ABSTRACT
The recent distance determination allowed precise estimation of the orbital parameters of Cyg X-1, which contains a massive 14.8 M☉ black hole (BH) with a 19.2 M☉ O star companion. This system appears to be the clearest example of a potential progenitor of a black hole + neutron star (BH–NS) system. We follow the future evolution of Cyg X-1, and show that it will soon encounter a Roche lobe overflow episode, followed shortly by a Type Ib/c supernova and the formation of a neutron star (NS). It is demonstrated that in majority of cases (≳ 70%) the supernova and associated natal kick disrupt the binary due to the fact that the orbit expanded significantly in the Roche lobe overflow episode. In the reminder of cases (≲ 30%) the newly formed BH–NS system is too wide to coalesce in the Hubble time. Only sporadically (∼1%) may a Cyg X-1-like binary form a coalescing BH–NS system given a favorable direction and magnitude of the natal kick. If a Cyg X-1-like channel (comparable mass BH–O star bright X-ray binary) is the only or dominant way to form BH–NS binaries in the Galaxy, then we can estimate the empirical BH–NS merger rate in the Galaxy at the level of ∼0.001 Myr−1. This rate is so low that the detection of BH–NS systems in gravitational radiation is highly unlikely, generating Advanced LIGO/VIRGO detection rates at the level of only ∼1 per century. If BH–NS inspirals are in fact detected, it will indicate that the formation of these systems proceeds via some alternative and yet unobserved channels.
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1. INTRODUCTION
Estimates for the rates of gravitational radiation (GR) sources from coalescing degenerate binaries are typically performed with population synthesis methods (e.g., Lipunov et al. 1997; Bethe & Brown 1998; De Donder & Vanbeveren 1998; Bloom et al. 1999; Fryer et al. 1999; Nelemans et al. 2001; Voss & Tauris 2003; or more recently Belczynski et al. 2010a). These studies attempt to explain the past evolution of the given observed binary or stellar population and put some constraints on the physics of stellar/binary evolution (e.g., Valsecchi et al. 2010). Nevertheless, there are often critical model parameters that are poorly constrained.
Recently, Bulik et al. (2011) have taken a different approach, examining specific binary systems with well-established parameters and investigating their future evolution. If a binary is chosen close to the end of its life (e.g., the formation of a double compact object) such a method has potentially great predictive power as many unknowns relating to its prior binary and stellar evolution can be avoided. In particular, Bulik et al. (2011) considered two high-mass X-ray binaries (HMXBs), IC10 X-1 and NGC300 X-1, and showed that these systems will soon form close black hole + black hole (BH–BH) systems that will merge within a Hubble time and produce strong GR signature. This provided an empirical lower limit of detection chances for the current GR instruments without direct reference to population synthesis methods.
In this study, we consider the future evolution of one of the most interesting binaries known in our Galaxy: Cyg X-1. Recently, the distance to this system was determined by radio parallax and other methods (Reid et al. 2011; Xiang et al. 2011), allowing the basic parameters of this binary to be firmly established (Orosz et al. 2011). This HMXB hosts one of the most massive (15 M☉) Galactic black holes (BHs) in a close orbit around a massive (20 M☉) O star. Since the companion is in the mass range for neutron star formation, we have selected this system to investigate yet another unobserved population of potential GR sources: black hole + neutron star (BH–NS) systems.
2. ESTIMATES
2.1. The Future Evolution of Cyg X-1
To evolve the system forward in time we use evolutionary prescriptions incorporated in the StarTrack population synthesis code (Belczynski et al. 2002). The evolution of the system is relatively simple and we do not need any population synthesis tools at this point. We start off with the best estimate of current binary parameters: the BH with the mass MBH = 14.8 M☉, the optical star with the mass Mopt = 19.2 M☉ and radius of Ropt = 16.2 R☉ and the orbital period Porb = 5.6 days (Orosz et al. 2011).
The optical companion is almost filling its Roche lobe and will start Roche lobe overflow (RLOF) in less than 0.2 Myr while still on the main sequence (MS; see Figure 1). The mass ratio is close to unity so we do not expect the common envelope evolution, but rather a stable RLOF phase (Belczynski et al. 2008b; Wellstein et al. 2001). However, the mass transfer rate may reach quite high values while the donor is moving through Hertzsprung gap (HG). The evolution of the system is presented in Figure 2. Mass transfer rate is calculated using physical properties of the donor and the system parameters (Belczynski et al. 2008a). The mass accretion onto the BH is calculated using the slim disk models (e.g., Abramowicz et al. 1988; Ohsuga et al. 2005; Ohsuga 2007) and thus can significantly exceed the classical Eddington limit (see Belczynski et al. 2008b for details). The BH increases its mass to MBH = 17.8 M☉, while the optical star loses most of its mass Mopt = 4.2 M☉ to become a massive helium core with a bit of H-rich envelope. Note that the majority of the mass lost from the donor is lost from the system (highly non-conservative case). The period of the system increases to Porb = 90 days. RLOF stops as the donor decreases in size due to the loss of its H-rich envelope. The massive helium or Wolf–Rayet (WR) star (RWR ≲ 1 R☉) is well within its Roche lobe Rlobe ≈ 60 R☉. After some wind mass loss (∼0.5 M☉) and about 2 Myr after the RLOF termination, the WR star explodes in a Type Ib/c supernova and forms an NS.
Figure 1. Radius evolution of an optical star in Cyg X-1. Current radius is found at R = 16.17 R☉ and that places the star at the end of its main sequence (dashed line) or at the beginning of the Hertzsprung gap (dot-dashed line). Since the star is very close to its Roche lobe (Rlobe = 17.24 R☉ for the orbital period Porb = 5.6 days) and since it has not yet started RLOF it means that the star is on the main sequence at the first intersection of its evolutionary track and radius line of R = 16.17 R☉. Once the star increases its radius by about 1 R☉, it will start RLOF while still on main sequence and the RLOF will last through the Hertzsprung gap (see Figure 2). The radius evolution is taken from the single-star models of Hurley et al. (2000).
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Figure 2. Evolution of Cyg X-1 through RLOF that will start in about 105 yr. Bottom panel: mass transfer rate from the massive donor star is very high. However, it is much lower while the donor is on the main sequence (1–3 × 10−5 M☉ yr−1; dashed line) as compared to the transfer during the Hertzsprung gap (10−3 − 10−4 M☉ yr−1; dot-dashed line). Mass accretion onto BH is factor of ∼3–5 lower than the transfer rate to account for the fact that the BH cannot accept all the transferred material. Note that the accretion rate is significantly higher than the typically employed Eddington rate (5 × 10−7 M☉ yr−1 for a 15 M☉ non-spinning BH) as we account for more realistic supercritical accretion (slim disk advection dominated accretion flow) onto a rapidly spinning BH with a = 0.9 (Gou et al. 2011). Middle panel: the donor loses most of its mass to become a 4 M☉ helium core with a small H envelope. Most of the donor mass (∼9 M☉) is lost from the system, while the BH increases its mass from 14.8 to 17.8 M☉. Top panel: period of the system changes from the currently observed 5.6 days to 90 days. System becomes wide after RLOF due to the non-conservative mass exchange and mass ratio reversal (most of the mass is accreted onto BH while the donor became the less massive component of the binary).
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There is about 2 M☉ of mass loss in the supernova and that is not enough to disrupt the system. However, the pre-supernova binary is rather wide (Porb = 104 days, and semimajor axis a = 250 R☉ due to additional orbit expansion caused by the mass loss from the WR star), and any large natal kick tends to disrupt the binary. In Table 1 we list the disruption and survival probabilities for two assumptions about the distribution of kick velocities. The "full" kicks are adopted from the velocity distribution of the Galactic single pulsars that is best described by a Maxwellian with one-dimensional σ = 265 km s−1 (Hobbs et al. 2005) and uniform distribution of orientation. As there may be some observational and theoretical evidence that natal kicks are smaller in close binary systems (see the discussion in Belczynski et al. 2010b), we also explore a "half-"kick model with kicks drawn from the same distribution but with σ = 132.5 km s−1. As expected for such a wide system, the binary disruption and formation of two single compact objects is most likely: 94% and 74% for full and half kicks, respectively. Less likely, but still quite probable is the formation of a wide BH–NS system: 5% and 25%. The least likely is the formation of the close BH–NS system with the coalescence time below Hubble time (13.47 Gyr): 0.2% and 0.8%. In order for the system to form a close BH–NS binary, the supernova explosion must produce a significant kick (Vkick ∼ 150 km s−1) that sends the NS toward the BH.
Table 1. BH–NS Formation Statisticsa
| Fate/Porbb | 104 days | 61.7 days | 5.6 days |
|---|---|---|---|
| SN disruption | 0.944 (0.743) | 0.914 (0.673) | 0.680 (0.446) |
| Wide BH–NS | 0.054 (0.249) | 0.082 (0.313) | 0.237 (0.500) |
| Close BH–NS | 0.002 (0.008) | 0.004 (0.014) | 0.083 (0.054) |
Notes. aFraction of Cyg X-1-like systems that after the second supernova will be disrupted and will form a wide BH–NS or close BH–NS (merger time shorter than the Hubble time). The fractions are given for the full natal kicks with σ = 265 km s−1 (or half-kicks with σ = 132.5 km s−1). bNumbers for Porb = 104 and 61.7 days correspond to physical system modeling with RLOF starting while the optical star is on MS and HG, respectively. The last model is unphysical, no RLOF was assumed and the orbital period was kept constant at Porb = 5.6 days through the evolution (see the discussion).
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Additionally, we have investigated an evolutionary scenario in which the RLOF starts early in the HG. Since at this point the optical star has a more massive core as compared with the above example, it will retain more mass through RLOF (only the H-rich envelope is transferred/lost). Since less mass is transferred/lost, the orbital expansion is not as dramatic and the final orbital period of the binary at the supernova explosion is Porb = 61.7 days (a = 174 R☉). This obviously leads to higher survival probabilities and a higher chance of the close BH–NS formation: 0.4% and 1.4% for the full kick and half-kick models, respectively.
2.2. Rate Estimates
Cyg X-1 has been detected because of its unusually strong X-ray brightness. Let us consider the question of the formation rate of Cyg X-1 like binaries in our Galaxy. The upper limit on the X-ray active phase in Cyg X-1 is set by the evolutionary time of the secondary, which is 10 Myr for a 20 M☉ star. We employ this upper limit. Had we adopted the more formal estimate of 5 Myr (it takes about 5 Myr for a massive star to form a BH that is already in the system and start an X-ray phase) the estimated detection rate of BH–NS inspirals in GR would increase by a factor of two. Given that we see just one such object and assuming that the current star formation rate is representative across the lifetime of the Galaxy we can infer the formation rate of Cyg X-1-like binaries to be one per 10 Myr, i.e., r ≈ 10−7 yr−1. Our simulations show that the chance of forming a merging BH–NS system is between 2 × 10−3 and 14 × 10−3 (see bottom row of Table 1) from a Cyg X-1-like progenitor. This means that the formation rate of BH–NS binaries from Cyg X-1-like progenitors will be only (2–14) × 10−10 yr−1. This means that only a handful (2–14) close BH–NS systems might have been formed over a 10 Gyr history of the Milky Way. For comparison the Galactic empirical neutron star + neutron star (NS–NS) merger rate is significantly higher: NS–NS (3–190) × 10−6 yr−1 (Kim et al. 2010).
The implied Advanced LIGO/VIRGO detection rate follows once we determine the range of these detector for a BH–NS system. Given the final mass of the BH (MBH = 17.8 M☉) and assuming the mass of an NS to be canonical, 1.4 M☉, we obtain the chirp mass of the newly formed system to be Mchirp ≡ (MBHMopt)3/5(MBH + Mopt)−1/5 = 3.8 M☉. Since the range for the Advanced detectors for NS–NS system with a chirp mass of 1.2 M☉ is 300 Mpc, we obtain the range for such BH–NS systems of 786 Mpc (the detection distance scales like ∝M5/6chirp). If we adopt the density of Milky-Way-like galaxies in the local universe to be 0.01 Mpc−3 (e.g., O'Shaughnessy et al. 2008), then within such a distance there should be 2.0 × 107 galaxies. Combining it with the Galactic rate we obtain the detection rate to be in the range (0.4–2.8) × 10−2 yr−1, or a detection every 36–250 years in the Advanced LIGO/VIRGO.
The observational uncertainties on component masses (≲ 2 M☉) do not play a significant role on our findings and qualitatively they do not change any of our results. The same applies to the stellar wind mass loss rate from the companion star as the 20 M☉ star loses only about 1 M☉ during its MS (e.g., Vink et al. 2001). The major uncertainties arise in the orbital evolution during RLOF in which ∼15 M☉ is lost/exchanged and supernova outcome (natal kicks).
3. DISCUSSION
So far we have discussed only one binary, Cyg X-1, as a potential progenitor of a BH–NS system. Tomsick & Muterspaugh (2010) list several known Galactic HMXBs with an NS and a companion massive enough to potentially produce a BH: Vela X-1 (24 M☉), XTEJ1855-026 (25 M☉), 4U1907+09 (28 M☉), and GX301 (40 M☉). The expansion of massive companions will eventually lead to an RLOF. These systems due to extreme mass ratio will evolve into common envelope that will result in a merger of an NS with its companion, aborting the BH–NS formation. Such outcome follows from the fact that the mass of an NS is so small in respect to the mass of the envelope of a companion that there is not enough orbital energy to eject the companion envelope (e.g., Webbink 1984). An extragalactic system LMC X-3 hosting a 10 M☉ BH was believed to have a companion massive enough to form an NS. However, recent spectral analysis that takes into account irradiation of the companion indicates that it is a B5 dwarf setting its mass at about 5 M☉ (Val-Baker et al. 2007) and makes it a WD progenitor.
HMXBs like Cyg X-1 are wind-fed, and thus their non-degenerate stars do not fill their Roche lobes. It is therefore curious that those binaries for which good system parameters are known tend to be close to RLOF. Cyg X-1 is an example: our analysis implies that the system will begin RLOF in 0.2 Myr, after a lifetime about 50 times longer than that. An even more extreme case is that of LMC X-1, in which an 11 M☉ BH is found in a 3.9 day orbit with a 32 M☉ companion. The companion currently fills over 90% of its Roche lobe, implying that the system will undergo RLOF within ≈0.1 Myr (Orosz et al. 2009). Note that the high mass of the companion implies that RLOF will result in unstable mass transfer and the likely formation of a common envelope system—this LMC X-1 is unlikely to produce any sort of double degenerate binary. There are only a few HMXBs with precisely known system parameters, so this tendency to be alarmingly near RLOF may be a statistical anomaly. Alternatively, it may be a selection effect: a system in which the companion is far from RLOF will have a smaller fraction of its stellar wind accrete onto the compact object, and thus have a lower X-ray luminosity.
Nevertheless, it may be worth entertaining the idea that this effect is neither a statistical glitch nor an observational bias, but that there is some unknown physical process that halts the growth of the companion star near the Roche lobe boundary. We note that the gravitational potential becomes shallow near the L1 point, and X-ray irradiation becomes a bigger effect for companion stars that present a relatively large cross section. Both of these effects may dramatically change the surface structure and stellar winds of the companion star as it approaches the Roche lobe, conceivably in a self-limiting way. Such winds might be observable through the strength, shape, and variability of emission lines associated with the wind. If such an effect is present, the evolutionary path of the binary system may prove to be quite different from what is commonly assumed, and what we have assumed above.
To evaluate the possible effect this might have on rates of GR sources, we consider how the future evolution of an HMXB like Cyg X-1 might proceed if it never reaches RLOF. As a limiting case, we constrain the orbital period to remain at its currently observed value of Porb = 5.6 days. In fact, we expect the orbital period to increase somewhat (even without RLOF) as mass is lost from the system in stellar wind. In this case, the survival through the supernova and the formation of the close BH–NS system is expected in 8.4% of cases and corresponds to an Advanced LIGO/VIRGO detection rate of ∼1 per decade. So, despite the fact that we have violated (in favor of producing close BH–NS systems) our current understanding of the stellar evolution, we still do not get enough BH–NS mergers to expect detection in gravitational waves.
Thus, we find that if indeed BH–NS mergers are observed as GR sources, their immediate precursors will not be systems like Cyg X-1 that are currently observed.
Authors acknowledge the hospitality of the Aspen Center for Physics and support from MSHE grants N N203 302835 (T.B. and K.B.); N203 404939, N N203 511238, and NASA Grant NNX09AV06A to the UTB Center for Gravitational Wave Astronomy (K.B.); and NSF-AST grant 0707627 (C.B.).