ASTRONOMY AND ASTROPHYSICS. D. Vanbeveren, J. Van Bever, and E. De Donder

Transcription

1 Astron. Astrophys. 317, (1997) ASTRONOMY AND ASTROPHYSICS The effect of binary evolution on the theoretically predicted distribution of WR and O-type stars in starburst regions and in abruptly-terminated star formation regions D. Vanbeveren, J. Van Bever, and E. De Donder Astrophysical Institute, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Received 7 March 1996 / Accepted 3 June 1996 Abstract. We first discuss in detail the massive close binary evolutionary model and how it has to be used in a population number synthesis study. We account for the evolution of case A, case B and case C systems, the effect of stellar wind during core hydrogen burning, hydrogen shell burning, the red supergiant phase and the WR phase, the effect of common envelope evolution in binaries with large periods, the consequences of spiral in in binaries with small mass ratio, the effect of an asymmetric supernova explosion on binary system parameters using recent studies of pulsar velocities, the evolution of binaries with a compact companion. The parameters entering the population model where close binaries are included, are constrained by comparing predictions and observations of the massive star content in regions of continuous star formation. We then critically investigate the influence of massive close binary evolution on the variation of the massive star content in starburst regions. We separately consider regions where, after a long period of continuous star formation, the star formation rate decreases sharply (we propose to call this an abruptly terminated star formation region) and we show that also in these regions WR/O number ratios are reached which are significantly larger than in regions of continuous star formation. The most important conclusion of the study is that within our present knowledge of observations of massive stars, massive close binary evolution plays an ESSENTIAL role in the evolution of starbursts and abruptly terminated star formation regions. Key words: stars: evolution binaries: close Wolf-Rayet stars: early-type 1. Introduction The variation of the O type star and WR star content in starburst regions has been discussed by Meynet (1995). However only single stars were considered. It was assumed somewhat arbitrarily that close binaries play no role in determining massive star numbers, although the author correctly pointed out that no definitive conclusion can be drawn without explicit computations. It is surprising that a similar conclusion but for regions of continuous star formation was drawn by Maeder and Meynet (1994). As illustrated by Vanbeveren (1995), the latter conclusion is based on an error in the formalism of Maeder and Meynet. A correct one then leads to the conclusion that binaries play a very significant role in the determination of the WR/O number ratio, not only in regions with small metallicity (the Magellanic Clouds) but also in the Galaxy. We therefore decided to investigate critically the assumption of Meynet for starburst regions. Close binary evolution can affect the WR and O type star population in regions of continuous star formation and in starbursts, because of three main reasons: the WR lifetime of binary components is different from the WR lifetime of single stars, WR stars in close binaries may have lower mass progenitors compared to single WR stars, the occurrence of the class of rejuvenated O type mass gainers which may be single or may have a compact companion depending on the physics of the previous supernova explosion; the evolution of this class not only makes the WR phase of starbursts longer, it may also produce a class of WR stars which do not have a single star counterpart. In Sect. 2 we discuss in detail the massive close binary evolutionary scenario and how it can be used in a population number synthesis model. Special attention is given to those cases which have been barely touched in the past but which can lead to observable WR binaries. The evolutionary computations used in the present paper are summarized in Sect. 3. Sect. 4 deals with our population model. The inclusion of binaries obviously complicates things and a number of parameters enter the population model which at present cannot be determined from first principles. In Sect. 5

2 488 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution and 6 we discuss observations which may help to constrain these parameters. In Sect. 7 we then combine single stars and close binaries in order to compute the evolution of starbursts as a function of metallicity. The computations are compared to observations of so called WR galaxies (Vacca and Conti, 1992). We separately consider the case of a region where after a long period of continuous star formation, the star formation rate decreases sharply. We will call this an abruptly terminated star formation region. It will be shown that this abruptly terminated star formation model also gives WR/O number ratios which are significantly larger than the ratio in regions of continuous star formation (Sect. 8). 2. The evolutionary model of massive close binaries A star is called a massive star (MS) when it experiences all nuclear burning phases non degenerately and finally explodes as a supernova (SN). The minimum mass M sn,min for this to happen depends somewhat on the detailed physics of the stellar interior and ranges between 8 M and 10 M. A close binary (CB) is called a massive close binary (MCB) when one or two components have a mass larger than M sn,min. Remark that although a CB may start its life as a non MCB, due to mass exchange the secondary may gain mass and the CB turn into an MCB. As illustrated by van den Heuvel and Heise (1972), we distinguish the following evolutionary phases of an MCB: the pre RLOF phase (RLOF = Roche lobe overflow) where one or both components are losing mass by stellar wind (SW), the first RLOF phase (time scale a few years or less) where the originally most massive component (= primary or mass loser or mass donor) loses most of its hydrogen rich mass layers and where possibly part or all of this lost mass is captured (accreted) by the secondary component (= mass gainer or accretor), the post RLOF phase where the primary is a hydrogen poor core helium burning (CHeB) star (possibly a WR star) or a core carbon burning (CCB) star and the companion looks like a normal OB type star, the SN explosion of the primary leaving behind either an OB type runaway with a compact companion (CC), or a single OB type runaway and a compact single star (a neutron star or black hole). In the latter case the OB type runaway further evolves like a single star. As the OB type companion in an OB+CC binary expands during its CHB phase and its ensuing hydrogen shell burning, the CC is dragged into the OB star envelope and spiral in occurs. The final product is either a very short period binary (period of hours) with a hydrogen poor CHeB star (possibly a WR star) with a CC, or a merged binary, i.e. a star, possibly a red supergiant, with a CC in its center (a Thorne Żytkow object, Thorne and Żytkow, 1977). Although the further evolution of such massive objects is still rather uncertain, in Sect. 2.5 we propose two possibilities where we predict the existence of weird hydrogen poor CHeB stars and, if the latter are massive enough, weird WR stars (i.e. WR stars with a CC in their center). We now consider in more detail the different evolutionary phases discussed above, leading to a general evolutionary model for MCBs that can be used in a population number synthesis model The pre RLOF phase We assume that prior to RLOF both stars in a binary evolve like single stars. The majority of the massive stars is characterized by three major expansion phases: the core hydrogen burning (CHB) phase, the hydrogen shell burning and the helium shell burning after CHeB. An MCB is called case A (resp. case B, case C) when its massive primary starts losing mass as a consequence of RLOF during the first (resp. second and third) expansion phase, and this obviously depends on the binary period. A massive star during hydrogen shell burning either has a radiative envelope, or a convective envelope. Since the reaction of a star to mass loss due to Roche lobe overflow depends on whether the envelope is radiative or convective (Paczynski and Sienkiewicz, 1972), case B is further subdivided into case Br and case Bc. Massive stars lose mass by SW. The SW is more or less spherical symmetric and this means that in binaries evolving through a SW phase, the period varies like (Hadjidemetriou, 1967) P P 0 = ( ) 2 M10 + M 20, (1) M 1 + M 2 (the index 0 refers to values at the onset of the SW phase). The evolution of MCBs depends on the adopted stellar wind mass loss rate formalisms during CHB, hydrogen shell burning and the red supergiant (RSG) phase. We assume that the rates in binary components are similar as the rates in single stars with the same mass. A single star with mass larger than M (a very massive star, VMS) becomes a luminous blue variable (LBV) at the end of CHB or the beginning of hydrogen shell burning. The LBV loses mass by stellar wind at a very large rate, although its value is highly uncertain. The evolution of a VMS depends obviously on this LBV mass loss formalism. When the initial mass of a single star is larger than some minimum value M s,min, the SW mass loss during its RSG phase (i.e. the first part of its CHeB phase) is large enough to remove most of the star s hydrogen rich layers. The star stops expanding, returns to the blue part in the HR diagram and becomes a WR star. It is clear that in a binary with primary mass larger than M s,min the evolution of case Bc and case C binaries may be significantly affected by this RSG mass loss process (the RSG scenario of MCBs, Vanbeveren, 1996). The value of M s,min is somewhat uncertain and depends on the adopted stellar wind mass loss rate formalism. Schaller et al. (1993) published an extended grid of single star evolutionary computations for the Galaxy, the LMC and the SMC where the effect of SW in pre WR phases is calculated using the formalism of de Jager et al. (1988). We call this scenario the standard single star scenario.

3 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution The LBV scenario of very massive close binaries When a star with mass larger than M is the primary of a binary (a very massive close binary, VMCB) with period such that RLOF starts during or after the LBV phase, the mass loss due to RLOF may be significantly reduced as a consequence of the LBV SW mass loss. If the SW mass loss is large enough the RLOF can be avoided and no binary interaction occurs there (the LBV scenario of VMCBs, Vanbeveren, 1991). This scenario will be used here and we will use 40 M as lower mass limit the RSG scenario of MCBs A single star with mass between M s,min and 40 M loses almost all its hydrogen rich layers by stellar wind mass loss during the RSG phase. This means that in binaries with such a star as primary where the period is large enough so that RLOF starts during or after the RSG phase (late case Bc, case C), the total mass loss due to RLOF may be largely reduced by the SW mass loss process (the RSG scenario of MCBs, Vanbeveren, 1996). The value of M s,min depends on the adopted single star scenario, i.e. 30 M with the standard single star scenario grid and 20 M with the alternative single star scenario. Fig. 1. Limiting periods between the different cases of close binary evolution. When going from bottom to top, the first line (1) gives the separation between case A and case Br, the second (2) gives P min for q = 0.6, the third line (3) gives P min for q =0.9, the fourth line (4) gives the separation between case Br and case Bc, the fifth line (5) gives the separation between case Bc and case C, and the last line (6) gives the limiting period in order to have interaction. For primary masses above 40 M, the LBV scenario is assumed to be operational and thus neither case B nor case C exists. When the effect of binaries is correctly included the WR and O star distributions are not well predicted with the standard single star scenario. Therefore Vanbeveren (1991, 1995, 1996) proposed a much larger SW mass loss rate during the RSG phase of a massive single star than the de Jager et al. rates (i.e. larger by more than a factor 10 for a 20 M star). Observations of red supergiants in the LMC seem to confirm this proposition (Feast, 1992). With this larger RSG stellar wind the WR/O number ratio is much better reproduced. This scenario will be referred as the alternative single star scenario. In Fig. 1 we show the limiting periods between case A, B and C as a function of primary mass for different initial binary mass ratios (= mass secondary/mass primary) on the basis of the foregoing discussion. We also give the limiting period between case Br and case Bc The first RLOF phase We list here the different processes we have to account for in our MCB evolutionary model Binaries with q 0.2 The classical Roche model loses its significance in binaries not belonging to one of the two categories discussed above but with an initial mass ratio q 0.2 (Sparks and Stecher, 1974; Paczynski, 1976). As the primary evolves and expands, the lower mass secondary is dragged into the envelope of the primary. The further orbital evolution is determined by frictional forces in the common envelope, a process generally referred to as spiral in (Livio and Soker, 1988; Taam and Bodenheimer, 1989, 1991). Although detailed 3 dimensional computations of this process do not exist yet, it is possible to get an idea of the possible remnant binaries after such a spiral in phase using the prescription discussed by Webbink (1984). Due to friction the orbital energy of the secondary is converted with some efficiency α CE into potential energy of the envelope of the primary. When it is assumed that the mass of the secondary M 2 remains constant, the variation of the distance a between both components is given by GM 10 M λr 10 a 0 = α CE ( GM1 M 2 2a GM ) 1 0 M 2, (2) 2a 0 M = the mass that can be removed from the primary when the orbital separation decreased from a 0 to a, and thus the remaining mass of the primary M 1 = M 10 M. The index 0 refers to values prior to the spiral in phase. An interpolation formula for r 1 (= the dimensionless Roche radius of the primary) has been given by Eggleton (1983), i.e. r 1 = q 2 3ln(1 + q 1 3 ). (3) Possible values for α CE range between 0.3 and 0.7 (Taam and Bodenheimer, 1989; Livio and Soker, 1988); however a fully satisfactory value does not yet exist. We therefore leave

4 490 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution α CE as a parameter entering the MCB model. Its influence on population synthesis will be discussed in Sect. 6. The prescription contains a parameter λ which is a measure of the degree of central concentration of the envelope of the primary. If the envelope is highly convective (case Bc, case C) λ is of order unity (we will use λ = 1). In case A and case Br however the primary envelope is in radiative equilibrium for which λ 0.5 (see also Bhattacharya and van den Heuvel, 1991). A primary in a case B (resp. case C) close binary stops expanding when it has lost most (resp. all) of its hydrogen rich layers, i.e. when its mass roughly equals the mass of the helium core formed during core hydrogen burning. The star quickly restores thermal equilibrium, contracts and moves to the blue part of the HR diagram (the He burning main sequence). It is clear that if the companion star did not yet spiral in into the helium core of the primary, then spiral in should stop. When however Eq. (2) (with the mass of the whole hydrogen envelope and M 1 = the mass of the helium core) predicts that either the Roche radius of the CHeB remnant or the Roche radius of the companion (or both) is smaller than the equilibrium radius of the CHeB star (resp. the companion), it is assumed that the binary merged. We will show that early case B binaries with q 0.2 always merge. Since the available orbital energy of the secondary is smaller in case A systems whereas the potential energy of the envelope of the primary is larger at the beginning of the spiral in process, it is logical to assume that case A binaries with mass ratio q 0.2 merge as well. The further evolution of merged binaries is unknown at present. We will present two possibilities and study the effect on our population synthesis results (Sect. 6) Case A and case Br MCBs with q > 0.2 Detailed evolutionary computations of case A and case Br MCBs with q > 0.2 were performed in the last two decades (see the reviews of Vanbeveren, 1991, 1994, 1996 and references therein). In order to decide how much mass lost by the primary is accreted by the secondary, we need to consider in detail the physics of accretion and the reaction of a star when it gains mass. The behavior of a mass accreting star can be studied from a phenomenological point of view using two assumptions which constitute two limiting cases, i.e. model I: accretion makes the whole star convective (Vanbeveren et al., 1994; Vanbeveren and de Loore, 1995), model II: if the outer layers of the mass gaining star are in radiative equilibrium, accretion does not destroy it (the standard accretion model, Neo et al, 1977). Before we propose a general model for mass accretion in MCBs, we account for the following remarks: a. Accretion of matter lost by a primary during its RLOF proceeds either by direct hit or through a (Keplerian) disk. The first mode occurs when the distance between both components is too small to form a disk, i.e. preferentially small period (early case B) and/or large mass ratio. Using the hydrodynamical calculations of Lubow and Shu (1975) and our set of binary evolutionary computations, Fig. 1 gives the limiting period P min between the two modes as a function of initial primary mass for different mass ratios. b. When accretion proceeds by direct hit, the accretion effects are very local (hot spot) and it is conceivable that accretion model II is appropriate here. c. When the binary period P is larger than P min, transferred matter first forms a disk. If the physics of these disks is similar to the physics of accretion disks during star formation, it could be that accreted matter settles down onto the mass gainer with a significantly lower entropy than the entropy of the outer layers of the star. This situation is unstable and convection is the result. We therefore propose that significant mixing of the mass gainer (and thus accretion model I) could be efficient when mass transfer proceeds through a disk. d. If prior to accretion a Keplerian disk is formed, it can readily be checked that only a small amount of mass has to be accreted by the gainer in order for the outer layers to rotate at break up velocity (e.g. Packet, 1981). If the rotation is able to penetrate deep enough in the star (i.e. if viscous shear is strong enough to sweep up a large part of the star, may be the whole star), the ensuing turbulent diffusion process (Zahn, 1994) will help to mix the massive star with the disk in an efficient way and therefore strengthen our proposal that large scale mixing in massive mass gainers is very efficient when during the mass transfer phase a disk is formed first. e. When the binary period is smaller so that the gas stream during RLOF hits the gainer directly, the transport of angular momentum is smaller. Furthermore in the absence of large scale turbulence or convection, the major contributor to the viscosity in the outer layers of a star is the radiative viscosity. Since this is very small, even if the mass transfer in this case is able to sweep up the outer layers of the gainer, it is hard to believe that the deeper layers will be dragged with the outer ones in an efficient way. Tidal interaction in such smaller period binaries may therefore find it quite easy to slow down these outer layers again. Taking account of the foregoing remarks, we are tempted to propose the following accretion model for a secondary in a massive case Br binary: when P P min, accretion proceeds by direct hit so that accretion model II applies. The accretor is not necessarily a long lasting rapid rotator, when P > P min, a disc is formed first. Accretion may induce efficient mixing of the accretor which becomes a rapid rotator. If during the RLOF phase significant mass loss from the system occurs on the Kelvin Helmholtz time scale (= the RLOF time scale), this can to our knowledge only happen through the following two processes: process a: material leaves the binary through the second Lagrangian point L 2 forming a ring around the binary,

5 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution 491 process b: material lost by the primary gains sufficient energy from the orbit by dynamical friction in order to be pushed out of the binary. For the purpose of the present paper we will use the formalism of Vanbeveren et al. (1979): the period relation before and after RLOF is given by ( ) 3 ( ) 3α+1 P M10 M 20 M1 + M 2 =, (4) P 0 M 1 M 2 M 10 + M 20 (the index 0 denotes values at the beginning of the RLOF). When matter leaves the binary through L 2 (process a), a comparison of formula 4 with the particle trajectory calculations of Flannery and Ulrich (1977) results in α values 6. When the process b is responsible for driving matter out of the binary, we can try to estimate α by using a similar description as the spiral in process of Webbink (1994). If β equals the fraction of the mass lost by the primary which is accreted by the secondary, an equation similar to Eq. (2) can be derived for the amount of mass which has to be removed from the system: ( GM 10 (1 β) M GM1 M 2 = α CE GM ) 1 0 M 20, (5) λr 0 2a 2a 0 M 2 = M 20 + β M,M 1 = M 10 M. The parameter λ describes the relative binding energy of the mass (1-β) M to the primary, R 0 is the Roche radius of the primary at the onset of the RLOF. When β = 0.5, also the foregoing prescription corresponds to large α values in formula 4 (i.e. α 6 when q = 0.9 and α 3 when q = 0.3). In order to study the effect of the parameters α and β on our population model, we first define q min (> 0.2) as a minimum value of the mass ratio above which β in case A and case Br MCBs is assumed to be constant. For binaries with q q min, we made our computations by assuming β = 0.5 (α = 3 and 6) and β = 1 (conservative). The results will be given for q min = 0.4 and 0.6. In between q min and 0.2 (where the spiral in process is at work and thus β = 0), we assume that β varies linear with q and we also use either α =3orα= 6. Remark that how β varies between q min and 0.2 does not significantly affect our results Case Bc and case C binaries If the primary mass is smaller than 40 M case Bc binaries exist and if the primary mass is smaller than M s,min,sodo case C binaries. When the primary fills its Roche lobe it has a deep convective envelope. The adiabatic response of such a star to mass loss due to RLOF is quite dramatic (Paczynski and Sienkiewicz, 1972): the larger the mass loss the faster the star s expansion and the mass loss becomes even larger, i.e. the mass loss process in case Bc, case C is a very fast dynamical process. The secondary which in most cases is a normal CHB star, is unable to accrete this mass and a common envelope forms. Also this phase can be modeled by means of the spiral in prescription (formula 5 with β = 0) where, since the envelope is highly convective, the parameter λ =1. Common envelope evolution stops when the primary has lost most of its hydrogen rich layers, i.e. at the beginning of CHeB when the atmospheric hydrogen abundance X atm for case Bc (just like in case Br) and at the beginning of CCB when X atm = 0 for case C. The star quickly restores thermal equilibrium and moves to the blue part of the HR diagram (the He or C burning main sequence). When as a consequence of the spiral in description discussed above one of the Roche radii (or both) of the components of the remnant binary is smaller than its equilibrium radius, it is assumed that the binary merged. The further evolution of such merged systems is unknown and we decided to ignore them. As will be discussed further on, case Bc and case C systems are a minority among MCBs so that ignore them has little effect on our results The post RLOF phase The evolution after RLOF or after spiral in, during the CHeB phase of the stripped primary in a case A or case B binary, depends on the SW mass loss formalism. The only CHeB remnants after RLOF in an MCB that are visible are the WR stars (a possible exception is Φ Per). WR stars consist of a core with a surface temperature of about K and a dense stellar mantle which redistributes the energy and emits it at considerably lower temperatures ( K). The stellar mantle is also responsible for the typical emission line spectrum of the WR. A SW mass loss rate formula for WR stars as function of luminosity has been proposed by Langer (1989), Vanbeveren (1991) and Hamann et al. (1995). They all give very similar results when applied in an evolutionary code. WR stars in binary components mostly have masses larger than 5M (however see Herbig et al., 1964) (the corresponding minimum CHB mass is called M b,min ; depending on the evolutionary details during the CHB phase of a massive star, its value ranges between 16 M and 18 M ). However this does not necessarily tell us that for CHeB binary remnants with mass lower than 5 M, the SW mass loss is zero; rather, it tells us that the SW mass loss has become too small to make a mantle which is sufficiently thick and extended to form a typical WR spectrum. Most of the observed WR stars have SW mass loss rates larger than 10 5 M /yr. One could conclude that a mass loss rate that high is necessary in order for a CHeB post RLOF remnant to show a WR like spectrum. When the SW formalisms discussed above are used, it is interesting to notice that once the mass of the CHeB star drops below 5M, the mass loss rate drops below 10 5 M /yr and the star does not any longer show a WR like spectrum although SW mass loss keeps going on. We therefore think that the SW mass loss formalism discussed above applies to all post RLOF MCB CHeB remnants without mass limitation. The period variation during the post RLOF phase is obviously also calculated by means of formula 1. Remark that since the primary remnant after case C mass loss is a He shell burning/core carbon burning star, the remaining lifetime up to the SN explosion is very short. Therefore the contribution of these systems to the WR + OB binary population should be very small.

6 492 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution 2.4. The SN explosion of the primary At the end of its life the primary of an MCB experiences an SN explosion, the final remnant being a compact star, a neutron star or a black hole. A small asymmetry of the SN ejecta is sufficient to give the compact star a substantial kick velocity v kick. The influence of an asymmetric SN explosion on system parameters has been studied by Sutantyo (1978) (see also Verbunt et al., 1990, and Wijers et al., 1992). We summarize here the basic formulae which are used in our model. It is convenient to use the following parameters: µ = M 1 0 M 20 M 10 + M 20, (6) m = M M 20 M 1 + M 20, (7) v = v kick, (8) v orb v 3 orb = 2πG( ) M 10 + M 20, (9) P δ = M 10 M 1, (10) θ is the angle between the direction of v kick and the initial orbital plane, and φ is the angle between the projection of v kick onto the orbital plane and the pre SN velocity vector of the exploding star with pre SN mass M 10. The index 0 refers to values prior to the SN explosion whereas M 1 = mass of the compact remnant after the SN explosion; we will use 1.4 M = the average mass of a neutron star. We also made some test calculations where for the most massive stars the compact remnant isa3m black hole. If we define γ = 1 2v ( 2 m 1 v2 ), (11) the requirement for a bound binary after the SN explosion is given by cosφcosθ <γ. (12) When it is assumed that the direction of the kick velocities is isotropic, the probability p for a bound orbit equals 1 if γ 1, p = 0.5(1 + γ)if-1 γ 1andp=0ifγ -1. The relation between the post and pre SN period in the bound case is given by P P 0 = m [2 m(1+2vcosφcosθ + v 2 )] 3, (13) or by Kepler s law a relation for the semi major axis a 1 = a 0 2 m(1+2vcosφcosθ + v 2 ). (14) The post SN eccentricity e can be expressed as e = 1 m a 0 a [(1 + vcosφcosθ)2 +(vsinθ) 2 ]. (15) The system velocity (= runaway velocity of the system = v rw ) in case the binary remains bound is given by v rw = with F (M 1 )= v orb M 1 + M 20 F (M 1 ), (16) ( µδ M 10 ) 2 2 µδm 1 M 10 vcosφcosθ +(M 1 v) 2. (17) When the binary is disrupted, the runaway velocity of the single OB type star equals its orbital velocity with respect to the center of mass just before the SN explosion, i.e. v rw = v orb M 10 M 10 + M 20. (18) In order to determine the post SN population, starting from a pre SN population of massive binaries, we obviously need the distribution function of kick velocities. Using recent measurements of pulsar proper motions (Harrison et al., 1993) and a new pulsar distance scale (Taylor and Cordes, 1993), Lyne and Lorimer (1994) obtained a pulsar velocity v p distribution which can very well be described by the following relation: f(v p )= v p 3/2 e 3vp/514. (19) We assume that these velocities reflect the kick velocity that a compact star may get as a consequence of an asymmetric SN explosion. Remark that this distribution implies an average kick velocity of 450 km/s which is substantially larger than any previous estimate. We therefore expect that a large number of binaries will become unbound during the SN explosion The post-sn phase The single OB type stars of disrupted binaries further evolve as normal single stars where we will use the two different single star evolutionary scenarios already mentioned earlier. If the SN explosion did not disrupt the binary, we distinguish the following possibilities: a. the OB star mass is larger than 40 M and it becomes an LBV star prior to the onset of spiral in. In this case the OB type star will lose its mass by SW (LBV scenario) and will evolve into a WR star without the interference of the CC. Since SW mass loss is assumed to be isotropic (the mass of the CC is probably too small in order to destroy this isotropy), it follows that the evolution of the binary period can be described by Eq. 1 where the index 0 refers to values at the onset of the LBV mass loss process. This type of evolution thus implies a period increase. Suppose a neutron star mass M 10 = 1.4 M and M 20 =40M. Then the corresponding WR mass (WN) has a mass M 2 =20M and thus P/P 0 4. b. M s,min the mass of the OB star 40 M. When the period is large enough the RSG scenario applies, i.e. the OB type

7 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution 493 Fig. 2a. The intermediate mass or massive close binary evolutionary model up to the end of core helium burning of the original primary. We consider three mass intervals; M 1 is the initial mass of the primary, M b,min is the minimum initial mass a primary of a close binary must have in order to evolve into a WR star after Roche lobe overflow (RLOF); M s,min is the minimum initial mass of a star above which stellar wind mass loss during the RSG phase is sufficient in order to remove most of the hydrogen rich layers, i.e. for a star to evolve into a WR star without the explicit help of the RLOF process. star evolves into a WR star as a consequence of large SW mass loss during the RSG phase of the star. The result is a WR+CC binary with a very large period ( 1000 days). For M s,min we obviously use in a consistent way either 20 M or 30 M (see Sect. 2.1). c. In all other cases the OB+CC binary evolves according to the spiral in prescription discussed earlier. It is used in order to investigate whether, starting from an OB+CC binary, a WR+CC system can be formed and thus to estimate how many WR+CC binaries COULD possibly exist. We obviously also compute the number of merged OB+CC binaries: Thorne Żytkow stars (Thorne and Żytkow, 1977). Their further evolution and the influence on population number synthesis is studied by considering two limiting possibilities: c.1. although the binary will merge, it is assumed that sufficient mass will be removed during the spiral in so that the post spiral in star is a WR with a CC in its center; we propose to call them weird WR stars. The remaining WR time scale is assumed to be equal to the WR time scale of a normal post RLOF WR star with the same mass; c.2. if the binary merges, we take the pre spiral in mass of the OB type star and we assume that its further evolution is similar to the evolution of a normal single star, although it has a CC in its center. In this case the number of weird WR stars obviously depends on the adopted single star evolutionary scenario. Is there a WR subclass which may correspond to these weird WR stars? It is clear that weird WR stars will not be observed as binaries. Since they are formed in a binary where the original primary exploded, they should be runaway WR stars. Even if the binary progenitor belonged to a cluster, because of the runaway status weird WR stars should have left the cluster; therefore they are not expected to belong to a cluster. The class of WN8 stars corresponds to this picture (Moffat, private communication). Furthermore WN8 stars look puffed up and this may be due to the presence of a CC in the deep interior. Although the foregoing is still very speculative (just like the existence of Thorne Żytkow stars), it is an idea which deserves further thinking Non interacting binaries Binaries with period larger than the maximum period in order to have a case C system will not interact. Their evolution can be described with the single star evolutionary scenario. Fig. 2 summarizes the detailed evolutionary scenario of interacting CBs in general, and interacting MCBs in particular. 3. The MCB evolutionary results An extended set of evolutionary computations for case Br MCBs with updated physics has been published in a series of papers

8 494 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution Fig. 2b. The evolutionary model of a massive close binary after the supernova explosion of the original primary; M sn,min is the minimum mass a primary must have in order for a supernova to occur; M s,min has the same meaning as in Fig. 2a but applied to the rejuvenated mass gainer of the binary; M CHeB is the mass of the hydrogen deficient core helium burning star after the spiral in phase. (e.g. de Loore and Vanbeveren, 1994a,b,c, Vanbeveren and de Loore, 1994). When a primary expands and reaches some critical radius rapid mass loss will occur, either Roche lobe overflow in the standard sense, or spiral in or common envelope evolution. We can safely state that independently of the kind of mass loss process or of the initial period of the binary, and thus independently of the moment when this mass loss starts, the stellar expansion and thus the rapid mass loss phase will stop when almost all hydrogen rich layers are removed or when both stars merge. When merging can be avoided it is therefore plausible to use a unique relation between the mass of the star before the mass loss process and the mass after. We use the following relations for respectively the Galaxy, the LMC and the SMC: M a =0.093M b 1.44 M a =0.085M b 1.52 M a =0.048M b 1.7. (20) The equilibrium radius R e (in R of a hydrogen deficient CHeB star after RLOF is given by R e = M M M , (21) with M 1 the mass of the CHeB star (in M ). Relations (20) and (21) hold for initial masses larger than 9M whereas relation (21) hardly depends on the metallicity. We use the latter in order to decide whether a binary will merge or not. The O lifetimes and WR lifetimes for binaries and single stars with the standard and with the alternative model are given by Vanbeveren (1995). The O lifetimes are based on the calibration of Humphreys and McElroy (1982). For binaries with P P min (Sect. 2.2.d) we use the time scales of secondaries after accretion computed with accretion model II, whereas when P > P min we assume that accretion implies full mixing (accretion model I). 4. The population synthesis model We start with a population consisting of a single star population and a binary population. In order to estimate the number of stars in different evolutionary phases, we need an MCB evolutionary model a single star evolutionary model the lifetimes of the evolutionary phases star and binary parameter distributions. The MCB evolutionary model and the single star evolutionary model are discussed in the previous sections. Although the IMF of massive single stars could be different from the IMF of primaries of MCBs (Vanbeveren, 1982), we will use the same power law for both, i.e. IMF M 2.7 (Scalo, 1986). In order to investigate the effect of the IMF, we will also present results when IMF M 2. For MCBs we use the same period distribution f(p) as for other binaries, i.e. f(p) 1/P (Popova et al., 1982; Abt, 1983) with a maximum period of 10 years.

9 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution 495 Table 1. The the O and WR fractions for regions of continuous star formation for different population models. The various O and WR classes are explained in the text. In model 1 we assume f = 0.8, q min = 0.4, the IMF exponent = -2.7, minimum mass for LBV = 40 M, β max =1, flat φ(q), efficiency factors α CE = 1 and mass of the compact star (after SN) = 1.4 M. Model 2 is similar to model 1 but β max = 0.5 and α = 3. Model 3 is similar to model 2 but for α = 6. Models 4,5,6 are similar to resp. model 1,2,3 but with the φ(q) of Hogeveen, models 7,8,9 are similar to resp. model 1,2,3 but with φ(q) q 0.5. Model 10 = model 3 but with q min = 0.6, model 11, 12 = model 3 but with resp.f=1and0.5, model 13 = model 3 but with IMF exponent = -2, model 14, 15 = model 13 but with resp. the φ(q) of Hogeveen and φ(q) q 0.5. Model 16 = model 3 but with α CE = 0.5 and model 17 = model 3 but with the mass of the compact star (after SN)=3M. Table 1a gives the results when the standard single star scenario is used whereas the evolution of the OB+CC mergers = single star evolution of the OB star (see text). Table 1b is similar to Table 1a but with the alternative single star scenario. Table 1c then gives the results for models 2,5 and 8 but with the alternative evolution of OB+CC mergers (see text). In Table 1c we also give a test calculation (model 2 ) for the case where the O type close binaries with mass ratio q < 0.2 who merge are included in the computations as explained in the text. For this model 2 we include the real fraction of O type stars which will experience such a merger phase (O merger/o) and we also add the fraction of WR stars resulting from a merged binary with q < 0.2 (WR merger/wr). The observed mass ratio distribution φ(q) of the O-type binaries peaks in the interval 0.8 q 1 (Garmany et al., 1980), and very few systems have q 0.2 although observational selection could play a significant role here (Hogeveen, 1991). We explore

10 496 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution the consequences of different φ(q) functions by presenting the results for the distribution φ 1 (q) proposed by Hogeveen (1991) and a φ 2 (q) q 0.5 describing the case where the frequency of MCBs increases with increasing q. Since due to small number statistics a flat mass ratio distribution down to q 0.2 can not be excluded, we will also calculate the results for a flat distribution φ 3 (q). In order to decide when a star is an O type star, or an early B type dwarf, giant or supergiant, we used the calibration listed by Humphreys and McElroy (1984). A star in a binary is considered as a WR star when it is a hydrogen deficient CHeB star with mass larger than 5 M. Summarizing, our population synthesis model contains the following parameters: IMF M ɛ with ɛ = 2.7 and ɛ =2, the mass ratio distribution of MCBs: we use φ 1 (q) or φ 2 (q) or φ 3 (q), the period distribution of MCBs f(p) 1/P, the efficiency factor during the different spiral in phases in MCBs, i.e. α CE1 for non-evolved binaries with initial mass ratio q 0.2, α CE2 for case Bc and case C binaries, α CE3 for OB + CC binaries, the value of q min for case A/case Br binaries above which β is assumed to be constant = β max ; for β max we use two values i.e. β max = 1 and β max = 0.5, the angular momentum loss (expressed in terms of a parameter α) during the non conservative RLOF of case A and case Br binaries; we use α = 3 and α =6, the minimum initial mass M s,min of single stars for WR formation and the corresponding single star evolutionary model (the standard scenario or the alternative scenario), the final fate of OB+CC binaries which merge during their ensuing spiral in phase, the fraction f of MCBs with period up to 10 years in the population. Remember that the period of interacting massive binaries ranges between 1000 days and 2000 days (Fig. 1) so that the fraction of interacting binaries is smaller than f. If (WR/O) s is the WR to O-type star number ratio of the single star population and (WR/O) b is the number ratio of the MCB population (where we properly account for the post SN O type stars and WR stars), it follows that the WR/O number ratio of the whole population consisting of a single star population and a binary population is given by WR O =(1 f)(wr/o) s +f(wr/o) b, (22) (see also Vanbeveren, 1995). 5. Observational restriction of the parameters in the population synthesis model If it can be assumed that stars formed in a starburst (or abruptly terminated star formation regions, see Sect. 8) obey the same Fig. 3. Starting from the mass ratio distribution of 17 observed WR+OB binaries the figure gives the probable mass ratio distribution of these 17 binaries at the moment the RLOF ended. distributions as stars formed in a region of continuous star formation, the various parameters entering the population model can be restricted. We list here the observations which are very important for this purpose. Within the solar neighborhood (about 3 kpc from the sun) about 40% (+/ 10% accounting for small number statistics, van der Hucht, 1994) of all WR stars are member of a binary with an OB type companion and with a period which is small enough so that RLOF occurred. Furthermore the WR/O number ratio 0.1. For 17 WR+OB binaries we know the mass ratio. Using the evolutionary computations of MCBs during CHeB we estimated the mass ratio of the system just after RLOF. The distribution is shown in Fig. 3. If the distribution is representative for all WR+OB binaries, this restricts the values of β for case Br systems (Sect. 6). From a sample of 60 O-type stars, Garmany et al. (1980) conclude that about 33% are primaries of unevolved close binaries with period P 100 days and mass ratio q 0.2 (+/ 13% accounting for small number statistics). Given f, the fraction of MCBs, we determine the real O type star sample consisting of real O type single stars, O type primaries of unevolved MCBs but also post SN O type secondaries, using the MCB evolutionary model, the single star evolutionary model and the population synthesis model discussed above. The value of f is varied so that the fraction (relative to all O type stars) of O type primaries of unevolved binaries with period P 100 days and mass ratio q 0.2 falls within 33 +/ 13%. Remark that the value of f depends on the adopted parameters in the evolutionary and population model and therefore enters implicitly into the population computer code. 6. Population number synthesis for Galactic regions of continuous star formation We first applied our population model to Galactic regions where star formation can be considered as continuous. The results for the O type stars and WR stars are summarized in Table 1 for

11 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution 497 the various parameters of the population model. We separately consider (O+OB) obs /O = fraction of unevolved O-type primaries of binaries with mass ratio q 0.2 and period P 100 days, O+CC/O = fraction of O type stars with a compact companion, O sb /O = fraction of O type single stars but with a binary history, i.e. which became single after disruption during the SN explosion, O s /O = fraction of single O type stars, formed as single stars, O m /O b = fraction of O type binaries with q 0.2 which are merging due to spiral in, (WR+OB) i /WR = fraction of WR+OB binaries where RLOF (interaction) occurred, (WR+OB) ni /WR = fraction of WR+OB binaries where RLOF (interaction) did not occur, i.e the binaries formed through the LBV scenario or through the RSG scenario, (WR+CC)/WR = fraction of WR stars with a compact companion, weird WR/WR = fraction of WR stars with a CC in the center of the stars (Thorne Żytkow WR stars), WR sb /WR = fraction of single WR stars but with a binary history, i.e. the descendants from single O type stars which became single after disruption during the previous SN explosion, WR s /WR = fraction of real single WR stars, WR/O = the overall WR/O number ratio. Fig. 4 gives the mass ratio distribution just after RLOF of the WR+OB binaries for different values of the population parameters. Conclusions: the stellar content in a population depends only marginally on the parameter α describing the angular momentum loss during non conservative RLOF in a case Br binary, i.e. on the evolution of the period of an MCB, also the parameter α CE2 has little effect on our results. The reason is illustrated in Table 2 where we show the period evolution of case Bc binaries. Case Bc remnants after a common envelope phase are WR + OB binaries with periods of the order of days where the OB components are not affected by accretion. However inspection of Fig. 1 reveals immediately that if indeed the period distribution of MCBs 1/P, then the contribution of case Bc (and case C) MCBs to the WR + OB population should be small. This explains why the population results hardly depends on α CE2, when we compare the predicted WR+OB mass ratio distribution and the observed one it follows that conservative case Br evolution can be excluded corresponding to conclusions in earlier studies. Best correspondence is achieved for β = 0.5 with a flat φ(q) or a φ(q) which peaks at unity. Of course β = 0.5 is an average, i.e. a linear decrease of β from nearly conservative in early case Br to β = 0 in late case Br gives very satisfying results as well. We consider the latter as more realistic than the case β is constant, Table 2. The period P (in days) after common envelope evolution of a binary with primary mass M 10 (in M ), mass ratio q and initial period = 750 days for resp. α CE = 1 and for α CE = 0.5. M 10 q P M 10 q P / / / / / / / / / / / /6.3 Table 3. The minimum period P 0min (in days) for a binary with mass ratio q 0.2 and primary mass M 10 (in M ) in order to avoid merging for resp. α CE = 1 and for α CE = 0.5; the period P (in days) after spiral in corresponding to P 0min is given as well. M 10 q P 0min P M 10 q P 0min P / / / / / / / / the models computed with the q distribution of Hogeveen (1991) never give satisfactory results, in order to obtain an O type binary frequency for periods 100 days and mass ratio q 0.2 which is comparable to the observed value, the total fraction of MCBs (with periods up to 10 years) must be very large and ranges between 75% and 100%; the corresponding interacting binary frequency 55% to 70%, a significant fraction of the single WR stars have had a binary history, i.e. they are descendants from a binary which was disrupted during the first SN explosion of the original primary, the WR+OB/WR number ratio critically depends on the adopted evolutionary scenario of massive single stars. In order to obtain a ratio between 0.3 and 0.5 (as observed) the alternative single star scenario fits best, corresponding to the conclusion of Vanbeveren (1995). As can be noticed from the tables the latter also gives the best value for the WR/O number ratio, the main effect of the two limiting models that describe the evolution of OB+CC binaries which merge during their ensuing spiral in phase is the number of weird single WR stars. The overall distribution of WR stars in a population does however not depend critically on the adopted model. Taking the average of the two implies errors which are not larger than 5 7%, as can be noticed from the tables, a significant fraction of all O type binaries may have a mass ratio q 0.2. In Table 3 we illustrate the period evolution in these binaries; the minimum period in order to avoid merging is given as well. These minimum periods are quite high which means that the contribution of MCBs with initial q 0.2 to (and thus the effect of α CE1 on) the WR binary population is low. Only systems with large period may survive. As an illustration, consider a 30 M +3M

12 498 D. Vanbeveren et al.: The effect of binary evolution on the theoretically predicted distribution Fig. 4. The mass ratio distribution of the WR+OB binaries just after RLOF predicted by the population model for regions of continuous star formation. The model numbers correspond to the definitions given in Table 1. binary with period P = 1000 days. As a consequence of spiral in, it evolves into a 12 M +3M WR + late B type dwarf binary with period P = 3 days (maximum friction efficiency i.e. α CE1 = 1). We propose that this system resembles HD 50896, a WR star with a low mass companion and a period of about 3 days (Firmany et al., 1980). Since in the present model, the low mass companion is not a compact star (neutron star or black hole), hard X ray radiation is not expected. Although the contribution of these small q systems to the WR binary population may be low, depending on the adopted mass ratio distribution, the number of merged binaries resembling single stars could be significant. How these merged stars further evolve is still a matter of faith; however they could significantly affect the estimated population of single WR stars. The results of the WR star numbers in the tables were computed by ignoring the further evolution of these mergers up to a possible WR phase. In order to illustrate the consequences of this assumption, we made some test computations by assuming that the merged stars can be modeled by the standard accretion process, i.e. the primary behaves as if the total mass of the low mass companion is accreted and after accretion further evolves as a single star where then obviously the appropriate single star evolutionary scenario is used. The results are given in Table 2 as well. As can be noticed, all WR fractions are typically 10% lower; however we have to add a new class of single WR stars where the evolution was governed by the merging process. Their fraction for the model considered The total fraction of single WR stars including the non interacting binaries now