Black Hole Ultracompact X-Ray Binaries as Galactic Low-frequency Gravitational Wave Sources: The He Star Channel

Black hole (BH) ultracompact X-ray binaries (UCXBs) are potential Galactic low-frequency gravitational wave (GW) sources. As an alternative channel, BH UCXBs can evolve from BH+He star binaries. In this work, we perform a detailed stellar evolution model for the formation and evolution of BH UCXBs evolving from the He star channel to diagnose their detectability as low-frequency GW sources. Our calculations found that some nascent BH+He star binaries after the common-envelope (CE) phase could evolve into UCXB-LISA sources with a maximum GW frequency of ∼5 mHz, which can be detected in a distance of 10 kpc (or 100 kpc). Once BH+He star systems become UCXBs through mass transfer, they would emit X-ray luminosities of ∼1038 erg s−1, making them ideal multimessenger objects. If the initial He-star masses are ≥0.7 M ⊙, those systems are likely to experience two Roche lobe overflows, and the X-ray luminosity can reach a maximum of 3.5 × 1039 erg s−1 in the second mass-transfer stage. The initial He-star masses and initial orbital periods of progenitors of Galactic BH UCXB-LISA sources are in the range of 0.32–2.9 M ⊙ and 0.02–0.19 days, respectively. Nearly all BH+He star binaries in the above parameter space can evolve into GW sources whose chirp masses can be accurately measured. Employing a population synthesis simulation, we predict the birthrate and detection number of Galactic BH UCXB-LISA sources evolving from the He star channel are R = 2.2 × 10−6 yr−1 and 33 for an optimistic CE parameter, respectively.


INTRODUCTION
Ultracompact X-ray binaries (UCXBs) are low-mass X-ray binaries (LMXBs) with ultra-short orbital periods (usually less than 60 minutes), consisting of an accreting compact object and a hydrogen-poor donor star (Nelson et al. 1986;Nelemans & Jonker 2010).UCXBs can help us to understand the angular momentum loss mechanisms (Ma & Li 2009;van Haaften et al. 2012a,b), the common-envelope (CE) evolution (Zhu et al. 2012), and the accretion process of compact objects (Lin & Yu 2018), thus they are ideal laboratories for testing stellar and binary evolutionary theory.Furthermore, UCXBs can emit continuous low-frequency gravitational wave (GW) signals, which could be detected by the space-borne GW detectors such as the Laser Interferometer Space Antenna (LISA; Amaro-Seoane et al. 2017, 2023), Tian-Qin (Luo et al. 2016;Wang et al. 2019;Huang et al. 2020), and Taiji (Ruan et al. 2020).Therefore, UCXBs are ideal objects pursuing multi-messenger investigations (Chen et al. 2020).
Thus far, the number of confirmed UCXBs (with an accurately measured orbital period ≤ 80 minutes) and candidates is around 45 (Armas Padilla et al. 2023).Based on the accurate detection of orbital periods, 20 sources were identified to be UCXBs in high confidence, which includes 11 persistent sources and 9 transient sources (in't Zand et al. 2007;Liu et al. 2007;Heinke et al. 2013;Cartwright et al. 2013;Pietrukowicz et al. 2019;Coti Zelati et al. 2021;Peng & Shen 2021;Armas Padilla et al. 2023).Among confirmed UCXBs, 19 sources were discovered to include a neutron star (NS).At present, there exist two black hole (BH) UCXB candidates.The first one is the luminous X-ray source X9 in globular cluster 47 Tucanae, which was thought to be a BH accreting from a white dwarf (WD) in a close orbit (Miller-Jones et al. 2015;Bahramian et al. 2017;Church et al. 2017;Tudor et al. 2018).Recently, Moutard et al. (2023) found some evidences that the compact object in UCXB 4U 0614+091 may be a BH by the simultaneous NICER and NuSTAR observations.Chen et al. (2020) estimated that the main-sequence (MS) channel can form 60-80 NS UCXB-LISA sources in the Galaxy.However, LISA can only detect ∼ 4 BH UCXBs evolving from the MS channel (Qin et al. 2023).If the ratio of the numbers between NS and BH UCXB-LISA sources is similar to that of the identified numbers between NS and BH UCXBs, the number of confirmed BH UCXBs evolving from the MS channel is ∼ 1, which is comparable to the present observation.Therefore, it is difficult to observe BH UCXBs.
Because of the compactness of UCXBs, their donor stars are generally thought to be partially or completely degenerate stars such as WDs or helium (He) stars (Rappaport et al. 1982;Podsiadlowski et al. 2002;Deloye & Bildsten 2003).Optical spectroscopic analysis of UCXBs can help us to identify the properties of donor stars (Nelemans et al. 2004(Nelemans et al. , 2006)).Employed X-ray, ultraviolet, and optical spectroscopy, chemical elements of accretion disks in some UCXBs may include He, C, N, O, Ne, and Si (Nelemans et al. 2010).Such a diversity of chemical compositions implied those donor stars should evolve to different nuclear-burning stages and interior degeneracy, which requires different evolutionary models to account for the formation of UCXBs (Sengar et al. 2017).It is generally thought that UCXBs in the Galactic field evolved from the following three channels: the WD channel, the evolved MS star channel, and the He star channel (Postnov & Yungelson 2006;Nelemans et al. 2010).
In the first channel, the progenitors of UCXBs are compact binaries consisting of a NS/BH and a lowmass WD, in which GW radiation drives mass transfer (Belczynski & Taam 2004;van Haaften et al. 2012a;Jiang et al. 2017;Yu et al. 2021).Sengar et al. (2017) performed complete numerical models for the formation of UCXBs evolved from a stable mass transfer from a WD to an accreting NS, and found that the WD channel can reproduce the observed properties of some UCXBs with high He abundances.Using an improved mass transfer hydrodynamics model, Bobrick et al. (2017) argued that only NS+He WD binaries with a donor-star mass less than 0.2 M ⊙ could form UCXBs through a stable mass transfer.van Haaften et al. (2012a) also found that the donorstar masses have to be less than 0.4 M ⊙ to form UCXBs from NS+CO WD binaries.
The second channel originates from a NS/BH accreting mass from a MS donor star that fills its Roche lobe.
If the MS star starts mass transfer late enough and the orbital-angular-momentum loss by magnetic braking is efficient, the system would evolve toward a UCXB.In this channel, UCXBs generally evolve from BH/NS+MS binaries whose initial orbital periods are shorter than the bifurcation period (van der Sluys et al. 2005;Sengar et al. 2017;Chen et al. 2020;Qin et al. 2023).In the UCXB stage, the donor star is most likely to evolve into a WD.It is noteworthy that mass transfer never ceases in the whole evolutionary stage except for those systems with a NS and a fine-tuning initial orbital period (Chen et al. 2020;Qin et al. 2023).
In the He star channel, the direct progenitors of UCXBs are BH/NS+He star binaries.It is generally believed that BH/NS+He star binaries are the evolutionary products of high-mass X-ray binaries, in which the hydrogen envelope of the He-star progenitors are fully ejected in the CE stage (Quast et al. 2019;Abdusalam et al. 2020;Götberg et al. 2020;Wang et al. 2021).Due to the close orbit, the GW radiation dominates the orbital evolution of the nascent BH/NS+He star system and triggers a mass transfer.The BH/NS accretes He-rich materials from the He star that fills the Roche lobe and the system appears as an UCXB (Savonije et al. 1986;Dewi et al. 2002;Heinke et al. 2013;Wang et al. 2021).Employing detailed stellar evolution models, Jiang et al. (2021Jiang et al. ( , 2023) ) found that NS/BH+He star binaries can evolve into double NS or BH+NS systems through stripped supernova explosions, and these double NS and BH+NS systems would evolve toward high-frequency GW events that can be discovered by aLIGO.Binary population synthesis (BPS) simulations indicated that binary systems consisting of a NS/BH and a naked He star can account for ∼ 50% − 80% UCXBs (Zhu et al. 2012).By a population-synthesis simulation, Lommen et al. (2005) proposed there are ∼ 200 BH+He star binaries and ∼ 540 NS+He star binaries in the Milky Way.
Compared with those dim WDs, He stars are close to MS stars in the Hertzsprung-Russell (H-R) diagram (Gräfener et al. 2002;Geier et al. 2010;Götberg et al. 2018).Therefore, BH+He star systems can be observed in the detached stage, and provide more evolutionary details such as the mass-transfer efficiency (Packet 1981;de Mink et al. 2007), and the stellar wind (Puls et al. 2008;Smith 2014;Vink 2017).In general, low-mass He stars are referred to as the hot subdwarfs (Han et al. 2007), while high-mass He stars are called Wolf-Rayet stars (Gräfener et al. 2002).At present, several sources consisting of a compact star and a He star have been reported.For example, LS V+22 25 (LB-1) was proposed to contain a stellar mass (≈ 8M ⊙ ) BH and a low-mass (0.5-1.7 M ⊙ ) stripped He star (Eldridge et al. 2020;Yungelson et al. 2020). PG 1432+159, HE 0532-4503, PG 1232-136, and PG 1743+477 were also thought to be candidates containing a BH and a He star with a very thin hydrogen envelope (Geier et al. 2010).However, no Xray emissions from these detached binaries were detected.HD49798 (Mereghetti et al. 2009), M101 ULX-1 (Liu et al. 2013), IC 10 X-1, and NGC 300 X-1 are most likely X-ray sources including He stars, in which the X-ray emissions originated from a NS/BH accreting from the stellar winds of He stars.Especially for the latter two sources, Tutukov & Fedorova (2016) proposed that the accretion disk and the limited X-ray luminosity of 10 38 erg s −1 can be interpreted by a stellar-mass BH accreting from a Wolf-Rayet companion through the wind-Roche lobe overflow (RLOF) mechanism (see also El Mellah et al. 2019a,b).However, both the stellar wind accretion and the RLOF models can explain the observations of galactic strong X-ray sources Cygnus X-3 and SS 433 (Lommen et al. 2005), which are also thought to consist of a compact object and a masstransferring He star.As a consequence, binary systems consisting of a compact object and a He star are potential galactic strong X-ray sources.Furthermore, they are also promising low-frequency GW sources in the Galaxy (Wang et al. 2021;Liu et al. 2023).Therefore, it is of great significance to study the evolution of BH binaries including He stars.
In this paper, we perform a detailed stellar evolution model for a large number of BH+He star binaries to investigate whether or not they can evolve toward UCXBs and low-frequency GW sources in the Milky Way.In Section 2, we describe the binary evolution code.Some detailed simulated results are shown in Section 3. The discussion and conclusion are presented in Sections 4, and 5, respectively.

BINARY EVOLUTION MODEL
The Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011Paxton et al. , 2013Paxton et al. , 2015Paxton et al. , 2018Paxton et al. , 2019) is a popular code in the stellar and binary evolution field.In this work, we use a binary updated version (r12115) of the MESA to model the formation and evolution of BH UCXBs.We assume that the CE stage would produce many detached BH-He star binaries, which are taken to be the evolutionary beginning point of detailed stellar evolution.For simplicity, the BH is considered a point mass with an initial mass of M BH,i = 8 M ⊙ .The code only models the nuclear synthesis of the He star and the orbital evolution of the binary.Therefore, the evolutionary fates of BH-He star binaries depend on the initial He-star mass (M He,i ) and the initial orbital period (P orb,i ) for a given input physics.We build a zero age main sequence He star model with solar metallicity (Wong & Schwab 2019), which consists of 98% helium and 2% metallicity (i.e.Y = 0.98, Z = 0.02).The lowest He-star mass is taken to be 0.32 M ⊙ , under which the center He burning would extinguish (Iben 1990;Han et al. 2002;Yungelson 2008).For each binary system, we run the MESA code until the time step reaches a minimum time-step limit or the stellar age is greater than the Hubble time (14 Gyr).
For the wind setting of the He star, the "Dutch" options with a scaling factor of 0.8 are used in the schemes including hot − wind − scheme, cool − wind − RGB − scheme, and cool − wind − AGB − scheme (Glebbeek et al. 2009).We use Type 2 opacities for extra C/O burning during and after He burning.Furthermore, the time step options with mesh − delta − coef f = 1.0 and varcontrol − target = 10 −3 are adopted.Our inlists are available at doi:10.5281/zenodo.10075413.
The fast wind of the He star is thought to carry away its specific orbital-angular momentum.The windaccretion efficiency of the BH via Bondi-Hoyle-like accretion is low (Tauris et al. 2017).For example, the wind-accretion efficiency is ∼ 0.003 for a binary with a 8.8 M ⊙ BH and a 6.0 M ⊙ He star (Jiang et al. 2023).Therefore, we ignore the wind accretion in the whole evolutionary process.Once the He star fills its Roche lobe, the mass transfer initiates from the donor star to the BH at a rate of Ṁtr .During the mass transfer, we adopt the accretion efficiency scheme given by Tauris & van den Heuvel (2006), i.e. α = 0, β = 0.5, and δ = 0, here α, β, and δ represent the fractions of mass loss from the He star in the form of fast wind, the ejected mass from the vicinity of the BH and from a circumbinary co-planar toroid, respectively.Therefore, the accretion rate of the BH is Ṁacc = (1−β) Ṁtr = 0.5 Ṁtr .
The mass-growth rate of the BH is limited by the Eddington accretion rate as where G is the gravitational constant, c is the speed of light in vacuo, M BH is the mass of the BH, κ = 0.2(1 + X) is the Thompson-scattering opacity of electrons (X is the hydrogen abundance of the transferred material, and X = 0 for a He donor star) the initial mass of the BH) is the energy conversion efficiency of the accreting BH (Podsiadlowski et al. 2003).Therefore, the mass-growth rate of the accreting BH is ṀBH = min(0.5Ṁtr , ṀEdd ).The excess materials in unit time ( Ṁtr − ṀBH ) are thought to be ejected at the vicinity of the BH in the form of isotropic winds, carrying away the specific orbital angular momentum of the BH.Thus, the angular-momentum-loss rate due to isotropic winds can be written as where a is the orbital separation of the binary, M He is the mass of the He star, and P orb is the orbital period.
In our model, orbital angular momentum loss via GW radiation and mass loss (fast wind and isotropic wind) are included.
During the inspiral of two components in BH binaries, the change of the mass quadrupole produces lowfrequency GW signals with a frequency of f gw = 2/P orb .When the systems evolve into a close orbit, the emitting GW signals may be detected by space-borne GW detectors such as LISA.For a 4 years LISA mission, the GW characteristic strain of our simulated BH binaries is given by (Chen 2020) (3) d is the distance of the GW sources.For simplicity, the chirp mass M can be expressed as In the numerical calculation, the corresponding binaries are thought to be LISA sources if the calculated characteristic strain exceeds the LISA sensitivity curve given by Robson et al. (2019).
It is worth noting that the chirp mass in Equation (4) should be applied in a detached system that its orbital decay is entirely caused by GW radiation.For BH UCXBs, mass transfer is inevitable to influence their orbital evolution.However, the mass transfer in compact BH binaries is driven by GW radiation and goes along a timescale close to that of GW radiation (van Haaften et al. 2012a,b).Therefore, the estimation of the chirp mass in Equation (4) remains reliable in semi-detached BH binaries.

SIMULATION RESULT
For a fixed initial BH mass and a given input physics, the evolutionary fates of BH+He star binaries depend on P orb,i and M He,i .As evolutionary examples, we model the evolution of 10 BH+He star binaries with different M He,i and P orb,i , which divide into three groups as follows: (1) group 1 with M He,i = 0.6 M ⊙ , P orb,i = 0.03, 0.06, 0.09, 0.11 days; (2) group 2 with M He,i = 0.32, 0.4, 0.6, 0.8 M ⊙ , P orb,i = 0.06 days.(3) group 3 with M He,i = 1.2, 1.8, 2.8 M ⊙ , P orb,i = 0.06 days.Groups 1 and 2 only experience one RLOF, while there exist two or more RLOFs for group 3. Some main evolutionary parameters of three groups are listed in Table 1.

Orbital evolution
Figure 1 shows the evolution of orbital periods with the stellar age for BH+He star binaries in three groups.Because of small donor-star masses, the donor stars in groups 1 and 2 fill their Roche lobes until their orbital periods are in the range of 13 to 48 minutes, thus these systems appear as ultracompact detached binaries in a long timescale.In group 1, those systems with P orb,i = 0.03, 0.06, 0.09 days can firstly be detected by LISA as low-frequency GW sources at a distance of 10 kpc, and 100 kpc (these two distances are the typical distances of the sources in the Galaxy and the Large/Small Magellanic Cloud)(the system with P orb,i = 0.03 days is visible by LISA at a distance of 10 kpc at the beginning of binary evolution), then experience RLOF and become UCXBs.The system with P orb,i = 0.11 days initiates the mass transfer after it is visible by LISA at a distance of 10 kpc, then it appears a UCXB that can be detected by LISA at a distance of 10 kpc, and 100 kpc (i.e.UCXB-LISA source).
In group 2, the system with M He,i = 0.8 M ⊙ is visible by LISA in a distance of 10 kpc at the beginning of binary evolution, then it fills its Roche lobe and appears as a UCXB-LISA source.Subsequently, continuous orbital shrinkage makes it a strong GW source that can be detected by LISA at a distance of 100 kpc.The other three systems with low donor-star masses first appear as low-frequency GW sources that can be detected by LISA at a distance of 10 kpc and 100 kpc, then evolve into UCXB-LISA sources.Because of the short orbital periods, the GW radiation drives the orbital periods of the systems in groups 1 and 2 to continuously decrease to a minimum of 6−12.4 minutes, which are very similar to the minimum orbital periods (8 − 10 minutes) obtained by Wang et al. (2021) in NS+He star systems.Most evolutionary tracks of groups 1 and 2 emerge "knee" features, which are consistent with the positions that the RLOF starts.This phenomenon originates from the orbital expansion caused by the mass transfer from the less massive He star to the more massive BH, which dilutes the orbital decay due to GW radiation.
In group 3, three systems first appear as low-frequency GW sources that can be detected by LISA at a distance of 10 kpc, then begin case BA mass transfer at the first solid circles.The orbital period of the system with M He,i = 1.2 M ⊙ continuously decreases after the mass transfer.However, the orbits of the other two systems with M He,i = 1.8, and 2.8 M ⊙ show a widening tendency.In theory, the orbit should widen when the mass is transferred from the less massive donor star to the more massive BH.In contrast, GW radiation causes the orbit to shrink.The evolutionary fate of the orbit would depend on the competition between the mass transfer and GW radiation.The orbital shrinkage of the system with M He,i = 1.2 M ⊙ originates from a low mass-transfer rate (see also Figure 2), which can not conquer the orbital decay caused by GW radiation.After the case BA mass transfer ceases, three systems start a case BB mass transfer soon (the solid circles are approximately in the same positions as those open circles, and the timescales between these two circles are 0.346, 0.389 and 0.340 Myr for M He,i = 1.2, 1.8, and 2.8 M ⊙ , respectively).Subsequently, two systems M He,i = 1.8, and 2.8 M ⊙ can not be detected by LISA at a distance of 10 kpc due to a further expansion of the orbits.Once case BB mass transfer ends, the continuous orbital shrinkage by GW radiation causes the system with M He,i = 1.8 M ⊙ to evolve toward LISA sources that can be detected at a distance of 10 kpc, and 100 kpc.The evolution of the systems with M He,i = 1, 2, and 2.8 M ⊙ stops due to a numerical difficulty after the case BB mass transfer ceases.

Evolution of mass-transfer rates
Figure 2 plots the evolution of the mass-transfer rates of BH X-ray binaries in three groups.At t rlof ≈ 0.007 − 54.9 Myr (see also Table 1), the He stars fill their Roche lobes.The mass transfer begins early for those massive He stars.Because of a positive correlation between the mass and the radius (R He ∝ M 0.654 He , see also Tauris & van den Heuvel 2006) for zero-age MS He stars, the more massive He star is easy to fill its Roche lobe.
For low-mass He stars with M He,i ≤ 0.8 M ⊙ , GW radiation dominates the orbital evolution in the early mass-transfer stage.Due to the shrinkage of the orbit, the GW-radiation timescale decreases, thus the masstransfer rate slowly enhances.After the mass transfer dominates the orbital evolution, the orbital period achieves a minimum, at which the mass-transfer rate emerges a maximum (10 −7 − 10 −6 M ⊙ yr −1 , Yungelson 2008).Subsequently, the He star evolves to more degenerate, and the correlation between the mass and the ra-  Ṁ⊙ yr −1 , which increases with the increase of M He,i .As the He abundance in the core drops below 0.1, it develops a carbon-oxygen (CO) core.Due to the contraction of the He star, the binary gradually becomes a detached system, and the first mass transfer ceases.
After the core He is exhausted, the He star begins the He-shell burning.With the continuous growth of CO core, the He star begins to expand and initiates the second mass transfer.The mass-transfer rate in the second stage is significantly higher than that of the first mass-transfer stage, and the duration is shorter.A high donor-star mass tends to produce a high mass-transfer rate and a short mass-transfer duration.For M He,i = 1.8 and 2.8 M ⊙ , the second mass-transfer rate can exceed 10 −6 Ṁ⊙ yr −1 .The orbital-expand effect caused by such a high Ṁtr conquers the orbital-shrinkage effect caused by GW radiation, and the binary orbits begin to widen (see also Figure 1), then the mass-transfer rate continuously decreases.The whole burning He-shell is almost stripped in the second mass-transfer stage, and the remaining CO core becomes highly degenerate.Because of a negative correlation between the mass and the radius of a degenerate star, the binary detaches again.
In fact, the BH+He star systems in group 3 may undergo a third RLOF, in which the CO WD fills the Roche lobe and triggers a mass transfer.Due to the limitation of the minimum time step, the duration of the third mass transfer is too short (≤ 1 yr) to show in Figure 2. It is noteworthy that the BH+He star system with M He,i = 2.8 M ⊙ emerges a transient ultra-high masstransfer rate (∼ 10 −5 Ṁ⊙ yr −1 ) in the first mass-transfer stage.The mass transfer proceeds in a thermal timescale (Ergma & Fedorova 1990;Yungelson 2008), thus the mass-transfer rate is very high in this stage, which is much higher than ṀEdd .With a rapid decrease of the convective-core mass, the He star rapidly contracts until it re-reaches thermodynamic equilibrium (Dewi et al. 2002).Subsequently, the mass transfer will continue steadily on the nuclear timescale.

Evolution of X-ray luminosities
Compared with detached binaries, BH UCXBs are ideal multimessenger sources for the detection in both GW and electromagnetic wave bands.
Xray observations of these GW sources could reveal more information about the history of binary evolution (Madhusudhan et al. 2008;Patruno & Zampieri 2008), and provide some constraints on the nature of the companion, the structure of accretion disk, the relatively accurate position (Liu et al. 2013;Motch et al. 2014;Yao & Feng 2019), and so on.In the mass transfer phase, the X-ray luminosity of the accretion disk surrounding a BH can be calculated by (Köding et al. 2002) where ǫ is the radiation efficiency of the accretion disk, Ṁcrit is the critical accretion rate depending on the transition between the low/hard and high/soft states.In the calculation, we take ǫ = 0.1, and Ṁcrit = 10 −9 M ⊙ yr −1 (Narayan & Yi 1995).
Figure 3 depicts the evolution of X-ray luminosities of BH UCXBs.It is noteworthy that BH UCXBs evolved from the He star channel produce relatively high X-ray luminosities of ∼ 10 38−39 erg s −1 , which is 2 − 5 orders of magnitude higher than those in the MS channel (∼ 10 33−36 erg s −1 , see also Qin et al. 2023).However, the timescales of BH UCXBs evolved from the He star channel are ∼ 1 − 10 Myr, which is much shorter than those (∼ 50 − 900 Myr) in the MS channel.All systems in groups 1 and 2 are visible LISA sources at a distance of 10 kpc or 100 kpc during the UCXB stage.Two systems with high donor-star masses (M He,i = 1.8 and 2.8 M ⊙ ) in group 3 can not be detected by LISA in the later stage of the second mass-transfer phase because of a rapid orbital expansion.When the binaries evolve to the minimum orbital period, the masstransfer rates reach the maxima, and the X-ray luminosities emerge peaks (∼ 10 39 erg s −1 ).For a high M He,i , BH UCXBs tend to become so-called ultraluminous Xray sources (ULXs, L X ≥ 10 39 erg s −1 ) (Feng & Soria 2011;Kaaret et al. 2017), in which the maximum X-ray luminosity is 3.5 × 10 39 erg s −1 .Recently, Zhou et al. (2023) found the first smoking gun evidence for the existence of He donor star in ULXs by Very Large Telescope Multi Unit Spectroscopic Explorer observations.Similarly, the population synthesis study also indicated that NS X-ray binaries containing the He stars can account for a large part of ULXs in Milky Way-like galaxies (Shao et al. 2019).

Detection of GW signals
Since the Laser Interferometer Gravitational-Wave Observatory (LIGO) first detected the high-frequency GW signal from the double BH merger event GW150914 (Abbott et al. 2016), the detection of GW opens a new window to understand the distant universe.GW signals provide us with more useful information about stellar and binary evolution.Figure 4 plots the evolution of BH+He star systems in the characteristic stain versus GW frequency diagram.Because of the same chirp mass, the evolutionary tracks of the four systems in group 1 overlap before RLOF.Due to short initial orbital periods, the nascent BH+He star system with P orb,i = 0.03, and 0.06 days can be instantly detected by LISA and Taiji at a distance of 10 kpc after the CE stage.It is impossible to form detached BH+WD systems via isolated binary evolution (Qin et al. 2023).Therefore, those sources with chirp masses similar to those of BH+He star systems should be the post-CE systems, and provide indirect evidence of CE evolutionary stages.When the mass transfer starts, these lowfrequency GW sources also appear as UCXBs, which are the ideal multimessenger detection sources.For a long detection distance of 100 kpc, those binaries have to spend a longer evolutionary timescale to evolve toward a shorter orbital period, thus appearing as GW sources only in a few Myr.BH X-ray binaries in Group 3 experience a rapid orbital expansion, thus two systems with M He,i = 1.8, and 2.8 M ⊙ are not potential GW sources in the later stage of the second mass transfer.However, the GW radiation will drive the orbit of the system with M He,i = 1.8 M ⊙ to continuously shrink after the mass transfer ceases, resulting in a low-frequency GW source that is detectable in a long timescale.Since the chirp masses are constant, the evolutionary tracks of detached systems are lines with a slope of 7/6 (h c ∝ f 7/6 gw according to equation 3).In groups 1 and 2, the maximum GW frequency of BH UCXBs is 5.6 mHz, while the maximum frequency can reach about 63.5 mHz for a detached BH binary in group 3 (see also Table 1).
To understand the progenitor properties of BH UCXBs, we model the evolution of a large number of BH+He star binaries.Figure 5 summarizes the initial contour of the progenitors of BH UCXBs in the P orb,i − M He,i diagram.All BH+He star binaries with initial parameters located between two solid curves can evolve toward UCXBs (with durations of ≥ 0.1 Myr) that can be detected by LISA at a distance of 10 kpc.The initial He-star masses and initial orbital periods of progenitors of Galactic BH UCXB-LISA sources are in the range of 0.32 − 2.9 M ⊙ and 0.02 − 0.19 days, respectively.Such an initial parameter space is slightly wider than that (0.32−1.2 M ⊙ and 0.01−0.1 days) for the progenitors of NS UCXB-LISA sources evolving from the He star channel (Wang et al. 2021).It is clear that the sys-  tems with relatively long initial orbital periods (P orb,i ≥ 0.05 days) and high donor-star masses (M He,i ≥ 0.7 M ⊙ ) may experience two RLOF stages.For massive He stars with M He,i ≥ 1.2M ⊙ , all BH binaries will experience two RLOFs.Some systems marked by crosses can also evolve into low-frequency GW sources, however, they can not become valid UCXBs with a duration of ≥ 0.1 Myr.Those BH+He star binaries that experienced two RLOF stages could produce a high GW frequency in the final evolutionary stage (see also M He,i = 1.8 M ⊙ in Figure 4), while these sources are not UCXBs because of the absence of mass transfer.For a detection distance of 10 kpc, these binaries are most likely as multimessenger sources in the first mass-transfer stage.The measurement of chirp mass is very significant in constraining the masses of two components.For a detached binary system, the chirp mass can be derived by where ḟgw is the GW-frequency derivative.Therefore, the measurement of chirp mass depends on the accuracy of the ḟgw .The space GW detectors are only able to detect ḟgw for those ultra-compact binaries with a large signal-to-noise ratio (SNR) and a small orbital period close to the minimum orbital period (Tauris 2018).
The minimum ḟ that LISA can measure is given by (Takahashi & Seto 2002;Tauris 2018): where S/N is the SNR of GW signals, and T is the mission duration of LISA. Figure 6 illustrates the evolution of ḟgw with the GW frequencies for our simulated three groups.Taking S/N = 10 and T = 4 yr, we have the detection limitation of GW-frequency derivative as ḟgw,min = 2.5 × 10 −17 Hz s −1 , which is plotted by the horizontal dashed lines in Figure 6.In the orbital shrinkage stages, the angular momentum loss rates by GW radiation continuously increase, resulting in a rapid increase of ḟgw .
The maximum ḟgw is close to the peak GW frequency, however, ḟgw sharply decreases to be 0 at the maximum GW frequency, then its sign turns into negative due to an orbital expansion.Except for the orbital expansion stage of two systems in group 3, other BH UCXBs could provide detectable ḟgw in a timescale of 1 Myr.All evolutionary curves of ḟgw in the climbing stage are approximate lines with a slope of n = 11/3, which implies a relation ḟgw ∝ f 11/3 gw .This is consistent with that the GW radiation dominates the orbital evolution of the binaries.According to equation ( 6), ḟgw ∝ f 11/3 gw for a constant chirp mass in detached binaries, in which the angular-momentum loss is fully contributed by GW radiation (Webbink & Han 1998;Piro 2019).Therefore, if a braking index as ḟgw ∝ f n gw is defined, its measurement can diagnose whether the binary emitting low-frequency GW signals is a detached system.
Figure 7 presents the parameter space of BH+He star systems whose chirp masses can be accurately measured.For BH UCXB-LISA sources evolving from the He star channel, only several systems evolved from the progenitors with M He,i = 0.7 − 1.5 M ⊙ and 2.4 − 2.6M ⊙ and relatively long P orb,i are difficult to detect ḟgw .For the MS channel, only BH UCXBs with initial orbital periods very near the bifurcation period have detectable ḟgw (Qin et al. 2023).Therefore, BH UCXB-LISA sources evolved from the He star channel are most likely to measure their chirp masses.

Origin of BH+He star binaries
Similar to NS+He star systems, BH+He star systems could also be descendants of high-mass X-ray binaries, in which the companion of the BH loses its hydrogen envelope through stellar wind or a masstransfer stage (Bhattacharya & van den Heuvel 1991;Dewi et al. 2002).Cyg X-3 was proposed to include Note.The columns list (in order): the initial He-star mass, the initial orbital period, the stellar age at the beginning of RLOF, the minimum orbital period, the stellar age at the minimum orbital period, the maximum frequency of GW, the maximum X-ray luminosity when binaries are LISA sources at a distance of 10 kpc and 100 kpc, the detection timescale that the system can be detected by LISA at a distance of 10 and 100 kpc.* The system can be detected by LISA at a distance of 100 kpc as a detached system.a 2 − 4.5 M ⊙ BH (or NS) and a 7.5 − 14.2 M ⊙ Wolf-Rayet donor star (Zdziarski et al. 2013), and it may be the progenitor of a Galactic double BH or BH-NS (Belczynski et al. 2013).In addition, massive He stars might be formed through quasi-chemically homogeneous evolution (Woosley & Heger 2006;Yoon et al. 2006;Cantiello et al. 2007).However, our simulations find that BH binaries with massive He stars (M He,i > 3.0 M ⊙ ) are hard to evolve into detectable UCXBs with a duration longer than 0.1 Myr.Jiang et al. (2023) performed 1D model of post-CE BH binaries with short orbital period (≤ 0.2 days) consisting of a BH and a massive He star (M He,i ≥ 3.3 M ⊙ ) that experiences stable mass transfer, and found that their mass-transfer timescales are shorter than 0.1 Myr, which is consistent with our results.
NS/BH+He star binaries could evolve from NS/BH+MS binaries through Case B or Case C mass transfer (Dewi et al. 2002).For those systems with a large mass ratio, the mass transfer is dynamically unstable.Subsequently, the binary systems enter a CE phase (Ivanova et al. 2013), and the evolutionary products of those systems that experienced Case B mass transfer in the CE phase are binaries with an unevolved He star and a NS/BH.Those NS/BH+He star binaries produced from Case B are so close that they initiate a Case BA mass transfer in the core He burning stage.It is clear that our simulated systems in the parameter space of the BH UCXB-LISA source are formed from Case B mass transfer.Case C mass transfer would produce an evolved He star and a NS/BH after the CE phase, subsequent mass transfer would start after core He is exhausted.

He star Channel and MS channel
There exists a bifurcation period for the formation of UCXBs in the MS channel, which is defined as the maximum initial orbital period forming UCXBs within a Hubble time (van der Sluys et al. 2005;Sengar et al. 2017;Chen et al. 2020;Qin et al. 2023).However, in the He star channel there is not a critical period similar to the MS channel because of short initial orbital periods.In the He star channel, the initial donor-star masses of the BH+He star system that can evolve into UCXB-LISA sources are in a wide range from 0.32 to 2.9 M ⊙ .However, this range is shortened to be 0.4 − 1.6 M ⊙ in the MS channel because the stars with radiative envelope will not experience magnetic braking (Qin et al. 2023).Certainly, the initial orbital period range of the He star channel is much narrower than that of the MS channel.In the MS channel, the maximum GW frequency emitting by BH UCXBs is ∼ 3 mHz (Qin et al. 2023), which is slightly smaller than that (5.6 mHz) in the He star channel.Because of the high compactness of He stars, BH+He star systems can evolve into LISA sources that can be detected at a distance of 100 kpc, while this phenomenon is impossible for the MS channel.
To understand the final evolutionary fates of the He stars, we plot the evolution of five BH-He star binaries   in an H-R diagram in Figure 8.The final luminosities and the effective temperatures of five He stars are log(L/L ⊙ ) ∼ −1 to −3, and 6300 − 31000 K, respectively.Such luminosity and effective temperature ranges are comparable to those of WDs.It is clear that those BH binaries with a massive donor star (1.2−1.8M ⊙ ) had already evolved into detached systems before the donor stars evolved into WDs by a contraction and cooling stage.Therefore, the He star channel could form detached BH-WD systems.It is noteworthy that the MS channel can only form mass-transferring BH-WD systems rather than detached BH-WD systems (Qin et al. 2023).Similar to the MS channel, BH binaries with a low-mass (0.32 − 0.6 M ⊙ ) He star can only evolve into semi-detached BH-WD binaries rather than detached systems.Because those WDs evolving from the He star channel lack a thin H envelope, their effective temperatures are much higher than those of WD evolving from the MS channel.

He star Channel and Dynamic Process Channel
In globular clusters and young dense clusters, compact BH-WD binaries might assembled through BHs capturing WDs in a dynamic process such as tidal captures.Subsequently, GW radiation causes their orbits to continuously shrink, and detached BH-WD binaries appear as low-frequency GW sources that can be detected by space-borne GW detectors.Once WDs enter the tidal radius of BHs, they will be disrupted by BHs, and these systems become UCXBs.For a semi-detached BH-WD binary with M BH = 8 M ⊙ and M WD = 0.2 M ⊙ , the estimated minimum GW frequency is 6.4 mHz (Qin et al. 2023).Because the tidal radius of the BH satisfies (Hopman et al. 2004) the tidal radius will be smaller if the WD captured by the BH is more massive.According to the Keplerian third law, it would yield a relatively high minimum GW frequency.However, the maximum GW frequency is 5.6 mHz for BH UCXBs evolving from the He star channel.Therefore, the GW frequency could be a probe to test the formation channel of BH UCXBs.

Influence of tidal effects
In the detailed binary evolution models, we ignore the tidal effects.In principle, the tidal coupling between the orbit and the He donor star can influence the orbital evolution of BH X-ray binaries.During the shrinkage stage of the orbit, the donor star spins up to corotate with the orbital rotation due to tidal coupling.This spinup indirectly consumes the orbital angular momentum, extracting orbital angular momentum from the binary system at a rate of where I is the momentum of inertia of the He star, Ω is the derivative of the orbital angular velocity.Ignoring the influence of the mass transfer, the change rate of the orbital angular velocity Ω satisfies where J and Jgw are the total angular momentum of the system and its loss rate via gravitational radiation.Therefore, the ratio between Jt and Jgw is where µ = M BH M He /(M BH + M He ) is the reduced mass of the binary system.Considering a BH binary with M BH = 8 M ⊙ , M He = 1 M ⊙ , and P orb = 0.1 days, we have µa 2 = 3.0 × 10 55 g cm 2 .Since the radius of the He star R He = 0.212R ⊙ (M He /M ⊙ ) 0.654 (Tauris & van den Heuvel 2006), the momentum of inertia of the He star can be estimated to be I = 0.4MHe R 2 He = 1.7 × 10 53 g cm 2 .According to equation ( 11), the rate of angular momentum loss via the tidal effects is approximately two orders of magnitude smaller than that via gravitational radiation.Therefore, the influence of the tidal effects on the orbital evolution is trivial.

Influence of BH Masses on the Parameter Space
The binary population synthesis simulations found that the newborn BHs have a mass range of 5−16 M ⊙ in BH binaries with normal-star companions, in which the BH-masses distribution emerges a peak at ∼ 7 − 8 M ⊙ in the Model A of Shao et al. (2019).In BH-He star X-ray binaries, the BH masses range from 5 to 20 M ⊙ , and are most likely to gather at ∼ 7 − 8 M ⊙ in the Model A (Shao & Li 2020).Based on these statistical results, we adopt a constant initial BH mass of 8 M ⊙ in the detailed binary evolution models.
For a same He star, a high BH mass naturally results in a high chirp mass, a high characteristic strain, and a long detection distance of BH UCXB-LISA sources.When M He,i = 0.6 M ⊙ and P orb,i = 0.06 days, our calculations found that the maximum GW frequencies in the UCXB stage are 2.55, 2.75, and 2.91 mHz for M BH,i = 15, 8, and 5 M ⊙ , respectively.During the evolution BH binaries, a high BH mass would produce an efficient angular momentum loss via GW radiation, and a high mass transfer rate.The rapid mass transfer from the less massive He star to the more massive BH drives the orbits of the systems to widen, resulting a relatively small maximum GW frequency.Therefore, there exist inverse correlation between the initial BH mass and the maximum GW frequency in BH UCXBs evolved from the He star channel.
When M BH,i = 5 M ⊙ , and M He,i = 1 M ⊙ , our simulations show that the initial orbital periods of BH+He star binaries that can evolve into BH UCXB-LISA sources are in the range of 0.04 − 0.08 days, which is slightly narrower than that (0.04 − 0.09 days, see also Figure 5) in M BH,i = 8 M ⊙ .If the BH has a high initial mass of 15 M ⊙ , the range of initial orbital periods turns into 0.04 − 0.1 days.Therefore, a high/low initial BH mass tends to widen/reduce the initial orbital period range.This tendency originates from the gravitational radiation is the dominant mechanism driving the orbit of BH+He star binaries to shrink.A high mass BH naturally produces a high rate of angular momentum loss, driving the BH+He star binaries with a slightly long period to evolve into BH UCXB-LISA sources.Furthermore, the initial He-star masses that can evolve into BH UCXB-LISA sources are in the range of 0.32 − 3.0 M ⊙ and 0.32 − 2.9 M ⊙ for M BH,i = 5, and 15 M ⊙ , respectively.Therefore, the influence of the initial BH masses on the initial parameter space is trivial.

Detectability of BH UCXBs as LISA sources
We employ a rapid binary evolution code developed by Hurley et al. (2000Hurley et al. ( , 2002) ) to study the birthrate of BH UCXB-LISA sources in the Galaxy.Based on the binary population synthesis (BPS) approach, the primordial binary samples are produced in the way of Monte Carlo simulations.Subsequently, a sample of 1 × 10 7 primordial binaries are evolved until the formation of BH+He star systems through the rapid binary evolution code.Similar to Chen et al. (2020), the initial input parameters and basic assumptions in the BPS simulation are as follows: (1) All primordial stars are thought to be members of binary systems orbiting in circular orbits.(2) The primordial primary mass distribution arises from the initial mass function derived by Miller & Scalo (1979), and it produces the secondary mass distribution by adopting a constant mass ratio (0 < q ≤ 1) distribution n(q) = 1.(3) The initial separations distribution is taken to be constant in loga for wide binaries with orbital periods longer than 100 yr, then changes into a uniform distribution for close binaries (Eggleton et al. 1989).( 4) The standard energy prescription proposed by is used to tackle the CE ejection process (Webbink 1984), in which the efficiency α CE of ejecting the envelope and the parameter λ describing the stellar mass-density distribution are merged as a degenerate parameter α CE λ.BH UCXB-LISA sources are thought to be produced if the parameters of the BH+He star systems are consistent with those of the progenitors of BH UCXB-LISA sources in Figure 5.The influence of the initial BH mass on the initial parameter space is trivial, thus its affect on the population synthesis simulations can be ignored.As a consequence, the criterion determined compact objects in the BPS simulation is just a BH, no matter whether their masses are 8 M ⊙ .
In Figure 9, we plot the evolution of the birthrates of BH UCXB-LISA sources evolving from the He star channel as a function of time when we take a constant star formation rate (SFR) of 5 M ⊙ yr −1 for Population I.The Monte Carlo simulations predict the birthrates of BH UCXB-LISA sources in the Galaxy to be R = 2.2 × 10 −6 , and 3.6 × 10 −7 yr −1 when the degenerate CE parameter α CE λ = 1.5, and 0.5 (α CE is the CE ejection efficiency, and λ is the stellar structure parameter, Webbink 1984), respectively.These two birthrates are 1 − 2 orders of magnitude higher than that (∼ 3.9 × 10 −8 yr −1 ) estimated in the MS channel (Qin et al. 2023), however, are one order of magnitude lower than those (3.1−11.9×10−6 yr −1 ) predicted for NS UCXBs evolving from the He star channel (Wang et al. 2021).
According to Table 1, the mean detection timescale △t LISA,10 ≈ 15 Myr for BH UCXB-LISA sources evolving from the He-star channel for a detection distance of 10 kpc.Therefore, we can estimate the detection number of BH UCXB-LISA sources formed by the He-star channel in the Galaxy to be N = R △ t LISA,10 ≈ 33, and 5 as α CE λ = 1.5, and 0.5, respectively.For a low degenerate CE parameter α CE λ = 0.5, the detection number of BH UCXB-LISA sources evolving from the He-star channel is similar to that from the MS channel (≈ 4, Qin et al. 2023).However, the detection number of BH UCXB-LISA source evolving from the He star channel is one order of magnitude higher than that from the MS channel for a conventional degenerate CE parameter α CE λ = 1.5.

CONCLUSION
BH UCXBs can emit both low-frequency GW signals and X-ray emission, making them intriguing multimessenger detection sources.In this paper, we employ the MESA code to simulate the formation and evolution of BH UCXBs evolving from the He star channel and diagnose their detectability in both X-ray and GW bands.Taking an initial BH mass of M BH,i = 8 M ⊙ , our main conclusions are summarized as follows: 1.The mass transfer of BH+He star binaries is sensitive to the masses of He stars.When M He,i ≥ 0.7 M ⊙ , those systems may experience two RLOFs.The mass-transfer rates in second RLOF stages can reach ∼ 10 −6 M ⊙ yr −1 .Such a high mass-transfer rate causes their orbits to rapidly expand, thus some systems can not appear as UCXB-LISA sources at a distance of 10 kpc in the later stage of the second mass-transfer phase.
2. The mass-transfer rates of BH UCXBs evolving from the He star channel are much higher than those from the MS channel, producing X-ray luminosities that are 2 − 5 orders of magnitude higher than those from the MS channel.Due to the short initial orbital periods, the BH+He star systems become UCXBs at the beginning of the mass transfer and emit an X-ray luminosity of ∼ 10 38 erg s −1 .The maximum X-ray luminosity in the first masstransfer stage can exceed 10 39 erg s −1 .Those BH binaries with massive He stars experience a second RLOF, and produce a maximum X-ray luminosity of 3.5 × 10 39 erg s −1 , which exceeds the threshold luminosity of ULXs.
3. The shortest orbital period in the BH UCXB stage is 6 minutes, which corresponds to a GW frequency of 5.6 mHz.Because of short initial orbital periods, our simulated BH+He star binaries are already BH UCXB-LISA sources within a distance of 10 kpc (or 100 kpc) at the onset of the first mass transfer.BH X-ray binaries with mas-sive He stars (M He,i = 1.8, and 2.8 M ⊙ ) will not be detected by LISA at a distance of 10 kpc in the later stage of the second mass-transfer phase due to a rapid orbital expansion.However, the continuous orbital shrinkage due to GW radiation causes the system with M He,i = 1.8 M ⊙ to evolve toward LISA sources after the second mass transfer ceases, emitting GW signals with a frequency of up to 63.5 mHz.
4. Compared with NS UCXBs from the He star channel (Wang et al. 2021), the progenitors of BH UCXB-LISA sources have a slightly wide initial parameter space.The initial He-star masses and initial orbital periods of the progenitors of Galactic BH UCXB-LISA sources are in the range of 0.32 − 2.9 M ⊙ and 0.02 − 0.19 days, respectively.Meanwhile, nearly all systems in this parameter space can evolve into BH UCXB-LISA sources whose chirp masses can be accurately measured.However, only a tiny fraction of BH UCXB-LISA sources possess measured chirp masses for the MS channel (Qin et al. 2023).
5. In the He star channel, those BH binaries with a massive He star (1.2 − 1.8 M ⊙ ) can evolve into detached BH-WD binaries, which can not be achieved in the MS channel.Meanwhile, the effective temperatures of those WDs evolving from the He star channel are much higher than those of WD evolving from the MS channel because of the absence of a thin H envelope.
6.The Monte Carlo BPS simulations predict the birthrates of Galactic BH UCXB-LISA sources evolving from the He star channel to be R = 2.2 × 10 −6 , and 3.6 × 10 −7 yr −1 when α CE λ = 1.5, and 0.5, respectively.In a 4-year LISA mission, the detection numbers of Galactic BH UCXB-LISA sources for the He-star channel are 33, and 5 for α CE λ = 1.5, and 0.5, respectively.
7. Compared with the MS channel, BH UCXB-LISA sources from the He star channel possess high Xray luminosities and high possibility of detecting the chirp masses.Therefore, BH UCXB-LISA sources evolving from the He star channel are ideal multimessenger objects that deserve to be pursued in the X-ray and GW community.

Figure 1 .
Figure 1.Evolution of BH+He star binaries in the orbital period vs. stellar age diagram for groups 1, 2, and 3.The solid circles and open circles denote the onset and end of mass transfer, respectively.The solid triangles and stars represent the onset that BH binaries can be detected by LISA at distances of d = 10 and 100 kpc, respectively.BH binaries can not be detected by LISA at a distance of 10 kpc at the open triangles.The horizontal dashed lines represent the threshold period (90 minutes) of UCXBs.

Figure 4 .
Figure4.Evolution of BH+He star binaries in groups 1, 2, and 3 in the characteristic strain vs. GW frequency diagram.The detection distances of the black and green curve groups are 10, and 100 kpc, respectively.To avoid overlap, the dashed, dotted, and dashed-dotted curves are slightly moved up and down in parallel in the top panel (actually, these three curves overlap with the solid curve).The blue, purple, and red curves denote the sensitivity curve of LISA(Robson et al. 2019), TianQin(Wang et al. 2019), and Taiji(Ruan et al. 2020) based on the numerical calculation of 4-year observations, respectively.

Figure 6 .
Figure 6.Evolution of GW frequency derivative of BH UCXBs in groups 1, 2, and 3 in the | ḟgw| vs. fgw diagram.The horizontal dashed line represents the minimum ḟgw that can be detected by LISA for S/N = 10 and T = 4 yr (see also equation 7).

Figure 7 .
Figure 7. Parameter space distribution that describes the detectability of ḟ for BH+He star systems in the initial orbital period vs. initial He-star mass diagram.The solid diamonds represent BH+He star systems whose ḟ can be detected by LISA during the UCXB stage.The solid curves represent the boundary that can evolve toward the valid UCXB stage (durations ≥ 0.1 Myr).The dashed curves represent the boundary that ḟ can be detected (the lower boundary coincides with the solid curve).

Figure 8 .
Figure 8. Evolutionary tracks of several BH+He star binaries with different initial donor-star masses and an initial orbital period of 0.06 days in the H-R diagram.The solid and open circles represent the onset and end of mass transfer, respectively

Figure 9 .
Figure9.Evolution of the birthrates of BH UCXB-LISA sources evolving from the He star channel as a function of time when we take a constant SFR of 5 M⊙ yr −1 for Population I.The solid, and dashed curves represent the evolutionary tracks when the degenerate CE parameter αCEλ = 0.5, and 1.5, respectively.
(Avila-Reese 1993;Yungelson 2008) in a decreasing mass-transfer rate(Avila-Reese 1993;Yungelson 2008).For highmass He stars with M He,i ≥ 1.2 M ⊙ , mass transfer dominates the orbital evolution due to high mass-transfer rates.The orbital periods increase or slowly decrease, producing decreasing mass-transfer rates.
Figure 2. Evolution of mass-transfer rate of BH binaries in groups 1, 2, and 3 in the mass-transfer rate vs. mass-transfer timescale diagram.t,and t rlof represent the stellar age and the time when the donor star begins RLOF, respectively.dius

Table 1 .
Some Important Evolutionary Parameters of BH+He Star Binaries in Groups 1, 2, and 3.