Population synthesis of ultracompact X-ray binaries in the Galactic Bulge
L. M. van Haaften, G. Nelemans, R. Voss, S. Toonen, S. F. Portegies Zwart, L. R. Yungelson, M. V. van der Sluys
AAstronomy & Astrophysics manuscript no. vanhaaftenetal2013˙ucxbpopsynt c (cid:13)
ESO 2018September 10, 2018
Population synthesis of ultracompact X-ray binariesin the Galactic Bulge
L. M. van Haaften , G. Nelemans , , R. Voss , S. Toonen , S. F. Portegies Zwart , L. R. Yungelson , andM. V. van der Sluys Department of Astrophysics / IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands, e-mail:
[email protected] Institute for Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya Str., 119017 Moscow, RussiaPreprint online version: September 10, 2018
Abstract
Aims.
We model the present-day number and properties of ultracompact X-ray binaries (UCXBs) in the Galactic Bulge. The mainobjective is to compare the results to the known UCXB population as well as to data from the Galactic Bulge Survey, in order to learnabout the formation of UCXBs and their evolution, such as the onset of mass transfer and late-time behavior.
Methods.
The binary population synthesis code
SeBa and detailed stellar evolutionary tracks have been used to model the UCXBpopulation in the Bulge. The luminosity behavior of UCXBs has been predicted using long-term X-ray observations of the knownUCXBs as well as the thermal-viscous disk instability model.
Results.
In our model, the majority of UCXBs initially have a helium burning star donor. Of the white dwarf donors, most have heliumcomposition. In the absence of a mechanism that destroys old UCXBs, we predict (0 . − . × UCXBs in the Galactic Bulge,depending on assumptions, mostly at orbital periods longer than 60 min (a large number of long-period systems also follows from theobserved short-period UCXB population). About 5 −
50 UCXBs should be brighter than 10 erg s − , mostly persistent sources withorbital periods shorter than about 30 min and with degenerate helium and carbon-oxygen donors. This is about one order of magnitudemore than the observed number of (probably) three. Conclusions.
This overprediction of short-period UCXBs by roughly one order of magnitude implies that fewer systems are formed,or that a super-Eddington mass transfer rate is more di ffi cult to survive than we assumed. The very small number of observed long-period UCXBs with respect to short-period UCXBs, the surprisingly high luminosity of the observed UCXBs with orbital periodsaround 50 min, and the properties of the PSR J1719–1438 system all point to much faster UCXB evolution than expected from angularmomentum loss via gravitational wave radiation alone. Old UCXBs, if they still exist, probably have orbital periods longer than 2 hand have become very faint due to either reduced accretion or quiescence, or have become detached. UCXBs are promising candidateprogenitors of isolated millisecond radio pulsars. Key words. binaries: close – stars: evolution – Galaxy: bulge – X-rays: binaries – pulsars: general
1. Introduction
Ultracompact X-ray binaries (UCXBs) are low-mass X-ray bina-ries with observed orbital periods shorter than ∼ ◦ × ◦ regions centered1 . ◦ to the North and South of the Galactic Center. One of thegoals of the GBS is to investigate the properties of populationsof X-ray binaries in order to constrain their formation scenarios,especially the common-envelope phase(s).The present study aims to predict and explain GBS resultsregarding the number and luminosities of UCXBs by meansof binary population synthesis and UCXB evolutionary tracks,thereby contributing to a better understanding of the formationand evolution of UCXBs. We also compare our results to the or-bital periods and chemical compositions of the known UCXBpopulation. In Sect. 2 we describe our assumptions on the starformation, stellar and binary evolution, and the observable char-acteristics of evolved UCXBs. The results follow in Sect. 3,where we present the modeled present-day population and itsobservational properties. In Sect. 4, we compare our results withthe population synthesis studies by Belczynski & Taam (2004a),Zhu et al. (2012a), and Zhu et al. (2012b), as well as to obser-vations, and we discuss various implications. We conclude inSect. 5. a r X i v : . [ a s t r o - ph . S R ] F e b . M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge
2. Method
The study of the evolution of the UCXB population consists ofseveral steps. First, the star formation history of the GalacticBulge and the binary initial mass function prescribe which typesof zero-age main sequence binaries form in the Bulge, andwhen they form. The binary population synthesis code
SeBa (Portegies Zwart & Verbunt 1996; Nelemans et al. 2001; Toonenet al. 2012) is used to simulate the evolution of this popu-lation of zero-age main sequence binaries. During the evolu-tion of the population, all UCXB progenitors are selected at acertain moment after the supernova explosion that leaves be-hind the neutron star or black hole. More detailed evolutionarytracks are used to trace the subsequent evolution. This yieldsthe present-day number and intrinsic parameters of the UCXBsin the Galactic Bulge. Finally, using long-term observations bythe
Rossi X-ray Timing Explorer
All-Sky Monitor (
RXTE
ASM)(Bradt et al. 1993; Levine et al. 1996) as well as the accre-tion disk instability model, the modeled UCXB parameters aretranslated into observational parameters using the results of vanHaaften et al. (2012c). The result is a prediction of the present-day observable population.
The binary population synthesis code
SeBa models the evolu-tionary transformations of a population of binary stars based ona distribution of initial binary parameters. It follows the evo-lution of stellar components using analytic formulas by Hurleyet al. (2000), taking into account circularization due to tidal in-teraction, magnetic braking, gravitational wave radiation, massexchange via Roche-lobe overflow, common envelopes, and em-pirical parameterizations of wind mass loss. For a more exten-sive description of
SeBa we refer to Portegies Zwart & Verbunt(1996), Nelemans et al. (2001), and Toonen et al. (2012).As is common, we use parameterizations to describe thecommon-envelope process. In this study, the density profile ofthe donor envelope is parametrized by λ = /
2, the e ffi ciencywith which orbital energy is used on unbinding the common en-velope by α CE = γ = / α CE λ = γ - and α CE -prescriptions areused as described in Toonen et al. (2012). A metallicity Z = . − . Because these parameters arevery uncertain, in Sect. 4 we will consider the e ff ect of varyingthe common-envelope parameters and the supernova kick veloc-ity distribution from our standard values. The star formation history of the Galactic Bulge can be approx-imated by a Gaussian distribution with a mean µ = −
10 Gyr anda standard deviation σ = . − . × M (cid:12) (Clarkson & Rich 2009; − − − − − − − S t a rf o r m a ti on r a t e ( M (cid:12) y r − ) σ = σ = Figure 1.
Star formation history of the Galactic Bulge as aGaussian distribution for mean µ = −
10 Gyr and two valuesof the standard deviation σ . The total mass of the stars formed is1 × M (cid:12) . Time = = −
13 Gyr.Wyse 2009). Star formation is assumed to start 13 Gyr beforepresent. For a narrow distribution the star formation is concen-trated around 10 Gyr in the past, but the σ = . σ = . σ = . × M (cid:12) of stellar mass is found tocontain 5 . × binary systems. We simulated 1 million bina-ries with a lower primary mass limit of 4 M (cid:12) (because systemswith a lower primary mass do not produce a supernova event ineither component), and another 4 . M (cid:12) , after it became clear that systemswith lower primary masses do not produce UCXBs. Using thebinary initial mass function we calculated to how many binariesin the full mass range of primaries (0 . − M (cid:12) ) this simula-tion corresponds. The resulting population was then multipliedby a factor (of 14 .
45) to scale to the entire Bulge population.For an analytic derivation of the binary initial mass functionwe refer to Appendix A.
We consider three UCXB-progenitor classes, each defined by thestellar type of the donor at the time it fills, or will fill, its Rochelobe:
Class 1.
White dwarf with a neutron star or black hole compan-ion (Tutukov & Yungelson 1993; Iben et al. 1995; Yungelsonet al. 2002),
Class 2.
Helium burning star with a neutron star or blackhole companion (Savonije et al. 1986; Iben & Tutukov 1987;Yungelson 2008),
Class 3.
Evolved main sequence star of about 1 M (cid:12) with a neu-
2. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge tron star or black hole companion (Tutukov et al. 1985; Nelsonet al. 1986; Fedorova & Ergma 1989; Pylyser & Savonije 1989;Podsiadlowski et al. 2002; Nelson & Rappaport 2003; van derSluys et al. 2005a; Lin et al. 2011).These classes include all the binary systems that may even-tually evolve into an UCXB (Belczynski & Taam 2004a; van derSluys et al. 2005a; Nelemans et al. 2010) – in some models in-volving accretion-induced collapse of a white dwarf or a neutronstar, the donor star has already transferred mass before the for-mation of the eventual accretor, a neutron star or a black hole,respectively. The detailed tracks follow the evolution of the he-lium burning donor starting immediately after its formation, andthe main sequence donor immediately after the supernova event.The white dwarf donor tracks start at the onset of Roche-lobeoverflow. In each of the detailed tracks the mass transfer is con-servative as long as the mass transfer rate does not exceed theEddington limit. If the mass transfer is faster than that, accre-tion at the Eddington limit is assumed, and the mass that is lostfrom the system carries the specific angular momentum of theaccretor.The initial system parameters (component masses and or-bital periods) and major events during the evolution towards anUCXB are described below for each class.
The evolution of UCXBs with a white dwarf donor can be di-vided into two main categories based on whether the primary(initially more massive star) or secondary component becomesa supernova. A supernova explosion of the secondary star ispossible when it gains mass by hydrogen accretion from theprimary (e.g. Tutukov & Yungelson 1993; Portegies Zwart &Verbunt 1996; van Kerkwijk & Kulkarni 1999; Portegies Zwart& Yungelson 1999; Tauris & Sennels 2000). A supernova ex-plosion of the secondary probably never produces a black hole,neither does the primary turn into a helium white dwarf after thesupernova, because it starts out too massive. Thus, all secondary-supernova systems have a carbon-oxygen or oxygen-neon whitedwarf donor and a neutron star accretor. Because the high stel-lar mass required for a supernova explosion of the primary isrelatively rare due to the steep initial mass function, in our simu-lations a significant fraction of the systems (13% of the carbon-oxygen white dwarf systems and 36% of the oxygen-neon whitedwarf systems) experience their supernova in the secondary star.Systems with a black hole accretor are rare, about 0 .
2% of allwhite dwarf systems. All black holes form from the primary andhave a ∼ . − . M (cid:12) carbon-oxygen white dwarf companion.Carbon-oxygen white dwarf systems are 1 . ∼
30 times more prevalent than helium white dwarfsystems (combining primary and secondary supernovae).
Supernova explosion of the primary
This category can be sub-divided by the predominant white dwarf composition: helium,carbon-oxygen or oxygen-neon.Evolution starts with a zero-age main sequence binary inwhich the primary is a massive star ( M (cid:38) M (cid:12) if the sec-ondary is to become a helium or carbon-oxygen white dwarf,and M (cid:38) M (cid:12) in the case of an oxygen-neon white dwarfcompanion) that evolves o ff the main sequence first. Systems The envelopes of some white dwarfs contain helium, but we referto them as carbon-oxygen white dwarfs because the envelope will belost relatively early. that eventually produce a helium white dwarf donor have ini-tial orbital periods ranging mainly from 1 to 100 yr. For sys-tems that produce a carbon-oxygen or oxygen-neon white dwarfdonor the orbital periods lie mostly between 0 . . . M (cid:12) , whereas for carbon-oxygen white dwarf donors thisrange is 2 . − M (cid:12) , where 2 . M (cid:12) is the maximum mass of singlestars that undergo the helium flash. A small fraction started witha higher initial mass. Progenitors of oxygen-neon white dwarfsecondaries have a mass between 7 and 11 M (cid:12) on the zero-agemain sequence and do not become a supernova due to severemass loss (e.g. Gil-Pons & Garc´ıa-Berro 2001). (There is someoverlap with the progenitor mass range of carbon-oxygen whitedwarfs – the end product depends on whether burning stops be-fore of after carbon ignition.) After the supernova explosion hasoccurred, the secondary evolves o ff the main sequence. As asubgiant, it initiates a common envelope with the neutron star,shrinking the orbit by another factor of a few tens. The corecools into a helium white dwarf ( (cid:46) . M (cid:12) ) or, after a heliumburning and helium giant stage, a carbon-oxygen white dwarf(0 . − . M (cid:12) ), or even, after carbon burning, an oxygen-neonwhite dwarf ( (cid:38) . M (cid:12) , Gil-Pons & Garc´ıa-Berro 2001). Orbitalangular momentum loss via gravitational wave radiation furthershrinks the orbit until the white dwarf eventually overfills itsRoche lobe, which happens at an orbital period of a few min-utes. Supernova explosion of the secondary
In this scenario, the to-tal binary mass needs to be at least 9 M (cid:12) if the primary becomesa carbon-oxygen white dwarf and 12 M (cid:12) if the primary becomesan oxygen-neon white dwarf. The primary transfers several so-lar masses to the secondary in a stable manner (avoiding a com-mon envelope – initially the secondary must have a mass of atleast 0 .
55 times the primary mass) while ascending the red giantbranch (e.g. Tauris & Sennels 2000). Eventually the core be-comes a helium star, or a carbon-oxygen or oxygen-neon whitedwarf. The secondary, which becomes the more massive compo-nent of the system, evolves o ff the main sequence and initiatesa common envelope. The orbit shrinks, and 30 −
70 Myr afterthe binary formation the secondary explodes as a supernova andproduces a neutron star. In systems which remain bound after thesupernova explosion, the primary will eventually reach Roche-lobe overflow as a relatively massive ( (cid:38) . M (cid:12) ) carbon-oxygenwhite dwarf or an oxygen-neon white dwarf ( (cid:38) . M (cid:12) ). Therelatively high initial mass of the primary precludes less mas-sive carbon-oxygen white dwarfs. The initial orbital period inthis scenario can be much shorter than in the primary-supernovascenarios, down to a few days. The initial stellar masses lie be-
3. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge tween 4 . ∼ M (cid:12) for the primary and 4 − M (cid:12) for thesecondary (if the former becomes a carbon-oxygen white dwarf)and about 5 − M (cid:12) for both the primary and secondary (if theformer becomes an oxygen-neon white dwarf). The onset of mass transfer from the white dwarf
Most whitedwarf–neutron star systems merge upon the onset of mass trans-fer. For a 1 . M (cid:12) neutron star companion, white dwarfs with amass higher than ∼ . M (cid:12) experience dynamically unstablemass transfer, assuming a zero-temperature (completely degen-erate) mass-radius relation for the donor (e.g. Yungelson et al.2002; van Haaften et al. 2012b). This assumption implies thatthese white dwarfs have cooled considerably by the time theyeventually fill their Roche lobe, although tidal heating and ir-radiation before the onset of mass transfer may counteract thisfor a short time. This leads to runaway mass loss on the dy-namical timescale of the donor, followed by accretion of partof the disrupted white dwarf via a disk around the neutron star(see e.g. van den Heuvel & Bonsema 1984; Fryer et al. 1999;Paschalidis et al. 2011; Metzger 2012). Furthermore, in systemswith a donor mass larger than ∼ . M (cid:12) (Yungelson et al. 2002;van Haaften et al. 2012b) (this value is only weakly sensitive toaccretor mass) the accretor will be unable to eject enough trans-ferred matter from the system by isotropic re-emission, wheremost arriving matter leaves the vicinity of the accretor in a fast,isotropic wind powered by accretion (Soberman et al. 1997;Tauris & Savonije 1999). This also leads to a merger. Therefore,systems that are unstable due to either a dynamical instabilityor insu ffi cient isotropic re-emission have been removed from thesample. These instabilities only occur in systems with a whitedwarf donor, because of the negative donor mass-radius expo-nent and the small donor size (hence, small orbit) at the onsetof mass transfer. Dynamical instabilities may occur in systemswith helium or main sequence donors if they have masses con-siderably higher than considered in this study (see e.g. Pols &Marinus 1994).In our simulation 97 .
4% of all white dwarf systems have adonor with a mass higher than 0 . M (cid:12) and do not survive theonset of mass transfer. This includes 99 .
1% of carbon-oxygen(solid line in Fig. 2) and all oxygen-neon (dashed line in Fig. 2)white dwarf systems. In about 80% of the surviving white dwarfdonor systems, the donor is a helium white dwarf (dotted line inFig. 2), in the remainder it is a carbon-oxygen white dwarf. Allsurviving systems experienced the supernova explosion in theprimary star and host a neutron star.If a white dwarf donor with a mass higher than the 0 . M (cid:12) isotropic re-emission limit has a non-degenerate surface layer,the system may not merge immediately upon the onset of masstransfer, but it will once this layer has been lost.Two thirds of the white dwarf donor systems start to trans-fer mass to the neutron star within 2 Gyr, but some systems takemuch longer (Fig. 3). This is the case for all white dwarf types.White dwarfs can take very long to start mass transfer dependingon the width of the initial orbit, since the orbital decay of bina-ries consisting of a neutron star and a white dwarf is caused bygravitational wave radiation only. Evolutionary tracks
For each donor composition, the evolutionafter the stage described in Sect. 2.3.1 follows the tracks de- If the white dwarfs are warm, their radius is larger at a given mass(Bildsten 2002; Deloye & Bildsten 2003) and therefore the orbital peri-ods we find are strictly lower limits. M (cid:12) )01020304050 N u m b e r o f s y s t e m s ( × ) Figure 2.
Total time-integrated number of UCXBs with a helium(dotted line), carbon-oxygen (solid line) or oxygen-neon (dashedline) white dwarf donor at the onset of mass transfer to a neutronstar (including merging systems). R a t e o f s y s t e m ss t a r ti ng R L O F ( M y r − ) Figure 3.
Delay time distribution between the zero-age mainsequence (ZAMS) and the onset of mass transfer to a neutronstar for UCXBs with a white dwarf (excluding merging systems,solid line), helium star (dashed line) and main sequence (dottedline) donor.scribed in van Haaften et al. (2012b). Initially, the white dwarfdonor has not yet cooled and therefore is larger than a zero-temperature white dwarf of the same mass. While the donorloses mass, its radius is held constant until it equals the zero-temperature radius of the same mass (this is justified by the rapidmass loss the donor initially experiences). From this point on, thezero-temperature radius (Zapolsky & Salpeter 1969; Rappaportet al. 1987) is used, which increases with further mass loss. Theinitial neutron star mass is taken to be 1 . M (cid:12) and its radius 12km (Guillot et al. 2011; Steiner et al. 2013). The evolution ofUCXBs with degenerate donor stars is governed by angular mo-mentum loss through gravitational wave radiation, which forcesmass transfer via Roche-lobe overflow.
4. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge − − − − − − − − l og - M a ss t r a n s f e rr a t e ( M (cid:12) y r − ) Figure 4.
Sample of helium star–neutron star UCXB tracks, forinitial helium star mass range 0 . − . M (cid:12) and initial orbitalperiod range 20 −
200 min by Nelemans et al. (2010) (solid lines),and our white dwarf donor track extensions (dashed lines).
The supernova in low-mass helium burning star systems occursin the primary, which has an initial mass M (cid:38) M (cid:12) . Most sys-tems start out with an orbital period between 0 . . − M (cid:12) experience hydrogen shell burning (Savonije et al.1986) and lose their hydrogen envelope in case B mass trans-fer (Kippenhahn & Weigert 1967). They fill their Roche lobeswithin another ∼
200 Myr, which is much earlier than UCXBswith white dwarf donors. In part this is due to the requirementthat the helium star has not yet turned into a white dwarf beforethe onset of mass transfer (thereby disqualifying itself from thehelium burning donor sample) which constrains the size of theinitial orbit. We do not find systems with a black hole accretor.We have used stellar evolutionary tracks for systems withan 0 . − . M (cid:12) helium star and a 1 . M (cid:12) neutron star at ini-tial orbital periods between 20 and 200 min (Nelemans et al.2010, table in electronic article), part of which is shown in Fig. 4.These tracks were made in the same way as the tracks for sys-tems with white dwarf accretors in Yungelson (2008). The donormetallicity Z = .
02. Because the donors at the end of thetracks are degenerate, we have extended these tracks by makinga smooth transition to the zero-temperature white dwarf evolu-tion. The tracks describe the orbital period, mass transfer rate,donor mass, and core and surface compositions as a function oftime. In Fig. 4, the helium stars live up to 400 Myr. After the on-set of mass transfer (vertical part of the tracks), the orbits shrinkuntil the period minimums, then expand towards the bottom rightof the figure. For each individual ‘zero-age’ UCXB system pro-duced by
SeBa , the track that best matches its donor mass andorbital period has been used. The initial orbital period is the period directly after completionof the common-envelope phase that leaves behind the helium star(Yungelson 2008).
Main sequence donors have mostly evolved from 1 . − . M (cid:12) secondaries that started transferring mass after orbital decay dueto magnetic braking (van der Sluys et al. 2005a). After the mag-netic field disappears (because the star becomes fully convectiveas a result of mass loss), gravitational wave radiation becomesthe dominating angular momentum loss mechanism, continuingthe orbital shrinking. In this scenario, the initial period and donormass need to fall within relatively narrow ranges in order to suf-ficiently evolve the main sequence star. Moreover, the magneticbraking must be su ffi ciently e ffi cient (van der Sluys et al. 2005b)which it probably is not (Queloz et al. 1998). Depending on theextent of hydrogen depletion in the stellar center, systems canreach a minimum orbital period between 10 and 80 min, where ∼
80 min is the lower limit for hydrogen-rich donors (Paczy´nski1981).We have used stellar evolutionary tracks by van der Sluyset al. (2005a) for binaries with an 0 . − . M (cid:12) main sequencestar and a 1 . M (cid:12) neutron star at initial orbital periods between0 .
50 and 2 .
75 days. The donor metallicity Z = .
01. These tracksdescribe the orbital period and mass transfer rate as a functionof time, as well as the core and surface compositions.
Figure 4 suggests that once the donor has become degenerate,UCXBs ‘uneventfully’ reach long orbital periods and very lowmass transfer rates. This is probably not the case. Instead, at lowmass transfer rate a thermal-viscous instability in the accretiondisk (Osaki 1974; Lasota 2001) can cause UCXBs with a suf-ficiently low mass transfer rate to become transient (Deloye &Bildsten 2003). This usually implies that these systems are vis-ible only during outbursts when the disk is in a hot state, whichis only a small fraction of the time, and not during the quies-cent state when the disk is cold and gaining mass. Furthermore,due to accretion of angular momentum, a neutron star accretorin an UCXB can be recycled to a spin period between one and afew ms (Bisnovatyi-Kogan & Komberg 1974; Alpar et al. 1982;Radhakrishnan & Srinivasan 1982). Combined with a low masstransfer rate, the magnetosphere may transfer angular momen-tum from the neutron star to the accretion disk, thereby accel-erating orbiting disk matter and counteracting accretion, knownas the ‘propeller e ff ect’ (Davidson & Ostriker 1973; Illarionov& Sunyaev 1975). As a result, the inner accretion disk, sourceof most X-ray radiation, can become disrupted by the magneto-sphere. See van Haaften et al. (2012b) for more details on thethermal-viscous disk instability and propeller e ff ect in UCXBs.Finally, at low donor mass, high-energy radiation from the neu-tron star, the magnetosphere and the accretion disk may evapo-rate the donor, or detach it from its Roche lobe (Klu´zniak et al.1988; van den Heuvel & van Paradijs 1988; Ruderman et al.1989a; Rasio et al. 1989). Hot, low-mass donors may su ff er froma dynamical instability caused by a minimum value of their massin the case of a constant core temperature (Bildsten 2002).Each of these mechanisms can potentially diminish the visi-bility of UCXBs. Because it is impossible to precisely quantifyat which stage of the evolution (if at all) these mechanisms be-come important, and to what degree, we do not remove UCXBsfrom the sample, instead we will discuss the implications on thepopulation of old UCXBs in the Discussion (Sect. 4).
5. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge
The present-day number and system parameters of UCXBs inthe Galactic Bulge can be found by evaluating the evolutionarystage of all simulated systems at the present time. The most in-teresting parameters are the orbital period, mass transfer rate andsurface composition, because these can be inferred from obser-vations. The orbital periods of observed systems can be foundfrom periodic modulations in the light curve or spectrum, al-though this is usually very di ffi cult for UCXBs (e.g. Nelemanset al. 2006; in’t Zand et al. 2007). Since the transferred matteroriginates from the surface of the donor, the occurrence and rel-ative abundance of elements in the donor can be inferred fromX-ray (Schulz et al. 2001), ultraviolet (Homer et al. 2002), andoptical (Nelemans et al. 2004) spectra, and more indirectly, type-I X-ray bursts (in’t Zand et al. 2005). The mass transfer rate can-not be directly determined observationally. However, becausethe energy output of an X-ray binary is for a large part providedby the gravitational energy release of the accreted matter, themass transfer rate strongly influences the luminosity of the sys-tem, which can be observed. We employ two methods of converting the modeled mass trans-fer rate to bolometric luminosity.An observational method is to match the modeled systems toreal systems and assume that the modeled system behaves simi-lar to the real system in terms of emission. We match a modeledUCXB to the real UCXB with the nearest orbital period. Therelevant parameter of the emission behavior is the fraction ofthe time a source radiates at a given bolometric luminosity, mea-sured over a su ffi ciently long timespan. We use 16-year obser-vations by the RXTE
ASM to determine this behavior for the 14known UCXBs (including two candidates) for which ASM datais available. Figure 5 shows this behavior for sources when theyare visible well above the noise level (van Haaften et al. 2012c).ASM X-ray luminosity was converted to bolometric luminosityusing an estimate by in’t Zand et al. (2007). At a given time,the luminosity of an UCXB is randomly drawn from either theindividual ASM observations that make up this time-luminositycurve, or (most of the time) from the faint-end extrapolations ofthe curves in Fig. 5 (van Haaften et al. 2012c). These extrapola-tions are constructed in such a way that the average luminosityof the luminosity distribution is equal to the time-averaged lu-minosity of the source as observed by the ASM. The amount oftime that a given source spends at a particular luminosity trans-lates into the number of sources in a population at the same lu-minosity.The second method, of a more theoretical nature, is to con-vert a system’s modeled mass transfer rate to luminosity, usingpredictions by the disk instability model (Sect. 2.4) in the caseof long-period UCXBs. According to this model the mass trans-fer rate must exceed a critical value in order to be stable andthe source to be persistent, i.e., visible at a relatively high lu-minosity (almost) all the time. A crude estimate for the criticalmass transfer rate in the case of an irradiated disk is given by in’tZand et al. (2007), based on Dubus et al. (1999); Lasota (2001);Menou et al. (2002)˙ M crit ≈ . × − f (cid:32) M a M (cid:12) (cid:33) . (cid:18) P orb h (cid:19) . M (cid:12) yr − (1) . . . . . . Estimated bolometric luminosity (erg s − )10 − − − − − F r ac ti ono f ti m e Orbital period (min)
Figure 5.
UCXB variability: fraction of time that a source emitsabove a given luminosity for 14 UCXBs, including two candi-dates with tentative orbital periods, adapted from van Haaftenet al. (2012c). The numbers associated with the curves indicatethe orbital periods in minutes.with M a the accretor mass, P orb the orbital period, and f is afactor accounting for the disk composition; f ≈ f ≈ L is assumed to be constantat L = GM a ˙ M a R a , (2)with G the gravitational constant and R a the accretor radius.Sources with a time-averaged mass transfer rate below thecritical value are assumed to be visible only during outburststages. The predictions by the thermal-viscous disk instabilitymodel regarding the degree of variability of sources is supportedby observations (van Paradijs 1996; Ramsay et al. 2012; Coriatet al. 2012). The duty cycle (fraction of the time the source is inoutburst) is DC = L avg L outburst , (3)where L avg is the time-averaged luminosity based on the theo-retical mass transfer rate, (Eq. 2), and L outburst is the luminosityduring outburst, derived by Lasota et al. (2008) L outburst ≈ . × (cid:18) P orb h (cid:19) . erg s − , (4)which is consistent with observations of outbursts in UCXBs(e.g. Wu et al. 2010). The period of this cycle is not relevanthere. We neglect the decay in the light curve after an outburst.Furthermore, we do not predict the luminosity of systems thatare in quiescence, which in fact has been assumed to be zero inthe above method.Both methods have advantages and shortcomings. The ASMobservations have a rather high lower limit in converted bolo-metric luminosity, of ∼ erg s − at 8 .
6. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge model method. The ASM data show that UCXBs with a simi-lar orbital period can behave rather di ff erently, for instance XTEJ1751–305 (42 . . . . . . The donor surface compositions of the modeled UCXBs are pre-dicted using the helium-star donor and main sequence donortracks, as well as the white dwarf types from the population syn-thesis model. These predicted compositions can be comparedto observations of real systems, in the Bulge and elsewhere.Donors that start mass transfer as a white dwarf can be heliumand carbon-oxygen white dwarfs (Sect. 2.3.1). In the latter casewe assume 30% carbon and 70% oxygen by mass, based onthe most common eventual compositions in the helium burningdonors (Sect. 3.4). Donors that start mass transfer as a heliumburning star can also produce helium-carbon-oxygen donors dueto an interrupted helium burning stage. For the subsequent tracksfor these systems, we use the mass-radius relation for degen-erate donors composed of a mixture of 60% helium, 30% car-bon and 10% oxygen, a choice based on the dominant tracks byNelemans et al. (2010) as will be discussed in Sect. 3.4. We notethat the degenerate tracks are not very sensitive to the composi-tion (as long as there is no hydrogen), so these simplifications arejustified. Matter processed in the CNO cycle has a high nitrogen-to-carbon abundance ratio, whereas helium burning converts thisto a low ratio. Consequently, the nitrogen-to-oxygen ratio is agood test for the formation channel because it can discriminatebetween a history as a helium white dwarf donor or a heliumburning donor (Nelemans et al. 2010).Based on the overview in van Haaften et al. (2012c), thecompositions of observed UCXBs can be summarized as be-ing roughly equally distributed over helium and carbon-oxygencompositions. There is no clear dependency on the orbital pe-riod, although helium composition may be more common amongsystems with a long orbital period ( (cid:38)
40 min) (van Haaften et al.2012c). The surface composition of very low-mass donors cor-responds to the (inner) core composition of the object before itstarted transferring mass.
3. Results
Convolving the star formation history (Fig. 1) with the delaytimes of the onset of mass transfer (Fig. 3) yields the birth rate − − − − − − − R a t e o f s y s t e m ss t a r ti ng R L O F ( M y r − ) WD, σ = . σ = . σ = . σ = . σ = . σ = . Figure 6.
Birth rate of systems reaching Roche-lobe overflowagainst time for UCXBs from the white dwarf donor channel(solid lines), helium-star donor channel (dashed lines) and mainsequence donor channel (dotted lines). Time = σ = . σ = . σ = . σ = .
5, these percentages are 97%, 100%and 77%, respectively. The main distinguishing feature betweenthe three classes (Sect. 2.3) is the most recent time at which masstransfer can begin. Initially wide systems from the white dwarfdonor channel can still start mass transfer at the present, whereasmain sequence donor systems and especially helium-star donorsystems cannot, unless they have formed relatively recently (starformation history width σ = . ffi ciently degenerate following the extinction ofnuclear fusion (this happens some time after the period mini-mum). The orbital period decreases in the case of helium burn-ing or main sequence donors, whereas the period increases withmass loss for systems with degenerate donors (in each channel).If main sequence donor systems become ultracompact, this typ-ically happens ∼ −
7. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge − − − − − − − N u m b e r o f U C X B s all WD, σ = . σ = . σ = . Figure 7.
Number of UCXBs from the white dwarf donor chan-nel (solid lines), helium-star donor channel (dashed lines) andmain sequence donor channel (dotted lines). Time = σ = . ff erent classes decline at di ff erent rates correspondingto their respective recent birth rates (Fig. 6). While the evolution of the population is interesting in itself, thepopulation today can be used to validate the results. In the caseof a star formation history distribution width σ = . ∼
80 min. Since evolution slows down atlonger periods, systems tend to ‘pile up’. Di ff erences in donorcomposition lead to di ff erent present-day orbital periods. This isthe case even among hydrogen-deficient compositions becauseduring most of the evolution, the donor mass is low enough forCoulomb physics to be important to the stellar structure, or evendominating degeneracy pressure. Coulomb interactions cause adonor that is composed of ‘heavy’ elements such as carbon andoxygen to have a smaller radius than donors with lighter compo-sition, such as helium, of the same mass (Zapolsky & Salpeter1969). A larger donor radius (at each mass) results in a longerorbital period at each mass, but also at each age (because lesstime is spent at a given orbital period). For σ = . σ = . For a donor mass-radius exponent ζ , the number of UCXBs at agiven period N ∝ P (11 / − ζ ) / (1 − ζ )orb (Deloye & Bildsten 2003). However, more time is spent at a given donor mass. F r e qu e n c y ( m i n − ) HeCO CO HeCOHe
Figure 8.
Present-day orbital period distribution for UCXBsfrom the white dwarf donor channel (solid lines), helium-stardonor channel (dashed lines) and main sequence donor channel(dotted lines). The elements next to the lines indicate the mostabundant element(s) at the surface of the donor, hence in thetransferred matter. The star formation history has width σ = . F r e qu e n c y ( m i n − ) HeCO CO HeCOHe
Figure 9.
Same as Fig. 8 except σ = . ff at the long-period end of several curves is due to the assumptionthat star formation suddenly starts 13 Gyr before present.Combining all donor compositions, the result is a current ∼ . × population of UCXBs, mostly at long orbital pe-riod (60 −
90 min). The total number of UCXBs in each class is3 . × (18%) with white dwarf donors, 1 . × (81%) withhelium-star donors, and 5 . × (0 . . × for σ = . . × for σ = . .
8% and 3 .
8% of the population, respectively).We note that these numbers are rather sensitive to assumptionsin the model, and could be lower by an order of magnitude, aswill be discussed in Sect. 4.
8. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge . . . . . . . . Bolometric luminosity (erg s − )0.010.1110 N u m b e r o f b r i gh t U C X B s Figure 10.
Present-day luminosity distribution of UCXBs inthe Bulge based on
Rossi XTE
All-Sky Monitor observations,after incorporating the accelerated evolution of the systems(Sect. 3.3.1). The star formation history has width σ = . σ = . As described in Sect. 2.5.1, in order to determine what we canobserve at high luminosity we have to convert the modeled pop-ulation to luminosities.
RXTE
All-Sky Monitor
In the first method, we apply the observations of known UCXBsby the
RXTE
ASM (Fig. 5, Sect. 2.5.1) to the modeled popula-tion (Figs. 8 and 9, Sect. 3.2). Modeled UCXBs with an orbitalperiod longer than 60 min are left out because of the absence ofknown real systems with such periods (i.e., they are assumednever to reach luminosities above ∼ erg s − ). The time-averaged luminosity of most UCXBs with orbital periods longerthan 40 min is approximately two orders of magnitude higherthan expected from the gravitational-wave model (van Haaftenet al. 2012c). This implies that either the observed sources areatypically bright, or that they show normal behavior, but evolvemuch faster than if driven only by gravitational wave losses (theimplications will be discussed in Sect. 4.2). In each case, dueto energy conservation, we need to reduce the number of brightsources at each orbital period by a factor that corresponds to theratio between the gravitational-wave luminosity and the actualobserved luminosity, given by van Haaften et al. (2012c, theirFig. 3). Figure 10 shows the resulting number of bright UCXBspredicted by the ASM data.For star formation history width σ = . σ = . ∼
80, because recent starformation causes more young systems to exist, which have notyet reached orbital periods of 60 min. The cut-o ff at the faintend of the histogram is an artifact of the assumed linear extrap-olation to the faint behavior. This also results in a relatively highminimum luminosity. In reality, especially sources with orbitalperiods longer than ∼
40 min are expected to be very faint (i.e.,much fainter than suggested by a linear extrapolation) at leastsome fraction of the time, which means the cumulative luminos-ity distribution flattens at faint luminosities, causing a tail at the . . . . . . . . Bolometric luminosity (erg s − )0.010.1110 N u m b e r o f b r i gh t U C X B s HeCO COHeCO
Figure 11.
Present-day luminosity distribution of UCXBs in theBulge based on the disk instability model (Sect. 2.5.1). Linecolor distinguishes between helium (thick black lines), carbon-oxygen (thick gray and thin black lines) and helium-carbon-oxygen (thin gray lines) donor compositions. The thick lines cor-respond to the white dwarf donor channel, the thin lines to thehelium burning channel. The star formation history has width σ = . σ = . The second method relies on converting theoretical masstransfer rates to luminosities using the disk instability model(Sect. 2.5.1), the result of which is shown in Figs. 11 and 12. Thetotal number of bright ( (cid:38) erg s − ) sources (either persistentor in outburst) is 34 for σ = . σ = . −
80 min).Figure 11 shows the luminosity distribution of brightsources. UCXBs with a luminosity between ∼ − erg s − are the ones with the shortest orbital periods, below ∼
20 min.From here, the number of sources at a given luminosity in-creases towards fainter luminosities because these sources havelonger period derivatives and lower time-averaged mass transferrates, until a sharp cut-o ff defined by the longest orbital periodat which sources are still considered persistent. Carbon-oxygendominated sources are persistent to longer periods and lower lu-minosities because accretion disks composed of carbon-oxygenare more stable than helium dominated disks, see Eq. (1). For σ = . (cid:38)
40 min), the duty cycle (Eq. 3)determines the number of sources in outburst. The duty cycledecreases below 10 − at orbital periods longer than 60 −
70 min,depending on donor composition. During their rare outbursts,sources are temporarily bright at ∼ − erg s − , as followsfrom Eq. (4). Their peak luminosities are used in Fig. 11.
9. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge N u m b e r o f b r i gh t U C X B s HeCO COHeCO
Figure 12.
Orbital period distribution of the predicted brightpopulation of UCXBs in the Bulge at the present based on thedisk instability model. For details see Fig. 11.Two peaks can be distinguished in the lines representingthe white dwarf channel in Fig. 11 (thick lines). The peak at ∼ erg s − consists of systems with orbital periods just longerthan the critical period because these still have a relatively highduty cycle. The second peak at ∼ . erg s − consists of long-period systems because these are very numerous and distributedover a relatively narrow interval of orbital periods. The duty cy-cle at a given orbital period is higher for helium burning systemsowing to their larger size and, because their average density isset by the orbital period, correspondingly larger mass. Hence,their time-averaged mass transfer rate at a given orbital period isalso higher.In Fig. 12 the orbital period distribution is shown for thesame population of bright sources as in Fig. 11. The jumps ofthese distributions correspond to the respective low-luminosityends of the distribution in Fig. 11. The cut-o ff period of persis-tent sources (at 30 −
40 min) lies at a longer orbital period for thesystems with a helium burning donor origin compared with sys-tems with a white dwarf origin. The reason is that these donorshave a higher temperature than originally white dwarf donors,and therefore the time-averaged mass transfer rate is higher atthe same period. This causes the disk to remain stable (and thesources to be persistent) up to a longer period. Again we seethat carbon-oxygen donor systems are persistent up to a longerorbital period than helium-dominated donor systems. Transientsystems with orbital periods longer than ∼
40 min are rarely inoutburst and at most a handful have a high luminosity at a giventime.
The helium-star donors have partially turned into carbon-oxygenwhite dwarfs during their evolution, depending on their massand evolutionary stage at the onset of Roche-lobe overflow (de-termined by the initial mass and orbital period). When the starstarts mass transfer after filling its Roche lobe, burning is ex-tinguished quickly (Savonije et al. 1986), and at this stage thecore mass fraction of helium varies between a few percent to al-most 100% (Nelemans et al. 2010). Figure 13 shows the surfaceabundances at the present day, assuming a narrow star forma-tion history ( σ = . . . . . . . Y . . . . . . C a r bon , oxyg e n s u rf ace m a ss fr ac ti on Figure 13.
Surface abundances of helium versus carbon (blacksquares) and oxygen (white triangles) for donors in the helium-star channel at the present time in the case of a su ffi ciently nar-row star formation history. These correspond to the core abun-dances at the end of the tracks by Nelemans et al. (2010). Thesurface area of a symbol is proportional to the number of systemsin the corresponding track.with less than 10% helium on their surface. Systems that startedout with a short orbital period generally have a higher heliummass fraction, because these had less time to burn helium be-fore the onset of mass transfer. The abundances depend on thetemperature at which helium and carbon burning takes place. Ahigher temperature causes a higher helium burning rate, produc-ing more carbon. Later, the carbon abundance reduces in favorof oxygen. The scatter in Fig. 13 is therefore due to di ff erencesin core burning temperature caused by di ff erent stellar masses.UCXBs produced via the white dwarf donor channel havedonors mostly composed of either helium material (producedin the CNO cycle), or carbon-oxygen. The ratio between bothtypes is about 3:1 but strongly depends on the e ffi ciency ofisotropic re-emission, which strongly a ff ects the number ofcarbon-oxygen white dwarf donor systems that survive the onsetof mass transfer (Sect. 2.3.1).Evolved main sequence donors with an initial mass of 1 . − . M (cid:12) reach a helium surface abundance Y ≈ . Z = .
01, which has been present from the start (vander Sluys et al. 2005a).Because the core material is exposed early on (e.g., at anorbital period shorter than ∼
20 min for the helium burning sys-tems, Nelemans et al. 2010), and the core is homogeneous dueto convection during its burning stages, the chemical composi-tion is expected not to change with increasing orbital period, andtherefore the same for the total and the observable population.
Even though variability behavior determines the number ofUCXBs in outburst and their luminosity, the collective luminos-ity of all UCXBs at a given orbital period is in principle not de-pendent on variability, because the time-averaged mass transferrate of an UCXB is a relatively straightforward function of or-bital period. During the evolution of an UCXB, its evolutionarytimescale increases with age and orbital period. This means that
10. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge l og E n e r gy r e l ea s e p e r un it o r b it a l p e r i od ( e r g m i n − ) Neutron star accretor Black hole accretorNeutron star accretor Black hole accretor
Figure 14.
Energy released per unit interval of orbital period foran UCXB with a zero-temperature helium white dwarf donorand a 1 . M (cid:12) neutron star accretor or a 10 M (cid:12) black hole ac-cretor. Note that the quantity on the vertical axis should not beinterpreted as a luminosity; the time unit represents change inorbital period rather than passing time.there exist many more systems at longer orbital period. On theother hand the time-averaged mass transfer rate decreases withage and orbital period, so a long-period source has a lower time-averaged luminosity. The total energy output of a source, or ofthe population as a whole (under the assumption of a constantstar formation rate), at a given orbital period is an indication ofat which orbital periods systems are likely to be observed.The amount of energy emitted by an UCXB per unit orbitalperiod is given by d E d P orb = L ˙ P orb = − GM a R a d M d d P orb (5)where E is the emitted energy and M d the donor mass. This re-lation is illustrated in Fig. 14. The donor mass decreases muchfaster at shorter orbital periods ( M d ∝ P − . (van Haaften et al.2012b), i.e., d M d / d P orb ∝ P − . ), and since the donor mass is thefuel for the luminosity, systems emit much more energy at shortorbital periods, not only per time interval (their luminosity) butalso per orbital period interval. For instance, an UCXB will emit ∼
12 times as much energy during its evolution from 20 to 21 minas it does between 60 and 61 min. The consequence is that theshort-period systems dominate the collective X-ray output of anUCXB population, unless the star formation rate decreases veryfast. Depending on the variability of systems and the sensitivityof the instrument used, this could very well result in short-periodsystems dominating the visible population.
4. Discussion
We predict ∼ . × of UCXBs in the Galactic Bulge, predom-inantly at orbital periods of (cid:38)
70 min, but also a few thousandsystems with orbital periods shorter than 60 min (but mostlylonger than 40 min). Based on
RXTE
ASM observations, about Not to be confused with the known population, which is composedby individual observations at di ff erent times. In the known population,long-period systems will eventually dominate provided they can be seenduring outbursts Table 1.
Size of the modeled UCXB population in the GalacticBulge for di ff erent model parameters. α CE λ Kick distribution Number of UCXBs ( × )2 Paczy´nski 192 Maxwellian 140 . . −
80 of these sub-hour UCXBs should be visible at high lu-minosities of (cid:38) erg s − (Fig. 10), depending on the starformation history. Also, ∼ −
50 bright UCXBs with orbitalperiods (cid:46)
30 min (i.e., persistent) should be visible above sucha luminosity based on gravitational energy release and the diskinstability model (Figs. 11 and 12). The combined common-envelope parameter α CE λ for mas-sive stars may be lower than the value of 2 that we used (Voss& Tauris 2003). Decreasing this value to 0 .
2, as well as us-ing a Maxwellian kick velocity distribution with a dispersionof 450 km s − , rather than the distribution by Paczy´nski (1990),would reduce the number of UCXBs formed by a factor of ∼ ∼ × , and the number of brightsystems becomes ∼ −
10. Table 1 shows the number of UCXBsin our model for various combinations of common-envelope ef-ficiency and neutron star kick velocity distribution. Furthermore,the slope of the initial mass function at high stellar mass is alsouncertain. A steeper slope (resulting from earlier studies such asKroupa et al. 1993) leads to a smaller fraction of massive starsand therefore fewer UCXBs. A di ff erent choice for the initialcomponent mass pairing may also reduce the number of UCXBsby an order of magnitude (Belczynski & Taam 2004a).We can distinguish several disagreements with observa-tions. First, no UCXBs with orbital periods longer than 60 minhave been discovered, faint or bright, in the Bulge or else-where. Second, no bright UCXBs with a short orbital period( (cid:46)
30 min) have been identified in the Galactic Bulge. Third,only three UCXBs with orbital periods between 40 and 55min, XTE J1807–294 (Markwardt et al. 2003), XTE J1751–305(Markwardt et al. 2002) and SWIFT J1756.9–2508 (Krimm et al.2007), are presumably located in the Bulge, based on their posi-tions in the sky, as their distances are not known.As for the predicted ∼ . × long-period systems, theprobable existence of three observed UCXBs with orbital peri-ods shorter than 55 min in the Bulge can be used to calibrate theformation rate of UCXBs, independent of population synthesis .This yields a much larger number of UCXBs than three for sys-tems with a period longer than 55 min, based only on the rapidincrease of the evolutionary timescale (set by gravitational waveradiation) with orbital period. For instance, UCXBs are expectedto reach an orbital period of 55 min within ∼ The observed UCXBs with orbital periods around 20 min are notclearly detected by ASM most of the time, even though they are pre-dicted to be persistent by the disk instability model. Incorporating thefact that they are bright less than 100% of the time, the predicted numberof bright systems becomes lower, depending on the precise luminositydistribution of these systems. 11. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge sitive instruments at ∼ − erg s − (e.g. Bildsten & Rutledge2001; Heinke et al. 2003; Belczynski & Taam 2004b).The three observed UCXBs that are located in the directionof the Bulge have undetermined donor compositions, thoughSWIFT J1756.9–2508 is thought to have helium composition(Krimm et al. 2007). This cannot be used to constrain the starformation history, as UCXBs with helium white dwarf donorscan form with a wide range of delay times. Belczynski & Taam (2004a) performed a population synthesisstudy of (primordial) UCXBs in the Galactic disk. The main dif-ference between their and our results is that they found a totalof 478 UCXBs with orbital periods shorter than 80 min in thedisk at the present epoch, about three orders of magnitude fewerthan our result, per unit star forming mass. It is not clear whatcauses this discrepancy, although these authors use di ff erent ini-tial binary parameters than we do (Appendix A), for example asteeper high-mass slope in the initial mass function that leads tofewer massive stars, relatively. Also, their initial primary massesleading to UCXBs span a narrower range. Of all UCXBs in theirsimulation, about 20% have a black hole accretor, all of whichform via the accretion-induced collapse of a neutron star. Thispercentage strongly depends on the assumed upper mass of aneutron star (2 M (cid:12) ), the mass retention e ffi ciency of the accret-ing neutron star and the evolutionary stage at which the commonenvelope happens. As in our study, these authors did not find any(surviving) UCXBs with a black hole that was formed directly inthe collapse of a massive star. They found that about 90% of theneutron star accretors form via the accretion-induced collapse ofan oxygen-neon-magnesium white dwarf, a scenario our modeldoes not produce. In our simulations, 81% of the UCXBs startmass transfer from a helium burning donor, compared to 40% inBelczynski & Taam (2004a) for the Galactic field. This is not un-reasonable given the uncertainties in e.g. the onset of mass trans-fer from a white dwarf, and di ff erences in assumptions betweenboth studies. The number of persistent sources predicted by thedisk instability model depends sensitively on the orbital periodseparating the persistent and transient sources, because most ofthe predicted persistent sources have orbital periods only slightlyshorter than this critical period. Details in accretion disk models(e.g. X-ray irradiation) and composition can make a large dif-ference. We predict about 0 .
02% of the UCXBs to be persistent(Sect. 3.3.2), a much smaller fraction than found by Belczynski& Taam (2004a) (2 . −
50 from thedisk instability model, using our standard parameters) can alsobe compared with the number of persistent UCXBs with whitedwarf donors (600 − ff erence in adoptedcut-o ff donor mass for persistent behavior. These authors foundthat UCXBs with donor masses lower than 0 . M (cid:12) are transientwhereas our limit lies around 0 . M (cid:12) . Using their limit, our es-timate would reduce to roughly ten, which scales within a factorof a few with their number given the stellar mass ratio betweenBulge (1 × M (cid:12) ) and Galaxy (an additional 4 − × M (cid:12) in the Disk, Klypin et al. 2002). The overprediction of bright,persistent UCXBs is therefore not unique to our study.Recently, Zhu et al. (2012b) performed a population synthe-sis study of Galactic UCXBs with neutron star accretors, andpredicted 5 − × systems in the Galaxy, depending onneutron star birth kicks. As in our study, the helium burningdonor channel was the most common. Notable di ff erences withour work are that these authors found a large number of UCXBswith a carbon-oxygen white dwarf origin, and a peak in the or-bital period distribution near 40 min. An important clue towards what may happen at long periodscomes from the long-term ASM data. The reason for the dif-ference in predictions by the ASM and disk instability model(Sect. 3.3) lies in the ASM observations that the UCXBs withorbital periods 40 −
55 min are approximately two orders ofmagnitude brighter than theoretically expected from the time-averaged mass transfer rate, assuming mass transfer is drivenexclusively by gravitational wave radiation in a binary with a(semi-)degenerate donor (van Haaften et al. 2012c). Assumingthat the observed systems have been displaying normal behav-ior during the 16 years of
RXTE observations, additional angularmomentum loss besides that from gravitational wave radiationwould cause a higher mass transfer rate at the same orbital pe-riod, and therefore a higher time-averaged luminosity (see alsoRuderman et al. 1989b). As mentioned in Sect. 2.4, an e ffi cientphysical mechanism for additional loss of angular momentumfrom the system is a wind from the donor, induced by irradi-ation from the accretion disk or millisecond pulsar. In blackwidow systems, which host a millisecond pulsar and a low-mass( (cid:46) . M (cid:12) ) companion in a <
10 h orbit (King et al. 2005),such donor evaporation has been observed (e.g. Fruchter et al.1988). This scenario has also been proposed to be happening tothe unusually light ( ∼ − M (cid:12) ) detached companion to the mil-lisecond pulsar PSR J1719–1438 (Bailes et al. 2011) via eitherthe white dwarf or helium burning donor channels (van Haaftenet al. 2012a) or the evolved main sequence donor channel (withan orbital period minimum at ∼
45 min, Benvenuto et al. 2012),though the latter scenario does not produce a carbon-oxygen richdonor.The recently discovered spin-powered millisecond gamma-ray pulsar PSR J1311–3430 system (Pletsch et al. 2012), with anorbital period of 93 . ∼ ff ect (Sect. 2.4) Zhu et al. (2012a) found a much larger number of UCXBs with ahelium burning donor than our study, but this is expected given the largedi ff erence in recent star formation between Bulge and Disk. These ASM luminosities are direct observations, unrelated to theextrapolations of the curves shown in Fig. 5 and discussed in Sect. 3.3.12. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge is the most promising mechanism to explain this non-detection.The rotational energy of the millisecond pulsar is su ffi cient tomake long-period UCXBs with very low mass transfer rates( ∼ − M (cid:12) yr − ) much fainter, since it causes arriving matterto be unbound (van Haaften et al. 2012b). The propeller e ff ectcould still allow for a (very) low rate of accretion that wouldprevent radio emission and make the sources visible in the ul-traviolet (owing to their low disk temperatures and possibly dis-turbed inner accretion disks). Furthermore, radio emission froma millisecond pulsar itself, once switched on after an interrup-tion in mass transfer, is capable of preventing accretion (Burderiet al. 2001; Fu & Li 2011).Using the Chandra X-Ray Observatory (Weisskopf et al.2002), the Galactic Bulge Survey has found 1234 X-ray sourcesin 8 . (Jonker et al. 2011) so far, most of which have not yetbeen identified. Although many are expected to be foregroundCataclysmic Variables or non-ultracompact X-ray binaries, thisnumber of systems found in approximately 5% of the total areaof the Bulge on the sky is at least consistent with a large pop-ulation of faint X-ray binaries. Also, a potentially large popula-tion of sub-luminous X-ray transients with neutron star accretorsexists near the Galactic Center (Sakano et al. 2005; Wijnandset al. 2006; Degenaar & Wijnands 2009, 2010). These systemshave (intrinsic) peak luminosities near ∼ − erg s − (in the2 −
10 keV range), and may include UCXBs, although the diskinstability model predicts peak luminosities (cid:38) erg s − forUCXBs. King & Wijnands (2006) found that the luminositiesof some very faint X-ray transients imply mass transfer rates of ∼ − M (cid:12) yr − , which is consistent with the behavior of oldUCXBs.The additional angular momentum loss increases the timederivative of the orbital period, and as a result the actual numberof systems at long periods in our prediction based on only gravi-tational wave radiation (Figs. 8 and 9) should be reduced by twoorders of magnitude. Since UCXBs with an orbital period shorter than ∼
30 min areexpected to be persistently bright, our overprediction of thesebinaries ( ∼ −
50 systems based on the disk instability model,depending on assumptions in the model, is about one order ofmagnitude more than the three observed Bulge UCXBs) canhave several causes: the population synthesis model producestoo many UCXBs, fewer UCXBs survive the onset of mass trans-fer, or short-period UCXBs are bright less than 100% of the time.It is uncertain whether the white dwarf donor mass limitof 0 . M (cid:12) for isotropic re-emission (based on the zero-temperature mass-radius relation and a consideration of the en-ergy necessary to eject matter) should be used as the thresholdfor the survival of a system. A di ff erent assumption in the detailsof the hydrodynamics of the onset of mass transfer could resultin either a lower or a higher limit. However, the number of sur-viving white dwarf donors is not very sensitive to small changesin the isotropic re-emission limit because of the small numberof donors around this value (Fig. 2). On the other hand, due tothe sensitivity of this donor mass limit to the actual and criti-cal mass transfer rates (van Haaften et al. 2012b), the number ofsurviving UCXBs could be significantly smaller if the isotropicre-emission e ffi ciently is less than unity, which seems plausible.However, systems could survive a mass transfer rate exceedingthe isotropic re-emission limit if an UCXB with a ∼ ∼ orbits)with a significant amount of matter orbiting in and around the bi- nary, perhaps in a circumbinary ring (Soberman et al. 1997; Ma& Li 2009) that would be removed at a later stage. The valueof isotropic re-emission limit can be tested only using short-period UCXBs, because for those the white dwarf donor channelis expected to dominate. In the total population (mostly long-period systems), the helium donor channel is more important.This channel does not experience the high mass transfer ratesthat characterize the onset of mass transfer from a white dwarfdonor. Near the period minimum the mass transfer rate remainsbelow approximately three times the Eddington limit (Fig. 4)Even though Fig. 7 shows that the contribution of the helium-star channel dominates, its importance has not been establishedobservationally yet, as no detached (short orbital period) heliumstar–neutron star binaries have been discovered so far (Nelemanset al. 2010). If helium burning stars would turn into white dwarfsbefore the onset of Roche-lobe overflow, over 90% would beunable to survive as a binary system.Given these uncertainties, our overprediction by approxi-mately one order of magnitude may simply be a consequenceof poorly known parameters in our simulation, and in this case no problematic discrepancy between our results and X-ray ob-servations would remain. If mass transfer would cease completely in most old UCXBs assuggested by the existence of PSR J1719–1438, then a fractionof the neutron stars they harbor would become visible to us as(binary) millisecond radio pulsars for several billion years. Thesame is probably true for UCXBs in environments closer to usthan the Bulge. The number of isolated or binary millisecondradio pulsars in the Bulge suggested by the number of UCXBswith (cid:38)
70 min periods we predict in our standard model, af-ter correcting for the factor of ten overestimation in the UCXBbirth rate identified above, is about 2 × , assuming the pulsarshave not yet turned o ff as a result of spinning down. The esti-mated Galactic population of millisecond pulsars, based on theobserved population, is 4 × (Lorimer 2008). Of the knownGalactic Disk population (i.e., excluding globular clusters) of ∼
100 radio pulsars with spin periods shorter than 10 ms, about60% have a companion too massive to be consistent with late-time UCXB evolution, based on the
ATNF Pulsar Catalogue inJanuary 2013 (Manchester et al. 2005). (These have descendedfrom hydrogen-rich low-mass and intermediate-mass X-ray bi-naries, Bhattacharya & van den Heuvel 1991; Tauris 2011.)Extrapolating this to the estimated Galactic population, ∼ × millisecond pulsars are left that have no or a very low-mass com-panion, roughly the same number we predict from UCXBs forthe Bulge alone (after normalizing the number of short-periodUCXBs).It seems likely that millisecond pulsars have evaporated theircompanion entirely and are left as isolated millisecond pulsarsgiven the reasonable match between the predicted number of oldUCXBs and the number of isolated millisecond radio pulsars,combined with the very small number of observed millisecondradio pulsars with companions with masses lower than 0 . M (cid:12) .Alternative formation channels for isolated millisecond pulsarsare spin up of a neutron star by a disrupted white dwarf com-panion (van den Heuvel 1984), and disruption of a millisecondpulsar binary during the supernova explosion of the donor starin a high-mass X-ray binary (e.g. Camilo et al. 1993). The num-
13. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge ber of neutron star–white dwarf mergers, however, seems toohigh to be consistent with the number of isolated millisecondpulsars that have formed. Based on Fig. 2, merging systems aremuch more common than surviving UCXBs, also after includ-ing UCXBs from the helium burning donor channel (see alsoIben et al. 1995). On the other hand, the number of millisecondpulsars that lose their companions when it explodes as a super-nova seems too small to be responsible for a large fraction ofthe isolated millisecond pulsars (Burgay et al. 2003; Belczynskiet al. 2010) – moreover the high-mass donor star may not livelong enough to spin up the neutron star to a spin period shorterthan ∼
10 ms.
5. Summary and conclusions
We modeled the present-day population of primordial ultracom-pact X-ray binaries in the Galactic Bulge with the purpose ofgaining insight in their formation and evolution. Both binaryevolution and accretion physics determine the observable pop-ulation, and we attempted to disentangle these in this study.We considered three main formation channels: systems that startRoche-lobe overflow by a white dwarf donor, a helium burningdonor or an evolved main sequence donor. Our simulations havenot produced UCXBs containing a black hole, because most sys-tems with a very massive primary merge during unstable masstransfer, and the small number that remains is expected to mergeduring the onset of mass transfer from a relatively massive whitedwarf to the black hole. Thus, all UCXB systems in our simula-tions have a neutron star accretor.The vast majority of UCXBs form via the helium burningdonor channel (81%) or the white dwarf donor channel (18%),and therefore their exposed cores are expected to show eithercarbon and oxygen in their spectra, or helium, as well as smallamounts of other reaction products. These two channels di ff er inthe delay time between the zero-age main sequence and the onsetof Roche-lobe overflow to the neutron star. In the white dwarfchannel this delay can be as long as the age of the Universe ormore (though for most systems it is less than a few billion years),whereas in the helium burning channel the delay is less than 1Gyr.The size and characteristics of the present-day populationare only marginally dependent on the assumed width, σ , of theGaussian distribution describing the star formation history if thisvalue is (cid:46) σ are smallcompared to the age of a 10 Gyr old system. A broad star forma-tion history allows for recent star formation and short orbital pe-riod UCXBs with a helium burning donor origin, because of theirshort delay time. With a narrow star formation history, short-period UCXBs must have a white dwarf origin and therefore canhave helium composition.Very short period UCXBs can have a helium or carbon-oxygen white dwarf donor, since these must have formed re-cently. Recent UCXB formation is dominated by the white dwarfdonor channel, even for σ = . ∼
30 min is particularly important, since those systems areprobably observable as persistently bright sources, and thereforewell suited to test and calibrate the simulations. We predict about40 bright sources, mostly of helium and carbon-oxygen compo-sition and with orbital periods shorter than 30 min. The UCXBswith the shortest periods ( (cid:46)
20 min) are more likely to havehelium composition. The observed number of bright UCXBs isabout ten times smaller than suggested by our model, whichreflects the uncertainties in the adopted star formation history, initial binary parameters, natal kick velocities of neutron stars,common-envelope parameters and the onset of mass transfer toa neutron star accretor.We predict about (0 . − . × UCXBs in the GalacticBulge, and we stress that such a large population is necessarybased on the simple argument that the evolutionary timescaleof UCXBs increases rapidly towards longer orbital periods, andtherefore the observed number of short-period UCXBs, in theBulge and also in the Galactic Disk, implies several orders ofmagnitude more UCXBs at long orbital periods ( >
60 min). Withdi ff erent model assumptions, this number could be up to an orderof magnitude lower.Irradiation of the donor star by the neutron star and accre-tion disk strongly influences UCXB evolution, at least at orbitalperiods longer than 40 min. These systems evolve much faster ,probably by ∼
100 times, than they would if their evolution wasdriven exclusively by angular momentum loss via gravitationalwave radiation, as assumed in this paper. UCXBs with orbitalperiods longer than 1 h have not been detected yet, which im-plies that, if existent, these systems are very faint in all electro-magnetic bands (and therefore cannot be considered true X-raybinaries). We suggest that the majority of these systems have or-bital periods on the order of 1 . − . ∼ . . M (cid:12) , and could very well have evaporatedtheir companions entirely, being left as isolated millisecond pul-sars.In a forthcoming paper we will model the population ofhydrogen-rich low-mass X-ray binaries in the Galactic Bulge. Acknowledgements.
LMvH, GN, RV, and SFPZ are supported by theNetherlands Organisation for Scientific Research (NWO). GN and RV are sup-ported by NWO Vidi grant . .
305 to GN. SFPZ is supported by NWOgrants . .
803 (Vici) and . . Appendix A: Binary initial mass function andnormalization of the simulation
In the initial binary system, the more massive component iscalled the primary. We use primary-constrained pairing to con-struct ‘zero-age’ binaries (Kouwenhoven et al. 2008). The pri-mary masses M primary of the zero-age main sequence binaries aredrawn from the stellar initial mass function (IMF) of primaries in massive star clusters that we derive from the results by Kroupa(2001), where M is the stellar mass and 0 . ≤ M / M (cid:12) ≤ < M secondary / M primary ≤ . M (cid:12) are accepted. The eccentricity e distribution is propor-tional to e between 0 and 1, and the semi-major axis a distribu-tion is inversely proportional to a (Popova et al. 1982; Abt 1983),up to 10 R (cid:12) (Duquennoy & Mayor 1991) – the lower limit is setby the requirement that the initial stellar radii fit inside the cir-cularized orbit.The specific binary fraction as a function of M is given bythe observationally practical definition (Reipurth & Zinnecker
14. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge B ( M ) ≡ N binary ( M primary = M ) N single ( M ) + N binary ( M primary = M ) = N binary ( M primary = M ) IMF ( M ) (A.1)where N single ( M ) is the distribution of single stars of mass M , N binary ( M primary = M ) the distribution of binary systems con-taining a primary of mass M , and IMF ( M ) the IMF of sys-tems (single stars and multiple systems combined) by Kroupa(2001). Based on observations summarized in Kouwenhovenet al. (2009); Kraus & Hillenbrand (2009); Sana et al. (2012)we approximate B ( M ) = +
14 log ( M ) (0 . ≤ M / M (cid:12) ≤ N binary ( M primary = M ) ∝ B ( M ) IMF ( M ) , N single ( M ) ∝ [1 − B ( M )] IMF ( M ) . (A.3)It follows that single stars are more common than binary sys-tems; there are 1 . M T = B ( M primary ) M primary + M secondary , (A.4)and the average star forming mass for each binary system formed(i.e., including mass from single stars) by¯ M T = (cid:90) M (cid:12) . M (cid:12) (cid:32) + B ( M )2 (cid:33) IMF ( M ) M d M ≈ . M (cid:12) (A.5)(the factor 1 / . M (cid:12) ), the average secondary mass(0 . M (cid:12) ) and the corresponding average mass in single starsper binary system (0 . M (cid:12) ). A lower limit of 0 . M (cid:12) increasesthe average mass per binary by ∼ (cid:90) M (cid:12) . M (cid:12) N binary ( M ) d M = × M (cid:12) ¯ M T ≈ . × . (A.6)Of all primaries, 1 .
3% have a mass higher than 8 M (cid:12) . For thesemasses, the power-law slope of the primary IMF (defined overlinear mass intervals), from which we draw primary masses,varies between − .
15 (for M = M (cid:12) ) and − . M = M (cid:12) ),compared to the estimate of − . N binary ( M primary = M ) is flatter than the IMFof systems IMF ( M ) because Eq. (A.2) is an increasing function(most low-mass stars are single whereas massive stars are usu-ally in binaries). The IMF for single stars only is steeper than − . This value is higher than the average mass of 0 . M (cid:12) of the IMFby Kroupa (2001) because single stars are excluded. A non-zero mass- independent binary fraction leads to an IMF of allstars combined that is steeper than the IMF of systems (Sagar & Richtler1991; Scalo 1998; Kroupa 2001). This does not a ff ect our method as weonly consider the IMF of all primary components. References
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15. M. van Haaften et al.: Population synthesis of UCXBs in the Galactic Bulge