Blue straggler formation via close binary mass transfer

Transcription

1 Mon. Not. R. Astron. Soc. 409, (2010) doi: /j x Blue straggler formation via close binary mass transfer P. Lu, 1,2 L. C. Deng 1 and X. B. Zhang 1 1 Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing , China 2 Graduate University of Chinese Academy of Sciences, Beijing , China Accepted 2010 July 13. Received 2010 July 13; in original form 2010 March 18 ABSTRACT Several mechanisms of blue straggler (BS) formation via mass transfer in close binaries are studied in this work. Detailed close binary evolutionary models including both Cases A and B are calculated. We follow the evolution of a binary of 1.4 M M and compare the evolutionary behaviour of both cases. The results show that Case B is as important as Case A in forming BSs. Moreover, BSs formed via Case B are generally bluer and even more luminous than those produced by Case A, which provides a clue as to the origin of luminous BSs such as those in M67. Based on the models calculated at 4.0 Gyr of age, a Monte Carlo simulation of the BS population in M67 is carried out using both Case A and Case B scenarios. Five BSs formed via mass transfer were predicted by our simulation and another three by dynamical merger processes. The simulation uses an artificially enlarged sample size for better statistics. The contributions from the different channels are estimated. We also investigate the influence of mass transfer efficiency β during Roche lobe overflow on close binary evolution. The result shows that a high value of β could reproduce the observed data better. The mechanism discussed in this work cannot predict all observed BSs in M67 because many factors are involved in such an old open cluster that affect the formation of BSs. However, we offer a better explanation of blue and luminous BSs in the binary evolution channel. Key words: blue stragglers galaxies: star clusters: individual: M67. 1 INTRODUCTION Blue stragglers (BSs) were first noticed in the globular cluster M3 (Sandage 1953). The existence of BSs has been proven by observations in all types of stellar systems. BSs remain on the main sequence for an exceedingly long lifetime compared to standard main-sequence stars. These massive stars lie in the blue and luminous region of the colour magnitude diagram (CMD) and appear unevolved when compared to the turn off of their population. Based on high-resolution images obtained by the Hubble Space Telescope, two distinct parallel sequences of BS populations were found by Ferraro et al. (2009) in the globular cluster M30. It has been suggested that BS formation in M30 was boosted because M30 may have undergone a core collapse 1 2 Gyr ago, during which both mass transfer in binaries and the collision rate would have been enhanced. The bluer sequence was suggested to have arisen from direct stellar collisions because it can be well fitted by collisional isochrones corresponding to ages of 1 2 Gyr and the redder one may have arisen from the evolution of close binaries that are probably still experiencing mass exchange. lupin@bao.ac.cn (PL); licai@bao.ac.cn (LCD); xzhang@bao. ac.cn (XBZ) Several mechanisms have been proposed to explain BS formation (see the review of Stryker 1993). In general, BSs can be formed by way of direct collisions between stars, mass transfer or coalescence in close binary systems. The direct collision hypothesis was originally presented by Hills & Day (1976). They proposed that the remnants of two mainsequence stars colliding with each other could produce a BS. Collision is a consequence of the dynamic evolution of host clusters, so it is believed to dominate the production of BSs in dense environments such as the cores of globular clusters, or even in the centre of open clusters such as M67 (Hurley et al. 2001, 2005; Glebbeek, Pols & Hurley 2008). McCrea (1964) proposed that BSs could also form by way of mass transfer in close binaries. The primary could transfer material to the secondary through the inner Lagrangian point after filling up its Roche lobe. The secondary could then become a more massive main-sequence star with a hydrogen-rich envelope provided by the primary. Thus, the lifetime of the secondary could be doubled as a result of mass transfer when compared to a normal star with the same final mass. Mass transfer in primordial binaries is believed to dominate the production of BSs in open clusters and in the field (Lanzoni et al. 2007; Dalessandro et al. 2008; Sollima et al. 2008). The mass transfer scenario can be divided into Cases A, B and C according to the evolutionary state of the primary at the onset of C 2010 The Authors. Journal compilation C 2010 RAS

2 1014 P. Lu, L. C. Deng and X. B. Zhang mass transfer (Kippenhahn & Weigert 1968). They are associated with the evolutionary stages, respectively, of hydrogen burning in the core on the main sequence, post main sequence before helium ignition, and during central He-burning and onwards. Hagai & Daniel (2009) discussed the possibility of BS formation in primordial and/or dynamical hierarchical triple star systems. The dynamical evolution of the triples through the Kozai mechanism and tidal friction can induce the formation of very close inner binaries. Then BSs can be produced by mass transfer or merger as a result of angular momentum loss in a magnetized wind or stellar evolution. It is widely accepted that more than one scenario can account for BSs at the present time. Pols & Marinus (1994) performed Monte Carlo simulations of close binary evolution in young clusters. They showed that BS production agrees quantitatively well with observations for clusters younger than about 300 Myr; however their simulations could not produce enough BSs for clusters of older ages (between 300 and 1500 Myr). Hurley et al. (2005) applied N-body models to simulate BSs in M67. They suggested that both mass transfer and cluster dynamics contribute to the formation of BSs. Binary coalescence from Case A evolution was estimated by Chen & Han (2008). They were able to predict high-luminosity BSs using coalescence models, but the results cannot match the statistics of M67 for any tuning of the parameters. BSs could, in principle, be formed by direct collisions between main-sequence stars, but that is unlikely to happen in open clusters, as the time-scale of collisions is too long (Press & Teukolsky 1977; Mardling & Aarseth 2001). Case A binary evolution in M67 has been studied in the past. Tian et al. (2006, hereafter TDHZ06) calculated a series of Case A models to simulate BSs in M67; the result produced too few BSs for the cluster (four BSs and 15 progenitors or descendants). They suggested that mechanisms of BS formation other than just Case A mass transfer are needed in order to understand the BS population in M67. In an earlier work (TDHZ06), we examined detailed evolutionary modelling of close binary systems in order to understand BS formation through the Case A mass transfer process. In order to get a general picture of BS formation, we now extend the work to include Case B mass transfer. We calculated a grid of detailed evolutionary models for binary evolution covering the same range of primary mass but with a larger orbital separation. We discuss and present the different properties of BSs formed via Cases A and B in Section 2. A Monte Carlo simulation for BSs in M67 is carried out based on these models and the result is shown in Section 3. Summaries and conclusions are given in the final section. 2 THE MODEL OF PRIMORDIAL BLUE STRAGGLERS We use an updated (Han, Podsiadlowski & Eggleton 1994; Pols et al. 1995, 1998; Han, Tout & Eggleton 2000) version of the stellar evolutionary code of Eggleton (1971, 1972, 1973) to compute close primordial binary evolution models. The code uses a self-adaptive non-lagrangian mesh. The radiative opacity library from Iglesias & Rogers (1996) and molecular opacities of Alexander & Ferguson (1994) are adopted in this code. Roche lobe overflow (RLOF) is treated as a boundary condition within the code, which is written as dm/dt = const. max [0, (R star /R lobe 1) 3 ], (1) where dm/dt is the mass transfer rate between the two components. R star is the radius of the donor and R lobe is the effective Roche lobe radius which is a function of mass ratio (Eggleton 1983) and is defined as the radius of a sphere which has the same volume as the corresponding lobe. We take const. = 500 M yr 1 to keep RLOF steady. The donor overfills its Roche lobe as necessary but never by much: ( Rstar R lobe 1) (Han et al. 2000). The merge process is complicated and the physics during the process is still uncertain, so our calculations are terminated once the two members come in contact with each other. Previous population synthesis studies adopted much simpler stellar evolution schemes. In principle, it is efficient to empirically estimate the basic quantities of stars during the course of evolution using analytic formulae to approximate the main characters of stellar evolution but at the cost of losing physical details and accuracy. Eggleton s stellar evolution code can provide details about the components of binary systems, but the computation of RLOF is focused only on the component losing mass. In our work, we want to follow the detailed evolution of both components, so that the evolutionary behaviours of the system can be modelled. To do that, we made minor modifications to the code. Specifically the mass-loss history of the donor is recorded, which is then used as an input parameter of mass gaining rate for subsequent calculations on the secondary. The mass exchange rate is used to coordinate the calculation of the two components. In this way, the evolution of both components can be synchronized. The accreted material from the primary is assumed to be simply deposited on to the surface of the secondary and distributed homogeneously all over the outer layers instantly. Because of this modification, our calculations can provide more detailed information on the evolution of close binaries. We assume the system conserves both mass and angular momentum, and the orbital eccentricity is set to e = 0 for all our models at zero age main sequence (ZAMS). Tidal evolution, magnetic braking and stellar spin are neglected in our calculations. It has been proven that tidal interaction can circularize the short-period binaries during the dynamical evolution of the system by previous studies (Zahn 1966; Tassoul 1988; Goldman & Mazeh 1991). The tidal circularization will be more crucial if mass exchange occurs for eccentric binaries. So it is reasonable to adopt an initial orbital eccentricity e = 0. The assumption of conservation of mass and angular momentum was only reasonable for a restricted range of intermediate-mass binaries as predicted by Nelson & Eggleton (2001). We also consider a non-conservative case by taking into account the influence of mass transfer efficiency to our calculations. Magnetic activity is detected in some low-mass binaries with late-type components. Magnetized stellar winds probably do not carry away much mass, but they are rich in angular momentum because of the magnetic linkage of binaries (Chen & Han 2008). This will be discussed in Section 3.1. Magnetic braking and tidal synchronization will affect the mass transfer rate and merger time-scale (Stepien 1995). For simplicity, we do not include the treatment of these effects in our models and the conservation assumption is adopted. Case B binary evolution indicates that mass transfer occurs when the primary evolves past the main sequence but before helium core ignition. We are unable to follow the RLOF on the red giant branch (RGB) because an instable mass transfer occurs at the base of RGB which makes the calculation very uncertain. We only carried out the calculations of Case B when RLOF occurs in the Hertzsprung Gap (HG). We use two different treatments to model the binaries with different initial mass ratios. For the binaries with larger initial mass ratios, the mass transfer is quite stable and both components can be calculated using Eggleton s code. We name these models Case B models in the following. In order to compare our Case B with the previous Case A models (TDHZ06), we present a binary system (1.4 M M R ) with the same primary-to-secondary

3 Binary evolution and blue stragglers 1015 Figure 1. The evolutionary tracks of two components ( M ) undergoing the Case B scheme. The dashed and solid lines are the evolutionary tracks of the primary and the secondary, respectively. The plus sign, open circles, open squares and filled squares show their positions, respectively, when mass transfer begins, the mass ratio equals 1, mass transfer terminates and the system is 4.0 Gyr old. mass-to-mass ratio but a different orbital separation as the Case A example. We also adopt a solar composition in our models (Z = 0.020, Y = 0.280). Fig. 1 shows the evolutionary tracks of the two components. As a result of mass transfer, the donor departs from its regular evolution (shown by the dashed line). Mass transfer begins when the primary overfills its Roche lobe, and slows as the primary evolves to the bottom of the RGB. Mass ratio quickly reverses during the first stage of mass transfer. A second episode of mass transfer begins when the primary evolves to the bottom of the RGB and ends at the top of the RGB because of star contraction. Evolution to the end of helium burning has not been followed in this example because, after the second episode, the primary evolves to a helium white dwarf and the code breaks down because of the high degree of degeneracy. As a result of mass accretion, the luminosity of the secondary evolves upward along the main sequence in the CMD as shown by the solid line. After the termination of mass transfer, it follows the regular evolutionary path of a single star with 88.7 per cent of the total mass of the system. In Table 1, we list five key epochs in the evolution of our example model. It should be noted that the parameters of the primary we list in the final epoch are at Gyr of age because the code breaks down at this point owing to the high degree of degeneracy of the white dwarf mentioned above. The remnant is a helium dwarf with 0.26 solar mass whose contribution to the system has already been considered in our calculation all the way to the last point on the track (Fig. 1). We ignore it only when its luminosity drops further down. Close binaries cannot be visually resolved at the distance of M67 in observations with the current technology. What we obtained from observations is the synthetic position of the two components on the CMD. So we need to take into account the contribution from both components and to derive their composite magnitude. Given the effective temperature, surface gravity from our models and solar composition, we obtained Johnson B and V magnitudes of the two components in a binary system by linear interpolations from the theoretical stellar spectra library (Lejeune, Cuisinier & Buser 1997, 1998). We get the synthetic magnitude by ( M i = M i,1 2.5 log M i,1 M i,2 2.5 ), (2) where M i, M i,1, M i,2 indicate the magnitudes in the i band of the synthetic model, the primary and the secondary. In Fig. 2, the synthetic evolutionary track of the example binary system is shown on the CMD. The luminosity and colour are dominated by the primary before mass ratio reversion. After the mass ratio equals 1, the secondary dominates the evolution and evolves to the BS region in CMD. Its evolution in (B V) 0 space is also shown in Fig. 3. It has a long enough lifetime on the blue side to be observed as a BS. We can notice some of the differences between Cases A and B. For Case A, the secondary took longer to accrete enough mass via mass transfer to dominate the magnitude and colour and to bring the system to the BS regime. This procedure is faster for Case B with a nearly equivalent total amount of mass transferred in the process. The secondary from Case B resides in the blue region longer as a main-sequence star following normal evolution. This phenomenon is more prominent in binaries with more massive primaries and will be discussed in a subsequent work. In Case B, binaries are more likely to be observed as bluer and more luminous than those in Case A. For the binaries with small initial mass ratios q < q crit (the secondary to the primary), the mass transfer rate can be extremely high and leads to dynamically instable RLOF. A common envelope (CE) Table 1. The results of the example binary (1.4 M M ). Epoch Age P a Mass lg(l/l ) lgt eff X C Y C Ṁ (10 9 yr) (d) (R ) (M ) (M yr 1 ) The columns are as follows: (1) the model serial number, (2) the age, (3) the period, (4) the orbital separation, (5) the mass, (6) the luminosity, (7) the effective temperature, (8) hydrogen abundance in the core, (9) helium abundance in the core and (10) mass transfer rate.

4 1016 P. Lu, L. C. Deng and X. B. Zhang Figure 2. Synthetic evolutionary track of the example binary. The solid line is the synthetic evolutionary track of the binary. The dashed and dotted lines are the evolutionary tracks of the primary and the secondary, respectively. The polygon outlined by the long dashed line indicates the locations of BSs in M67. All the symbols are the same as in Fig. 1. time is recorded as t contact. Mergers are then constructed according to the combined mass and composition of the components at the onset of mass transfer. There are several ways of constructing mergers. The lifetimes of mergers on the main sequence (MS) and their positions on the CMD at 4.0 Gyr can be very different. We considered an extreme case of full mixing for the coalescence process. The composition of two components is considered to be mixed homogeneously without any mass-loss. The mergers are replaced by a ZAMS model with the corresponding composition and total mass of the system. We use dynamically merged models hereafter to identify these merger products. The ZAMS models were then evolved to 4.0-Gyr-t contact to indicate the positions of binaries with small initial mass ratios at 4.0 Gyr. In order to make it possible to simulate the contribution of close binary evolution to the number of BSs in clusters, we compute a grid of Case B models that can experience mass exchange during 4.0 Gyr. The donor mass is from 1.3 M to 1.6 M and the companion mass is from 0.2 M to the mass of the corresponding donor. Mass intervals are 0.1 M for both the donors and their companions. Those binary models with small initial mass ratios are replaced by ZAMS stars possessing the total mass of the system right after mass transfer begins for the reasons as discussed above. Those with larger mass ratios are still calculated using Eggleton s code. Orbit separations of previous Case A models (TDHZ06) are extended to 9.0 R in our calculations to cover all possible cases of binary models which experience mass exchange at the onset of HG. All of the models are shown in Fig. 4. As we predicted, BSs formed via the Case B scheme lie in a more luminous and bluer region than those formed via Case A. Each group of Case B binaries with the same primary mass and mass ratio is found concentrated in the same position on the blue side of the CMD despite having different orbital separations. For example, of the five open triangles on the Figure 3. Evolution of synthetic colour of the example Case B binary. The long dashed line indicates the colour of turn-off stars in M67. All the symbols have the same meaning as in Fig. 1. is formed because the secondary quickly fills its Roche lobe after the onset of mass transfer. The CE may be ejected or the binary may merge. The remnants could be a BS if both components are main-sequence stars. The physics during the process is still uncertain. As the main focus of the present work is mass transfer rather than coalescence, it is assumed that these binaries with small initial mass ratios will merge right after mass transfer has begun. The calculation stops when the donor has fully filled its Roche lobe. The Figure 4. The grid of all our models at 4.0 Gyr. The solid line is the 4.0 Gyr theoretical isochrone. The dashed line is the equal-mass photometric binary sequence. The long dashed lines are the ZAMS and the ZAMS shifted by 0.75 mag. The open triangles are our Case B models. The open circles are the merged models and the plus signs are Case A models from TDHZ06. The dotted line shows the upper boundary of the simulated primordial BSs formed via Case A in TDHZ06.

5 Binary evolution and blue stragglers 1017 left, each consists of several Case B models. This is understandable because the interval of orbital separation is small (0.1 R ) and all these BSs have terminated RLOF. Their positions in CMD are determined by the total mass accreted by the secondary which is not sensitive to orbital separation but rather to mass ratio. This is different from Case A because most BSs from Case A are still experiencing mass transfer, and the onset of RLOF is dependent on initial orbital separation. The results of our earlier work on Case A binary evolution models (TDHZ06) show a population of binary systems which still experiences mass transfer of 0.75 mag above the ZAMS in the BS region, which could well reproduce the redder BS sequence predicted by Ferraro et al. (2009). We have a slightly different explanation for the bluer BS sequence in their work. We can see in Figs 2 and 3 that the synthetic model remains on the redder BS sequence during RLOF. However, there is a jump in colour at 3.7 Gyr when RLOF terminates. The companion possesses most of the mass of the system (see Table 1) and evolves as a normal massive main-sequence star on the bluer BS sequence. We do not include stellar collisions but we did model some merger products from small initial mass ratio binaries. We see in Fig. 4 that most of our Case B models can also reproduce a bluer BS sequence which fits the merger sequence well. We also plot ZAMS and the ZAMS shifted by 0.75 mag isochrones as a comparison with Ferraro et al. (2009). It is clearly shown in Fig. 4 that the sequence of BSs formed from mass transfer and mergers are offset from the ZAMS at the high-luminosity end. 3 MONTE CARLO SIMULATIONS OF THE PRIMORDIAL BSS IN NGC M67 It is possible to simulate BSs in a cluster with a grid of all our models at the corresponding age. We use M67 as an example to examine our models using the Monte Carlo method. Previous studies showed several discrepancies in the age of M67 ranging from 3.2 ± 0.4 Gyr (Bonatto & Bica 2003) to as high as 6.0 Gyr (Janes & Phelps 1994). Sandquist (2004) claimed that the main features of the cluster in the CMD up to the subgiant branch can be reproduced well by a 4-Gyr isochrone, with an uncertainty lower than 0.5 Gyr. Hurley et al. (2005) also investigated the behaviour around 4 Gyr using N-body simulations of this cluster. The metallicity of M67 is generally believed to be solar (Hobbs & Thorburn 1991; Friel & Janes 1993) while some other studies show a little difference (Carraro et al. 1996; Fan et al. 1996). We adopt log(age) = 9.60 and solar metallicity for M67. A single starburst period is assumed in this work. Based on the previous studies of the basic parameters of M67, we present a numerical simulation to investigate BS formation via RLOF in primordial binaries. The initial mass functions (IMF) of the primaries, distribution of initial mass ratios and distribution of initial orbital separations are in agreement with previous work (TDHZ06). (1) We adopt the IMF of Kroupa, Tout & Gilmore (1991) as in TDHZ06 for consistency. The IMF was also adopted in Hurley et al. (2001, 2005) as the generating function for binaries. [ M(X) = (1 X)2 (1 X) (1 X) where X is a random number uniformly distributed between 0 and 1. M(X) is the binary mass in units of M. M(X) is limited between 0.2 and M by the assumption that the single-star population of M67 has a coverage of initial mass from 0.1 to 50.0 M. ] (3) Table 2. Models of Case B and merger + Case A (TDHZ06). Only a sample of the table is included here. The full table is available in the electronic version of the paper (see Supporting Information). M 1 M 2 a V B V β (M ) (M ) (R ) (2) The mass ratio distribution treatment is quite controversial. We adopt a uniform distribution of mass ratio as given by Hurley et al. (2001). [ 0.1 ] 1 >q>max M(X) 0.1, 0.02(M(X) 50.0). (4) (3) We assume the distribution of separations is constant in log α (α is separation) from Pols & Marinus (1994). The lower limit of separation is the minimum size of the Roche lobe that just fits a zero-age star, and an upper limit of 50 au used by Hurley et al. (2005) is adopted. Hurley et al. (2005) suggested an initial configuration of single stars and primordial binaries which can reproduce the observed parameters of M67 well. Such a configuration is also adopted in the previous work (TDHZ06; Chen & Han 2008) to study the BS population in M67. This configuration is also adopted in the current work. We do not consider the influence of dynamic evaporation since we only investigate primordial binaries which may form BSs. These primordial binaries must be more massive than current MS turn-off mass, which is much greater than the average mass of the cluster (0.6 M for M67). Dynamic evaporation is unlikely to influence the result by much. 3.1 Results and discussions A Monte Carlo simulation is performed using the distribution of initial parameters listed above. We obtained M V and (B V) 0 by a linear interpolation in the grid of models of 4.0 Gyr of age. The models are described in Fig. 4 and in Table 2. 1 We assume that mass transfer does not occur in the binaries whose initial parameters from Monte Carlo are beyond the range of the grid. To make the discussions more convenient, we would like to define a few terms that will be used in the following text. 1 The complete table and the evolutionary tracks of all the models are available in electronic form at see also Supporting Information.

6 1018 P. Lu, L. C. Deng and X. B. Zhang Figure 5. The simulated CMD for the primordial BSs in M67. The left-hand panel represents β = 1.0 and the right-hand panel β = 0.5. The solid line is the 4.0 Gyr theoretical isochrone. The dashed line is the equal-mass photometric binary sequence. The filled circles are our simulated BSs. The open triangles are the results from TDHZ06 and the open squares are from Hurley et al. (2005). The plus signs are the observed BSs from Deng et al. (1999) with a distance modulus of m M = 9.55 and a reddening of E(B V) = (Carraro et al. 1996; Fan et al. 1996). The dotted line shows the upper bound of the simulated primordial BSs formed via Case A in TDHZ06. Photometric binary: all binaries with the two components physically bound, whose photometric properties are given by direct integration of the light of the two stars. Progenitor of BS: a close binary system undergoing mass transfer and evolving towards a BS. Descendent of BS: a star evolving away from its BS phase. The result for M67 is shown in the left-hand panel of Fig photometric binaries experiencing mass transfer are obtained from our simulation and their parameters are listed in Table 3. The number seems to be less than in TDHZ06 who obtained 19 photometric binaries (open triangles in Fig. 5) experiencing mass transfer. The reason could be that the binaries undergoing Case A RLOF have a larger range of primary masses: from 1.1 to 1.6 M at 4.0 Gyr. However, most of them with primary mass less than 1.4 M only contribute to the progenitor population of BSs which cannot be distinguished from photometric binaries below turn-off from observation. In our Case B models, binaries with primary mass less than 1.3 M have not experienced mass transfer at 4.0 Gyr and those with a primary more massive than 1.5 M have already ended their evolutions. Only those models with primary mass Table 3. The parameters of the photometric binaries from our Monte Carlo simulations. M 1 q a V B V Comments V (B V ) Comments (M ) (M 2 /M 1 ) (R ) BS, Case A BS, Case A BS, Case A BS, Case A BS, Case B BS, Case B BS, Case B BS, Case B BS, Case B BS, Case B BS, Case B BS, Case B BS, merge BS, merge BS, merge BS, merge BS, merge BS, merge PBS, Case B PBS, Case B DBS, Case B BS, Case B DBS, Case B BS, Case B DBS, Case B DBS, Case B DBS, Case B DBS, Case B The columns are as follows: (1) the initial mass of the donor; (2) the initial mass ratio; (3) the initial orbital separation; (4) the Johnson V magnitude; (5) B V colour; (6) the comment on photometric binary property: PBS-progenitor of a BS, DBS-descendant of a BS; (7) (9) are the same as (4) (6), respectively, but for β = 0.5.

7 Binary evolution and blue stragglers 1019 between 1.3 and 1.5 M contribute to the present BS population and nearly all of them stay in the BS region. Our simulation is based on only those Case A and Case B models with a primary mass between 1.3 and 1.6 M. Among the 19 photometric binaries obtained in TDHZ06, only four were in the region of BSs. 13 out of the 19 photometric binaries were progenitors of BSs and the remaining two were identified as descendants of BSs. Our simulation predicts eight BSs, one progenitor and three descendants. Among the eight BSs, two formed via Case A, three by way of Case B and three from dynamical merger models. Two of the three most luminous BSs given in the blue region in Fig. 5 formed via Case B while the other was produced by merging. The total number of BSs is nearly twice as much as in TDHZ06 when only the Case A mechanism was considered. The results of Hurley et al. (2005) using an N-body simulation are also plotted in Fig. 5 for comparison. They predicted 11 single BSs produced by mergers of MS stars in primordial binary systems, nine formed via the onset of Case A mass transfer and eventual coalescence due to angular momentum loss. Our number of simulated BSs undergoing RLOF is less than their prediction. The reason could be that they include dynamic evolution in N-body simulation and a BSE algorithm (Hurley, Tout & Pols 2002) to deal with binary evolution. Dynamic evolution could affect binary orbital separation by encounters. Nearly all their simulated BSs have experienced encounters with a single star or another system. We are just concerned with detailed binary evolution in our work and do not consider dynamic evolution. Mass transfer efficiency, β, defined as the ratio of mass accreted by the secondary to matter lost by the primary is still a major issue in the evolutionary calculations of close binaries. This has been a very uncertain parameter till now. A single constant has been adopted for many calculations (e.g. de Greve & de Loore 1992; Chen & Han 2002a, 2009). As a comparison, we recalculate all our models using a constant mass transfer efficiency of 0.5, except for Case A. The non-conserved case is shown in the right-hand panel of Fig. 5. By taking β = 0.5, material accreted by the secondary obviously decreases, which causes a decrease in luminosity when compared with the conservative case. On the other hand, decreased accreted material slows down the evolution of the secondary, especially the time spent on the main sequence in other words, the time spent as a BS. This means that binaries with more massive initial primaries may still exist as BSs at the age of 4.0 Gyr. This result is shown in Fig. 5 (right-hand panel). Compared with previous results, more BSs are obtained and most of them are concentrated in the lowluminosity area around the main sequence turn off. The parameters of BSs are listed in Table 3. Comparing these two cases with observations, the set of β = 1.0 could reproduce the observed luminous BSs better. Our conclusions are somewhat affected by the way we construct mergers. However, the influences are mainly limited to binaries with a small initial q value. The main feature affecting the result is the composition of mergers (which will influence the lifetimes of mergers on MS and their positions on CMD at 4.0 Gyr) whereas the fully mixed model is an extreme case. Any alternative will lead to a shorter MS lifetime. Assuming all low q mergers are fully mixed, Fig. 4 shows low q binaries with 1.4 M initial primaries in open circles; they are very bright and blue at 4.0 Gyr. Their time-scale of t contact is about 3.4 Gyr and their composition right after full mixing is about X 0.60, Y Those binaries with 1.3 M initial primary mass have not come into contact and those with 1.5 M initial primaries have ended their evolutions already. BSs formed via mergers are mainly formed by low q binaries with a primary mass between 1.3 and 1.4 M. If we preserve the core of the primary instead when we construct the mergers, the helium content of the mergers will be significantly enhanced in cores because the primaries have already evolved to HG before contact. The result could be that mergers formed with 1.4 M primaries also end their evolution on the MS. Thus their contribution to BSs can be significantly reduced. BSs were obtained by a linear interpolation of three parameters. They are in order as follows: orbital separation, mass ratio and primary mass. For better statistics, an artificially enlarged sample is used in the simulation. In order to investigate the distribution of BSs on the CMD from our models, an enhanced number of primordial binaries (N = ) is given and the same simulation method as above is used. The number of primordial binaries is the same as in TDHZ06 for the purpose of comparison. We now compare the results of TDHZ06 to the present work. The result is shown in the left-hand panel of Fig. 6. In TDHZ06, 11 per cent of the 1984 photometric binaries are BSs and most of these photometric binaries stay between the turn off and about 2 mag below the turn off of a 4.0 Gyr old theoretical isochrone. TDHZ06 also defined an upper bound for magnitude and colour of BSs formed via Case A; this profile is shown in Fig. 6 with a short dashed line. By taking into account Case B scenario, the BS population from our simulation exceed their upper limitation. We select BSs using a criterion of M V < and (B V) 0 <0.533 which is the V magnitude and B V colour of the turn off of the 4.0 Gyr isochrone. Among the primordial binaries, only those in the range of our grid of models can contribute to BS population in the current work. By limiting the initial primary mass to M, the number of initial primordial binaries is reduced from to Another limitation, initial orbital separation of 9.0 R, cuts the number down further to Finally, 810 photometric binaries were obtained after the interpolation as given in Table out of the 810 photometric binaries are in the BSs region. Among the 507 BSs, about 51 per cent are formed via Case B, 17 per cent from mergers and 32 per cent from Case A. By taking β = 0.5 for our Case B models, we obtained 809 BSs using the same enhanced primordial binaries and the same selecting criterion of BSs. The result is shown in the right-hand panel of Fig. 6. In order to compare our simulations to observations, a sample of 24 photometrically selected BSs (Deng et al. 1999) is also plotted in Fig. 6 designated by plus signs. Three luminous BSs in M67 (S1434, S1066 and S0968) could be reproduced by our simulation with an enhanced number of primordial binaries; however by lowering the number of primordial binaries to N = , only few luminous BSs can be produced. The reason could be that the binaries with initial primary mass around 1.4 M are contributing to luminous BSs; but there are no data for q>0.466 or q < with a primary mass of 1.5 M in our grid at the age of 4.0 Gyr (since those models have ended their evolutions before 4.0 Gyr). 80 primordial binaries with primary mass between 1.4 and 1.5M, q > or q < are ruled out because the interpolation in this area cannot be done, but actually some of them could produce luminous BSs or descendants of BSs. Their positions on the CMD could be above the long dashed line in Fig. 6. Among all the BSs made by enhanced primordial binaries, about 17 per cent could be the result of dynamic mergers by our simplified treatment introduced in Section 2. We do not include the remnants of binary coalescence from Case A. The coalescence in Case A was discussed in Chen & Han (2008). The result showed mergers from conservative evolution are located in the region of high luminosity.

8 1020 P. Lu, L. C. Deng and X. B. Zhang Figure 6. The simulated CMD with enhanced primordial binaries: β = 1.0 on the left-hand panel and β = 0.5 on the right-hand panel. The points are the BSs formed by our models. The long dashed line is the lower bound of possible positions for those BSs formed by 1.5M >M 1 > 1.4M and q > The solid line, short dashed line, dotted line and the plus signs have the same meaning as in Fig. 5. However, by using primordial binaries, they obtained only one BS with high luminosity. The most luminous BS obtained from our calculation is 0.5 mag below F81, which is the most luminous BS in M67, with (B V) 0 = 0.10, V= It cannot be explained well by the mass transfer scheme in this work or any other scenarios. Chen & Han (2008) investigated binary coalescence and, with the assumption of a cluster age of 3.8 Gyr and a coalescence time-scale of 10 9 yr, their coalescence models can produce BSs as bright as F81. But the assumptions are still debatable since the physics in coalescence is not well understood. Such a luminous BS could also be formed in triple system as discussed by Hagai & Daniel (2009). Very close inner binaries can be also induced by triples and thus lead to the production of BSs by mass transfer or merger. We do not take the effects of dynamic evolution into account. BSs generated from our models are in the range of primary mass from 1.3 to 1.5 M. Consider the fact that low-mass stars tend to escape while the massive ones tend to sink into the centre because of tidal forces. Dynamic evolution of clusters could also result in the collision of binary binary or binary single systems which would cause binaries to be tighter or farther apart and would cause exchange interactions. This could also affect the number of BSs. Only seven of 20 BSs evolved from unperturbed primordial binaries in Hurley et al. (2005). The dynamic environment was instrumental in producing approximately half the BS population. 4 SUMMARY AND CONCLUSIONS We investigated BSs generated from mass transfer scenarios in our work including both Cases A and B. In reality, BSs are believed to be produced via various mechanisms (Piotto et al. 2004). No single mechanism can account for all the BSs observed in a cluster. In general, BSs could be formed through direct collisions, which are considered to be crucial in dense stellar environment (Fregeau et al. 2004), or by mass transfer in primordial binaries, which is believed to dominate BSs formation in a sparse environment (Mathys 1991). Based on the scheme presented by McCrea in 1964, we use Eggleton s stellar evolution code to calculate a grid of close binary evolution models undergoing both Case A and Case B RLOF. Example systems are shown in Figs 2 and 3 to compare close binary evolution behaviour in Case A and Case B mass exchange scenarios. BSs formed via Case B show bluer colours, and they stay in BS region for a longer time than in Case A. Binaries undergoing both scenarios can stay in BS region even longer and constitute BSs in a given population (clusters). Based on the grid of all our models at 4.0 Gyr of age, a Monte Carlo simulation of M67 was made. Two cases with different mass transfer efficiency are considered. The selection of a high value of β can reproduce luminous BSs well while low-value case can match the number of BSs with low luminosity well. Both results show that the number of BSs could be significantly increased by taking into account Case B. BSs from our simulation can cover the region of all observed luminous BSs in M67 except for F81, which is therefore excluded from the simple binary mass exchange channel. We consider only BSs formed via the Case A and Case B schemes in this work. It is difficult to associate the contribution of the mass transfer scenario in primordial binaries with the number of BSs in old open clusters because both the primordial binary population and the dynamic environment play essential roles in generating BSs. The dynamic processes destroy BSs while creating new ones through capture and exchange processes (Hurley et al. 2005). Magnetic activity, as discussed in Section 2, can also affect the number of BSs in old clusters since, while magnetized stellar winds probably do not carry away much mass, they could remove a lot of angular momentum. There are a number of papers including the treatment of angular momentum loss (AML) (Li, Han & Zhang 2004; Demircan et al. 2006; Micheal & Kevin 2006; Stepien 2006). Chen & Han (2008) also examine the effect of AML in M67. They deal with low-mass binaries with primary mass between 0.5 and 1M. Their result indicates that AML in low-mass binaries could be much more important in this old open cluster for the region with low luminosity and range of colours.

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