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The Astrophysical Journal, 683:346Y356, 2008 August 10
# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.

A

MODELS FOR LOW-MASS X-RAY BINARIES IN THE ELLIPTICAL GALAXIES NGC 3379 AND NGC 4278: COMPARISON WITH OBSERVATIONS
T. Fragos,1 V. Kalogera,1 K. Belczynski,2 G. Fabbiano,3 D.-W. Kim,3 N. J. Brassington,3 L. Angelini, R. L. Davies,5 J. S. Gallagher,6 A. R. King,7 S. Pellegrini,8 G. Trinchieri,9 S. E. Zepf,10 A. Kundu,10 and A. Zezas3
Received 2007 December 21; accepted 2008 March 6
4

ABSTRACT We present theoretical models for the formation and evolution of populations of low-mass X-ray binaries ( LMXBs) in the two elliptical galaxies NGC 3379 and NGC 4278. The models are calculated with the recently updated StarTrack code, assuming only a primordial galactic field LMXB population. StarTrack is an advanced population synthesis code that has been tested and calibrated using detailed binary star calculations and incorporates all the important physical processes of binary evolution. The simulations are targeted to modeling and understanding the origin of the X-ray luminosity functions ( XLFs) of point sources in these galaxies. For the first time we explore the population XLF in luminosities below 1037 ergs sю1, as probed by the most recent observational results. We consider models for the formation and evolution of LMXBs in galactic fields with different CE efficiencies, stellar wind prescriptions, magnetic braking laws, and IMFs. We identify models that produce XLFs consistent with the observations both in shape and absolute normalization, suggesting that a primordial galactic field LMXB population can make a significant contribution to the total population of an elliptical galaxy. We also find that the treatment of the outburst luminosity of transient systems remains a crucial factor for the determination of the XLF, since the modeled populations are dominated by transient X-ray systems. Subject headingg binaries: close -- galaxies: elliptical and lenticular, cD -- stars: evolution -- X-rays: binaries s: Online material: additional figures, machine-readable table 1. INTRODUCTION Alow mass X-raybinary( LMXB)isaRochelobeYoverflowing, mass-transferring binary system with a compact object accretor, either a black hole ( BH ) or a neutron star ( NS), and a low-mass ( k1 M ) donor. Since the late 1980s it has been suggested that LMXBs should exist in early-type galaxies ( E and S0) and that they might even dominate the X-ray emission ( Trinchieri & Fabbiano 1985; Fabbiano 1989; Kim et al. 1992). The stellar populations in these galaxies are typically old and homogeneous. Massive stars have already evolved to compact objects, and LMXBs are probably the only sources with X-ray luminosities above 1036 ergs sю1. Uncontroversial detection of LMXBs in early-type galaxies became possible only this last decade with Chandra's increased angular resolution ( Fabbiano 2006; Sarazin et al. 2000). The spectra of individual X-ray sources are consistent with those expected from LMXB models and the LMXBs observed in the Milky Way and M31 ( Humphrey & Buote 2006; Irwin et al. 2003). For many galaxies observed with Chandra, the XLFs have been derived, and they can usually be fitted with a single or a broken power law. The detections limit for these surveys is usually a few times 1037 ergs sю1. Kim & Fabbiano (2004) derived XLFs for 14 early-type galaxies, and they included completeness corrections. Each XLF is well fitted with a single power law with cumulative slope between ю0.8 and ю1.2. The composite XLF of these galaxies, however, is not consistent with a single power law. There is a prominent break at (5 ф 1:6) ; 1038 ergs sю1, close to the Eddington luminosity (LEdd ) of a helium-accreting NS-LMXB. This break might be hidden in the individual XLFs due to poor statistics (see also Sarazin et al. ґ 2000; Kundu et al. 2002; Jordan et al. 2004; Gilfanov 2004). Other recent studies by Jeltema et al. (2003), Sivakoff et al. (2003), ґ and Jordan et al. (2004) suggested a break of the XLF at a higher luminosity ($1039 ergsю1). The exact position and the nature of these breaks are still somewhat controversial, as the correct interpretation of the observed XLFs relies significantly on the proper completeness correction when looking at luminosities close to the detection limit and small number statistics at the high end of the XLF. Recent Chandra observations ( Kim et al. 2006a) have yielded the first low-luminosity XLFs of LMXBs for two typical old elliptical galaxies, NGC 3379 and NGC 4278. The detection limit (corresponding to the lowest luminosity source that is detected) in these observations is $3 ; 1036 ergs sю1, which is about an order of magnitude lower than in most previous surveys of earlytype galaxies. The observed XLFs of the two ellipticals extend only up to 6 ; 1038 ergs sю1 and are well represented by a single power law with a slope (in a differential form) of 1:9 ф 0:1. 346

1 Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208; tassosfragos@northwestern.edu, vicky@ northwestern.edu. 2 Department of Astronomy, New Mexico State University, 1320 Frenger Mall, Las Cruces, NM 88003; kbelczyn@nmsu.edu. 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138; gfabbiano@cfa.harvard.edu, kim@cfa.harvard.edu, nbrassington@ head.cfa.harvard.edu , azezas@cfa.harvard.edu. 4 Laboratory for High Energy Astrophysics, NASA Goddard Space Flight Center, Code 660, Greenbelt, MD 20771; angelini@davide.gsfc.nasa.gov. 5 Denys Wilkinson Building, University of Oxford, Keble Road, Oxford OX1 3RH, UK; rld@astro.ox.ac.uk. 6 Astronomy Department, University of Wisconsin, 475 North Charter Street, Madison, WI 53706; jsg@astro.wisc.edu. 7 University of Leicester, Leicester LE1 7RH, UK; ark@star.le.ac.uk. 8 ` Dipartimento di Astronomia, Universita di Bologna, via Ranzani 1, 40127 Bologna, Italy; silvia.pellegrini@unibo.it. 9 INAFY Observatorio Astronomico di Brera, via Brera 28, 20121 Milan, Italy; ginevra.trinchieri@brera.inaf.it. 10 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-2320; zepf@pa.msu.edu.


LMXB MODELS IN NGC 3379 AND NGC 4278 When Chandra observations are compared with optical images from Hubble or other ground-based telescopes, it is generally found that a significant fraction of the LMXBs are inside globular clusters (GCs). On average 4% Y5% of the GCs in a given galaxy are associated with a LMXB (with Lx > $1037 ergs sю1), while the fraction of LMXBs located in GCs varies from 10% to 70% depending on the type of the galaxy and its GC specific frequency. At present the origin and the properties of these systems, both in GCs and the field, are not yet well understood. It has been noted that XLFs at high luminosities for each subgroup (GCs and field) do not reveal any differences within the statistics of the samples considered ( Fabbiano 2006; Kim et al. 2006b; Kundu ґ et al. 2007; Jordan et al. 2004; Sarazin et al. 2003). However, in more recent studies, Fabbiano et al. (2007) and Voss & Gilfanov (2007) independently found that the two XLFs (GCs and field sources) show significant differences at low luminosities ( below 1037 ergs sю1), pointing to a different LMXB formation mechanism in GCs. The natural question that arises is whether (1) all LMXBs were formed in GCs through dynamical interactions and some eventually escaped or some GCs dissolved in the field, or (2) field LMXBs were born in situ through binary evolution of primordial binaries. The formation rates associated with these two possibilities are not understood well enough to give accurate predictions and are based on the relative numbers in the samples. Juett (2005) has shown that the observed relationship between the fractionof LMXBs found inGCs and the GC-specific frequencyin early-type galaxies is consistent with the galactic field LMXB population being formed in situ. Similarly, Irwin (2005) compared the summed X-ray luminosity of the LMXBs to the number of GCs in a galaxy; in the case of all LMXBs having formed exclusively in GCs, the two should be directly proportional regardless of where the LMXBs currently reside. Instead, he found that the proportionality includes an additive offset, implying the existence of a LMXB population unrelated to GCs. In the past, semianalytical theoretical models have been introduced for the study of the LMXB population in early galaxies. White & Ghosh (1998) studied the connection between the star formation rates of normal galaxies, i.e., galaxies without an active nucleus, and the formation rate of LMXBs and millisecond pulsars, assuming that all LMXBs are formed from primordial binaries. Considering a time-dependent star formation rate, they showed that the general relativity timescales relevant to the evolution of primordial binaries to LMXBs and to millisecond pulsars lead to a significant time delay of the peak in the formation rate of these populations after the peak in the star formation rate. In a follow-up work Ghosh & White (2001), using several updated star formation rate models, calculated the evolution of the X-ray luminosity of galaxies. They found that different star formation models lead to very different X-ray luminosity profiles, so the observed X-ray profiles can be used as probes of the star formation history. Finally, they compared their models with the first Chandra deep imaging observations and concluded that these first results were consistent with current star formation models. Piro & Bildsten (2002) argued that the majority of LMXBs in the field of elliptical galaxies have red giant donors feeding a thermally unstable disk and that they stay in this transient phase for at least 75% of their life. The very luminous X-ray sources (Lx > 1039 ergs sю1) detected in Chandra surveys have been suggested to be X-ray binaries with highly super-Eddington mass inflow near the accreting component. In elliptical galaxies these objects have been suggested by King (2002) to be microquasar-like, as these galaxies contain no high-mass X-ray binaries ( King 2002). More recently, Ivanova & Kalogera (2006) also argued that this

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bright end of the XLF is most likely dominated by transient LMXBs with BH accretors during outburst and that it can be used to derive constraints on the BH mass function in LMXBs; they also showed that the standard assumption of a constant transient duty cycle ( DC) across the whole population seems to be inconsistent with current observations. Semianalytical population synthesis ( PS) models of LMXBs have also been constructed for late-type galaxies. Wu (2001) created a simple birth-death model, in which the lifetimes of the binaries are inversely proportional to their X-ray luminosity, and calculated the XLFs of spiral galaxies. His models reproduce certain features, such as the luminosity break in the observed XLFs of spiral galaxies. The position of this break depends on the star formation history of the galaxy, and Wu suggested that it can be used as a probe of the galaxy's merger history. In addition, the presence of faint primordial X-ray binaries in an old galactic component is addressed, predicting for the first time that this population of X-ray binaries should be observable in old galaxies such as the two ellipticals NGC 3379 and NGC 4278. The formation of LMXBs in GCs via dynamical interactions is less well studied, since apart from the binary stellar evolution, one has to also take into account the complex cluster dynamics. Bildsten & Deloye (2004) considered a semianalytical model for accretion from degenerate donors onto NSs in ultracompact binaries and showed that binaries with orbital periods of 8Y10 minutes and He or C/O white dwarf ( WD) donors of 0.06 Y0.08 M naturally provide the primary slope (ю0.8 for cumulative form) typically derived from XLFs of elliptical galaxies. Ultracompact systems are predicted to form in the dense GC environment and have relatively short persistent lifetimes (<3 ; 106 yr), but they form continuously through dynamical interactions. Ivanova et al. (2008) presented PS studies of compact binaries containing NSs in dense GCs. They used StarTrack as their PS modeling tool in addition to a simplified treatment for the dynamical interactions. Their models produced a mixed population of LMXBs, with red giant and MS donors, and ultracompact X-ray binaries; relative formation rates can be comparable, but the different subpopulations have very different lifetimes. In this paper we use advanced PS simulations to investigate the plausibility of an important contribution being made to the XLFs of these two galaxies by a primordial galactic field LMXB population. In x 2 we describe briefly the physics included in our PS code and explain in detail the way we are constructing the modeled XLFs, as well as the treatment of transient LMXBs. We discuss the results of our simulations in x 3: the modeled XLFs from different models, a statistical comparison with the observed XLFs of the elliptical galaxies NGC 3379 and NGC 4278, and an analysis of the dependence of the modeled XLF properties on the PS parameters. Finally, in x 4 we discuss the implication of our findings and the caveats entailed by our methods. 2. LMXB POPULATION MODELS For the models presented in this study we focus on LMXBs formed in the galactic field as products of the evolution of isolated primordial binaries. The standard formation channel ( Bhattacharya & van den Heuvel 1991; Tauris & van den Heuvel 2006) involves a primordial binary system with a large mass ratio; the more massive star evolves quickly to the giant branch and the system goes into a common envelope (CE) phase. During this phase, the less massive star, which is still dense and unevolved, orbits inside the envelope of the primary and is assumed to remain intact. The orbit of the system changes dramatically, however, as orbital energy is lost due to friction between the unevolved star and the envelope of the giant. Part of the lost orbital energy is used to


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TABLE 1 Ga l a xy P ro p er ti es Parameter Distance ( Mpc) .................................. Age (Gyr) ........................................... Metallicity ([ Fe/ H ]) .......................... Mass (M) .......................................... GC specific frequency ....................... NGC 3379 10.57 9.3 0.16 8.6 ; 1010 1.2 NGC 4278 16.07 10.7 0.14 9.4 ; 1010 6.9 References Tonry et al. (2001) Terlevich & Forbes (2002) Terlevich & Forbes (2002) Cappellari et al. (2006) Ashman & Zepf (1998)

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expel the envelope of the giant star. The fraction of the lost orbital energy that is used to heat up the envelope of the giant star and finally expel it defines the CE efficiency factor CE. The CE phase results in a binary system with an unevolved low-mass main-sequence ( MS) star orbiting around the core of the massive star in a tighter orbit. The massive core soon reaches core collapse to form a compact object, either a NS or a BH, and the binary orbit is altered due to mass loss and possible supernova kicks. If the binary does not get disrupted or merge in any of the stages described above, angular momentumYloss mechanisms, such as magnetic braking, tides and gravitational wave radiation, will further shrink the orbit and the low-mass companion may evolve off the MS. The companion star eventually overflows its Roche lobe, transferring mass onto the compact object and initiating the system's X-ray phase. An alternative formation channel for NS-LMXBs is through the accretion-induced collapse of a WD accretor into a NS. These systems have generally very low X-ray luminosity and do not affect the LMXB population in the luminosity range that we are interested in this paper. 2.1. Synthesis Code: StarTrack We perform the simulations presented here with StarTrack ( Belczynski et al. 2002, 2008), a advanced PS code that has been tested and calibrated using detailed mass transfer calculations and observations of binary populations, and incorporates all the important physical processes of binary evolution: (1) The evolution of single stars and noninteracting binary components, from ZAMS to remnant formation, is followed with analytic formulae ( Hurley et al. 2000). Various wind mass-loss rates that vary with the stellar evolutionary stage are incorporated into the code, and their effect on stellar evolution is taken into account. (2) Throughout the course of binary evolution, the changes in all the orbital properties are tracked. A set of four differential equations is numerically integrated, describing the evolution of orbital separation, eccentricity, and component spins, which depend on tidal interactions as well as angular momentum losses associated with magnetic braking, gravitational radiation, and stellar wind mass losses. (3) All types of mass-transfer phases are calculated: stable driven by nuclear evolution or angular momentum loss and thermally or dynamically unstable. Any system entering the Roche lobe overflow ( RLOF ) is assumed to become immediately circularized and synchronized. If dynamical instability is encountered the binary may enter a CE phase. For the modeling of this phase we use the standard energy balance prescription. (4) The SN explosion is treated taking into account mass loss as well as SN asymmetries (through natal kicks to NSs and BHs at birth). The distribution of the SN kick magnitudes is inferred from observed velocities of radio pulsars. For this project we use the distribution derived by Hobbs et al. (2005), which is a single Maxwellian with ј 265 km sю1. It is, however, assumed that NS formation via electron capture or accretion induced collapse does not lead to SN kicks. (5) Finally, the X-ray luminosity of accreting binaries with NS and BH primaries ( both for wind-fed

and RLOF systems) is calculated. For RLOF-fed systems there is a distinction made between persistent and transient (systems that undergo thermal disk instability), while wind-fed systems are always considered as persistent X-ray sources. The mass transfer is conservative up to the Eddington limit for persistent X-ray binaries, while transients are allowed to have slightly super-Eddington luminosity (up to 3 ; LEdd ) ( Taam et al. 1997). In all cases we apply appropriate bolometric corrections ( bol ) to convert the bolometric luminosity to the observed Chandra band. A much more detailed description of all code elements, treatments of physical processes and implementation is provided in Belczynski et al. (2008). 2.2. Model Parameters for NGC 3379 and NGC 4278 In this study we focus on trying to understand the XLF characteristics of the two elliptical galaxies NGC 3379 and NGC 4278, observed with Chandra and reported by Kim et al. (2006a). In the development of our models we incorporate our current knowledge about the characteristics of the stellar population in these galaxies (see Table 1). The observationally determined parameters of the stellar populations, such as their age and metallicity, or their total stellar mass, are similar for NGC 3379 and NGC 4278. This allows us to develop the same models in our simulations for both of them. Terlevich & Forbes (2002) estimated the ages and metallicities of 150 elliptical and late-type spiral galaxies using published high-quality spectral line indices. For NGC 3379 they are reporting an age of 9.3 Gyr and a metallicity of Ѕ Fe/ H ј 0:16, while for NGC 4278 the corresponding values are 10.7 Gyr and Ѕ Fe/ H ј 0:14. The two galaxies have very similar optical luminosity and assuming the same light to mass ratio, they should also have similar masses. Cappellari et al. (2006) used I -band observations from the Hubble Space Telescope to calculate the total stellar mass of the two galaxies and they found them to be 8:6 ; 1010 and 9:4 ; 1010 M for N3379 and N4278, respectively. The ratio of the integrated LMXB X-ray luminosity to the optical luminosity is 4 times smaller for NGC 3379, which also has 6 times lower GC specific frequency compared to NGC 4278 (see Kim et al. 2006a; Ashman & Zepf 1998). There are, however, a number of parameters in our models for which we do not have any direct guidance from observations. We have no information about the star formation history of the two galaxies, and thus we assume a -functionYlike star formation episode at time t ј 0. Also unknown are the initial mass function ( IMF ) and the distributions of orbital separation and eccentricity for the primordial binary systems. We adopt two different IMFs: Scalo/ Kroupa and Salpeter, while for the distributions of the orbital properties we follow the standard assumptions described in Belczynski et al. (2002). Other parameters that can affect the final LMXB population are the binary fraction of the host galaxy, the magnetic braking law adopted, and the CE efficiency ( CE ) The specific parameters we used to model the ellipticals NGC 3379 and NGC 4278 are listed in Table 2.


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LMXB MODELS IN NGC 3379 AND NGC 4278

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TABLE 2 Model P ar ameters for NGC 3379 and NGC 4278 Parameter Star formation ........................... Population age (Gyr) ................ Metallicity ................................. Total stellar mass (M )............. Binary fraction (%) ................... IMF ........................................... CE efficiency (%)...................... Magnetic braking ...................... Notation Value -function at t = 9 Y10 0.03 9 ; 1010 50 Scalo/ Kroupa or 20 Y100 Rappaport et al. or Ivanova & 0

they occasionally go into an outburst. The fraction of the time that these systems are in outburst (Toutburst ) defines their DC: DC Toutburst : Toutburst Ч Tquiescent П 1ч

Z Mц Fbin
CE

Salpeter (1983) Taam (2003)

We note that in our calculations we combine CE and k into one CE parameter, where k is a measure of the central concentration of the donor and the envelope binding energy (see, e.g., the appendix in Justham et al. 2006). In the rest of the text, whenever we mention the CE efficiency CE, we refer in practice to the product CE ; k, effectively treating k as a free parameter (see Belczynski et al. 2008 for details). Justham et al. (2006) argued that k-values for high-mass stars are very small (<0.1) and concluded that unreasonably high values of CE are needed for CE ejection. However, one must take into account that k is highly uncertain primarily because of important uncertainties in defining the core-envelope boundary in the donor. As clearly shown by Dewi & Tauris (2000) and especially by Tauris & Dewi (2001) k-values for a 20 M (10 M ) are uncertain by factors of $70 ($20). Therefore, choosing a fixed value of k for donors of a certain mass and evolutionary stage is a gross oversimplification. For these reasons we combine CE and k into one model parameter and explore reasonable values of it. 2.3. Models for the X-Ray Luminosity Function In our models we keep track of all the binary properties, in cluding the mass-transfer rate (M ), as a function of time for populations of accreting NS and BH. We use the mass-transfer rates to identify the persistent and transient sources in our simulation results. Binaries for mass transfer rate higher than the critical rate Mcrit for the thermal disk instability (van Paradijs 1996; King et al. 1996; Dubus et al. 1999; Menou et al. 2002) are considered persistent sources, and their X-ray luminosity (Lx ) is calculated directly from the mass transfer rate as Lx ј GMa M ; Ra

Observations of Galactic LMXBs show that transient systems spend most of their life in the quiescent state, hinting at a DC below 20% ( Tanaka & Shibazaki 1996). The outburst luminosity and the DC of transient systems is not well understood and cannot be calculated from first principles. Instead, we have to rely primarily on empirical constraints and simple theoretical arguments. In our analysis we consider a number of different treatments of these parameters for transients, which we describe in what follows. As a first approximation it has been suggested that transient LMXBs emit at their Eddington luminosity (LEdd ) when they are in outburst. In a different approach Portegies Zwart et al. (2005) derived an empirical correlation between the outburst luminosity of Milky Way transient LMXBs with BH accretors and their orbital period P: P : П 2ч Lx ј bol ; min 2 ; LEdd ; 2 ; LEdd 10 hr We can generalize this relation to all transient LMXBs in galaxies other than our own, but we note that there has not been any observational work that shows that NS-LMXBs follow a similar trend. A more physical treatment is to assume that in the quiescent state the compact object does not accrete (or accretes an insignificant amount of mass) and matter from the donor is accumulated in the disk. In the outburst state all this matter is accreted onto the compact object emptying again the disk. Taking into account as well that the X-ray luminosity probably cannot exceed LEdd by more than a factor of 2 (see Taam et al. 1997), we end up with a definition of the outburst luminosity as GMa Md 1 Lx ј bol ; min 2 ; LEdd ; П 3ч Ra DC In the equation above, DC is unknown. Dobrotka et al. (2006) studied accretion disk models for cataclysmic variables that are thought to experience the same thermal disk instability (dwarf novae). They found a correlation between the DC of the system and the rate at which the donor star is losing mass Md . The exact relation of these to quantities depends on the values of the disk's viscosity parameters, but the general behavior can be approximated by DC ј Md crit M 2 : П 4ч

bol

where the radius of the accretor (Ra ) is 10 km for a NS and 3 Schwarzschild radii for a BH, gives a conversion efficiency of gravitational binding energy to radiation associated with accretion onto a NS (surface accretion ј 1:0) and onto a BH (disk accretion ј 0:5), and bol is a factor that converts the bolometric luminosity to the X-ray luminosity in the Chandra energy band (0.3Y8 keV ). For RLOF-accreting BHs this conversion factor is estimated to be bol ј 0:8 ( Miller et al. 2001), while for RLOF-accreting NSs bol ј 0:55 ( Di Salvo et al. 2002; Maccarone & Coppi 2003; Portegies Zwart et al. 2004). The two correction factors and bol are applied in both persistent sources and transient sources in outburst. In the context of the thermal disk instability model, mass transferring binaries with M < Mcrit are considered transient sources, meaning that they spend most of their life in a quiescent state (Tquiscent ), in which they are too faint to be detectable, and

Plugging equation (4) into equation (3) we eliminate the DC dependence and get an expression for the outburst luminosity of a transient system that depends only quantities which are directly calculated in our population modeling: Lx ј
bol

; min 2 ; L

Edd

2 GMa Mcrit ; : Ra Md

П 5ч

The accretion disk models by Dobrotka et al. (2006) assume accretion onto a compact object with a hard surface and it is not obvious that the same results will apply for accretion onto a BH.


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TABLE 3 Populati on Synt hesis M od els Model 1........................................ 2........................................ 3........................................ 4........................................ 5........................................ 6........................................ 7........................................ 8........................................ 9........................................ 10...................................... 11...................................... 12...................................... 13...................................... 14...................................... 15...................................... 16...................................... 17...................................... 18...................................... 19...................................... 20...................................... 21...................................... 22...................................... 23...................................... 24...................................... 25...................................... 26...................................... 27...................................... 28......................................
CE

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In order to take into account all the available information, empirical and theoretical, about LMXB transient behavior, we treat BHs and NS-LMXBs differently and define the outburst luminosity as 8 P > > min 2 ; LEdd ; 2 ; LEdd ; for BH acc; > < 10 hr Lx ј bol 2 > > > min 2 ; LEdd ; GMa Mcrit ; for NS acc: : Ra Md П 6ч We note that for BH-LMXBs, we adopt a single DC value for simplicity and lack of other information, although we have no clear physical reason to believe that all BH systems have the same DC. It is believed that the DC of BH systems is smaller than that of NS systems, on the order of 5% ( Tanaka & Shibazaki 1996). We found that in all our models NS accretors greatly outnumber BH accretors. BH systems only have an important contribution at high luminosities, where the error bars in the observed XLFs (see Kim et al. 2006a) are too large to give us any tight constraints for our models. As we show in x 3.1, the treatment described in equation (6) gives us the best agreement with observations. To construct the XLF we consider a snapshot of the whole population at the time we are interested in and we identify the LMXBs as transient or persistent. If a system is transient we decide whether it is in outburst or in quiescence according to its DC and either assign an outburst luminosity or discard the system as quiescent and hence too faint to contribute to the XLF. We then construct the XLF by calculating the cumulative X-ray luminosity distribution of the sources that are detectable ( persistent and transient in outburst). We note that the age of the elliptical galaxies NGC 3379 and NGC 4278, and hence their LMXB population, is known only to within $1 Gyr. Consequently, we cannot just choose a unique snapshot of the population. Instead, we construct the XLF by considering the time window of 9Y10 Gyr, divided into time slices of 1 Myr. We construct the XLF at each of these time slices, and we take the average to represent the XLF that corresponds to the time window of interest. By doing this, we also improve the statistics of our model sample. It is computationally impossible to evolve enough binaries to correspond to the total initial number of binaries in an elliptical galaxy ($109 binaries). For each model we evolve 106 binaries, which takes about 2 months of CPU time on a modern processor. We then normalize to the total mass of the galaxy in question, taking into account the initial binary fraction and the IMF. 3. RESULTS 3.1. Exploring the Parameter Space One of the implicit weaknesses of PS models is the large number of free parameters that one can vary and fine-tune in order to get the desirable result. There ar