A&A 443, 231-241 (2005)
D. Lommen1,2 - L. Yungelson1,3,4 - E. van den Heuvel1 - G. Nelemans5 - S. Portegies Zwart1,6
1 - Astronomical Institute "Anton Pannekoek'', University of Amsterdam, and Center for High Energy Astrophysics, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
2 - Sterrewacht Leiden, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
3 - Institute of Astronomy of the Russian Academy of Sciences, 48 Pyatniskaya Str., 119017 Moscow, Russia
4 - Isaac Newton Institute, Moscow Branch, 12 Universitetskii Pr., Moscow, Russia
5 - Astronomy Department, IMAPP, Radboud Universiteit Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands
6 - Informatics Institute, University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands
Received 6 February 2005 / Accepted 7 July 2005
Cygnus X-3 is a strong X-ray source ( erg s-1) which is thought to consist of a compact object accreting matter from a helium star. We analytically find that the estimated ranges of mass-loss rate and orbital-period derivative for Cyg X-3 are consistent with two models: i) the system is detached and the mass loss from the system comes from the stellar wind of a massive helium star, of which only a fraction that allows for the observed X-ray luminosity is accreted, or ii) the system is semidetached and a Roche-lobe-overflowing low- or moderate-mass helium donor transfers mass to the compact object, followed by ejection of its excess over the Eddington rate from the system. These analytical results appear to be consistent with evolutionary calculations. By means of population synthesis we find that currently in the Galaxy there may exist 1 X-ray binary with a black hole that accretes from a Wolf-Rayet star and 1 X-ray binary in which a neutron star accretes matter from a Roche-lobe-overflowing helium star with mass . Cyg X-3 is probably one of these systems.
Key words: accretion, accretion disks - stars: individual: Cyg X-3 - stars: binaries: close - stars: binaries: general - stars: Wolf-Rayet
Cygnus X-3 was discovered as an X-ray source by Giacconi et al. (1967). It is a strong X-ray source ( erg s-1, assuming a distance of 9 kpc), and the X-rays are expected to be due to accretion of matter onto a compact object (c.o.), presumably a black hole (BH) or a neutron star (NS) (see, e.g., Predehl et al. 2000; Kitamoto et al. 1987). The X-ray and infrared (IR) emission show a periodicity of 4.8 h, which is believed to be the orbital period P of the system (see, e.g., Parsignault et al. 1972). Van den Heuvel & de Loore (1973) suggested that Cyg X-3 consists of a NS with a helium (He) star companion, as a later evolutionary product of a high-mass X-ray binary. Tutukov & Yungelson (1973a) independently considered a NS accompanied by a He star as a stage in an evolutionary scenario leading from a pair of main-sequence stars to a binary NS. There is too much interstellar obscuration towards the source to observe it optically, but observations in the IR wave bands in the 1990's by van Kerkwijk & coauthors (1992, 1993, 1996) identified a Wolf-Rayet (WR) spectrum with Cyg X-3. Both the observations of van Kerkwijk and coauthors and high-resolution spectroscopy by Fender et al. (1999) revealed hydrogen depletion of the mass donor. Furthermore, phase-to-phase variations in the X-ray spectra can be explained by a strong (factor 10-100) overabundance of carbon, nitrogen, or oxygen (Terasawa & Nakamura 1994), consistent with classification of the Cyg-X-3 companion as a WN-type star. This added credibility to van den Heuvel & de Loore's and Tutukov & Yungelson's prediction.
The aims of this paper are twofold.
Tavani et al. (1989) and later Mitra (1998,1996) argued that the companion in Cyg X-3 should be a very-low-mass ( ) He star. However, this is hard to reconcile with the high IR luminosity of the system (Hanson et al. 2000). Our calculations (Sects. 3.2 and 4) and population synthesis (Sect. 5) also exclude a donor less massive than .
Terasawa & Nakamura (1994) found, from the ionisation structure of the wind in Cyg X-3, that the mass of the wind-supplying component has to be moderate: .
Schmutz et al. (1996) conclude that the variations in the profiles of several near-IR emission lines are due to the orbital motion of the WR star and derive a mass function for the donor . For the range of assumed Wolf-Rayet masses 5 to 20 and a range of possible inclinations , they get a mass in the range for the c.o., from which they infer that it is a BH.
Hanson et al. (2000) found a sinusoidal absorption feature originating in the wind in the 2.06 m spectral region of Cyg X-3, which allowed them to derive a mass function . They considered two options: an origin of the absorption in the accretion disk or other material centred on the compact object, or association of absorption with the donor. The first option is consistent with low- or moderate-mass () donors, but requires a low orbital inclination of the system (). Association of the absorption feature with the donor limits the mass of the WR companion to if the accretor is a NS. For BH accretors the mass of the secondary may be as high as .
Stark & Saia (2003) studied the modulation of X-ray emission lines from Cyg X-3. Based on a discussion of the location of the regions of emission of highly-ionised silicon, sulfur, and iron, they assume that the iron line is produced in the wind captured by the c.o. or in an accretion disk around it. They then use the fact of non-detection of a modulation of the iron lines to derive an upper limit to the mass function for the accretor: . For an accretor of or , the minimum mass of the donor is then and , respectively. We furthermore note that in the case of Roche-lobe overflow (RLOF) a mass ratio is inconsistent with the observed increase in the period (see below), which then also implies an inclination in this case.
The early models of Cyg X-3 that assumed an elliptic orbit for the system (Ghosh et al. 1981) may be discarded now, since no signs of an apsidal motion were found in 30 yr of observations (Singh et al. 2002). One implication of not discovering any apsidal motion is the irrelevance of values found by Ghosh et al. (1981) for the orbital inclination, which are often used in the literature.
To summarise, an ambiguity still exists in the interpretation of the radial-velocity curves of Cyg X-3, mostly related to different locations of spectral features that serve as the basis for radial-velocity determinations. However, at the moment it seems likely that Stark & Saia (2003) really measure emission originating in the vicinity of the c.o. Then their results suggest a rather moderate mass for the companion to the c.o., if the c.o. is a neutron star or a stellar-mass black hole.
The period P of Cyg X-3 has been extensively monitored over the years (e.g., van der Klis & Bonnet-Bidaud 1981; Kitamoto et al. 1987), and is found to be increasing on a relatively short time scale of 106 yr. There are also indications of a second derivative on the order of -10-10 yr-1 to the period (van der Klis & Bonnet-Bidaud 1989; Kitamoto et al. 1995). A summary of the estimates of for Cyg X-3 is presented in Table 1.
Table 1: The values for of Cyg X-3, derived by fitting two different models to the observations. A parabolic ephemeris assumes , whereas in the cubic ephemeris a second derivative of the period unequal to zero is also taken into account.
The mass-loss rate for Cyg X-3 was estimated from IR observations, usually using the Wright & Barlow (1975, W & B) model for the emission of a spherical, homogeneous, constant-velocity, isothermal wind. Stars have an accelerating wind with a temperature gradient, but W & B note that observations show spectrum flattenings in the near-IR similar to those predicted by their constant-temperature, constant-velocity model.
Waltman et al. (1996) and Miller-Jones et al. (2005) use another method to estimate . They also assume the mass outflow to be spherically symmetric and then use the fact that a (post-outburst) jet becomes observable with different delays after the burst at different frequencies (see Waltman et al. 1996, for details). Note that this method gives in the wind, not in the jet itself, which is assumed to be much smaller.
The estimated mass-loss rates for Cyg X-3, varying from yr-1 up to yr-1, are presented in Table 2. All estimates except one assume spherical symmetry. Note that deviations from spherical symmetry will most probably result in a higher mass-loss rate in the estimates from time delays (Waltman et al. 1996; Miller-Jones et al. 2005), whereas deviations from spherical symmetry in the other cases will result in a lower effective mass-loss rate from the system (see, e.g., Koch-Miramond et al. 2002).
If the increase in the orbital period is considered to be the result of a high-velocity wind from the system that takes away specific angular momentum of the donor (e.g., Kitamoto et al. 1995; Ergma & Yungelson 1998), the formula yields yr-1, where is the total mass in the system.
Table 2: Values for from the literature. We only show values that are obtained from observations of the mass loss from the system, hence not those inferred from the evolution of the orbital period. All estimates except Ogley et al. (2001)d assume spherical symmetry. See the main text for details.
In principle the observed mass loss may be a combination of a direct wind and re-ejection of transferred mass. However, since in the case of Roche-lobe overflow (RLOF), the mass-loss rate considerably exceeds the mass-loss rate of a direct wind (see Sect. 3.2), we consider only the extreme cases, one being only wind mass loss (with only so much mass transfer as to allow the c.o. to accrete at the Eddington rate), the other being no direct wind and only mass transfer from the He star followed by re-ejection from the compact star.
|Figure 1: Mass-loss rates of He stars: a) wind mass-loss rate for a homogeneous He star; b) wind mass-loss rate for a He star at He terminal-age main sequence; c) rough estimate of the mass-loss rate of a star that overflows its Roche lobe on a thermal time scale after completion of core He burning.|
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Observed population I WR stars are more massive than (see, e.g., Nugis & Lamers 2000); lower-mass He stars will probably not show a WR spectrum, nor will they produce a wind that could explain the mass-loss rate observed in the Cyg-X-3 system. Nugis & Lamers do not report any WR stars with mass-loss rates below yr-1, whereas Miller-Jones et al. (2005) find that the mass-loss rate from Cyg X-3 may be below yr-1. This might be an indication that the WR spectrum observed from Cyg X-3 is due to the re-ejection of Roche-lobe-overflowed material that mimics the WR phenomenon. Another explanation, however, may be that the outflow is not spherically symmetric in the case of Cyg X-3 (Sect. 2).
Evolutionary calculations of Paczynski (1971), Tutukov & Yungelson (1973b), and Iben & Tutukov (1985)
showed that He stars with
expand during core He burning and later evolutionary stages.
Also He stars with
hardly expand before carbon (C) ignition in their cores, later stages are so short that they may be neglected.
For intermediate-mass He stars, inspection of the summary figure of
Dewi et al. (2002, their Fig. 1) shows that in a binary with P = 0.2 days containing a NS and a
the latter may overflow its Roche lobe in the He-shell-burning stage if
or in the core-C-burning stage if
However, the expected number of systems in the Galaxy that experience RLOF in the C-burning stage is negligibly small since this stage is very
and we are left with (0.8-5.8)
He stars, overflowing their
Roche lobe in the He-shell-burning stage (so-called BB case of evolution).
If RLOF occurs after core He burning is completed, the mass-exchange
time scale is on the order of the thermal one:
|Figure 2: Schematic representation of possible system configurations that are consistent with and as observed in Cyg X-3. RLOF solutions give a very narrow range of possible combinations of and (the two narrow strips in the lower left; the gap is due to the presumed gap between the masses of NSs and those of BHs). For wind-accretion solutions, the range of possible combinations of and is much larger (hatched areas). Population synthesis shows that RLOF systems with NSs are formed in sufficiently large numbers to produce an observable Cyg-X-3 system. On the other hand, Roche-lobe overflowing BH systems are formed so rarely with P similar to the period of Cyg X-3 that the probability of observing them is negligible. See Figs. 5 and 4 and text.|
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To verify the inferences in the previous section,
we carried out several evolutionary calculations of
semidetached systems consisting of
He stars accompanied by a c.o. We assumed that the
latter can accrete matter at
and that the excess of
the transferred mass is lost from the system, taking away the specific angular
momentum of the accretor.
Prior to RLOF, mass loss by stellar wind was computed according to formulae (2). Accretion prior to RLOF was neglected.
In the RLOF stage wind
mass loss directly from the He star was
neglected (see Fig. 1).
The adopted ranges of initial masses were
for the He stars and
for the c.o.'s.
Computations were carried out using P. Eggleton's evolutionary code (priv. comm. 2003, see also Eggleton & Kiseleva-Eggleton 2002, and references therein).
A selection of the results
are presented graphically in Fig. 3.
The systems had the following combinations of component masses at
the onset of mass transfer:
(Fig. 3a, b),
(Fig. 3c, d), and
(Fig. 3e, f).
In all the computed systems the He stars started RLOF at
|Figure 3: and as function of for systems with Roche-lobe overflowing He-star donors and compact accretors. Masses at the onset of mass transfer are: (3.0, 5.0), (1.46, 1.4), (1.0, 1.4) (from top to bottom). Roche-lobe overflow starts at day. In all panels thick solid lines show results of computations. In panels for thin solid lines show the limits of observed in Cyg X-3. Dotted curves show rough estimates for and based on Eq. (3), derived with the approximations to R and L at the terminal-age He main-sequence from Hurley et al. (2000).|
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We find that the systems with He-star donors traverse the range of observed for Cyg X-3 in 102 yr. We do note, however, that the typical value of in the RLOF phase decreases with the mass of the donor (for a given mass of the compact star) and that the time spent close to the observed range increases for lower-mass systems. A system that at the onset of mass transfer consists of a He star and a c.o. (Figs. 3c, d) spends about yrs in the range observed in Cyg X-3, and stays some yrs at values less than twice those in the observed range. A system of a He star and a c.o. stays within the observed range throughout RLOF (Fig. 3e, f).
We carried out a population synthesis to determine the current number of He+c.o. binaries in the Galaxy. The details of the population synthesis are briefly described in Appendix B. We used the approximations of Pols (1993) to the computations of Paczynski (1971) and Habets (1986) to estimate the core-He-burning times. In Figs. 4 and 5 we plot the masses of the components and the orbital periods of the He+c.o. systems that have He components in the core-He-burning stage. We find that there are currently 200 He+BH and 540 He+NS binaries in the Galaxy.
As noted in Sect. 3.2.1 we expect an accretion disk in the Cyg-X-3 system.
The systems shown in Figs. 4 and 5 may form a disk through wind accretion if the wind matter carries enough angular momentum
as realised first by Illarionov & Sunyaev (1975); see, e.g., Livio (1994) for later work on this subject.
|Figure 4: Current population of core-He-burning He+BH binaries in the Galaxy. Upper panel - distribution in plane; middle panel - distribution in plane; lower panel - distribution in plane. The dash-dotted vertical lines in the upper and lower panel show the lower-mass boundary for He stars identified as WR stars. Systems that satisfy the disk-formation criterion for wind-fed objects (Eq. (4)) are marked by open circles; the subset of them with is marked by stars.|
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As already noted by Iben et al. (1995) and Ergma & Yungelson (1998), with , Cyg X-3 may have a wind-fed disk if . The latter value fits well into the model range of expected black-hole masses in wind-fed systems with disks and orbital periods close to that of Cyg X-3 (the star symbols in Fig. 4). An additional reason why only a WR system with days shows up as a WR X-ray binary may have to do with the velocity profile of the wind. In such a close system the wind will, at the orbit of the compact object, not yet have reached its terminal velocity (of for example 1500 km s-1), whereas in the wider systems it may have, such that no disk forms in the wider systems.
|Figure 5: Current population of core-He-burning He+NS binaries in the Galaxy. The dash-dotted vertical line shows the lower-mass boundary for He stars identified with WR stars. The four dashed lines show the critical periods below which according to criterion (5) disk formation through wind accretion is possible, if v1000 = 1, 1.5, 2, and 3 (highest to lowest), respectively.|
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Figure 5 shows the population of He+NS binaries, which can be divided into two subpopulations. The first and larger subpopulation consists of systems with day and . The short period is due to their previous common-envelope (CE) phase; the large number of low-mass He-star systems is due to the initial-mass function and to the fact that low-mass He stars live much longer than higher-mass He stars. The other systems form through a "double spiral-in'' in which two giants go through a CE phase, producing two He stars. This formation channel only occurs for nearly-equal-mass binaries, and since the most massive He star collapses into a c.o., there is a lower limit to the mass of the other He star (see Brown 1995).
As shown by Illarionov & Sunyaev (1975) and Ergma & Yungelson (1998), an accretion disk will form in a He+NS binary if the rotation
period of the NS is longer than the equilibrium period that was
established during the CE episode that accompanied the formation of
the He star. Assuming that the wind-mass-loss rate may be described by a formula
(e.g. Nelemans & van den Heuvel 2001), one derives as the disk-formation
One should note, however, that the mass-loss rates for He stars quoted above were derived from observational data on WR stars. Data on mass-loss rates for lower-mass He stars are not available. It is therefore not clear whether may be extrapolated to below . Hence, the validity of criterion (5) below this mass is uncertain.
Another possibility for Cyg X-3 is that the system contains a He star of which transfers mass in the case BB of RLOF. Then the WR spectrum arises in re-ejected matter. Also in the RLOF case, we can estimate the number of systems we should currently observe as WR X-ray binaries. As mentioned in Sect. 4, the RLOF systems that can provide a close to the observed range must initially have had a low-mass (1.5 ) He-star donor. From the population-synthesis calculations we expect 500 such systems in the Galaxy at any time. Typically these systems live yrs, of which they spend 105 yrs in the phase of Roche-lobe overflow. We thus expect only of order 1 such system in the Galaxy to be in the phase of Roche-lobe overflow at any time.
As indicated by Fig. 5, the bulk of these systems has h. The typical mass-transfer rates in these systems are in the range yr-1, and most of the transferred mass will be lost from the system through re-ejection. These rates are consistent with the lowest observational estimates of for Cyg X-3.
Our population synthesis shows that there are several core-He-burning He+NS binaries and a few dozen core-He-burning He+BH binaries with He-star masses in our Galaxy. If we assume that all matter that passes through the so-called accretion radius (G the gravitational constant, c the speed of light) is accreted by the c.o., and that the gravitational potential energy of the accreted matter is converted into luminosity, all wind-accreting He-star binaries with in our model population will have an intrinsic luminosity 1036 erg s-1 and should be observable as He-star X-ray sources. In this we also assume that Eq. (2) holds down to low-mass He stars.
Apart from Cyg X-3, a few WR+c.o. candidates are reported: e.g., HD 197406/WR 148 (Marchenko et al. 1996), HD 191765/WR 134 (Morel et al. 1999), HD 104994/WR 46 (Marchenko et al. 2000). However, it is still unclear whether the companions to these WR stars really have a relativistic nature, as the systems lack the X-ray luminosity expected in such a case. A low can be reconciled with, e.g., a spinning pulsar, that deflects the flow.
The fact that we do not observe several tens of He-star X-ray binaries in the Galaxy may be due to self-absorption of the X-ray photons by the wind of the donor. It turns out that for the binaries in our population-synthesis sample, the minimum column density between the c.o. and Earth due to the He-star wind depends mainly on . This column density is 10 g cm-2 for all sources with , rendering these sources unobservable in X-rays at energies below 20 keV. We do note that INTEGRAL has discovered several sources at energies >20 keV, of which 40 are still unidentified (Ubertini 2005). These hard sources might well be the missing He+c.o. binaries.
The derivation for the column density also applies to RLOF-accreting sources that spherically symmetrically throw out overflowing matter in excess of the Eddington rate. We saw in Sect. 3 that the mass-transfer rate for RLOF systems is yr-1, well above the Eddington rate for a solar-mass c.o. ( yr-1 for a 1.4 NS accreting pure He) and good enough for a minimum column density of 102 g cm-2. This may support the suggestion of a model for Cyg X-3 in which the excess matter is thrown out of the system equatorially instead of spherically symmetrically (Sect. 3), together with a very low inclination for the system.
Bauer & Brandt (2004) and Clark & Crowther (2004) find that the luminous X-ray source IC10 X-1 [ erg s-1] in the starburst galaxy IC10 is spatially coincident with WNE star [MAC92] 17-A (notation adopted from Crowther et al. 2003). Assuming [MAC92] 17-A to be the most probable optical counterpart to IC10 X-1, Clark & Crowther (2004) fit a model with a stellar temperature of 85 000 K, log ( , yr-1, and a terminal wind velocity of 1750 km s-1 to the observed He II and N V 4603-20 emission. They infer a mass for the WR star of , using the WR mass - luminosity relations of Schaerer & Maeder (1992). Allowing for clumping, Clark & Crowther find that is equivalent to a homogeneous mass-loss rate of yr-1.
Bauer & Brandt (2004) note that IC10 X-1 is quite similar to Cyg X-3 in terms of X-ray luminosity, spectrum, and variability. Thus, IC10 X-1 may be the first extragalactic example of a short-lived WR X-ray binary similar to Cyg X-3. Identification of the optical counterpart to IC10 X1 with a massive WR star suggests that the system is wind-fed.
Note, however, that in the field of IC10 X-1 there are three other candidate optical counterparts, with O- or B-spectral types, to the X-ray source. Clark & Crowther (2004) suggest that their wind mass loss is insufficient to explain the observed X-ray luminosity with wind accretion. From this Clark & Crowther argue that, if one of those three candidates is the optical counterpart to IC10 X-1, the system is Roche-lobe-overflow fed similar to LMC X-4 or LMC X-3.
The variable X-ray and radio source SS 433 was also suggested to be a WR X-ray binary (van den Heuvel et al. 1980; Fuchs et al. 2004,2002a). Van den Heuvel et al. suggested that SS 433 contains an evolved early-type star or a WR star, based on the nature of its stationary spectrum, the size of the emitting region, the necessary presence of a strong wind for the production of the IR emission, and the large outflow velocity of the wind.
In one of the latest attempts to identify the optical counterpart of SS 433, Fuchs et al. (2002b,a) compared its mid-IR spectrum to WR stars of the WN subtype. They found the spectrum of SS 433 to resemble that of WR 147, a WN8+B0.5V binary with colliding wind. Using the formulae of Wright & Barlow (1975) and taking wind clumping into account, Fuchs et al. (2004) obtain yr-1, compatible to a strong late-WN wind.
Fuchs et al. (2004) proposed that the material surrounding the c.o. forms a thick torus or envelope around it rather than a classic thin accretion disk. They argue that the material is ionised by X-rays emitted from the vicinity of the c.o. and expelled by radiation pressure, which results in the imitation of a WR star. As mentioned above, this model needs further elaboration, especially the formation of the WR spectrum and the self-absorption of the X-rays.
On the other hand, King et al. (2000) suggested that SS 433 is a mass-transferring system in which the formation of a CE may be avoided if radiation pressure expels the transferred matter in excess of the Eddington rate, i.e., a re-ejection model with a hydrogen-rich donor. For this model the donor mass must be in the range of . The model has received support from the discovery of A-type super-giant features in the spectrum of SS 433 observable at certain orbital phases (Hillwig et al. 2004; Gies et al. 2002). Estimated masses of the components are and , fitting the King et al. model. As noted by Fuchs et al. (2004), if the results of Gies et al. (2002) and Hillwig et al. (2004) are confirmed, one needs to resolve an apparent contradiction of the simultaneous presence of A-star and WR-star features in the spectrum.
If the presence of an A-type star is not confirmed, however, it may appear that SS 433 really is a WR X-ray binary, the second one known in the Galaxy after Cyg X-3.
The first possibility is a system consisting of a massive () helium (i.e., WR) star and a BH around which a disk is formed through wind accretion. Population synthesis predicts that at any time 1 such system with an orbital period similar to that of Cyg X-3 is present in the Galaxy, provided that the wind velocity near the orbit of the c.o. is 1000 km s-1. In this case the system will have a lifetime of several times 105 yrs (the lifetime of the He star), and the secular orbital-period increase is simply due to stellar-wind mass loss.
The second possibility is a system consisting of a He star with a mass and a NS, which is powered by mass transfer due to RLOF at a rate in the range yr-1. Population synthesis predicts that also 1 such system with P < 10 h may be present in the Galaxy at any time. In this case the system will have a lifetime of about 105 yrs, and the secular orbital-period increase is due to the combined effects of the mass transfer and subsequent mass loss - at a rate close to the transfer rate - from the accretion disk around the NS.
In view of the population-synthesis results, we deem both configurations equally likely for Cyg X-3. We note that, though the first of the solutions implies the presence of a "real'' WR star in the system, its mass is probably not extremely high, . Thus, both solutions are consistent with the conclusion of a relatively moderate mass for the companion in Cyg X-3, which follows from identification of the emission region with the vicinity of the compact object (Stark & Saia 2003; Fender et al. 1999).
We now speculate on the fate of these configurations. In the "wind'' case, if the WR star loses sufficient mass, it might terminate as a NS, such that a system consisting of a BH plus a young radio pulsar would emerge. In view of the likely mass of for the BH (a value well within the range of BH masses in confirmed BH binaries, see, e.g., McClintock & Remillard 2004), disruption of the system in the supernova explosion seems unlikely. Alternatively, the WR star might collapse to a black hole, producing a double-BH binary.
The fate of the "RLOF'' configuration is most likely a binary consisting of a massive white dwarf (composed of CO or ONeMg), together with a recycled neutron star. Several such systems are known in our galaxy as fast-spinning binary radio pulsars with massive white-dwarf companions in circular orbits (see, e.g., Stairs 2004).
We are grateful to P. Eggleton for providing the latest version of his evolutionary code. We would like to thank our colleagues at the "Anton Pannekoek'' Astronomical Institute and at the Institute of Astronomy of the Russian Academy of Sciences for useful discussions. In particular we would like to thank T. Maccarone for discussions on X-ray binaries, J. Miller-Jones for sharing preliminary results on Cyg X-3, and K. van der Hucht, A. de Koter, and N.N. Chugai for useful discussions on the properties of WR stars. LRY acknowledges the warm hospitality of the Astronomical Institute "Anton Pannekoek,'' where part of this work was accomplished. LRY is supported by RFBR grant 03-02-16254, the Russian Ministry of Science and Education Program "Astronomy'', NWO, and NOVA. SPZ is supported by KNAW.
The total orbital angular momentum of the system is
|Figure 6: Summary of the evolution of the mass of single stars or components of wide binaries ( top panel) and of stars in binaries experiencing RLOF ( bottom panel). As a function of initial mass, the lines give: the mass at the end of the main-sequence ( , solid line), the initial mass of the He-core ( , dotted line), the mass just before the supernova explosion or the formation of the white dwarf ( , dashed line), and the mass of the final c.o. ( , black-white-black line). The bottom panel shows the masses for a primary that loses its hydrogen envelope soon after the end of the main sequence, before the He-core burning starts.|
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We used the Nelemans et al. (2004) realisation of the SeBa program (Portegies Zwart & Verbunt 1996) to carry out our population-synthesis calculations. The effects of stellar wind and c.o. formation are shown in Fig. 6. The most important assumptions about the evolution of massive stars in binaries relevant to this paper are as explained in Portegies Zwart & Yungelson (1999); Nelemans et al. (2001); Portegies Zwart & Yungelson (1998) with the following two differences: