars cannot be usefully comparedto the numbers of pulsars in the model, but the fractions of pulsarsystem types can be compared. Field systems include all field pul-sars with P < ms and B < × G (Manchester et al.2005) . “BD” refers to brown dwarf systems; M comp < . M ⊙ .“MS” identifies likely main-sequence companions, identified bybeing eclipsing systems with M comp > . M ⊙ (Freire 2005).“He WDs” refer to all non-eclipsing systems with . < M c < . M ⊙ . “Heavy WDs” include all MSPs with M c & . M ⊙ ,unless the companion is known to be a NS. “ > d” refers to thatsubset of all MSPs with orbital periods longer than 10 days. “High- B ” pulsars are taken to be MSPs with spin-period P & ms. Toimprove the statistics for our high- B systems we count all high- B pulsars spun up within the past 1 Gyr, and divide this number by afactor of 10 to account for their shorter lifetimes.Several clear trends with the central density of globular clus-ters can be seen in the observations. Single MSPs rise smoothlyfrom a rate of 0.25 of all MSPs in the field, to 0.7 of all MSPs inclusters. This is well-described by our model, suggesting that theformation of single MSPs may be reasonably well understood. Theformation of MSPs with “brown dwarf” companions, with very lowmasses, exhibit a complicated density dependence, with few in thefield, a quarter of all MSPs in standard-density clusters, and declin-ing fractions in higher density clusters. This indicates that their for- (cid:13) , 000–000 inaries with NS in globular clusters Table 11. “Observable” pulsars at 11 Gyr.model low- B High- B All psrs Single Binary All psrsCompanion BD WD MS He WD heavy WD recently M c . . M ⊙ & . M ⊙ recycledPeriod < . d < . d > d . ÷ d . ÷ dstandard . ± . . ± . . ± . . ± . . ± . . ± . . ± . . . ± . . ± . metal-poor . ± . . ± . . ± . . . ± . . ± . . . ± . . . ± . . ± . . . ± . high-den . ± . . ± . . ± . . . ± . . ± . . ± . . . ± . . ± . . . ± . med-den . ± . . ± . . . ± . . ± . . ± . . . ± . . ± . . . ± . . . ± . low-den . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . low- σ . ± . . ± . . ± . . . ± . . . ± . . . ± . . ± . . . ± . . ± . long- t rh . ± . . ± . . ± . . . ± . . . ± . . . ± . . ± . . . ± . . . ± . BF05 . ± . . ± . . ± . . . ± . . ± . . ± . . ± . . . ± . . . ± . fast-MB . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . . ± . CE-reduced . ± . . ± . . . ± . . . ± . . ± . . . ± . . . ± . . . ± . . . ± . oldkicks . ± . . ± . . ± . . . ± . . ± . . ± . . ± . . . ± . . . ± . 47 Tuc . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . . ± . Terzan 5 . ± . . ± . . ± . . ± . . ± . . . ± . . ± . . ± . . ± . The numbers of pulsars that can be detected in a GC (including MSPs only if they gain mass via MT from a non-degenerate donor, or via DCE). Recenthigh- B pulsars are those which were formed or mildly recycled less than years ago. Mass-transferring NS-MS binaries where a MS star is & . M ⊙ areexcluded, as they are likely LMXBs/qLMXBs. “Companion” indicates the type of the companion; “ M c ” is the companion mass, and “Period” is the orbitalperiod. For all GC models, except 47 Tuc and Terzan 5, the numbers are scaled per 200 000 M ⊙ stellar population mass at the age of 11 Gyr; for 47 Tuc thenumbers are given for 47 Tuc’s mass, taken as M ⊙ ; for Terzan 5, 370 000 M ⊙ . Table 12. Detected pulsars and simulations.Cluster All pulsars Single BD He WD MS heavy WD > d ecc high-BHigh-den model . ± . . ± . 11 0 . ± . 03 0 . ± . 05 0 . ± . . . ± . 08 0 . ± . 00 0 . ± . 10 0 . ± . All high-den 19 0.71 0.06 0.0 0.12 0.12 0.06 0.18 0.12Terzan 5 model . ± . . ± . 06 0 . ± . 03 0 . ± . 06 0 . ± . 03 0 . ± . 03 0 . ± . 03 0 . ± . 07 0 . ± . All Ter5 49 0.43 0.08 0.31 0.12 0.06 0.08 0.16 0.0447Tuc model . ± . . ± . 06 0 . ± . 04 0 . ± . 09 0 . ± . 06 0 . ± . 12 0 . ± . 10 0 . ± . 14 0 . ± . All 47Tuc 37 0.43 0.19 0.30 0.11 0 0.03 0.08 0.03Standard model . ± . . ± . 18 0 . ± . 04 0 . ± . 17 0 . ± . 10 0 . ± . 10 0 . ± . 10 0 . ± . 20 0 . ± . all stand 8 0.25 0.25 0.50 0.0 0 0.25 0.25 0.0Med-den model . ± . . ± . . . ± . . . ± . . . ± . . . ± . . . ± . . . ± . . . ± . ∼ pfreire/GCpsr.html as of late 2007. See text ( § 8) for details. mation requires a high density environment, but also suggests thattheir binary evolution can be disrupted in high density conditions.Our simulations do not reproduce this trend very well; likely this isbecause our lack of a proper equation of state for brown dwarfs pre-vents us from accurately tracking the late evolution of NS-MS bi-naries and the eventual possible destruction of the companion. Wemay produce too many heavy WDs and too few He WDs, whichboth have only mild density dependences. The fraction of eccen-tric MSPs, perhaps surprisingly, shows little density dependence(in our simulations and in real systems). The production of MSPswith main-sequence companions and of high- B MSPs are sharplydependent on the central density of the cluster, in agreement withthe suggestions by Freire (2005) and Lyne et al. (1996) that theirproduction requires multiple binary interactions, and direct impact of NSs with other stars, respectively. Overall, our predicted num-bers are in quite good agreement with the observations. As predicted in § × M ⊙ of stellar population evolved in different clusters– only 3 primordial DNSs were formed. In all three systems bothNSs were formed via ECS, and none of the systems have mergedwithin a Hubble time.Let us estimate the rate of dynamical DNS formation. Thecharacteristic time for a single NS to have an encounter with aNS-binary can be estimated using eq. 10. The binary compan- c (cid:13) , 000–000 N Ivanova et al. ion in a NS-binary is usually less massive than the NS. If an ex-change occurs, the binary separation in the post-exchange binary a is larger than the binary separation a in the pre-exchange binary, a ∼ a m NS /m , where m is the mass of the replaced companion(Heggie et al. 1996). Binary eccentricities in hard binaries follow athermal distribution, with an average eccentricity e ∼ / (Heggie1975). The maximum a that leads to a merger within 11 Gyr, if thepost-exchange eccentricity is 2/3, is about . R ⊙ . The maximummass for a NS-companion before the exchange is about the mass ofa turn-off MS star – . M ⊙ . Then the maximum binary separationin a pre-exchange binary that leads to the formation of a mergingDNS is a = 4 . R ⊙ .From our simulations we find that the relative fraction of NSbinaries in the core f NS − bin , core = N NS − bin , core /N objects , core is0.175 per cent for the standard model, and 0.6 per cent for ourTerzan 5 model. Then the characteristic time for a NS to havean encounter that may lead to the formation of a merging DNS is τ DNS , m ≈ × Gyr in the case of a standard cluster, in otherwords, about 500 NSs are needed to form one merging DNS within11 Gyr. This assumed that every encounter will lead to the exchangeand that all NS-binaries have the separation of ∼ R ⊙ . In the caseof Terzan 5, the chance of forming a merging DNS, per NS, is 25times bigger than in a standard cluster – the formation rate is highlydependent on the cluster density, as the power 2, and the number ofretained NSs increases as the escape velocity increases. However,most NS-binaries have larger binary separations than ∼ R ⊙ , andeven though encounters are more frequent in this case, they do notlead to the formation of a merging DNS directly. Subsequent en-counters of the wide DNS with other stars would be necessary toincrease the DNS’s eccentricity, and thus reduce its merging time.Only 14 DNSs are formed dynamically during 11 Gyr in all ofour 70 cluster models. Since the formation rate of DNSs is highlydensity dependent, most of those DNSs were formed in the clustermodels of Terzan 5. Only two of these 14 DNSs merged within11 Gyr, while another two DNSs merged within 14 Gyr, a Hubbletime. 5 of the 14 DNSs stayed in the cluster for 11 Gyr, while theremaining 9 were destroyed within a Gyr after their formation.We note that the formation rate of DNSs is strongly con-nected to the formation rate of LMXBs, as the main building mate-rial of DNSs is NS-binaries that otherwise become LMXBs. Aswe show in § 7, we produce LMXBs in numbers comparable tothose observed, but we underestimate the formation rates for core-collapsed clusters. In core-collapsed clusters it is especially clearthat our simplified models do not properly treat the cluster dy-namics and therefore we cannot be very quantitative about thetotal number of DNSs that should be produced by all GCs, al-though only a small fraction of GCs are core-collapsed. We con-clude therefore that GCs are rather unimportant for the formationof merging DNSs and can at most provide a small ( − )contribution to the production of short γ -ray bursts in old hostgalaxies as estimated by Grindlay et al. (2006). Such systems canonly be produced in core-collapsed clusters, like M15, where sucha system is observed (Anderson et al. 1990; Jacoby et al. 2006).Also, recent observations seem to indicate that majority of shortGRBs are associated with young hosts (Berger et al. 2007). If infact short hard GRBs originate from NS-NS mergers (although seeBelczynski et al. 2007), then their rates and occurrence in younghosts is understood in terms of NS-NS formation without dynam-ical interactions, i.e. in field populations (Belczynski, Stanek &Fryer 2007). 10 SUMMARY In our study of binaries with neutron stars in globular clusters weconsidered in detail the problem of the formation of LMXBs andMSPs. We predict from our simulations that most retained neu-tron stars in GCs must come from an electron capture supernovaformation channel (c.f. the field, where most NSs are from core-collapse supernovae). A typical GC could contain at present (wherethe adopted cluster age is 11 Gyr) as many as ∼ NSs, of whichhalf are located in the halo; a massive GC like 47 Tuc could havemore than a thousand NSs.Analyzing encounters with NSs, we find that for NSs locatedin the core, about half formed a binary through an exchange en-counter and only a few per cent formed a binary through a physicalcollision with a giant or via tidal capture. The relative importance oftidal captures to physical collisions increases as the velocity disper-sion decreases. Although there are fewer binaries formed throughDCE and TC than through binary encounters, the production ofmass-transferring systems is roughly the same for all three chan-nels. We note as well that many physical collisions lead to mergersbetween NSs and other stars in the core.We derived a “collision number” from our globular clustermodels which appears to scale linearly with the LMXB produc-tion frequency from our simulations, although only when the coredensity alone is varied. Variations of other GC dynamical propertieswith a fixed core density lead to a large scatter. This may explain theobserved deviations from a linear dependence between the collisionnumber and the number of LMXBs in non-core-collapsed clusters.We predict that the numbers of qLMXBs will have no dependenceon metallicity, in contrast to the numbers of bright LMXBs.Our rates of LMXB formation predict 7.5 UCXBs in allgalactic GCs, which is consistent with the observed number. ForqLMXBs, we expect about 180 systems, which agrees with therange (100-200) estimated from observations (Pooley et al. 2003;Heinke et al. 2003, 2005a). From our list of clusters which are ex-pected to have one or more LMXBs (bright or in quiescence) –47 Tuc, Terzan 5, NGC 1851, NGC 6266, NGC 6388, NGC 6440,NGC 6441, NGC 6517, M54, NGC 7078 – all clusters which havebeen studied with Chandra (all but NGC 6517) show at least oneLMXB or qLMXB. Our results do not support the idea that theobserved luminosity function in extragalactic globular clusters isprincipally due to UCXBs.The resulting retention fraction of ∼ − (or ∼ NSsper 200,000 M ⊙ ), obtained in our simulations, seems to be firm, asthe numerically derived formation rates of LMXBs, which stronglycorrelate with the retention fraction, are consistent with those ob-served. In our GC models, we achieved such a retention fractionby producing NSs via different channels of electron capture super-nova. If the natal kick velocity distribution will be revised in futurefrom that used in this paper (Hobbs et al. 2005), then the necessityof ECS will decrease. If, however, it should be shown that AIC doesnot occur, then the retention fraction that is required to match theobservations of LMXBs might have to be larger by a factor of 2.We find from our simulations that if all possible channels ofNS formation and all possible mechanisms for their spin-up leadto MSP formation, then we overproduce MSPs. However, we stillneed these channels to produce observed LMXBs. We propose thathigh B -field MSPs (which are short-living) can be formed not onlyduring core-collapse supernovae, but also due to physical collisionsor accretion in a post-AIC system. We find that NSs which accreteand spin up during CE events overproduce bMSPs in the cluster ha-los from primordial binaries of intermediate masses. Such bMSPs c (cid:13) , 000–000 inaries with NS in globular clusters would be present in low-density clusters and have not yet been seen.In the case of the NS-WD LMXBs, we propose that the MT in suchsystems does not lead to the formation of radio bMSPs. The ratesof UCXB formation (verified by observations of UCXB LMXBs)predict large numbers of ultracompact bMSPs in GCs which arenot detected.Excluding the systems discussed above, as well as those whichare still actively accreting their donor’s material and are seen in-stead as LMXBs, we obtain lower limits of “detectable” bMSPs.The predicted numbers are in rather good agreement with the ob-servations – ± MSPs in 47 Tuc and ± in Terzan 5. Thefraction of isolated pulsars is comparable to that observed and islarger in Terzan 5 than in 47 Tuc.Comparing the population census of our models with the ob-servations of all detected pulsars to date in GCs, we find goodagreement for all types of pulsars – single, those with brown dwarfcompanions, with MS companions, with light or heavy WD com-panions, and with large periods.We do not find very efficient formation rates for double NSs– only a dozen were formed in 70 simulated models. These ratesincrease with the square of the GC core density. We conclude thatDNS formation is most likely to occur in massive, very dense, andpreferentially core-collapsed GCs, as suggested by the identifica-tion of only one double NS so far, in the core-collapsed GC M15.In conclusion we outline several important issues that must beaddressed for further progress in studies of NSs in globular clusters: • Common envelope events will occur in primordial binaries ofintermediate mass that produce NSs via EIC. What is the commonenvelope efficiency, and is the NS spun up by accreting the mate-rial? • What is the result of mass accretion onto a NS after it has ex-perienced a merger, either in a binary due to unstable mass transfer,or during a collision? • What is the final fate of a mass-transferring NS-WD binary? • How does the evolution of a mass-transferring NS-MS binaryproceed when the companion’s mass reaches . . M ⊙ ? • What is the dependence of qLMXB numbers on the clustermetallicity? This holds the potential to determine the cause of theLMXB metallicity dependence (see § ACKNOWLEDGMENTS We thank Chris Deloye and Norman Murray for helpful discus-sions and an anonymous referee for comments that helped toimprove the paper. This work was supported by a Beatrice D.Tremaine Fellowship to NI. FAR was supported by NASA GrantsNNG06GI62G and NNG04G176G. JMF acknowledges supportfrom Chandra theory grant TM6-7007X and Chandra PostdoctoralFellowship Award PF7-80047. KB acknowledges support from KBN grant 1P03D02228. COH acknowledges support from a Lind-heimer Postdoctoral Fellowship, and from Craig Sarazin throughChandra grant G07-8078X. Simulations were performed on CITA’sSunnyvale cluster, funded by the Canada Foundation for Innovationand the Ontario Research Fund for Research Infrastructure. REFERENCES Altamirano D., Casella P., Patruno A., Wijnands R., van der KlisM., 2007, ArXiv e-prints, 708Anderson S. B., Gorham P. W., Kulkarni S. R., Prince T. A., Wol-szczan A., 1990, Nature, 346, 42Arzoumanian Z., Chernoff D. F., Cordes J. M., 2002, ApJ, 568,289Barkat Z., Reiss Y., Rakavy G., 1974, ApJL, 193, L21Belczynski K., Kalogera V., Bulik T., 2002, ApJ, 572, 407Belczynski K., Kalogera V., Rasio F. A., Taam R. E., Zezas A.,Bulik T., Maccarone T. J., Ivanova N., 2008, ApJ Supp, 174Belczynski K., Perna R., Bulik T., Kalogera V., Ivanova N., LambD. Q., 2006, ApJ, 648, 1110Belczynski, K., O’Shaughnessy, R., Kalogera, V., Rasio, F., Taam,R., & Bulik, T. 2007, ArXiv e-prints, 712, arXiv:0712.1036, Sci-ence, submittedBelczynski,K. Stanek, K.Z. & Fryer C.L. 2007, ApJ, submittedBellazzini M., Pasquali A., Federici L., Ferraro F. R., Pecci F. F.,1995, ApJ, 439, 687Berger, E., et al. 2007, ApJ, 664, 1000Bhattacharya D., van den Heuvel E. P. J., 1991, PhysRep, 203, 1Biggs J. D., Bailes M., Lyne A. G., Goss W. M., Fruchter A. S.,Biggs J. D., 1994, MNRAS, 267, 125Bildsten L., 2002, ApJL, 577, L27Bildsten L., Deloye C. J., 2004, ApJL, 607, L119Bregman J. N., Irwin J. A., Seitzer P., Flores M., 2006, ApJ, 640,282Breton R. P., Roberts M. S. E., Ransom S. M., Kaspi V. M., DurantM., Bergeron P., Faulkner A. J., 2007, ApJ, 661, 1073Brown E. F., Bildsten L., Rutledge R. E., 1998, ApJL, 504, L95Buras R., Janka H.-T., Rampp M., Kifonidis K., 2006, A&A, 457,281Camilo F., Lorimer D. R., Freire P., Lyne A. G., Manchester R. N.,2000, ApJ, 535, 975Camilo F., Rasio F. A., 2005, in Rasio F. A., Stairs I. H., eds,ASP Conf. Ser. 328: Binary Radio Pulsars Pulsars in GlobularClusters.p 147Campana S., Colpi M., Mereghetti S., Stella L., Tavani M., 1998,A&AR, 8, 279Chapman, R., Levan, A. J., Priddey, R. S., Tanvir, N. R., Wynn,G. A., King, A. R., & Davies, M. B. 2007, Astronomical Societyof the Pacific Conference Series, 372, 415Clark G. W., 1975, ApJL, 199, L143Colpi M., Possenti A., Gualandris A., 2002, ApJL, 570, L85Davies M. B., Benz W., Hills J. G., 1992, ApJ, 401, 246Deloye C. J., 2007, ArXiv e-prints, 710Deloye C. J., Bildsten L., 2003, ApJ, 598, 1217Dessart, L., Burrows, A., Ott, C. D., Livne, E., Yoon, S.-C., &Langer, N. 2006, ApJ, 644, 1063Deutsch E. W., Margon B., Anderson S. F., 2000, ApJL, 530, L21Dewi J. D. M., van den Heuvel E. P. J., 2004, MNRAS, 349, 169Di Salvo T., Burderi L., 2003, A&A, 397, 723Dieball A., Knigge C., Zurek D. R., Shara M. M., Long K. S., c (cid:13) , 000–000 N Ivanova et al. Charles P. A., Hannikainen D. C., van Zyl L., 2005, ApJL, 634,L105Drukier G. A., 1996, MNRAS, 280, 498Dubus, G., Lasota, J.-P., Hameury, J.-M., & Charles, P. 1999, MN-RAS, 303, 139Eggleton P. P., Kiseleva-Eggleton L., 2001, ApJ, 562, 1012Ford E. B., Kozinsky B., Rasio F. A., 2000, ApJ, 535, 385Fregeau J. M., Cheung P., Portegies Zwart S. F., Rasio F. A., 2004,MNRAS, 352, 1Fregeau, J. M., & Rasio, F. A. 2007, ApJ, 658, 1047Freire P. C. C., 2005, in Rasio F. A., Stairs I. H., eds, Binary RadioPulsars Vol. 328 of Astronomical Society of the Pacific Confer-ence Series, Eclipsing Binary Pulsars, p 405Fruchter A. S., Goss W. M., 1995, Journal of Astrophysics andAstronomy, 16, 245Fryer C. L., 2004, ApJL, 601, L175Galloway D. K., Chakrabarty D., Morgan E. H., Remillard R. A.,2002, ApJL, 576, L137Giersz M., 2006, MNRAS, 371, 484Gnedin O. Y., Zhao H., Pringle J. E., Fall S. M., Livio M., MeylanG., 2002, ApJL, 568, L23Grindlay J., Portegies Zwart S., McMillan S., 2006, NaturePhysics, 2, 116Grindlay J. E., 1993, in Smith G. H., Brodie J. P., eds, The Globu-lar Cluster-Galaxy Connection Vol. 48 of Astronomical Societyof the Pacific Conference Series, X-Raying Stellar Remnants inGlobular Clusters, p 156Grindlay J. E., Camilo F., Heinke C. O., Edmonds P. D., Cohn H.,Lugger P., 2002, ApJ, 581, 470Guerrero J., Garc´ıa-Berro E., Isern J., 2004, A&A, 413, 257G¨urkan M. A., Freitag M., Rasio F. A., 2004, ApJ, 604, 632G¨urkan M. A., Rasio F. A., 2003, in Piotto G., Meylan G., Djor-govski S. G., Riello M., eds, New Horizons in Globular ClusterAstronomy Vol. 296 of Astronomical Society of the Pacific Con-ference Series, The Radial Distribution of Millisecond Pulsars in47 Tuc, p 300Harris W. E., 1996, AJ, 112, 1487, revision 2003Heggie D. C., 1975, MNRAS, 173, 729Heggie D. C., Hut P., McMillan S. L. W., 1996, ApJ, 467, 359Heinke C. O., Grindlay J. E., Edmonds P. D., 2005a, ApJ, 622,556Heinke C. O., Grindlay J. E., Edmonds P. D., Cohn H. N., LuggerP. M., Camilo F., Bogdanov S., Freire P. C., 2005b, ApJ, 625,796Heinke C. O., Grindlay J. E., Lugger P. M., Cohn H. N., EdmondsP. D., Lloyd D. A., Cool A. M., 2003, ApJ, 598, 501Heinke C. O., Wijnands R., Cohn H. N., Lugger P. M., GrindlayJ. E., Pooley D., Lewin W. H. G., 2006, ApJ, 651, 1098Hessels J. W. T., Ransom S. M., Stairs I. H., Kaspi V. M., FreireP. C. C., 2007, ApJ, 670, 363Hobbs G., Lorimer D. R., Lyne A. G., Kramer M., 2005, MNRAS,360, 974Holman M., Touma J., Tremaine S., 1997, Nature, 386, 254Hurley J. R., Pols O. R., Tout C. A., 2000, MNRAS, 315, 543Hut P., 2006, ArXiv Astrophysics e-printsHut P., Murphy B. W., Verbunt F., 1991, A&A, 241, 137Innanen K. A., Zheng J. Q., Mikkola S., Valtonen M. J., 1997, AJ,113, 1915in’t Zand J. J. M., Jonker P. G., Markwardt C. B., 2007, A&A,465, 953Ivanova N., 2006, ApJ, 636, 979Ivanova N., Belczynski K., Fregeau J. M., Rasio F. A., 2005a, MNRAS, 358, 572Ivanova N., Belczynski K., Kalogera V., Rasio F. A., Taam R. E.,2003, ApJ, 592, 475Ivanova N., Fregeau J. M., Rasio F. A., 2005b, in Rasio F. A.,Stairs I. H., eds, ASP Conf. Ser. 328: Binary Radio Pulsars Bi-nary Evolution and Neutron Stars in Globular Clusters, p 231Ivanova N., Heinke C. O., Rasio F. A., Taam R. E., Belczynski K.,Fregeau J., 2006, MNRAS, 372, 1043Ivanova N., Rasio F., 2004, in Revista Mexicana de Astronomiay Astrofisica Conference Series Compact Binaries in GlobularClusters. pp 67–70Ivanova N., Rasio F. A., 2005c, in Burderi L., Antonelli L. A.,D’Antona F., di Salvo T., Israel G. L., Piersanti L., Tornamb`eA., Straniero O., eds, AIP Conf. Proc. 797: Interacting Binaries:Accretion, Evolution, and Outcomes Formation and evolution orcompact binaries with an accreting white dwarf in globular clus-ters. pp 53–60Ivanova N., Rasio F. A., Lombardi J. C., Dooley K. L., ProulxZ. F., 2005d, ApJL, 621, L109Ivanova N., Taam R. E., 2003, ApJ, 599, 516Ivanova N., Taam R. E., 2004, ApJ, 601, 1058Jacoby B. A., Cameron P. B., Jenet F. A., Anderson S. B., MurtyR. N., Kulkarni S. R., 2006, ApJL, 644, L113Janssen T., van Kerkwijk M. H., 2005, A&A, 439, 433Jord´an A., Cˆot´e P., Ferrarese L., Blakeslee J. P., Mei S., MerrittD., Milosavljevi´c M., Peng E. W., Tonry J. L., West M. J., 2004,ApJ, 613, 279Katz J. I., 1975, Nature, 253, 698Kawai Y., Saio H., Nomoto K., 1987, ApJ, 315, 229King A. R., Beer M. E., Rolfe D. J., Schenker K., Skipp J. M.,2005, MNRAS, 358, 1501King A. R., Pringle J. E., Wickramasinghe D. T., 2001, MNRAS,320, L45Kitaura F. S., Janka H.-T., Hillebrandt W., 2006, A&A, 450, 345Kozai Y., 1962, AJ, 67, 591Krimm H. A., Markwardt C. B., Deloye C. J., Romano P.,Chakrabarty D., Campana S., Cummings J. R., Galloway D. K.,Gehrels N., Hartman J. M., Kaaret P., Morgan E. H., Tueller J.,2007, ArXiv e-prints, 709Kroupa P., 2002, Science, 295, 82Kundu A., Maccarone T. J., Zepf S. E., 2002, ApJL, 574, L5Kundu, A., Maccarone, T. J., & Zepf, S. E. 2007, ApJ, 662, 525Kuranov A. G., Postnov K. A., 2006, Astronomy Letters, 32, 393Levan A. J., Wynn G. A., Chapman R., Davies M. B., King A. R.,Priddey R. S., Tanvir N. R., 2006, MNRAS, 368, L1Livio M., Soker N., 1988, ApJ, 329, 764Lombardi J. C., Proulx Z. F., Dooley K. L., Theriault E. M.,Ivanova N., Rasio F. A., 2006, ApJ, 640, 441Lyne A. G., Biggs J. D., Harrison P. A., Bailes M., 1993, Nature,361, 47Lyne A. G., Manchester R. N., D’Amico N., 1996, ApJL, 460,L41+Maccarone T. J., Kundu A., Zepf S. E., 2004, ApJ, 606, 430Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2005, AJ,129, 1993Mazeh T., Shaham J., 1979, A&A, 77, 145Miller M. C., Hamilton D. P., 2002, ApJ, 576, 894Miyaji S., Nomoto K., Yokoi K., Sugimoto D., 1980, PASJ, 32,303Muno M. P., Clark J. S., Crowther P. A., Dougherty S. M., deGrijs R., Law C., McMillan S. L. W., Morris M. R., NegueruelaI., Pooley D., Portegies Zwart S., Yusef-Zadeh F., 2006, ApJL, c (cid:13) , 000–000 inaries with NS in globular clusters c (cid:13)000