Wind Mass-loss Rates of Stripped Stars Inferred from Cygnus X-1

Cygnus X-1 is a high-mass X-ray binary (HMXB) in the Cygnus OB3 association, which hosts a star in orbit with a BH (e.g., Bolton 1972, 1975; Webster & Murdin 1972; Hutchings et al. 1973). The BH accretes matter from the stellar wind; this accretion powers X-ray radiation (e.g., Davidson & Ostriker 1973; van den Heuvel 1975; Conti 1978; Petterson 1978) and a jet (e.g., Bisiacchi et al. 1974; Marti et al. 1996; Stirling et al. 2001). Orosz et al. (2011) inferred the BH and stellar companion masses of Cygnus X-1 to be 14.8 ± 1.0M_⊙ and 19.2 ± 1.9M_⊙, respectively. Revised measurements of the distance to Cygnus X-1 (Miller-Jones et al. 2021) indicate that both objects are significantly more massive. The temperature and luminosity of the optical companion are estimated to be T_eff = 31.1 ± 0.7 kK and $\mathrm{log}(L/{L}_{\odot })=5.63\pm 0.07$ with a mass of ${M}_{\mathrm{opt}}={40.6}_{-7.1}^{+7.7}{M}_{\odot }$, where we quote the median value and the 68% confidence interval boundaries (Miller-Jones et al. 2021). The mass of the BH is estimated as ${M}_{\mathrm{BH}}={21.2}_{-2.3}^{+2.2}{M}_{\odot }$. The binary has an almost circular orbit with a semimajor axis $a={0.244}_{-0.013}^{+0.012}$ au and eccentricity $e={0.0189}_{-0.0026}^{+0.0028}$ (Miller-Jones et al. 2021). The BH is inferred to be nearly maximally spinning, with a dimensionless spin of at least 0.95, according to both disk continuum and reflection line fitting studies (Gou et al. 2011; Fabian et al. 2012; Miller-Jones et al. 2021; Zhao et al. 2021). Via optical spectroscopy, it has been found that the ratios of the surface abundances of both helium and iron to hydrogen are about twice the respective values for the Sun (Shimanskii et al. 2012). As we discuss below, these observations taken together present a challenge for models of massive stellar binary evolution.

In this paper, we describe the constraints that these observations place on the Cygnus X-1 formation channel (Section 2). We explore how the HeMS phase of this channel provides a constraint on the wind mass-loss rates of massive stars (Section 3). We present the likely future of the system and its implications for gravitational-wave detections (Section 4). Finally, we discuss some caveats and questions raised by the observations of Cygnus X-1 (Section 5).

We assume the following formation channel for Cygnus X-1 (see Figure 1 for illustration). Two stars are born in a binary. The more massive star (the primary) evolves more quickly and expands first. The companion (the secondary) is close enough for the late main-sequence primary to commence mass transfer. The primary is stripped of its envelope, leaving an exposed helium core. The core continues nuclear fusion until it collapses and forms a BH in orbit with the still core hydrogen burning MS companion. In the sections below, we describe the observations and theoretical analyses that support this channel.

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2.1. Eccentricity, Peculiar Velocity, and Fallback

2.2. Black Hole Progenitor Mass

2.3. Black Hole Spin and Progenitor Spin

2.4. Companion Mass and Abundance

We compared the observed mass, luminosity, and temperature of the optical companion against analytic fits to stellar tracks of Hurley et al. (2000) as implemented in the COMPAS rapid population synthesis code (Stevenson et al. 2017; Vigna-Gómez et al. 2018). The observations are consistent with a MS star that is about 70%–80% through its core hydrogen burning phase, ignoring the impact of rotation.

These stellar tracks are for stars with regular hydrogen-rich atmospheres. Using them ignores the possible impact of nonstandard surface abundances, and so corresponds to the assumption that only a thin surface layer has a significant overabundance of helium, rather than a uniform distribution of enriched material throughout the star. Stars in later stages of the MS with the relevant mass and metallicity should have, at most, a very thin convective layer at the surface (e.g., Maeder et al. 2008; Kippenhahn et al. 2012). Therefore, mixing is expected to be relatively inefficient; in the absence of large-scale convection, the Rayleigh–Taylor instability is likely to be suppressed by temperature inversion in the accreted material (Kippenhahn et al. 1980; Braun & Langer 1995). The resulting thermohaline mixing will mix the helium-rich material through the companion only on timescales longer than the expected few × 10⁴ yr since the formation of the BH if the bulk of the enriched material was accreted at or shortly before the BH formation.

The evolutionary channel shown in Figure 1 assumes that the optical companion has not overflowed its Roche lobe. While previous studies explored this possibility, perhaps with intermittent Roche lobe overflow followed by longer periods when the binary is detached as in the present state, these models were typically based on assumptions that the companion is less massive than the BH (Podsiadlowski et al. 2003), which is inconsistent with present observations. Indeed, if we assume that mass transfer onto a black hole is almost entirely nonconservative because of the Eddington limit, the binary's semimajor axis a evolves as

$$\begin{eqnarray}&&\displaystyle \frac{\dot{a}}{a}=-2\displaystyle \frac{{\dot{M}}_{\mathrm{opt}}}{{M}_{\mathrm{opt}}}\left[1-\left(\gamma +\displaystyle \frac{1}{2}\right)\displaystyle \frac{{M}_{\mathrm{opt}}}{{M}_{\mathrm{opt}}+{M}_{\mathrm{BH}}}\right],\end{eqnarray} \tag{ 1 }$$

where γ is the specific angular momentum of the ejected material in units of the binary's specific orbital angular momentum J/(M_opt + M_BH). The binary can widen as a result of such mass transfer only if γ ≲ 1 for the observed component masses. However, in the common assumption of isotropic re-emission from the accretor, γ = M_opt/M_BH ≈ 2. The ejected material would have to carry much lower specific angular momentum in order for the binary to be able to detach once Roche lobe overflow from the companion commences, which seems unlikely, lending support to our proposed channel.

The estimated mass and lifetime of the companion already enable us to roughly infer the initial mass of the BH progenitor. We assume that both stars are born and start fusion at the same time, and we use a fit (Farr & Mandel 2018) to the Brott et al. (2011) and Köhler et al. (2015) stellar models for the MS lifetime of nonrotating massive stars. For this simple estimate, we ignore the impact of binary interactions, consistent with the assumption of largely nonconservative mass transfer. The BH progenitor should have had an initial mass of ∼55–75M_⊙ in order to complete its evolution while leaving behind an MS companion of mass M_opt at ∼70%–80% of its MS lifetime.

The surface abundances of the companion are nonstandard for massive MS stars—and a challenge to explain even in the context of binary interaction. We discuss the abundances in more detail in Section 5. Here, we focus on the iron abundance, as this is key to the analysis of line-driven winds in the following section. Shimanskii et al. (2012) find that the iron abundance of the companion is 2.2 times the solar iron abundance, although precise measurements are challenging due to the complexity of the system. Assuming Z_⊙ = 0.014 for solar metallicity (Asplund et al. 2009), this corresponds to an effective metallicity of Z ≈ 0.03. On the other hand, Daflon et al. (2001) find a slightly sub-solar iron abundance of $\mathrm{log}\epsilon (\mathrm{Fe})=7.33\pm 0.12$ in HD 227460, which is a B0.5V star in the same Cygnus OB3 association as Cygnus X-1. This could indicate that Z ≈ 0.01 is a better estimate of the initial iron abundance, and the iron abundance of the companion in Cygnus X-1 has been enhanced during the collapse of the primary to a BH. Therefore, we explore an initial metallicity range 0.01 ≤ Z ≤ 0.03 in the following section.

Hereafter, we assume that, after the mass transfer episode induced by the BH progenitor, the primary is left with no hydrogen layer on top of the He core. This assumption is consistent with the channel proposed by Qin et al. (2019), in which the rapidly spinning BH is ultimately formed through the collapse of a Wolf-Rayet (WR) star (see, however, discussion in Section 5).

In order for Cygnus X-1 to form through the channel depicted in Figure 1, two things must be true. First, the HeMS star must be born with sufficient mass to give rise to the observed BH mass even after losing mass through WR winds during the HeMS. However, at a given metallicity, there is a maximum to the He core mass that can be formed at the end of the MS: as more massive stars have higher wind mass-loss rates, terminal-age MS He core masses asymptotically approach a maximum as a function of zero-age MS. This maximum He core mass is plotted in Figure 2. Because the BH progenitor is stripped of its hydrogen envelope at the end of the MS for the binary evolutionary channel we assume, we can directly constrain the amount of mass loss during the HeMS phase. The WR winds then must not remove more than the difference between this maximum mass and the final BH mass. We refer to this as the " ${M}_{\mathrm{HeMS}}\leqslant {M}_{\mathrm{HeMS},\max }$ " condition.

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The maximum He core mass plotted in Figure 2 is very sensitive to the MS wind prescriptions; for further discussion, see Renzo et al. (2017), Neijssel et al. (2019), and Miller-Jones et al. (2021). The analysis of the remnant mass from massive single stars is particularly sensitive to luminous blue variable winds (Belczynski et al. 2010; Miller-Jones et al. 2021), which can remove the hydrogen envelope of the star and hasten the onset of the WR phase (Conti 1975); therefore, increasing the potential remnant mass can be achieved by decreasing either luminous blue variable winds or WR winds. The plot in Figure 2 assumes the default COMPAS luminous blue variable mass-loss rate of 1.5 × 10⁻⁴ M_⊙ yr⁻¹ (Belczynski et al. 2010) for stars approaching the Humphreys–Davidson limit (Humphreys & Davidson 1994). The convective overshooting parameter employed in stellar evolution calculations is another source of uncertainty, with greater overshooting leading to larger cores (Brott et al. 2011). To explore the impact of convective overshooting, we simulated the He core mass at terminal-age MS for stars with different metallicities starting from a zero-age MS mass of 150 M_⊙ using the stellar evolution code MESA (v12115, Paxton et al. 2011). We find that varying the overshooting parameter by an order of magnitude between f = 0.02 and f = 0.2 (Qin et al. (2019) used f = 0.11) with the step overshoot scheme changes the He core mass at terminal-age MS by between 1% and 13% for the range of metallicities in Figure 2. Finally, the COMPAS single stellar evolution models are based on the fitting formulae of Hurley et al. (2000) to the evolutionary tracks of Pols et al. (1998), which involve extrapolations to higher masses than those covered by the initial range of models and may underestimate the He core masses of massive stars. We therefore use the following additional constraint, which bypasses these sources of uncertainty in the maximum He core mass.

In our assumed channel, the secondary MS companion must not overflow its Roche lobe onto the HeMS primary. We write this constraint as (Eggleton 1983)

$$\begin{eqnarray}&&{R}_{2}(t)\leqslant a(t)\displaystyle \frac{0.49{q}^{2/3}(t)}{0.6{q}^{2/3}(t)+\mathrm{ln}(1+{q}^{1/3}(t))},\end{eqnarray} \tag{ 2 }$$

where m₂(t) and R₂(t) are the mass and radius of the companion as a function of time, m₁(t) is the mass of the BH or its progenitor, $q(t)=\tfrac{{m}_{2}(t)}{{m}_{1}(t)}$ is the mass ratio, and a(t) is the orbital separation of the binary.

The orbital separation widens due to wind mass loss. In the limit of fast, noninteracting winds, the widening is described by

$$\begin{eqnarray}&&\displaystyle \frac{\dot{a}}{a}=-\displaystyle \frac{\dot{{M}_{\mathrm{tot}}}}{{M}_{\mathrm{tot}}},\end{eqnarray} \tag{ 3 }$$

where M_tot = m₁ + m₂ is the total mass of the system. Because winds widen the binary and remove mass from the HeMS primary faster than from its MS companion, they increase the size of the secondary's Roche lobe over time. Thus, even though the secondary is not overflowing its Roche lobe now, it may have done so in the past if the mass-loss rate was high. If we evolve the system back in time, the requirement that the secondary never overflows its Roche lobe imposes an alternative upper limit (Axelsson et al. 2011) on the maximum WR wind mass-loss rate. We refer to this as the "no RLOF" condition. Although the noninteracting wind assumption, describing the widening of the binary (Equation (3)), may be an oversimplification for such short-period systems (MacLeod & Loeb 2020), it allows us to conservatively estimate the constraints imposed by the existence of Cygnus X-1 and presented below.

We parameterize the mass-loss rate through WR winds, modeled with the prescription proposed by Belczynski et al. (2010) and based on the works of Hamann et al. (1995), Hamann & Koesterke (1998), and Belczynski et al. (2010), with a multiplicative parameter f_WR, following Barrett et al. (2018) (see Appendix for a definition and discussion). We constrain the allowed parameter space of the wind strength by rewinding the evolution of the binary from the current state. Because the BH formed recently in our model, we set the luminosity and temperature of the MS companion at the end of the HeMS phase of the primary equal to the current inferred luminosity and temperature of the observed MS secondary. The mass and age of the secondary inferred from temperature and luminosity vary slightly for different metallicities. As we argued earlier, the BH is expected to lose negligible mass during collapse, so we set the mass of the HeMS primary at the end of that phase equal to the inferred BH mass. We assume that the secondary was 99.7% Roche lobe filling at the end of the primary's HeMS phase (Miller-Jones et al. 2021). ⁹

The reverse evolution of the MS and HeMS stars is followed using the analytic fits to the stellar tracks of Pols et al. (1998) as presented in Hurley et al. (2000). The winds of the MS star are given by Vink et al. (2001). The HeMS winds are parameterized with the multiplicative factor f_WR as described above and in the Appendix. The orbital response to mass loss is given by Equation (3). We go back in the evolution for a HeMS lifetime (note that the HeMS lifetime depends on how massive the HeMS star initially was, which depends on the wind strength we assume) and check that the ${M}_{\mathrm{HeMS}}\leqslant {M}_{\mathrm{HeMS},\max }$ condition is satisfied at the start of the HeMS phase and the no RLOF condition is satisfied throughout this phase.

Figure 3 shows the upper limits on the wind strength, parameterized as f_WR, imposed by these two conditions as a function of metallicity. Both conditions show that the wind strength has to be reduced from the nominal value f_WR = 1 throughout the range of metallicities we have explored. The strongest constraint is placed by the ${M}_{\mathrm{HeMS}}\leqslant {M}_{\mathrm{HeMS},\max }$ condition, which is subject to uncertainties in the MS wind strengths, overshooting, and the single stellar evolution fits of Hurley et al. (2000). This yields an upper limit f_WR ≲ 0.4 at Z = 0.01 and ≲0.05 at Z = 0.03. However, even if we lift this constraint, the no RLOF condition still places a strong constraint on the allowed mass-loss rate: f_WR ≲ 0.45 at Z = 0.01 and ≲0.15 at Z = 0.03. We also explore the impact of BH mass and Roche lobe filling factor measurement uncertainties and find that the constraints on f_WR change by ≲0.1 over the range consistent with observations (Miller-Jones et al. 2021).

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The revised mass of the companion star in Cygnus X-1 makes it a potential candidate for a future BH and raises the intriguing prospect that this system could form a merging binary BH, connecting HMXBs with gravitational-wave sources. We model the future evolution of the system and find that it is unlikely to form a merging binary BH.

Belczynski et al. (2011) used earlier, lower estimates of the mass of the BH and companion in Cygnus X-1 in order to analyze the future evolution of this system. They predicted that imminent mass transfer from the nearly Roche lobe filling companion will significantly reduce the companion mass, leaving behind a core that could only form a neutron star, not a BH. They further estimated that the natal kick has a 30% probability of unbinding the binary, and there was only a ∼1% probability that the ensuing neutron star–black hole binary would merge within 14 Gyr through gravitational-wave emission.

Here, we update their predictions based on revised observations, using the COMPAS population synthesis code (Stevenson et al. 2017; Vigna-Gómez et al. 2018) for binary evolution calculations. We estimate the initial masses and orbital separation as in Section 3. We assume a Roche lobe filling factor of 0.997 and a metallicity of Z = 0.02. The median values for the BH mass and the luminosity and temperature of the MS companion from Miller-Jones et al. (2021) then yield M_BH = 21.2M_⊙, M_opt = 38.9M_⊙, a = 51.3R_⊙, and τ_MS = 0.81, i.e., the MS companion is 81% through its core hydrogen burning phase by this time. Once the MS companion overflows its Roche lobe, the mass transfer phase could brighten Cygnus X-1 up to close to its Eddington luminosity of a few × 10³⁹ erg s⁻¹. Once the envelope is removed, the companion will appear as a Wolf-Rayet (WRs) star in an HMXB in a phase lasting for ∼ 3 × 10⁵ yr. In the default model described below, we apply WRs mass loss with f_WR = 1 to such a star.

If we follow a treatment similar to Belczynski et al. (2011), in which the He core mass is determined by the mass of the star at the end of MS (after stripping in this case), we find the companion will collapse into a neutron star. Even if the binary survives the neutron star natal kick, which happens in 7% of all binaries in our models, it will generally be too wide to merge through gravitational-wave emission, with only a 0.12% probability that it will merge in 14 Gyr, similar to the findings of Belczynski et al. (2011). During this time, it may be detectable in radio pulsar surveys as a neutron star–black hole binary, although the nonrecycled pulsar will likely only be observable as a radio source for a few tens of Myr. However, this treatment may significantly underpredict the core masses of stars that donated mass on the MS.

In our revised model, we address the potentially underestimated core mass of stripped MS donors under the assumption that this core mass is determined only by the mass of the star at the end of the MS. We account for the substantial amount of helium synthesized by the companion before interaction with the following crude approximation. We consider a constant rate of helium production during the MS, so the mass of helium produced is M_He,MS = τ_MS M_c,final, where M_c,final is the final helium core mass that would be achieved at the end of the MS phase for the given stellar mass, in the absence of mass transfer. Keeping track of the helium mass synthesized before stripping, M_He,MS, leads to higher core masses at the end of the MS phase.

With this correction, we predict that in the absence of BH natal kicks, Cygnus X-1 will become a bound binary BH, but one that is too wide to merge within 14 Gyr. If we incorporate the COMPAS prescription for BH kicks based on the "delayed" model of Fryer et al. (2012), this changes to a 38% probability of surviving the kick and forming a bound binary BH, with a 4% probability that this binary will merge within 14 Gyr through gravitational-wave emission due to a fortuitous kick. However, the kick prescription for low-mass black holes that do not undergo complete fallback (Fryer et al. 2012) is rather uncertain, with conflicting evidence on the natal kick magnitudes of low-mass BHs (Mandel 2016; Mirabel 2017; Repetto et al. 2017; Atri et al. 2019; Wyrzykowski & Mandel 2020).

We also consider the impact of varying WRs mass loss. Motivated by the results reported in Figure 3 for Z = 0.02, we reduce f_WR to 0.2 from the value of 1 considered above. This increases the remnant mass of the current optical companion from ∼2.9 M_⊙ to ∼5.7 M_⊙, and the binary's probability of remaining bound after the supernova from 38% to 62%. With our adjusted prescription for the helium core mass of the stripped companion and f_WR = 0.2, Cygnus X-1 has a 5% probability to merge within 14 Gyr as a binary BH.

We thus find that it is possible that a small fraction of HMXBs like Cygnus X-1 could form merging binary BHs, although this conclusion is sensitive to the treatment of mass transfer from MS donors in population synthesis models and to the natal kick distribution of relatively low-mass BHs. If systems like Cygnus X-1 do become progenitors of gravitational-wave events, this would impact the predicted spin distribution of merging binary black holes (Kushnir et al. 2016; Zaldarriaga et al. 2018; Fuller & Ma 2019; Bavera et al. 2020; Belczynski et al. 2020). Gravitational-wave observations could ultimately address this possibility by resolving the spin distribution with more events (e.g., Farr et al. 2017). Meanwhile, wide, nonmerging binary BHs could potentially be observable through microlensing (Eilbott et al. 2017).

We show that the current properties of the Cygnus X-1 system imply a reduction in WR wind mass-loss rates for exposed HeMS stars. These results depend on several key assumptions.

We assumed that the optical companion did not experience Roche lobe overflow in its past. This assumption is consistent with the challenge of detaching from mass transfer once it commences given that the companion has roughly twice the mass of the BH, as explained in Section 2.4. However, it is somewhat surprising that several HMXBs with well-measured properties—Cygnus X-1, LMC X-1, and M33 X-7—share not only a high BH spin but also a similar evolutionary state. Selection effects favor observing bright, long-lived systems, i.e., those with massive main-sequence donors that are close to Roche lobe filling (enabling more efficient accretion). There may also be an evolutionary stalling point, increasing the number of systems in this phase.

The latter scenario could indicate that the systems do manage to detach and resume mass transfer multiple times. While this would negate our wind mass-loss rate conclusions, it would imply that much less angular momentum is carried away during nonconservative mass transfer onto a BH than we expected. Assuming nonconservative mass transfer (valid if accretion onto a BH is Eddington-limited) from a donor that is twice as massive as the accretor, the specific angular momentum of the material ejected from the binary in units of the binary's specific orbital angular momentum must be γ < 0.85 in order to avoid a decrease in the size of the Roche lobe (see Equation (1) for the change in orbital separation). For comparison, isotropic re-emission from the BH corresponds to γ = 2. Conversely, if γ = 2, the companion could still disengage from mass transfer if its radius shrinks faster than the size of the Roche lobe in response to mass loss. This would require the adiabatic logarithmic derivative of radius with respect to mass to exceed $\zeta \equiv d\mathrm{log}R/d\mathrm{log}M\gt 1.54$, which may be possible for stars in the late phase of their MS evolution that have already lost some mass.

It is also possible that the primary is not fully stripped during the mass transfer episode, but retains about ∼0.1 M_⊙ of its hydrogen envelope (e.g., Yoon et al. 2010; Bersten et al. 2014; Götberg et al. 2017, 2018; Yoon 2017; Laplace et al. 2020). The changed surface abundance could lead to reduced mass-loss rates until the remaining hydrogen is completely removed. However, it is not clear whether retaining an envelope of a fraction of a solar mass could be sufficient to prevent WR-like winds. In any case, whether WR wind mass-loss rates must be lower than anticipated or whether stars that experience mass transfer in binaries are only partially stripped, the impact on binary evolution is similar: there is less mass loss than previously assumed. In fact, our WR wind reduction factors can be broadly interpreted as constraints on winds from stripped stars, whether they are naked helium stars or retain a small hydrogen-rich envelope.

Naked helium cores can expand significantly in the last stages of their lives, potentially leading to another mass transfer episode from the BH progenitor late in the evolution. However, the degree of expansion is very mild for stars with initial masses ≳20 M_⊙ at near-solar metallicities (Yoon et al. 2010; Hirai 2017; Laplace et al. 2020), so Cygnus X-1 is unlikely to have experienced such mass transfer.

As a consequence of its mass accretion history, Dray & Tout (2007) posit that the secondary may be overluminous relative to single stars of the same total mass; however, see also Hellings (1983), who concludes that the MS accretors quickly return to single-star models, and Braun & Langer (1995), who reach the opposite conclusion and find that accretors are underluminous. Since we use single star evolutionary tracks to estimate the properties of the secondary star, this can affect our wind constraint from the no-RLOF condition. However, since we choose a stellar model that matches the observed radius of the secondary at the present day, we do not anticipate the impact to be significant.

Finally, as Figure 3 shows, the level of reduction in the winds is sensitive to the assumed metallicity of Cygnus X-1. We now discuss this in more detail.

Shimanskii et al. (2012) report that helium, carbon, oxygen, aluminum, sulfur, and iron are overabundant by [X/H] = 0.23–0.43 dex compared to the solar values (Anders & Grevesse 1989). Nitrogen, neon, and silicon have an even higher overabundance of [X/H] = 0.69–0.94 dex. These values appear robust against variations due to orbital motion and Roche lobe filling factors, although some hydrogen and helium lines are sensitive to variations in the wind (Shimanskii et al. 2012).

Previous accretion from the BH progenitor could significantly alter the chemical profile on the surface of the companion. For example, the detailed models of Qin et al. (2019) predict the observed enhancement of companion nitrogen abundances as a consequence of late main-sequence mass transfer from the BH progenitor. In addition to direct accretion, which is expected to enhance helium and nitrogen abundances, the deposited angular momentum can lead to a dramatic spin-up of the MS star (e.g., Packet 1981), although spin-up to near-breakup frequencies may suppress subsequent accretion. The surface could then be enhanced by helium and CNO-elements due to rotational mixing (Przybilla et al. 2010; Heger & Langer 2000; Meynet & Maeder 2000). The rotational mixing might also make the star overluminous compared to a nonrotating model (Langer 1992). The optical companion is observed to be tidally locked at present, with an inferred ratio of the rotational to orbital frequency of 1.05 ± 0.10 Miller-Jones et al. (2021). Assuming a present-day rotational period of 5.6 days, the rotational frequency is a third of the Keplerian (breakup) frequency at the stellar equator. Alternatively, as discussed above (and contrary to the channel assumed in this work), the MS companion may have been partially stripped by mass transfer onto the BH or its progenitor after the initial mass transfer phase from the primary, revealing deeper layers of the star.

Although these mechanisms could be responsible for the overabundance of some of the elements, they have difficulty in explaining the overabundance of late-stage burning elements such as silicon and iron. This implies that these elements were primordially enhanced or were deposited from the progenitor of the BH in the final stages of its evolution.

High primordial abundances imply that both stars in the binary had a high metallicity at birth, which therefore requires a very strong reduction in the mass-loss rate (a factor of ∼10) following the constraints described in Section 3. On the other hand, a weak explosion induced by the collapse of the core can lead to an ejection of a small fraction of the outer part of the envelope at very low velocities. Because most of the envelope is assumed to fall back into the BH, the ejected material will be barely above the escape velocity—and could be efficiently accreted by the companion in a RLOF-like manner. If the heavy elements synthesized at the center are efficiently mixed up to the outer regions before the inner slower material starts falling back, these elements can accrete onto the surface of the secondary. Such efficient mixing of heavy elements has been observed in supernovae such as SN 1987A and Cassiopeia A (e.g., Utrobin et al. 1995; Fesen et al. 2006) and has been reproduced in 3D supernova explosion simulations (e.g., Hammer et al. 2010; Wongwathanarat et al. 2015, 2017), while Liu et al. (2015) and Hirai et al. (2018) explore the contamination of an MS companion by supernova ejecta. However, it is not clear whether similar degrees of mixing can be induced in failed supernovae that form BHs rather than neutron stars, and the abundance pattern of the Cygnus X-1 companion merits further investigation.

Regardless of whether we assume that the observed companion metallicity of Z = 0.03 (Shimanskii et al. 2012) is primordial or use the lower metallicity of Z = 0.01 based on HD 227460 (Daflon et al. 2001), we conclude that the observed properties of Cygnus X-1 require a reduction in WR winds to ∼5%–40% of their previously assumed values in the context of our assumed evolutionary channel.

Recent theoretical modeling of mass loss from stripped stars (Vink 2017; Sander et al. 2020; Sander & Vink 2020) points to reduced mass-loss rates compared to earlier literature. Moreover, these models suggest a steep dependence on the Eddington factor, which can change significantly during the lifetime of the stripped star. This indicates that extrapolating the empirical WR mass-loss rates to the entire duration of the stripped star life is misleading. Our results are qualitatively consistent with these findings. The reduced mass-loss rates could also be attributed to strong wind clumping, which is expected to occur in line-driven winds due to radiative instabilities (Owocki et al. 1988; Sundqvist et al. 2018). Clumping of winds has been indirectly observed for massive MS stars in X-ray binaries (El Mellah et al. 2018; Lomaeva et al. 2020), and stripped stars may also experience high degrees of clumping.

We further find that HMXBs like Cygnus X-1 form through a different evolutionary channel than the bulk of merging binary black holes; for a review, see, e.g., Mandel & Farmer (2018). However, a fortuitous natal kick accompanying the birth of the secondary BH could lead Cygnus X-1 to merge as a BH binary within 14 Gyr. Gravitational-wave observations may be able to constrain the contribution of this channel to the formation of merging binary BHs through spin measurements.

Simulations in this paper made use of the COMPAS rapid binary population synthesis code, which is freely available at http://github.com/TeamCOMPAS/COMPAS. We thank Alexander Heger, Hagai Perets, Jerry Orosz, Stephen Justham, Selma de Mink, and Jakub Klencki for discussions, and Robert Izzard and Silvia Toonen for comments on a preliminary draft of the manuscript. C. J. N. thanks the University of Birmingham for financial support. J. C. A. M.-J. and I. M. are recipients of Australian Research Council Future Fellowships (FT140101082 and FT190100574, respectively). A. V. G. acknowledges funding support by the Danish National Research Foundation (DNRF132).

In this appendix, we describe the particular parameterized formalism we used to model wind-driven mass loss from stripped stars. While more recent theoretical models are available (e.g., Vink 2017; Sander et al. 2020), this provides us with a convenient framework for investigating the empirical constraints placed by Cygnus X-1.

Hamann et al. (1995) formulated a prescription for the wind mass-loss rate for stripped or WR stars as a function of the mass and luminosity of the stripped star. Hamann & Koesterke (1998) expanded on this work by reducing winds by a factor of $\sqrt{D}$ where D is the wind clumping factor (Moffat et al. 1988; Nugis et al. 1998). Vink & de Koter (2005) further introduced a metallicity dependence to the winds. Combining the effects of clumping and metallicity leads to the prescription

$$\begin{eqnarray}&&{({dM}/{dt})}_{\mathrm{WR}}=\mathop{\underbrace{\displaystyle \frac{1}{\sqrt{D}}{10}^{-11.95}{\displaystyle \frac{L}{{L}_{\odot }}}^{1.5}}}\limits_{1}\mathop{\underbrace{{\displaystyle \frac{Z}{{Z}_{\odot }}}^{0.86}}}\limits_{2}\ {M}_{\odot }\ {\mathrm{yr}}^{-1},\end{eqnarray} \tag{ A1 }$$

where term 1 is the result of Hamann et al. (1995) and Hamann & Koesterke (1998), and term 2 is from Vink & de Koter (2005). Setting D = 100, i.e., reducing winds by a factor of 10, recovers Equation (9) of Belczynski et al. (2010) and is consistent with the winds of Yoon et al. (2010).

We follow Barrett et al. (2018) in scaling the prescription of Belczynski et al. (2010) by a multiplicative factor f_WR in order to parameterize the uncertainty in the wind mass-loss rates:

$$\begin{eqnarray}&&{({dM}/{dt})}_{\mathrm{WR}}={f}_{\mathrm{WR}}\times {10}^{-13}{\displaystyle \frac{L}{{L}_{\odot }}}^{1.5}{\displaystyle \frac{Z}{{Z}_{\odot }}}^{0.86}\ {M}_{\odot }\ {\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ A2 }$$

Note that this is not the same f_WR as used in Yoon et al. (2010), because ours already assumes a reduction of the original wind prescription of Hamann et al. (1995). The default assumption of f_WR = 1 corresponds to the default models of Belczynski et al. (2010), Yoon et al. (2010), Stevenson et al. (2017), and Qin et al. (2019).

It is challenging to interpret our constraints on f_WR in terms of a wind clumping factor D. A direct interpretation of f_WR = 0.2 would imply a clumping factor of D = 2500 in the model of Equation (A1). Nugis et al. (1998) and Nugis & Lamers (2000) used a clumping factor ranging from 10 to 30; D could be as high as 16 according to Hamann & Koesterke (1998). More recent theoretical models by Sander et al. (2017) use depth-dependent clumping factors and suggest a maximum value at infinity of D_∞ = 10 to be consistent with observations of electron scattering wings. In later works, however, Sander et al. (2020) and Sander & Vink (2020) propose 50 as a maximum upper limit for D_∞, by comparison with previous theoretical works (Gräfener & Hamann 2005) and O/B star analyses (e.g., Bouret et al. 2012; Mahy et al. 2015). A clumping factor of 2500 seems extraordinary compared to previous models. As mentioned in Section 5, additional constraints on the clumpiness of stellar winds from massive stars can be obtained from X-ray binaries (e.g., Grinberg et al. 2017; El Mellah et al. 2018; Lomaeva et al. 2020).

In light of the above, our reduction of f_WR is probably best interpreted as an overall constraint on mass loss from massive stripped stars (Vink 2017), perhaps indicating a different dependence on metallicity or luminosity (see Sander et al. 2020; Sander & Vink 2020), rather than a specific change in the clumping.