Evolutionary constraints on the masses of the components of the HDE 226868/Cyg X-1 binary system (original) (raw)

Journal Article

N. Copernicus Astronomical Centre, ul. Bartycka 18, 00-716 Warsaw, Poland

Search for other works by this author on:

Received:

18 December 2004

Accepted:

04 January 2005

• PDF
• Split View
• • Article contents
  • Figures & tables
  • Video
  • Audio
  • Supplementary Data

Navbar Search Filter Mobile Enter search term Search

Abstract

Calculations carried out to model the evolution of HDE 226868, under different assumptions about the stellar wind mass-loss rate, provide robust limits on the present mass of the star. It has to be in the range 40 ± 5 M⊙ if the distance to the system is in the range 1.95–2.35 kpc and the effective temperature of HDE 226868 in the range 30 000–31 000 K. Extending the possible intervals of these parameters to 1.8–2.35 kpc and 28 000–32 000 K, one gets for the mass of the star the range 40 ± 10 M⊙. Including into the analysis observational properties such as the profiles of the emission lines, rotational broadening of the absorption lines and the ellipsoidal light variations, one can estimate also the mass of the compact component. It has to be in the ranges 20 ± 5 M⊙ and 13.5–29 M⊙ for the cases described above. The same analysis (using the evolutionary models and the observational properties listed above) yields a lower limit to the distance to the system of ∼2.0 kpc, if the effective temperature of HDE 226868 is higher than 30 000 K. This limit to the distance does not depend on any photometric or astrometric considerations.

1 Introduction

Cyg X-1 was the first binary system in which the presence of a black hole was suggested (Bolton 1972; Webster & Murdin 1972). For about a decade, it was the only object of that type. Many, sometimes exotic, models and scenarios were devised to avoid the presence of a black hole and to replace it with a neutron star. With the advent of subsequent black hole candidates (LMC X-3, A620−00), the motivation for such efforts substantially diminished. At present, there are no doubts about the presence of a black hole in the system. However, in spite of three decades of investigations, there is still substantial uncertainty concerning the masses of both components. The mass function is known rather precisely. Its most recent value was given by Gies et al. (2003): f(_M_x) = 0.251 ± 0.007 M⊙. There are also no doubts that the optical component must be close to filling its Roche lobe. Gies & Bolton (1986a,b), analysing the emission lines of the stellar wind, the rotational broadening of the absorption lines of the optical component and the photometric _V_-band light curve, found that the fill-out factor must be greater than 0.9 and its best value is in the range 0.95–1. The value of the inclination of the orbit is less certain. From the analysis mentioned above (rotational broadening of the absorption lines of the optical component and the ellipsoidal light variations in the _V_-band light curve), the authors estimated the inclination of the orbit to be _i_= 33°± 5°. On the other hand, from polarimetric measurements (in three colours) Dolan & Tapia (1989) found the inclination _i_= 62°+5−37, which, while different, is not inconsistent with the value of Gies & Bolton.

Paczyński (1974) has shown that based only on the lack of X-ray eclipses and the fact that the optical component cannot be larger than its Roche lobe, one can obtain lower limits to the masses of both components as functions of the distance to the system. Paczyński found that the mass of the compact component has to be larger than 3.2 M⊙ if the distance is greater than 1.3 kpc. He noticed also that, based on stellar evolution considerations, the distance cannot be larger than about 2.1 kpc. The distance to Cyg X-1 is not very well established, but, certainly, it is greater than 1.3 kpc. Margon, Bowyer & Stone (1973) and Bregman et al. (1973) analysed interstellar reddening in the direction of HDE 226868 and found that its distance is about 2.5 kpc. Wu et al. (1982), from a study of the UV colours of HDE 226868, concluded that its distance must be greater than 1.9 kpc. Bregman's value was confirmed by Ninkov, Walker & Yang (1987), who got the distance of 2.5 ± 0.3 kpc from the equivalent width of the Hγ line. HDE 226868 lies only 1° from the centre of NGC 6871 (the core of the Cyg OB3 association) and seems to be a member. However, the estimates of the distance to the association (or its core) gave mixed results. Crawford, Barnes & Warren (1974) used three methods (UBV photometry, the Hertzsprung–Russell (H-R) diagram fitting and the equivalent width of the Hβ line) and obtained the distance to the association of about 2 kpc. Humphreys (1978) estimated the photometric distance to Cyg OB3 to be about 2.3 kpc. Janes & Adler (1982) quote in their catalogue the distance to NGC 6871 as only 1.8 kpc. The value of 1.8 kpc as the distance to Cyg OB3 was obtained again by Garmany & Stencel (1992), who used the H-R diagram fitting method. The discrepancy between the estimates of the distances to HDE 226868 and to Cyg OB3 cast doubt on the membership of Cyg X-1 in NGC 6871. Fortunately, the extensive photometry and spectroscopy by Massey, Johnson & DeGioia-Eastwood (1995) seemed to clarify the situation. They estimated the distance to NGC 6871 as 2.14 ± 0.07 kpc, which was in reasonable agreement with the independent (Hγ width) estimate of the distance to HDE 226868 (Ninkov et al. 1987). Let us note that a very similar value of the distance (_d_≈ 2.15 kpc) was derived also by Gies & Bolton (1986a) as a byproduct of their careful analysis of a large set of observational data for the HDE 226868/Cyg X-1 binary system. Their result (see the point _M_x= 16 M⊙ and _M_opt= 33 M⊙ in their fig. 10) was obtained quite independently of any photometric considerations. We should also note, however, that while Herrero et al. (1995) do not discuss explicitly the value of the distance in their atmosphere modelling paper, their final model implies the value ∼1.8 kpc. Finally, the question of Cyg X-1 membership in NGC 6871 was definitely solved in a recent paper by Mirabel & Rodriguez (2003). Comparing the high-precision VLBI astrometry for Cyg X-1 and Hipparcos astrometry for the members of Cyg OB3, the authors convincingly demonstrated that Cyg X-1 shares, quite precisely, common proper motion with the Cyg OB3 association. The velocity of Cyg X-1 with respect to the Sun is 70 ± 3 km s−1. The relative space velocity of Cyg X-1 with respect to Cyg OB3 is only 9 ± 2 km s−1, which is typical of the random velocities of stars in expanding associations (Blaauw 1991).

After the work by Massey et al. (1995), several more estimates of distances to both Cyg OB3 and HDE 226868 were made and the results still exhibit a substantial scatter. Malysheva (1997), using different photometry than Massey et al. (1995), obtained for the distance to Cyg OB3 again _d_≈ 1.8 kpc. Dambis, Mel'nik & Rastorguev (2001) calculated the trigonometric distance to Cyg OB3 as a median value of the Hipparcos distances to the 18 individual member stars and found _d_= 2.3 (+1.4,−0.6) kpc. In the case of Cyg OB3 their trigonometric distance agreed very well with the photometric distance of Blaha & Humphreys (1989), although they found that, typically, Hipparcos trigonometric distances to OB associations are ∼12 per cent smaller than the photometric distances of Blaha & Humphreys. A recent photometric distance estimate made by Mikołajewska (private communication) gave the result _d_= 1.95 ± 0.10 kpc. Lestrade et al. (1999) used VLBI astrometry to estimate the distance to Cyg X-1 and obtained _d_=1.0–2.3 kpc.

All this extensive effort leads to the conclusion that the distance to the HDE 226868/Cyg X-1 binary system cannot be significantly different from 2 kpc. Therefore, I believe that the best choice will be to follow Massey et al. (1995) and use _d_= 2.15 ± 0.07 (1σ error) or ±0.2 (3σ error) kpc as the distance to Cyg X-1. As we shall see at the end of the paper, this choice will be, quite independently, supported by evolutionary considerations. The above value is used throughout the rest of this paper (except when I assume the distance to be a free parameter). In those parts of the discussion where the distance is assumed to be a free parameter, I shall consider the value 1.8 kpc as a reasonable lower limit to the distance of Cyg X-1.

2 The Model-Independent Lower Limits to the Components Masses

In this section, I shall repeat Paczyński's (1974) analysis, using the present-day data. This analysis is based only on solid observational facts and is, therefore, model independent. Let us remind readers that these solid facts include: (i) the value of the mass function, (ii) the lack of X-ray eclipses, (iii) the obvious fact that the optical component cannot be larger than its Roche lobe, and (iv) the spectral type and the photometry of the optical component (used to derive its effective temperature and the bolometric correction).

The most recent determination of the mass function is f(_M_x) = 0.251 ± 0.007 M⊙ (Gies et al. 2003), which differs only slightly from the earlier values f(_M_x) = 0.244 ± 0.005 M⊙ (Brocksopp et al. 1999) and f(_M_x) = 0.252 M⊙ (Gies & Bolton 1982). The corresponding projected radius of the optical component orbit is _a_1 sin _i_= 8.36 ± 0.08 R⊙. The spectral type of HDE 226868 was determined by Walborn (1973) as O9.7 Iab. This classification was confirmed by Gies & Bolton (1986a), by Ninkov et al. (1987) and by Herrero et al. (1995) and was used in most of the literature. On the other hand, Massey et al. (1995) obtained, from their new massive spectroscopy of OB associations, the somewhat earlier spectral type ON9 Ifa+. According to the calibration of effective temperatures for late O-type supergiants derived by Vacca, Garmany & Shull (1996, hereafter VGS), one has _T_e= 32 740 K for an O9 Ia star and extrapolation for an O9.7 Ia gives _T_e= 30 690 K. On the other hand, Herrero et al. (1995) used atmosphere models to reproduce the observed spectrum of HDE 226868 and obtained _T_e= 32 000 K as the best fit for the effective temperature of this star.

The calibrations of both VGS and Herrero et al. were based on pure H–He models of the atmospheres that neglected the effects of metallic lines blanketing. It is well known by now that taking into account the line blanketing leads to a lower effective temperature for a given spectral type (Martins, Schaerer & Hillier 2002; Markova et al. 2004; Repolust, Puls & Herrero 2004). The most recent calibration of effective temperatures for O-type supergiants (accounting for the line blanketing effects) was derived by Markova et al. (2004). Their scale gives generally lower effective temperatures than the VGS scale (by up to 10 000 K for the earliest spectral types). However, for the spectral type O9.7 I, both temperature scales converge to the value _T_e= 30 700 K. It seems that, at the present state of knowledge, the most reasonable approach would be to assume that _T_e= 30 700 K is the best estimate of the effective temperature of HDE 226868. To take into account the uncertainty of this estimate, in the further discussion I shall consider a broad interval of _T_e= 28 000–32 000 K, as the range of the possible values of the effective temperature of HDE 226868.

As the effective temperature–bolometric correction relation is not affected by the line blanketing (Martins et al. 2002), it is possible to take this relation from VGS tables. For the effective temperature 30 700 K, the appropriate value of the bolometric correction is BC =−3.06. The corresponding unreddened colour index should be (B_−_V)o=−0.26 (Schmidt-Kaler 1982). The most recent photometry of HDE 226868 by Massey et al. (1995) gives _V_= 8.81 and _B_−_V_= 0.83. Therefore the reddening is E _B_−_V_= 1.09 and the unreddened V magnitude is _V_o= 5.43 (following Massey et al., I adopt _RV_= 3.1). With the effective temperature and the bolometric correction given above, we can then express the radius and luminosity of HDE 226868 as functions of distance:

(1)

(2)

Following the procedure of Paczyński (1974), we can now calculate the lower limits to the masses of both components as functions of the distance to the system (under the assumptions listed at the beginning of this section). The results of these calculations are given in the first part of Table 1.

Table 1

Parameters of the components of the HDE 226868/Cyg X-1 binary system as functions of the assumed distance d for three values of the effective temperature of the optical component.

Let us notice that the changes, compared with Paczyński's table 1, are mostly due to substantial changes in the effective temperature (Paczyński took _T_e= 25 000 K) and the corresponding change in the bolometric correction for HDE 226868. Let us also notice that the effects of the changes in the effective temperature and the bolometric correction, to a large degree, cancel each other when we consider the radius of HDE 226868 and the lower limits to the masses of both components. They introduce, however, a dramatic (by a factor of ∼2.5) increase of the luminosity of HDE 226868 (for a given distance). As a result, HDE 226868 appears to be a very bright star: its optical luminosity is ∼1–2 × 1039 erg s−1 (two orders of magnitude higher than the typical X-ray luminosity of Cyg X-1).

Undoubtedly, we now know the calibration of effective temperatures for O-type supergiants much better than 30 years ago. However, still, there remains some uncertainty, clearly demonstrated by the recent major revision introduced by accounting for line blanketing effects (as discussed earlier). To investigate the effect of this uncertainty, I assumed that the effective temperature of HDE 226868 lies in the range 28 000–32 000 K and calculated the corresponding lower limits to the masses of both components (using the modified versions of equations 1 and 2). The results of these calculations are given in the second and third part of Table 1. All sets of results are illustrated in Fig. 1.

Figure 1

The lower limits to the masses of the optical (A) and compact (B) components as functions of the assumed distance, d. Thick solid lines correspond to the most likely value of the effective temperature of HDE 226868 (_T_e= 30 700 K). The broken lines and the dash-dotted lines correspond to the effective temperatures 28 000 K and 32 000 K, respectively. The thin solid vertical line indicates the most likely value of the distance (_d_= 2.15 kpc). The broken vertical lines indicate ±1σ errors in the distance estimate. The ±3σ error range corresponds to the distance interval 1.95–2.35 kpc.

Comparing both sections of the table (or looking at Fig. 1), we may notice that the uncertainty of the effective temperature does not significantly affect the lower limits to the masses.

Summarizing the considerations of this section, we may state that the masses of the compact and the optical components must be greater than ∼8 M⊙ and ∼29 M⊙, respectively (if the distance to the system is in the range 2.15 ± 0.07 kpc). Let us remind readers that these values are model independent and therefore very hard to contest. If one wants to be more conservative and use the 3σ error as the uncertainty of the distance estimate, then these lower limits become ∼7 M⊙ and ∼24 M⊙, respectively. Even adopting the smallest value ever claimed for the distance (_d_= 1.8 kpc), one still obtains the limits ∼6 M⊙ and ∼19 M⊙, respectively. One should add that our method gives only approximate estimates, since (as noted by Gies 2004, private communication) it assumes isotropic optical emission, which is not the case for a tidally distorted star. The resulting inaccuracies should not, however, be significant.

The lower limits, presented in Table 1 and Fig. 1, become the actual values if the fill-out factor is equal to 1.0 and the inclination of the orbit corresponds to the grazing eclipse orientation. The fill-out factor cannot be much different from 1.0 (Gies & Bolton 1986a,b). However, the inclination is a source of larger uncertainty. If the inclination is close to 62° (as suggested by the polarimetry, Dolan & Tapia 1989), then we are close to the grazing eclipse situation and the masses are close to the lower limits obtained in this section, i.e. 8–9 M⊙ and 30–36 M⊙, respectively. If (as seems more likely − see the discussion in Section 5) the inclination is rather close to 33° (advocated by Gies & Bolton 1986a, and still not excluded by polarimetry), then the masses are substantially higher and probably close to the values suggested by Gies & Bolton (16 ± 5 M⊙ and 33 ± 9 M⊙, respectively). I shall return to this point in the further discussion.

3 The Evolutionary Status of HDE 226868

Quite independent constraints on the masses may be obtained from analysis of the evolutionary status of the optical component. First, let us note that we know relatively well its luminosity. At the distance of 2.15 ± 0.07 kpc, the luminosity must be _M_bol≈−9.29 ± 0.07[equivalent to log(_L_/L⊙) = 5.62 ± 0.03 in Table 1]. Including the uncertainty of the effective temperature and assuming 3σ error in the distance estimate, the range for the luminosity becomes _M_bol≈ (−8.8)–(−9.6)[equivalent to log(_L_/L⊙) = 5.40–5.75 in Table 1]. The real range is somewhat wider because one should include also the errors in the estimates of E B_−_V (about ±0.05) and _R_V (about ±0.1). The final range therefore becomes _M_bol≈ (−8.5)–(−9.9). One may note that this range corresponds to _MV_≈ (−5.7)–(−6.7), which agrees quite well with _MV_≈−6.5 ± 0.2 obtained by Ninkov et al. (1987) from the equivalent width of the Hγ line.

Secondly, let us consider the evolutionary phase of HDE 226868. Even without the specific evolutionary calculations, it is relatively straightforward to argue that it must be a core hydrogen burning configuration. This comes from the fact that HDE 226868 must be in a relatively stable phase of its evolution. We observe no measurable variations of the orbital period of the system. The 3σ upper limit for the relative change of the orbital period | d ln P/d_t_| is ∼ 2–3 × 10−5 yr−1 (Gies & Bolton 1982). This implies that the present evolutionary time-scale of HDE 226868 must be substantially longer than ∼3–5 × 104 yr. This excludes the possibility of post-main-sequence evolution, as the expected evolutionary time-scale is, in that case, of the order of 103 yr. Ziółkowski (1977), investigating the post-main-sequence evolution of a 25-M⊙ star (of similar radius but smaller luminosity than HDE 226868) approaching its Roche lobe, found that its radius increased from 95 per cent of Roche lobe filling to 100 per cent in only ∼600 yr. After filling the Roche lobe, it took only ∼200 yr for the mass transfer rate to exceed the Eddington limit by three orders of magnitude. For HDE 226868, which is more luminous and, probably, more massive (35–45 M⊙, as we shall see later), the relevant time-scales should be even shorter. If HDE 226868 were expanding that fast (5 per cent radius increase in less than 600 yr), we should observe a noticeable rise in its stellar wind strength and the X-ray luminosity of the system (both parameters are sensitive to the fill-out factor if it is close to 1). Moreover, the chance of observing the system during such a short time window (just a few hundred years, before it gets extinguished as an X-ray source due to hyper-Eddington accretion) is very small. Therefore, we have to assume that HDE 226868 (similarly to the optical components of other massive X-ray binaries; see Ziółkowski 1977) must still be burning hydrogen in its core.

Exactly the same conclusion may be obtained directly as the unique outcome of the numerical modelling of the evolution of HDE 226868. This topic will be discussed in the next section.

4 The Evolutionary Calculations for HDE 226868

4.1 The general description

I computed evolutionary tracks for the core hydrogen burning phase of stars with initial masses in the range 40–80 M⊙. The Warsaw evolutionary code developed by Bohdan Paczyński and Maciek Kozłowski and kept updated by Ryszard Sienkiewicz was used. The initial chemical composition _X_= 0.7 and _Z_= 0.03 is adopted. The opacity tables incorporating OPAL opacities (Iglesias & Rogers 1996) as well as molecular and grain opacities (Alexander & Ferguson 1994) were used. The nuclear reaction rates are those of Bahcall & Pinsonneault (1995). The equation of state used was Livermore Laboratory OPAL (Rogers, Svenson & Iglesias 1996). I neglected semiconvective mixing, as it is not important during the evolutionary phase considered (most of the models of interest had central hydrogen content _X_c≳ 0.2). Similarly any overshooting at the border of the convective core was neglected (it is even less important).

The calculations were carried out under the assumption that the evolution starts from the homogeneous configurations. This means that the consequences of the fact that some of the matter of the star, possibly dumped from the progenitor of the present black hole, could have somewhat modified chemical composition, were neglected. It means also that the consequences of the fact that some nuclear evolution (hydrogen burning) could, possibly, take place while the mass of the star was smaller (prior to mass transfer) were neglected as well. It seems that both simplifications do not significantly alter the outcome of the evolutionary calculations. I shall return to this point in the later discussion.

4.2 The stellar wind mass loss

The most uncertain element of the calculations of the early evolution of massive stars is the mass loss due to the stellar wind. The uncertainty of the estimate of its rate is the single most important factor influencing the outcome of the calculations. The observations seem to indicate that there is a substantial scatter of the mass-loss rates (typically, by a factor of 2, but sometimes this factor can reach up to 5) among stars of similar luminosities and effective temperatures (see, e.g. De Jager, Nieuwenhuijzen & van der Hucht 1988). I decided to use the prescription given by Nieuwenhuijzen & de Jager (1990) in the form of the formula derived by Hurley, Pols & Tout (2000, hereafter HPT). Bearing in mind that the formula gives the mass-loss rate estimate with an accuracy that is probably not better than within a factor of 2, I introduced the multiplying factor _f_SW applied to the HPT formula. For each initial mass of the star three evolutionary sequences were calculated with the value of the parameter _f_SW equal, in sequence, to 0.5, 1 and 2. In this way, the uncertainty of the theory of evolution could, hopefully, be taken into account. It might be interesting to note, at this point, the astonishingly good agreement between the HPT formula and the observational determination of the mass-loss rate for HDE 226868: for the most likely values of the parameters of the star, the HPT value agrees with the observed one (, Gies et al. 2003, see below) to an accuracy of about 5 per cent (this is, definitely, better than the precision of both the HPT formula and of the observational estimate).

I should remind readers at this point that the supergiant components in some high-mass X-ray binaries are significantly undermassive for their luminosities (Ziółkowski 1977). In some systems, like Cen X-3, this undermassiveness is very serious and requires much stronger mass loss than the normal stellar wind (Ziółkowski 1978). However, most likely, this is not the case for HDE 226868. Its parameters may be fully explained by evolution with the normal stellar wind (as will be demonstrated in the further discussion).

4.3 The evolutionary tracks

Some of the obtained evolutionary tracks in the H-R diagram are shown in Fig. 2. The careful reader might notice that the luminosities of my models are by about 0.1 dex (∼25 per cent) smaller than the evolutionary tracks of Schaller et al. (1992). This difference should be attributed mainly to the fact that I used a later edition of the opacity tables containing higher values of the opacities (the opacity tables always evolve in the direction of growing opacity – never the other way). The correctness of the new opacities was confirmed by stellar pulsation calculations – see e.g. Pamyatnykh (1999). Part of the luminosity difference is due to the fact that I used higher metallicity (_Z_= 0.03 instead of _Z_= 0.02). And, finally, part of the luminosity difference (all factors work in the same direction) is due to the fact that my stellar winds are stronger (again due to higher metallicity).

Figure 2

The evolutionary tracks in the H-R diagram. The tracks are labelled with the initial mass of the star (in solar units). The solid lines describe the tracks computed with the stellar wind mass-loss rates according to the HPT (Hurley, Pols & Tout 2000) formula. The broken lines and the dash-dotted lines describe the tracks computed with the mass-loss rates smaller by a factor of 2 and larger by a factor of 2, respectively. The slanted dotted lines correspond to the position of HDE 226868 for different assumed values of its distance (the assumed value of the distance in kpc is given at the right end of each line). The vertical dotted lines correspond to the effective temperatures of HDE 226868 equal (from left to right) to 32, 31, 30 and 28 × 103 K. The most likely position of HDE 226868 lies within the large parallelogram (± 3 σ error in distance).

As may be seen from a quick look at Fig. 2, the initial (zero-age main-sequence) mass of HDE 226868 had to be in the range 35–55 M⊙. The masses of the models corresponding to the present-day state of HDE 226868 (inside the 3σ parallelogram) are in the range 32.4–50.5 M⊙. The central hydrogen content in these models is between 0.126 and 0.265 and their evolutionary age (since the beginning of central hydrogen burning) is between 2.7 and 4.3 × 106 yr.

To obtain evolutionary models which satisfactorily reproduce the present-day state of HDE 226868, one has to match not only the luminosity and the effective temperature but also the rate of the stellar wind mass loss. The first observational determination of this parameter for HDE 226868 was done by Hutchings (1976) who, analysing the visual spectrographic data, got the value . Persi et al. (1980) estimated the rate of the mass outflow from the infrared emission of the expanding circumstellar envelope and got the value . The more recent estimate of Herrero et al. (1995) was based on the fits of the Hα profile and led to the conclusion that the mass-loss rate lies between 2 and 6 × 10−6 M⊙ yr−1. They noted that the inaccuracy of the fits is probably due to the fact that the stellar wind from HDE 226868 is focused towards the black hole (for a discussion of the ‘focused’ wind model see Gies & Bolton 1986b; Miller et al. 2002). In their test modelling of the atmosphere of HDE 226868, Herrero et al. were using the value and for their final model they chose . The most recent estimate by Gies et al. (2003) gave the value for the low/hard state (which is a typical state of Cyg X-1) and for the high/soft state (which is less frequent in this source). Taking all this into account, I assumed that the observed rate of mass outflow from HDE 226868 is with an error of a factor of about 2, i.e. I assumed that the observed rate of mass outflow is between 1.3 and 5.2 × 10−6 M⊙ yr−1. Let me remind readers that our theoretical evolutionary models were using mass-loss rates in the range 0.5–2 times the rate given by HPT. Altogether, this means that the model for which the mass-loss rate calculated with the HPT formula would be a factor of up to 4 smaller or a factor of up to 4 larger than the nominal observational value is still considered to be an acceptable match. This is, probably, more than sufficient allowance for the uncertainty of the mass-loss rates. The parameters of some of these models are given in Table 2. The positions of the acceptable models in the mass−luminosity diagram (for the most likely value of the effective temperature _T_e= 30 700 K) are shown in Fig. 3. Since the luminosity of a model of HDE 226868 can be (for an assumed effective temperature) directly translated into the distance to the system (see Table 1), Fig. 3 can also be considered as the mass−distance diagram. The corresponding distance scale is shown on this picture. The line corresponding to the lower mass limit−distance relation obtained in Section 2 (see Table 1 and Fig. 1) is also shown. One may notice that the deduced mass of HDE 226868 depends mainly on the assumed distance to the binary system, and only very weakly on the assumptions about the mass-loss rates. For the assumed effective temperature (_T_e= 30 700 K) this mass has to be in the range 35–45 M⊙ (if the distance to the system is in the range 1.8–2.35 kpc). One may also conclude (with the help of the lower mass limits derived in Section 2) that the distance to the system cannot be larger than ∼2.5 kpc (still for the assumed effective temperature).

Table 2

Parameters of the selected evolutionary models of the binary system HDE /Cyg X-1.

Figure 3

The positions of the evolutionary models of HDE 226868 (with an assumed effective temperature _T_e= 30 700 K) in the mass−luminosity and mass−distance diagram (‘mass’ means here the present mass of the star). The scale of the distances (in kpc) is shown with the thin broken vertical lines (the most likely value of the distance is 2.15 ± 0.2 kpc, 3 σ error). The triangles, circles and squares correspond to the models with the stellar wind mass-loss rates calculated with the HPT formula multiplied by the factors 0.5, 1 and 2, respectively. Only the models with acceptable rates of mass loss (in the range 0.25–4 times the nominal observed value, ) are shown. The solid line (without circles) describes the relation between the distance and the lower limit for the mass of HDE 226868 obtained in Section 2. The location of crossing of this relation by the sequences of the evolutionary models indicates that the distance to the binary system cannot be larger than ∼2.5 kpc (for the assumed effective temperature). The oval indicates the range of the models of the optical star for which viable models of the binary system could be constructed (see the discussion in Section 5.2).

Similar diagrams as Fig. 3 may be constructed for other possible values of the effective temperature of HDE 226868. Qualitatively, they look similar to Fig. 3. For the effective temperature _T_e= 28 000 K the mass of the star has to be in the range 29–35 M⊙ and the upper limit for the distance to the system is ∼2.1 kpc. For the effective temperature _T_e= 32 000 K the mass of the star has to be in the range 37–50 M⊙ and the upper limit for the distance to the system is ∼2.7 kpc.

5 Discussion

5.1 The mass of HDE 226868

The obvious and fairly strong conclusion derived from the evolutionary calculations is that HDE 226868 had to be quite massive in the past and is still very massive at present. To put it very briefly, it has to be very massive because it is very bright (while still burning hydrogen in the core). For the most likely range of parameters (30 000–31 000 K for the effective temperature and 2.08–2.22 kpc for the distance) the present mass of HDE 2268668 has to be in the range 37–44 M⊙. Increasing the interval of the possible values of the distance to 1.95–2.35 kpc increases this range only slightly (to 33.5–47 M⊙). Extending the interval of the possible values of the effective temperatures to 28 000–32 000 K leads to the range of possible masses 31–50 M⊙. Finally, extending the range of the possible distances down to 1.8 kpc (the smallest value quoted in the literature) results in the final range of the possible masses of HDE 226868: 29–50 M⊙. The value of 29 M⊙ seems to be a firm lower limit for the present mass of HDE 226868.

One may ask whether it is possible to construct substantially less massive evolutionary models by taking a more liberal estimate of the observational uncertainties or accounting for some simplifications of our models. The answer is negative. Decreasing the effective temperature of HDE 226868 would result in lower luminosities and so would lead to less massive configurations. However, the value _T_e= 28 000 K, taken as the lower limit to the effective temperature in the above consideration, is probably already too low (the true value is probably in the range 30 000–31 000 K). Decreasing the minimum distance from 1.95 to 1.8 kpc decreased the minimum possible masses of the models by only ∼2 M⊙, as was mentioned above. Let us now consider the possible consequences of some simplifications of our evolutionary calculations. I neglected the possible dumping of matter on HDE 226868 from the progenitor of Cyg X-1 (and so ‘rejuvenation’ of the star) during the first mass-loss/exchange phase. Most likely, there was no significant accretion on HDE 226868 because any substantial mass transfer in such a close binary and in so early stage of evolution (when both stars had no distinct cores) would lead to the rapid build-up of a common envelope and the coalescence of both components. So, most likely, the first mass transfer was mainly the mass loss from the system. If there were no serious accretion, then the nuclear evolution prior to the mass transfer is properly accounted for in our calculations. If (which is rather unlikely) there was a substantial mass dumping on HDE 226868, then the nuclear evolution before the mass transfer is negligible because the substantially less massive star was evolving much more slowly [large mass gain essentially resets the evolutionary clock to zero-age main sequence (ZAMS)].

One may wonder whether smaller metallicity (_Z_= 0.02 instead of _Z_= 0.03) would not be a better choice for HDE 226868. To check the possible effects of such an alternative choice, I calculated several evolutionary tracks for stars with _Z_= 0.02. As might be expected, the changes in the results were small. For the models reproducing the ‘best’ parameters of HDE 226868, the masses decreased from about 41.7 M⊙ to about 39.9 M⊙ (only about 4 per cent).

There remains the rather remote possibility that the chemical composition of HDE 226868 is not normal, i.e. that it contains less hydrogen than a normal Population I star. For example, Herrero et al. (1995) concluded, from fitting models of the atmosphere, that helium might be overabundant by a factor of about 2. The evolutionary calculations presented here indicate that the stellar wind mass loss from HDE 226868 was not severe enough for hydrogen depleted layers to show up on the surface. However, if a large amount of matter with significantly decreased hydrogen content was dumped on HDE 226868 from the progenitor of Cyg X-1, then the star might contain less hydrogen than the assumed value of 0.7. The lack of CNO anomalies (Dearborn 1977) testifies against this being the case. However, to check the consequences of such a (rather unlikely) situation, I calculated several evolutionary sequences for stars with the initial chemical composition _X_= 0.5, _Z_= 0.03 (this corresponds to an overabundance of helium by a factor of 2). As could be easily expected, the models with similar luminosities were now less massive, but the change was not very substantial. For the ‘best-fitting’ models, the masses decreased from about 41.7 M⊙ to about 35.3 M⊙ (only about 15 per cent). However, I would like to stress again that there are no good reasons to expect such abnormal hydrogen content in HDE 226868.

The only way to produce an evolutionary model with mass ≲20 M⊙ is to decrease drastically the distance to the system. I constructed one evolutionary track that produced a 19.5-M⊙, configuration at the effective temperature 30 700 K with the mass-loss rate (the initial mass of the star was 32.5 M⊙). However, the price was very high: the distance had to be decreased to ∼0.9 kpc and the rate of the mass loss during the evolution had to be multiplied by a factor of ∼20 with respect to the HPT formula. Any of these conditions, even taken separately, seems extremely unlikely.

There is a clear conflict between the above considerations and the mass estimate given by Herrero et al. (1995). They used atmosphere models to reproduce the observed spectrum of HDE 226868. Then, they used their fit to determine the effective temperature, the surface gravity, the radius and the helium abundance of the star. Their mass estimate (_M_∼ 18 ± 4 M⊙) is a consequence of the surface gravity and the determination of the radius.

I see no solution of this conflict. I may only repeat that the evolutionary calculations for such an early evolutionary phase are very robust and rather difficult to dispute (it is, essentially, the mass–luminosity relation for main-sequence stars). I may also add that the evolutionary results presented above are roughly consistent with the model of Gies & Bolton (1986a), based on an extensive analysis of the large and diversified collection of observational data.

5.2 The mass of Cyg X-1

In the previous section we concluded that the present mass of HDE 226868 has to be in the range 29–50 M⊙ (assuming the widest possible ranges of the distances and the effective temperatures). The mass of its companion, black hole Cyg X-1, does not come out directly from the evolutionary calculations. However, it can be calculated (with the help of some observational constraints) once we select a chosen evolutionary model of HDE 226868. Once the selection is made, we know the mass _M_opt and the radius _R_opt of the optical component. We can also calculate the distance, with the help of equation (1) or (2) (or the corresponding expressions for other effective temperatures). Subsequently, we can use two equations to solve for the inclination of the orbit i and the mass ratio _q_=_M_opt/_M_x. One of these equations makes use of the mass function,

(3)

The other relates the radius of the star to the size of the orbit,

(4)

where _R_RL is the radius of the Roche lobe around HDE 226868, _f_RL is the fill-out factor (_f_RL=_R_opt/_R_RL), A is the orbital separation of the binary components and _a_1 is the radius of the orbit of HDE 226868 (see Section 2).

Inserting the observational data, equations (3) and (4) can be written as

(5)

(6)

Once a given evolutionary model of HDE 226868 is selected from the grid of the acceptable models and a value of the parameter _f_RL is assumed, equations (5) and (6) can be solved for i and q. Knowing q we can immediately calculate also the mass of compact component _M_x. In principle, this procedure can be applied to any combination of the evolutionary model and of the value of _f_RL. In fact, however, not every evolutionary model of HDE 226868 (acceptable if we consider the optical component alone) permits the construction of a consistent model of the binary system. This is because of the observational constraints on the value of the inclination i and, especially, because of the strong observational constraints on the value of the fill-out factor _f_RL. As demonstrated by Gies & Bolton (1986a,b), in order to explain quantitatively the He emission lines produced in the stellar wind from HDE 226868, the fill-out factor _f_RL has to be larger than 0.9 and, most likely, not smaller than 0.95 (perhaps the best value would be around 0.98). On the other hand, Gies & Bolton demonstrated that the observed rotational broadening of the photospheric absorption lines of HDE 226868 and the observed amplitude of the ellipsoidal light variations impose substantial constraints on the values of the mass ratio, the fill-out factor and the inclination. The rotational broadening depends mainly on the mass ratio and the measured broadening indicates the mass ratio in the range ∼2–2.5. The amplitude of the ellipsoidal light variations is determined mainly by the fill-out factor and the inclination. For the assumed values of _f_RL equal 0.9, 0.95 and 1, the resulting inclination is ∼38°, ∼ 33° and ∼28°, respectively.

Constructing models of the binary system to be consistent with observations, I assumed that for any adopted value of _f_RL, the calculated inclination should be within ±5° of the corresponding values quoted above. I started with different models of the optical component, acceptable from the point of view of stellar evolution, as described in Section 4.3. Then, I assumed the value of _f_RL equal to 0.95 and solved equations (5) and (6) to find q, i and _M_x. Subsequently, I tried higher values of _f_RL. The higher values of _f_RL produced solutions with lower (in many cases unacceptably low) values of the inclination. The dependence of q, i and _M_x on the assumed value of _f_RL may be seen from many sequences of binary models, presented in Table 2. For all sequences included in the table, I present only two limiting models for the lowest and the highest value of _f_RL for which an acceptable model could still be obtained. If there is only one entry for a given sequence (as is the case for _M_opt= 39.67 M⊙ and _M_opt= 47.37 M⊙), it means that only one value of _f_RL produced an acceptable solution (the value of _f_RL was varied with the increment of 0.01).

I classified, as acceptable, the models satisfying the following criteria:

• 1
  _d_= 1.8–2.35 kpc;
• 2
  _T_e= 28 000–32 000 K;
• 3
  ;
• 4
  _f_RL≥ 0.95;
• 5
  _i_= 28°+ (1 −_f_RL) × 100°± 5° (this corresponds to _i_≈ 28°–38° for _f_RL= 0.95 and _i_≈ 23–33° for _f_RL= 1.00, as advocated by Gies & Bolton 1986a).

Parameters of selected acceptable models are given in the second part of Table 2. For each assumed value of the effective temperature, I present two sequences of models corresponding to the lowest and the highest current mass of the optical component. The exception to this rule is the sequence of models for _T_e= 30 700 K and _M_opt= 41.67 M⊙ (_d_= 2.15 kpc). The initial evolutionary parameters (_M_0 and _f_SW) of the optical component model for this sequence were adjusted so as to obtain the perfect fit with the ‘best’ values of the observational parameters of HDE 226868 (effective temperature, luminosity and the rate of stellar wind mass loss). All models from this sequence (for _f_RL= 0.96–1.00) are successful from the point of view of our criteria.

There are several conclusions that can be drawn from the collection of the obtained models of the binary system (only some of these models are shown in Table 2). The first concerns the mass of the compact component. Assuming the widest possible intervals of the distances and the effective temperatures: 1.8–2.35 kpc and 28 000–32 000 K, the mass of the compact component must be in the range 13.5–28.5 M⊙. For the most likely intervals of these parameters: 1.95–2.35 kpc and 30 000–31 000 K, this range narrows to 15.5–25 M⊙. The second conclusion is related to the distance to the binary system. It appears that for a given (assumed) effective temperature, consistent models are possible only for some, relatively narrow, interval of distances (the distances outside of this interval would require unacceptably low or unacceptably high values of the inclination). For the effective temperatures equal to 28, 30, 30.7, 31 and 32 × 103 K, the corresponding intervals are 1.80–1.88, 1.98–2.15, 2.04–2.22, 2.07–2.27 and 2.21–2.35 kpc, respectively. It is certainly encouraging that the most likely range of the effective temperatures (30 000–31 000 K) requires the distance interval (1.98–2.27 kpc) that almost exactly coincides with the independent estimate of the most likely distance range (1.95–2.35 kpc). The obtained relation between the distance and the effective temperature of HDE 226868 means that, if in the future the distance will be known more precisely (e.g. from future astrometric space missions), then it will be possible to set constraints on the effective temperature. For example, if it is found that the distance to the system is greater than 2.1 kpc, it would mean that the effective temperature of HDE 226868 has to be higher than 30 000 K. Also the opposite is true. If it is found (e.g. from better models of the atmospheres) that the effective temperature of HDE 22868 is greater than 30 000 K, it would mean that the distance to the system must be larger than 2.0 kpc (and this conclusion would not depend on any photometric or astrometric considerations).

Finally, the third conclusion confirms the earlier results (based on evolutionary calculations alone) limiting the present mass of HDE 226868 to the range 29–50 M⊙ and the initial mass to the range 33.5–55 M⊙.

Let us note that for a black hole mass in the range 15–25 M⊙, the state transitions occur in Cyg X-1 at the luminosity level equal to 0.025–0.015 of the Eddington luminosity (assuming _d_= 2.15 kpc and using the flux values given by Zdziarski et al. 2002). These values lie in the range 0.007–0.03 found for other X-ray binaries (Maccarone 2003).

To summarize this section, the evolutionary considerations provide much stronger lower limits to the masses of both components than the model-independent analysis, presented in Section 2. In addition, they yield some independent constraints on the distance to the system.

6 The Possible Evolutionary Scenario

Let us speculate a little bit about the possible evolutionary past of our binary system. It had to start as a very massive system– the initial primary had to complete its evolution by the time the secondary (HDE 226868) reached its present evolutionary state, i.e. in less than ∼3–4 × 106 yr. The initial masses of the components were probably ∼80–100 and ∼40–50 M⊙. The massive primary (the progenitor of a black hole) was shedding the mass mainly in the form of the stellar wind (there was probably little, if any, mass transfer to the companion). The mass of the primary decreased, at the end of its evolution (before the collapse), to ∼15–25 M⊙. The low spatial velocity of Cyg X-1 with respect to its parental association indicates, as argued by Mirabel & Rodriguez (2003), that the final collapse to a black hole proceeded with very little (≲1 M⊙) mass ejection. It could be, even, a prompt collapse with no accompanying supernova explosion at all (‘formation of a black hole in the dark’). The secondary (the progenitor of HDE 226868) was also losing mass in the form of the stellar wind, although at a lower rate. This mass loss took place both before and after the collapse of its companion. This process decreased the mass of the secondary to the present value of ∼35–45 M⊙. In this way, the present-day binary consisting of a massive supergiant of ∼40 ± 5 M⊙ (with a central hydrogen content of ∼0.2–0.3) and a black hole of ∼20 ± 5 M⊙ was formed.

The above evolutionary history is only one of the possible scenarios. As noted by Gies (private communication), the above scenario implies that about half of the total initial mass of the binary system was lost in the form of the stellar wind. This implies, in turn, that the initial orbital period was ≲3 d, which means that the system was very tight. One cannot exclude the possibility that the initial masses were smaller, the initial orbital period longer and the system passed through the mass transfer and the common envelope phase.

7 Conclusions

• (i)
  The calculations modelling the evolution of HDE 226868, under different assumptions about the stellar wind mass-loss rate, provide robust limits on the present mass of the star. For the most likely intervals of the values of the distance and of the effective temperature: 1.95–2.35 kpc and 30 000–31 000 K, the mass of HDE 226868 is 40 ± 5 M⊙. Extending the intervals of these parameters to 1.8–2.35 kpc and 28 000–32 000 K, one obtains the mass of the star in the range 29–50 M⊙.
• (ii)
  Including the additional constraints resulting from the observed properties of the binary system HDE 226868/Cyg X-1, one can estimate the mass of the black hole component. For the most likely values of the parameters mentioned in item (i) above this mass is 20 ± 5 M⊙. For the extended intervals of the parameters the mass is in the range ∼13.5–29 M⊙.
• (iii)
  The distance to the binary system has to be in the ranges 1.80–1.88, 1.98–2.15, 2.04–2.22, 2.07–2.27 and 2.21–2.35 kpc, for the effective temperature of HDE 226868 equal to 28, 30, 30.7, 31 and 32 × 103 K, respectively.

Acknowledgments

I would like to thank A. Zdziarski for careful reading of the manuscript and for many helpful comments and stimulating discussions. I would like also to thank D. Gies who, acting as a referee, made several comments and suggestions which helped to improve this paper. This work was partially supported by the State Committee for Scientific Research grants No 1 P03D 018 27 and No PBZ KBN 054/P03/2001.

References

,

1995

,

Rev. Mod. Phys.

,

67

,

781

,

1991

, in , eds, NATO ASIC Proc. 342,

The Physics of Stars Formation and Early Stellar Evolution

.

Kluwer

, Dordrecht, p.

125

,

2001

,

Sov. Astron. Lett.

,

27

,

58

DOI:

,

1977

,

Astrophys. Lett.

,

19

,

15

,

1989

,

ApJ

,

344

,

830

DOI:

,

1982

,

ApJ

,

260

,

240

DOI:

,

1986

,

ApJ

,

304

,

371

DOI:

,

1986

,

ApJ

,

304

,

389

DOI:

et al. ,

2003

,

ApJ

,

583

,

424

DOI:

,

1978

,

ApJS

,

38

,

309

DOI:

,

2000

,

MNRAS

,

315

,

543

(HPT)DOI:

,

1976

,

ApJ

,

203

,

438

DOI:

,

1996

,

ApJ

,

464

,

943

DOI:

,

1982

,

ApJS

,

49

,

425

DOI:

,

1997

,

Sov. Astron. Lett.

,

23

,

585

,

1973

,

ApJ

,

185

,

L113

DOI:

,

1995

,

ApJ

,

454

,

151

DOI:

,

2003

,

Sci

,

300

,

1119

DOI:

,

1987

,

ApJ

,

321

,

425

DOI:

,

1999

,

Acta Astron.

,

49

,

119

,

1996

,

ApJ

,

456

,

902

DOI:

,

1982

, in , eds,

Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group VI, Vol. 2b

.

Springer

, Berlin, p.

31

,

1996

,

ApJ

,

460

,

914

(VGS)DOI:

,

1973

,

ApJ

,

179

,

L123

DOI:

Hammerschlag-Hensberge

G.

,

1982

,

PASP

,

94

,

149

DOI:

,

2002

,

ApJ

,

578

,

357

DOI:

,

1977

,

Eighth Texas Symposium on Relativistic Astrophysics

. Annals New York Acad. Sci.,

302

,

47

,

1978

, in , ed.,

Nonstationary Evolution of Close Binaries

.

Polish Scientific Publishers

, Warsaw, p.

29

© 2005 RAS

Citations

Views

Altmetric

Metrics

Total Views 887

617 Pageviews

270 PDF Downloads

Since 1/1/2017

Citations

97 Web of Science

×

Email alerts

Citing articles via

• Latest

• Most Read

• Most Cited

More from Oxford Academic