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A&A
Volume 636, April 2020
Article Number A99
Number of page(s) 7
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/201937253
Published online 27 April 2020

© ESO 2020

1. Introduction

The detection by LIGO/Virgo of gravitational waves (GW) resulting from merging binary neutron stars (BNS), merging binary black holes (BBH), and merging mixed binary systems (BNSBH) opened a new window to explore the Universe. One of the striking results (striking at least as far as the evolution of massive binaries is concerned) is that the LIGO/Virgo data support the existence of BHs with a mass up to (and even larger than) 40 M⊙. We note that these high masses were predicted in the recent past (De Donder & Vanbeveren 2003, 2004; Belczynski et al. 2010) by evolutionary simulations of massive binaries during which core helium burning (=the Wolf–Rayet [WR] phase) the effect of stellar wind mass loss was calculated using a formalism obtained with a WR-atmosphere code that includes clumping and with a metallicity dependency as predicted by the radiatively driven wind theory (e.g., Hamann & Koesterke 1998, 2000, see also Appendix A).

The LIGO-discovery of the first massive BBH merger (GW150914) was announced in 2015. We note that before 2015, binary population number (BPN) simulations of double compact star binaries predicted GW150914-like events (e.g., Voss & Tauris 2003; Dominik et al. 2012, 2013; Mennekens & Vanbeveren 2014). Since 2015, the number of such BPN simulations has increased significantly.

One of the main uncertainties in the BPN prediction of BBH and BNSBH mergers is whether BH formation is accompanied by a supernova (SN) explosion and whether this SN explosion produces a BH kick that is large enough to affect overall BBH and BNSBH formation. This is the main scope of the present paper.

The proper motion of pulsars has been intensively studied since the early 1990s (e.g., Lyne & Lorimer 1994; Hansen & Phinney 1997; Arzoumanian et al. 2002; Pfahl et al. 2002; Hobbs et al. 2005) and this resulted in the now generally accepted conclusion that neutron stars (NSs) may receive kicks at birth (natal kicks – NKs) with an average in the range of 200–500 km s−1, when they are formed in core-collapse SNe. Possible exceptions are NSs born in a prompt (fast) electron-capture SN where the resulting kick is expected to be small (Nomoto 1984, 1987; Podsiadlowski et al. 2004).

Arguments favoring or disfavoring BH kicks based on direct physical principles are inconclusive (Ozel et al. 2010; Farr et al. 2011; Belczynski et al. 2012). Individual binaries containing a BH companion have been investigated by many authors (e.g., Brandt et al. 1995; Nelemans et al. 1999; Remillard et al. 2000; Gualandris et al. 2005; Willems et al. 2005; Dhawan et al. 2007; Fragos et al. 2009; Wong et al. 2010, 2014, 2012). A common conclusion is that the BHs in these binaries must have either been formed without, or with rather modest, NKs.

Repetto et al. (2012) (see also Repetto et al. 2017) considered the observed distribution of distances above the Galactic plane of low-mass X-ray binaries containing a BH component and compared this with binary population synthesis simulations by adopting different initial NK-distributions. They conclude that a distribution similar to that of neutron stars (e.g., Hansen & Phinney 1997) seems to be preferred. For a critical discussion of the Repetto et al. results, see Belczynski et al. (2016).

Woosley (2019) present detailed evolutionary computations of massive galactic helium stars that have lost all of their hydrogen at the beginning of core helium burning (e.g., remnants after Roche-lobe overflow (RLOF) if these stars were binary components equal to He-zero age main sequence (He-ZAMS) stars). To account for mass loss by Wolf–Rayet-like stellar winds, the author used the prescription of Yoon (2017) (see Appendix A). An important conclusion resulting from these computations is that most of the hydrogen free helium stars with He-ZAMS mass between 9 M⊙ and 60 M⊙, which correspond to end-core-helium burning mass between 7 M⊙ and 30 M⊙ and to progenitor H-ZAMS mass between 30 M⊙ and 120 M⊙, make BHs. It is unclear whether some of them might explode and for those that explode whether the explosion is symmetric or asymmetric.

To further study the latter, the following may be a possibility. The 9–60 M⊙ He-ZAMS mass range is also the mass range of the WR stars in the known galactic WR+O binaries. If the WR star collapses into a BH, a BH+O binary is formed, which may become a wind-fed high-mass X-ray binary (a BH HMXB like Cyg X-1). In the present paper, we consider the following: There are about 70 WR+OB binaries, probable WR+OB binaries in the solar neighborhood1 (van der Hucht 2001), and only one high-mass BH+OB X-ray binary. We explore whether this is compatible with the assumption that the WR stars collapse into a BH and that the collapse does not disrupt the binary. Section 2 describes the methodology. In Sects. 3 and 4, we demonstrate that the aforementioned assumption is incorrect. Section 5 then deals with an alternative.

2. Method

We consider the population of observed WR+O binaries in the solar neighborhood and selected those where we have a reasonable estimate of the masses of both components. Using our evolutionary computations of hydrogen deficient post-RLOF massive core helium burning stars2 and given the mass of the WR star, we calculated the expected mass of the WR star at the end of core helium burning. In order to do this, we assumed that in case the WR star is of the nitrogen sequence (a WN star) (of the carbon sequence, a WC star, respectively) the star is at the beginning of the core helium burning phase (beginning of the WC phase defined as the moment during core helium burning where due to stellar wind mass loss 3-alpha burning products appear at the surface, respectively). It is important to mention that this assumption does not significantly affect the overall conclusions of the present paper. If the final WR star collapses into a BH without a SN explosion (i.e., the mass of the BH is equal to the final WR mass) a BH+O binary is formed, which may evolve into a (wind-fed) BH HMXB. Given the mass-spectral type-luminosity class of the O-type companion, we simulated its further evolution by interpolating in the Geneva tracks described in Ekstrom et al. (2012). We note that these evolutionary computations hold for massive stars that rotate at rates which are similar to the rotation rates of O-type components in WR+O binaries (Shara et al. 2017). We now proceed as follows. A massive binary system consisting of an OB star and a BH may become a HMXB when part of the stellar wind matter that is lost by the OB star is gravitationally trapped by the BH. Losses of gravitational energy of this material at the time of accretion by the BH then produce the X-ray radiation.

To obtain X-rays, however, the presence of an accretion disk is required in the case of a BH (Shapiro & Lightman 1976; Iben et al. 1995). A Keplerian disk is formed when the specific angular momentum of the accreted matter exceeds that of the largest possible stable orbit around the BH, corresponding to a radius RK in which RK is approximately three times the Schwarzschild radius. This condition is satisfied when

R A ≥ ( 1 − R A ) 8 7 ( R K R ) 1 7 ( M M BH ) 3 7 $$ \begin{aligned} \frac{R}{A} \ge \left(1-\frac{R}{A}\right)^{\frac{8}{7}}\left(\frac{R_K}{R}\right)^{\frac{1}{7}}\left(\frac{M}{M_{\rm BH}}\right)^{\frac{3}{7}} \end{aligned} $$(1)

or equivalently

R A ≥ a 1 + a $$ \begin{aligned} \frac{R}{A} \ge \frac{a}{1+a} \end{aligned} $$(2)

with a = ( R K A ) 1 8 ( M M BH ) 3 8 $ a=\left(\frac{R_K}{A}\right)^{\frac{1}{8}}\left(\frac{M}{M_{\mathrm{BH}}}\right)^{\frac{3}{8}} $, where R and M are the radius and the mass of the optical component, respectively, A is the semimajor axis of the binary, and MBH is the mass of the BH. When the optical OB companion of a BH does not fill its Roche lobe, the latter accretes material by capturing a fraction of the stellar wind gas of the former. To estimate the X-ray luminosity in these wind-fed systems, we used the model of Davidson & Ostriker (1973), which involves accretion of matter according to Bondi & Hoyle (1944).

We identified the X-ray luminosity LX with the gravitational energy release of the accreted matter

L X = G M BH M ˙ accr R K , $$ \begin{aligned} L_\mathrm{X} = G\frac{M_{\rm BH}\dot{M}_{\rm accr}}{R_K}, \end{aligned} $$(3)

where G is the gravitational constant and Ṁaccr is the amount of matter accreted from the stellar wind by the BH in a unit of time, given by

M ˙ accr = ε ( M BH M ) 2 δ 2 ( 1 + δ ) 3 / 2 M ˙ $$ \begin{aligned} \dot{M}_{\rm accr}= \varepsilon \left(\frac{M_{\rm BH}}{M}\right)^2\frac{\delta ^2}{(1+\delta )^{3/2}}\dot{M} \end{aligned} $$(4)

with δ = R 2 A ( v esc v wind ) 2 $ \delta=\frac{R}{2A}\left(\frac{\mathit{v}_{\mathrm{esc}}}{\mathit{v}_{\mathrm{wind}}}\right)^2 $. Here, vesc and vwind are the escape velocity of the OB-star and wind velocity at the location of the BH, respectively. The stellar wind mass loss rate of the OB-type star, Ṁ was calculated using the formalism proposed by Vink et al. (2000), which corresponds to the mass loss rate formalism used in the Geneva evolutionary code.

As shown by Groenewegen & Lamers (1989), vwind can be described by a β-law v wind = v ∞ ( 1 − R A ) β $ \mathit{v}_{\mathrm{wind}}=\mathit{v}_\infty\left(1-\frac{R}{A}\right)^{\beta} $, where v∞ is the wind velocity at infinity and β is between 0.7 and 1.5. Our results were computed with β = 1 but the LX values differ by no more than a factor of two when other values of β were used and this difference is small enough not to affect our basic conclusions. The ratio v esc v ∞ $ \frac{\mathit{v}_{\mathrm{esc}}}{\mathit{v}_\infty} $ was also taken from the formalism proposed by Vink et al. (2000).

The factor ε in Eq. (4) accounts for a self-limiting effect on the X-ray luminosity due to its radiation pressure. Eddington (1926) estimated it as

ε = ( 1 − L X L Edd ) 2 $$ \begin{aligned} \varepsilon = \left(1-\frac{L_\mathrm{X} }{L_{\rm Edd}}\right)^2 \end{aligned} $$(5)

with L Edd = 4 π c G M BH σ e = 65335 L ⊙ 1 + X M BH M ⊙ $ L_{\mathrm{Edd}} = \frac{4\pi cGM_{\mathrm{BH}}}{\sigma_{\mathrm{e}}}=\frac{65335\,L_{\odot}}{1+X}\frac{M_{\mathrm{BH}}}{M_{\odot}} $, where X represents the hydrogen abundance in the accreted material, c is the speed of light, and σe is the electron scattering coefficient. The mass accretion rate ṀEdd producing LEdd through its gravitational energy release satisfies

L Edd = G M BH R K M ˙ Edd . $$ \begin{aligned} L_{\rm Edd} = G\frac{M_{\rm BH}}{R_K}\dot{M}_{\rm Edd}. \end{aligned} $$(6)

Then, when solving for LX, one obtains

L X = α ( 1 + 1 + α ) 2 L Edd $$ \begin{aligned} L_\mathrm{X} = \frac{\alpha }{\left(1+\sqrt{1+\alpha }\right)^2}L_{\rm Edd} \end{aligned} $$(7)

with α = 4 δ 2 ( 1 + δ ) 3 / 2 ( M BH M ) 2 M ˙ M ˙ Edd $ \alpha = 4 \frac{\delta^2}{(1+\delta)^{3/2}}\left(\frac{M_{\mathrm{BH}}}{M}\right)^2\frac{\dot{M}}{\dot{M}_{\mathrm{Edd}}} $.

To illustrate our method, it is helpful to consider a fictitious but typical WR+O binary: a WN+O6V system with masses 19 + 32 M⊙ and with an orbital period of 2, 4, 10, and 15 days. The pre-RLOF progenitor of the WN star has a ZAMS mass of 40 M⊙, which implies that the WN star is ∼5.5 Myr old (see Vanbeveren et al. 1998b,a). At the end of core helium burning (∼0.5 Myr later), the WR star is a WC-type star with mass = 11 M⊙, whereas the orbital period has increased due to the spherical wind and reaches values of 2.8, 5.6, 14.1, and 21.1 days, respectively. According to the spectral type and luminosity class of the O-type companion, the star is on the Geneva 32 M⊙ evolutionary track somewhere between 0 and 2 Myr. This means that due to mass accretion during the previous RLOF, the O star is rejuvenated and looks like a 32 M⊙ star that is at most 2 Myr old, even though the lifetime of the binary is 5.5 Myr. We began our simulations by starting from the 2 Myr point on the 32 M⊙ evolutionary track and we computed the temporal evolution of the X-ray luminosity using the formalism described above and assuming that the 11 M⊙ WC star collapses into a 11 M⊙ BH without a SN explosion and thus without a natal kick. The results are shown in Fig. 13. First, it is important to note that all of our wind-fed systems have LX >  1035 erg s−1, which means that they would have been detected by present (and previous) all-sky-monitors if the sources were located within 3–4 kpc from the Sun (prof. Wijnands, prof. van der Klis, priv. comm.). We note that high-mass X-ray binaries with X-ray luminosities similar to the ones of the BH-systems predicted here (around 1035 erg s−1 or larger) are concentrated close to the Galactic plane, and they are seen throughout our Galaxy. So, extinction is not expected to play a role in their detectability. In particular, the hard part of their X-ray spectrum (above 10 keV), which tends to dominate in the case of BHs, is not expected to suffer any interstellar extinction. The results can be interpreted as follows: the average core helium burning lifetime of the WR+O binary is 400 000 ± 100 000 yrs. Figure 1a, which reveals that the X-ray lifetime of the wind-fed system is 4–5 Myr, then illustrates that for every WR+O binary within 3–4 kpc from the Sun with masses 19 + 32 M⊙ and with a period of two days, we expect to observe 4–5 Myr/300 000–400 000 yr = 10–17 BH+OB HMXBs. When we repeated this for the same WR+O binary but with an orbital period of ten days, it follows from Fig. 1d that we expect 1 BH+OB HMXB at most. It is important to note that for a WR+O period of 15 days, we essentially expect no wind-fed HMXBs. By considering the observed WR+O binary population within 3–4 kpc from the Sun and by performing the previous exercise for all of them, we can get an indication of how many BH+OB HMXBs are expected within 3–4 kpc if it is assumed that the WR stars in all of the WR+O binaries collapse into BHs without a SN explosion. This is discussed in the next section.

thumbnail Fig. 1.

Temporal evolution of the X-ray luminosity (as a function of O-star apparant, rejuvenated age; also see the text) of an O+BH binary with orbital parameters indicated in the figure.

3. The WR+O binary population in the solar neighborhood

Using the WR catalogs of van der Hucht (2001, 2006) and of Crowther (2017), Table 1 collects the double-lined spectroscopic WR binaries in the solar neighborhood for which reasonable estimates of the minimum masses of both components are known. The procedure as outlined by Vanbeveren et al. (2018) allows one to propose most probable masses of both components. In summary, applying the mass-spectral type and luminosity class relation proposed by Vanbeveren et al. (1998a) for the O-type component yields the mass-range estimates of Table 14. We note that very similar results are obtained when the calibration of stellar parameters of Galactic O stars of Martins et al. (2005) is used. The combination of the observed mass functions and the inclination angle ranges (we used the values given in the catalogues listed above when available) explains the adopted masses in Table 1. Using the method outlined at the beginning of the previous section then allows one to propose the masses of the WR components and the system periods at the end of core helium burning. The following two remarks are appropriate. First, the minimum mass of the WR components is ∼9 M⊙, corresponding to an initial hydrogen ZAMS mass of ∼25–30 M⊙. Second, in accounting for the study of Woosley (2019), it cannot be excluded that most of the WR components end their life as a BH.

Table 1.

Binary star physical properties.

4. The expected BH+O wind-fed HMXB population in the solar neighborhood

We calculated the expected BH+O wind-fed HMXB population in the solar neighborhood starting from the WR+O sample of Table 1 and assuming that at the end of core helium burning the WR star collapses into a BH without a SN explosion. The further evolution of the O component, and thus the further evolution of the BH+O binary, is interpolated from the Geneva tracks. The onset of X-ray radiation and the evolution of the X-ray luminosity was then computed using the method from Sect. 2. Table 1 lists the lifetime where the system is visible as a wind-fed HMXB with LX >  1035 erg s−1. As outlined in Sect. 2, division by the average WR lifetime = 400 000 yrs gives an indication of how many wind-fed BH+O HMXBs are expected for the WR+O binary considered. The table illustrates the conclusion that if the WR components in the 17 systems collapse into a BH without a SN explosion, we expect 44 wind-fed BH+O HMXBs in the solar neighborhood. If the 17 systems are representative for the whole population of WR binaries in the solar neighborhood (∼80 according to the WR catalogs of van der Hucht 2001, 2006 and of Crowther 2017), the number of expected wind-fed BH+O HMXBs is even more than 200. Since only one system is observed (Cyg X-1), the discrepancy is enormous.

5. Two possible solutions

5.1. Most WR stars end their life as a NS

Binary interaction (RLOF or the common envelope process) removes the hydrogen rich envelope of the mass loser on a very short timescale: the thermal timescale in the case of a RLOF and the dynamical timescale in the case of the common envelope process. At the end of RLOF, most of the mass losers are hydrogen depleted stars at the beginning of core helium burning, that is, in the case of massive binaries, the hydrogen depleted WR phase always starts at the beginning of core helium burning. Together with the effects of WR-like stellar wind mass loss, this has a significant effect on the evolution of the convective helium burning core, in general, and on the evolution of the compactness of the core (the compactness parameter as introduced by O’Connor & Ott 2011) in particular. Detailed calculations of the variation of the compactness parameter in massive helium burning stars have been presented by Sukhbold et al. (2016) and Ebinger et al. (2019; see also references therein). Although these computations hold for massive single stars, we think that they illustrate that it cannot be excluded that many of the WR stars in our Galactic WR+O binaries end their life as a NS. Since these NSs are expected to be accompanied by a asymmetric SN explosion, many of the WR+O binaries may be disrupted and it is clear that this would (at least partly) solve our problem.

5.2. During their formation, BHs receive a kick

The large number of expected wind-fed BH+O HMXBs discussed in the previous section obviously critically depends on the assumption that the collapse of the WR stars into a BH is not accompanied by a SN explosion. The discrepancy with observations may therefore be indicative that BH formation is accompanied by a SN explosion and that the BH receives a (natal) kick. If the natal kick is large enough to disrupt the binary, obviously no wind-fed HMXB is expected. However, the binary does not need to be disrupted in order to reduce the number of expected wind-fed HMXBs drastically. The simulations shown in Fig. 1 (Sect. 2) illustrate that very few wind-fed HMXBs are expected from BH+O binaries with a period > 15 days (see also the evolutionary models of the WR+O binaries in Table 1). So, a natal kick velocity distribution that assures that most of the BH+O binaries resulting from the WR+O sample have a period larger than 15 days significantly reduces the expected number of wind-fed HMXBs, and it explains the fact that only one (Cyg X-1) is observed. To get an idea as to the order of magnitude of the kick velocities that are capable of doing this, we used the formularium given below.

We define

m= M 1 + M 2 M BH + M 2 , $$ \begin{aligned}&m = \sqrt{\frac{M_1+M_2}{M_{\rm BH}+M_2}}, \end{aligned} $$(8)

v rel = || v kick || v orb , $$ \begin{aligned}&{ v}_{\rm rel} = \frac{||\boldsymbol{v}_{\rm kick}||}{{ v}_{\rm orb}}, \end{aligned} $$(9)

v orb = G M 1 + M 2 A , $$ \begin{aligned}&{ v}_{\rm orb} = \sqrt{G\frac{M_1+M_2}{A}} , \end{aligned} $$(10)

where M1 and M2 are the masses of both components before the collapse and A is the distance between both components before the collapse. Additionally, vrel thus represents the magnitude of the kick velocity relative to the orbital velocity. If P is the binary period before the SN explosion, PPS is the binary period after the SN explosion, and θ and ϕ are the angles that describe the direction of the kick velocity vkick with respect to the orbital velocity vorb before the collapse and SN explosion of M1. Then, following Brandt & Podsiadlowski (1995) (see also Vanbeveren et al. 1998a)

c = ( P m P PS ) 2 3 = 2 − m ( 1 + 2 v rel cos θ cos ϕ + v rel 2 ) $$ \begin{aligned} c = \left(\frac{P\sqrt{m}}{P_{\rm PS}}\right)^{\frac{2}{3}}=2-m(1+2{ v}_{\rm rel}\cos \theta \cos \phi +{ v}_{\rm rel}^2) \end{aligned} $$(11)

and after some straightforward calculus

v rel = − cos θ cos ϕ + − 1 + ( cos θ cos ϕ ) 2 + 2 − c m · $$ \begin{aligned} { v}_{\rm rel} = -\cos \theta \cos \phi +\sqrt{-1+(\cos \theta \cos \phi )^2+\frac{2-c}{m}}\cdot \end{aligned} $$(12)

For the WR+O binaries listed in Table 1, we calculated the vkick value needed to obtain a BH+O binary with a period of 15 days (we also give the value to disrupt the binary). For most of the WR+O binaries, m is close to one at the end of the core helium burning phase of the WR component. We intend to give indicative results and therefore we simplified the previous formulae by replacing m with one. It is important to note that in this case, an asymmetrical SN explosion (and a resulting substantial kick) is still possible due to the neutrino momentum flux (see e.g., Brandt & Podsiadlowski 1995). Results for this case m = 1 are given in Table 2 for cos θ cos ϕ = 1 (corresponding to a minimum kick, vkick, min, that is needed to obtain the desired result) and cos θ cos ϕ = 0 (since cos θ cos ϕ ranges between −1 and +1, the vkick corresponding to cos θ cos ϕ = 0 can be considered as an average value, ⟨vkick⟩). Similarly, a maximum value vkick, max can be calculated for cos θ cos ϕ = −1. The ⟨vkick⟩ values of Table 2 allow us to conclude that the paucity of observed BH+O wind-fed HMXBs in the solar neighborhood can at least partly be explained when the collapse of WR stars into a BH is accompanied by an asymmetric SN explosion, resulting in a moderate kick imparted to the BH. We also notice that the ⟨vkick⟩ values that are needed to disrupt most of the WR+O binaries during the collapse of the WR star into a BH are very similar to the average value of the kick velocity distribution of NSs. This means that when BHs are formed with kicks similar to those of NS (as suggested by Repetto et al. 2012, 2017), then most of the WR+O binaries are disrupted when the WR star collapses into a BH.

Table 2.

Required kick velocities.

6. Final remarks

We end the paper with the following remarks.

1. Both solutions discussed in the previous section would explain, in a straightforward manner, the fact that the solar neighborhood only contains 1 WR+BH binary (Cyg X-3).

2. The number of merging BBH expected from isolated binary evolution would obviously be significantly reduced.

3. A kick may lead to an eccentric system and one may wonder if one could get a periodically wind-fed system even when the binary period is 15 days or larger. However investigating the effects of eccentricity and of circularization is beyond the scope of the present paper since it does not affect the main conclusion, that is, the population of high-mass X-ray binaries in general; those with a BH component, in particular, seem to indicate that many Galactic WR stars in WR+O binaries end their life with an asymmetrical SN explosion.

1

Here, 3–4 kpc from the Sun is defined as the solar neighborhood.

2

We used the Brussels stellar evolutionary code as it was described in Vanbeveren et al. (1998a). One of the main uncertainties that significantly affects the core helium burning evolution of massive stars and binary components is the stellar wind mass loss rate formalism used in the code. The latter is discussed in the Appendix A.

3

The final rise of the X-ray luminosity is due to the fact that the 32 M⊙ star enters the hydrogen shell burning phase, thus implying a rapid increase in the radius and the stellar wind mass loss rate (the Vink-rates) of the star. Due to the fact that this phase is very short, it does not affect our population number results. The simulations stop when the star fills its Roche lobe.

4

In case the luminosity class is not known assuming class V applies to the minimum mass and assuming class I applies to the maximum mass.

Acknowledgments

We thank the anonymous referee for valuable comments and suggestions.

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Appendix A: The mass loss rate formalism during the core helium burning phase of hydrogen depleted massive stars

To calculate the evolution of hydrogen depleted massive stars (singles and binaries), since 1998, we (the Brussels team) have used (Vanbeveren et al. 1998c,b,a) a mass loss rate formalism during core helium burning that is based on the clumping corrected mass loss rates of WR stars that were available at that time (Hamann & Koesterke 1998) but also on the WN/WC number ratio of WR-binaries (a mass loss rate that is too high predicts a WN/WC number ratio that is too small and vice versa). We proposed

log 10 M ˙ [ M ⊙ yr − 1 ] = log 10 L [ L ⊙ ] − 10 . $$ \begin{aligned} \log _{10}\dot{M} [{M}_{\odot }\,\mathrm{yr} ^{-1}] = \log _{10}L [{L}_{\odot }] - 10. \end{aligned} $$(A.1)

Recently, the mass loss rates of Galactic WN and of Galactic WC stars and WR stars of the oxygen sequence (WO stars) were reconsidered accounting for the impact of revised distances from Gaia DR2 (Hamann et al. 2019; Sander et al. 2019). In Fig. A.1a, we compare our linear relation with the linear

thumbnail Fig. A.1.

Stellar wind mass loss rate as a function of luminosity of hydrogen depleted massive stars. Panel a: solid blue line: Brussels. Dashed red line: WNE Potsdam. Dotted gray line: WC Potsdam. Panel b: solid blue line: Brussels. Dashed red line: Tramper WC Y = 0.98. Dotted gray line: Tramper WC Y = 0.6. Panel c: dashed red line: Potsdam. Dotted gray line: Yoon fWR = 1. Panel d: solid blue line: Brussels. Dashed green line: Warsaw.

regression lines of Hamann et al. and of Sander et al.. The figure illustrates the statement that the core helium burning evolutionary results predicted with our mass loss rate formalism should be more than satisfactory, in general, for the scope of the present paper in particular. The latter is strengthened by Fig. A.1b where we compare our formalism with the formalism for WC and WO stars proposed by Tramper et al. (2016). And finally Fig. A.1c compares the Potsdam formalism for early type WN stars (WNE stars) with the one advocated by Yoon (2017, fWR = 1). As can be noticed, the latter may significantly underestimate the true WNE-rates.

The relation between the initial mass of a massive primary in an interacting binary and its mass at the end of core helium burning (just prior to the formation of a NS or a BH) critically depends on the adopted stellar wind mass loss rate formalism during core helium burning. The relation computed with our mass loss rate formalism was presented for the first time by De Donder & Vanbeveren (2003, 2004) and it is shown once more in Fig. A.2. The figure illustrates that when the mass of the BH in Cyg X-1 is confirmed to be 15 M⊙, then the initial mass of the BH-progenitor should be at least 60 M⊙. Moreover, massive BHs, such as those predicted by the LIGO/Virgo data, are no surprise in low metallicity regions. We would like to end this appendix with a word of caution. Number synthesis simulations that aim to predict the populations of double compact star binaries (NS+NS, NS+BH, BH+BH) rely on the evolution during core helium burning and thus on the WR-type stellar wind mass loss rate formalism. In Fig. A.1d, we compare our formalism with the one used by the Warsaw team (Belczynski et al. 2010). The difference is quite significant and may explain, at least partly, why some of the results of Warsaw differ from those of Brussels.

thumbnail Fig. A.2.

Relation between the initial mass of a massive primary in an interacting binary and its mass at the end of core helium burning (just prior to the formation of a NS or a BH). Solid blue line: Solar. Dashed red line: SMC. Masses in M⊙.

All Tables

Table 1.

Binary star physical properties.

In the text
Table 2.

Required kick velocities.

In the text

All Figures

thumbnail Fig. 1.

Temporal evolution of the X-ray luminosity (as a function of O-star apparant, rejuvenated age; also see the text) of an O+BH binary with orbital parameters indicated in the figure.
In the text
thumbnail Fig. A.1.

Stellar wind mass loss rate as a function of luminosity of hydrogen depleted massive stars. Panel a: solid blue line: Brussels. Dashed red line: WNE Potsdam. Dotted gray line: WC Potsdam. Panel b: solid blue line: Brussels. Dashed red line: Tramper WC Y = 0.98. Dotted gray line: Tramper WC Y = 0.6. Panel c: dashed red line: Potsdam. Dotted gray line: Yoon fWR = 1. Panel d: solid blue line: Brussels. Dashed green line: Warsaw.
In the text
thumbnail Fig. A.2.

Relation between the initial mass of a massive primary in an interacting binary and its mass at the end of core helium burning (just prior to the formation of a NS or a BH). Solid blue line: Solar. Dashed red line: SMC. Masses in M⊙.
In the text

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