Binary stars are responsible for many unusual astrophysical phenomena including some important explosive cosmic events. The stability criteria for rapid mass transfer and commonenvelope evolution are fundamental to binary star evolution. They determine the mass mass ratio and orbital distribution of systems such as Xray binaries and merging gravitationalwave sources. We use our adiabatic massloss model to systematically survey metalpoor and solarmetallicity donor thresholds for dynamical timescale mass transfer. The critical mass ratios q SUBadSUB are systematically explored and the impact of metallicity and nonconservative mass transfer are studied. For metalpoor radiativeenvelope donors q SUBadSUB are smaller than those for solarmetallicity stars at the same evolutionary stage. However q SUBadSUB do the opposite for convectiveenvelope donors. Nonconservative mass transfer significantly decreases q SUBadSUB for massive donors. This is because it matters how conservative mass transfer is during the thermal timescale phase immediately preceding a delayed dynamical mass transfer. We apply our theoretical predictions to observed highmass Xray binaries that have overfilled their Roche lobes and find a good agreement with their mass ratios. Our results can be applied to study individual binary objects or large samples of binary objects with binary population synthesis codes.
['Binary stars', 'Stellar mass black holes', 'Neutron stars', 'High mass x-ray binary stars', '154', '1611', '1108', '733', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']
Adiabatic Mass Loss in Binary Stars. V. Effects of Metallicity and Nonconservative Mass TransferApplication in High Mass XRay Binaries
{'Adiabatic Mass Loss in Binary Stars. V. Effects of Metallicity and Nonconservative Mass Transfer Application in High Mass X-ray Binaries': "Hongwei Ge , 1, 2, 3, 4 Christopher A Tout , 5 Xuefei Chen , 1, 2, 3, 4 Song Wang , 6, 7 Jianping Xiong , 1, 2, 3 Lifu Zhang , 1, 4 Qingzhong Liu, 8 and Zhanwen Han 1, 2, 3, 4  \n1 Yunnan Observatories, Chinese Academy of Sciences, Kunming, 650216, People's Republic of China  \n2 Key Laboratory for Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, People's Republic of China  \n3 International Centre of Supernovae, Yunnan Key Laboratory, Kunming 650216, People's Republic of China  \n4 University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China  \n5 Institute of Astronomy, The Observatories, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK  \n6 Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, People's Republic of China  \n7 Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, People's Republic of China 8 Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, ChineseAcademy of Sciences, Nanjing, People's Republic of China", 'ABSTRACT': 'Binary stars are responsible for many unusual astrophysical phenomena, including some important explosive cosmic events. The stability criteria for rapid mass transfer and common-envelope evolution are fundamental to binary star evolution. They determine the mass, mass ratio, and orbital distribution of systems such as X-ray binaries and merging gravitational-wave sources. We use our adiabatic massloss model to systematically survey metal-poor and solar-metallicity donor thresholds for dynamical timescale mass transfer. The critical mass ratios q ad are systematically explored and the impact of metallicity and nonconservative mass transfer are studied. For metal-poor radiative-envelope donors, q ad are smaller than those for solar-metallicity stars at the same evolutionary stage. However, q ad do the opposite for convective-envelope donors. Nonconservative mass transfer significantly decreases q ad for massive donors. This is because it matters how conservative mass transfer is during the thermal timescale phase immediately preceding a delayed dynamical mass transfer. We apply our theoretical predictions to observed high-mass X-ray binaries that have overfilled their Roche lobes and find a good agreement with their mass ratios. Our results can be applied to study individual binary objects or large samples of binary objects with binary population synthesis codes.  \nKeywords: Binary Stars(154) - Stellar mass black holes(1611) - Neutron stars(1108) - High-mass X-ray binaries(733)', '1. INTRODUCTION': 'Binary stars lie at the heart of many vital astrophysical phenomena (e.g., Tauris & van den Heuvel 2023). Different types of explosive cosmic events are produced by binary star objects, such as double black holes (DBHs, e.g., Mandel & Broekgaarden 2022), double neutron stars (DNSs, e.g., Gallegos-Garcia et al. 2023), dou-  \nCorresponding author: Hongwei Ge  \ngehw@ynao.ac.cn  \nble white dwarfs (DWDs, e.g., Li & Chen 2024), type Ia supernovae (e.g., Liu et al. 2023) and X-ray binaries (e.g., Tauris & van den Heuvel 2023). The study of binary evolution brings new opportunities for insights into fundamental physics (e.g., Amaro-Seoane et al. 2023) such as testing general relativity, production of heavy chemical elements, and discovering new double compact systems in the next few decades. Despite the importance of binary evolution, there remain unsolved fundamental questions of particular importance are mass transfer stability (Webbink 1984) and common envelope (CE) evolution (Paczynski 1976).  \nA series of recent study indicate that mass transfer in massive binary systems (Ge et al. 2015) and binaries with red/asymptotic giant branch (R/AGB) compaions (Ge et al. 2020a,b) is more stable than previously found. With these mass transfer thresholds as physical inputs, an isolated binary BH formation scenario consisting of a stable mass transfer during the second mass transfer phase (instead of CE) has been gaining popularity in the gravitational wave (GW) community (Neijssel et al. 2019). Besides the mass transfer physics, the supernova natal kicks (Mandel et al. 2021) are also critical for massive binary evolution. Hence, mass transfer is probably active in some systems in eccentric orbits. As for WD binaries, mostly formed from R/AGB progenitors, a first stable mass transfer plus a second CE channel may dominate the formation of DWDs (Li et al. 2023). These objects are low-frequency GW sources detectable with Tianqin and LISA detectors (Luo et al. 2016; AmaroSeoane et al. 2017).  \nMany observed stellar phenomena, including binaries, are dominated by metal-poor environments (Ge et al. 2023). Examples include blue stragglers (Sandage 1953), blue metal-poor stars (Preston & Sneden 2000), Galactic halo stars (Li et al. 2018), metal-poor thick disk stars (Wu et al. 2021), ultra-luminous X-ray sources (ULXs, Zampieri & Roberts 2009), partially stripped-envelope stars (Klencki et al. 2022), massive stellar-mass black holes (BHs) and their mergers (Vink et al. 2021), and metal-poor inner Galactic stars (Ardern-Arentsen et al. 2024). Nonconservative mass transfer in binary evolution, mass and angular momentum loss, plays an important role in binary evolution (Chen et al. 2024). It may substantially impact predictions about the production rates of various transients, such as X-ray binaries and the progenitors of coalescing compact binaries (Willcox et al. 2023).  \nWe extend our systematic studies of stars to metalpoor donors and nonconservative mass transfer. With our adiabatic mass loss model, we only provide critical mass ratio thresholds (Ge et al. 2010) assuming conservative mass transfer in the first place. However, the advantage is that the donor response to rapid mass transfer is independent of the binary orbital evolution. So our studies can easily resolve the nonconservative mass transfer cases. Our preliminary results have been included in binary population synthesis and applied to study objects such as DWDs (Li et al. 2023). After applied the new thresholds, the results support the observational DWD merger rate distribution per Galaxy and the space density of DWDs in the Galaxy.  \nIn Section 2 we describe methods to build an adiabatic mass-loss model and solve for the critical mass ra-  \ntio under conservative and nonconservative mass transfer. Using massive 20 M ⊙ stars as examples, in Section 3 we study the impacts of metallicity and nonconservative mass transfer on the critical mass ratios for dynamical timescale mass transfer in metal-poor Z = 0 . 001 and solar-metallicity Z = 0 . 02 stars 1 . In Section 4 we present our model grids and thresholds for unstable mass transfer. In Section 5 we collect observed high-mass Xray binaries (HMXBs) with known binary parameters and compare mass ratios and orbital periods with our theoretical predictions. In Sections 6 and 7 we discuss and summarize our systematic studies of the thresholds for dynamical timescale mass transfer.', '2. METHODS': "We use our adiabatic mass-loss model to study the responses of donor stars undergoing rapid mass transfer. Ge et al. (2010) decribed the detailed method and numerical implementation. The stellar evolution code is updated and extended by Ge et al. (2015, 2020a, 2023) and is based on the Cambridge Stars Code (Pols et al. 1995) originally developed by Eggleton (1971).  \nWe extend our previous studies of dynamical timescale mass transfer thresholds for nonconservative cases. This extention is essential for the delayed dynamical instability of donor stars with a deep radiative envelope. We first recall the key points to calculate the critical initial mass ratio q ad for the dynamical timescale mass transfer. The classical method (Webbink 1985) is to compare the slope of the mass-radius relation of both the stellar radius R and the Roche radius R L , such as equations 2 to 11 of Ge et al. (2010). The slopes are expressed by the so-called mass-radius exponents,  \nζ ad ≡ ( d log R d log M ) ad , (1)  \nand  \nζ L ≡ d log R L d log M , (2)  \nwhere ζ ad < ζ L implies instability on a dynamical timescale mass transfer and ζ ad ≥ ζ L implies stable mass transfer. However, runaway mass transfer increases gradually and some donors may have a loose and low-density envelope. So, instead of using the surface radius R of the donor star, we use an inner radius R KH to calculate the mass-radius exponent, ζ KH ad = d ln R KH /d ln M | ad . This innner radius R KH determines when the mass-loss rate ˙ M (see A9 of Ge et al. 2010) reaches a thermal timescale rate ˙ M KH = M/t KH .  \nWe assume a donor star overfills its Roche lobe at some initial mass and radius (the n th model from evolutionary sequences counting from the zero-age mainsequence, ZAMS). The critical initial mass ratio q i ≡ M donor /M accretor = M d /M a for dynamical timescale mass transfer is to be determined. We define a mass function µ = M d / ( M d + M a ) = q/ (1 + q ) which lies between 0 and 1. For the adiabatic mass-loss sequences, we enumerate models by b ∈ { 1 , 2 , ..., B } . Initially, b = 1, the initial mass is M i and radius R i . The initial separation A i is such that R i = R L , the Roche-lobe radius of the donor. For model b , the remaining total mass M b and the innner radius R b KH are found by our adiabatic mass-loss (Ge et al. 2010). The mass-radius exponent ζ KH ad can thus be found with models b -1 and b . It is independent of whether the mass transfer is conservative or not. To find the critical initial mass ratio, we need to know the mass-radius exponent ζ L , which depends on how conservative the mass transfer is.  \nThe inner Lagrangian Roche-lobe radius R L of the donor (Eggleton 1983) can be expressed by  \nR L A = r L ( q ) = 0 . 49 q 2 / 3 0 . 6 q 2 / 3 +ln(1 + q 1 / 3 ) . (3)  \nDifferentiating equation (3) we have  \nd ln r L d ln q = 1 3 [ 2 ln ( 1 + q 1 / 3 ) -q 1 / 3 / ( 1 + q 1 / 3 ) 0 . 6 q 2 / 3 +ln ( 1 + q 1 / 3 ) ] . (4)  \nAccording to Ge et al. (2020b), the outer Lagrangian Roche-lobe radius of the donor R out L = r out L ( q ) A can be found from  \nr out L ( q ) = r L ( q ) + [ 0 . 179 + 0 . 01 ( q 1 + q )] × ( q 1 + q ) 0 . 625 for q ⩽ 1 , (5)  \nand  \nr out L ( q ) = r L ( q ) + [ 0 . 179 + 0 . 01 ( q 1 + q ) -0 . 025 ( q -1 q )]( q 1 + q ) 0 . 625 q -0 . 74 for q ⩾ 1 . (6)  \nIn terms of Lograngian surfaces, the physical radius of the donor outer lobe is R L 3 for q > 1 and R L 2 for q < 1.  \nFor conserved mass and angular momentum, Ge et al. (2023) found  \nR L ( µ b ) R i = r L ( µ b ) r L ( µ i ) ( µ i µ b ) 2 ( 1 -µ i 1 -µ b ) 2 , (7)  \nwhere µ b = µ i M b /M i and µ = q/ (1 + q ). For nonconservative mass transfer, a fraction α of the total mass  \nlost by the donor is transferred to its companion. So a fraction 1 -α of the donor mass lost carries away the donor's specific angular momentum. We further suppose that a faction β of the transferred mass is accreted so that a faction (1 -β ) of the transferred mass is ejected from the accretor with γ times its specific angular momentum. So for mass-loss model b , the mass ratio  \nq = M b M i /q i + αβ ( M i -M b ) , (8)  \nand the separation varies as  \nA b A b -1 =1 + 2( M b -1 -M b ) M b × [ α (1 -qβ ) -q 1 + q ( γq +0 . 5)(1 -αβ ) ] . (9)  \nWhen mass is only lost from the accretor and carries away its specific angular momentum, we have α = 1 and γ = 1. We set β = 1 (fully conservative), 0.5 (nonconservative), and 0 (fully nonconservative) in this study. Starting from an initial guess, µ i = 0 . 5, we use a bisection method to calculate the initial mass function, µ b , satisfying both R b L = R b KH and ζ b L = ζ b KH . By tracing b = 1 to b = B for the whole adiabatic mass loss process, we can get the minimum value of µ min . So, the critical initial mass ratio is calculated finally with q ad = µ min / (1 -µ min ).  \nSimilar to Ge et al. (2015, 2020a) we build parallel donor star sequences with isentropic envelopes alongside standard donor stars with standard convective envelopes. In such stars, convective envelopes are replaced by fully isentropic envelopes. With this the limitations of an adiabatic approximation in a thin layer just below the photosphere are overcome. For such stars we calculate critical initial mass ratios ˜ q ad .  \nWe demonstrate our method for a metal-poor star of 20 M ⊙ and 167 . 2 R ⊙ in Figure 1. The Roche lobe radius becomes much smaller when the mass transfer is more nonconservative. This allows a more shallow decent of Roche lobe radius for nonservative mass transfer. Hence, critical mass ratios q ad become smaller rather than larger for nonservative mass transfer.", '3. MASSIVE STARS OF 20 M ⊙ WITH VARYING METALLICITIES': "We choose 20 M ⊙ massive donor stars as examples to demonstrate the impacts of both metallicity and mass transfer physics. A metal-poor star behaves as if more massive than a similar mass solar-metallicity star(Figure 2). Such behavior is similar for intermediatemass donor stars (Ge et al. 2023). A metal-poor massive  \n/s8857  \nR  \n/  \nR  \nFigure 2. The left panel shows the evolution for 20 M ⊙ stars with different metallicities in the Hertzsprung-Russell diagram. The right panel presents their helium core masses M He , where core hydrogen mass fraction first reaches 0.15, as a function of their radii. Solid and dashed lines are stars with metallicities of Z = 0 . 001 and Z = 0 . 02.  \n<!-- image -->  \n/s8857  \nFigure 1. Example critical mass ratios q ad for conservative and nonconservative mass transfer. The initial parameters are labeled. The black solid line marks the radius response during the adiabatic mass loss process. The blue dashed line is the inner radius R KH . The dash-dotted lines are the Roche lobe radii under conservative (black), semiconservative (violet), and fully nonconservative (cyan) mass transfer.  \nstar has a slightly higher central temperature T c and develops a larger helium core mass M He at the end of the main sequence. However, the convective core moves offcenter and shrinks for a while until helium ignites in the center ofer which M He increases to exceed that of a Z = 0 . 02 star of the same radius (Figure 2).  \nThe stellar radius R can be used to represent the evolutionary state of a star and because it determines when Roche-lobe overflow begins. The initial orbital period  \n)  \n)  \n)  \n)  \n)  \nL  \nL  \nL  \nL  \nL  \n/  \n/  \n/  \n/  \n/  \nL  \nL  \nL  \nL  \nL  \n(  \n(  \n(  \n(  \n(  \ng  \ng  \ng  \ng  \ng  \no  \no  \no  \no  \no  \nl  \n<!-- image -->  \nl  \nl  \nl  \nl  \n/s8857  \nM  \n/  \ne  \nH  \nM  \nFigure 3. The impact of metallicity on helium core mass M He (left panel) and the radius R (right panel) as functions of a 20 M ⊙ star's age. Solid and dashed lines are Z = 0 . 001 and Z = 0 . 02 stars.  \n<!-- image -->  \nP orb , or separation A , of a binary system only weakly depends on mass ratio but strongly on donor radius (Ge et al. 2015, Section 2). Metal-poor ( Z = 0 . 001) massive stars have longer core hydrogen burning lifetime (e.g., Schaller et al. 1992) and smaller radii R than those of same mass but Z = 0 . 02 star (Figure 3).  \nGe et al. (2020a) presented critical mass ratios q ad for 0 . 1 ≤ M/M ⊙ ≤ 100 stars with solar metallicity. They confirmed that donor stars with radiative and convective envelopes behave differently during adiabatic mass loss. Intermediate-mass (Ge et al. 2020a, 2023) and massive stars Ge et al. (2020a) while on the main sequence (MS), Hertzsprung gap (HG), and just before the base of the giant branch (BGB) have a similar structure, including an outer radiative envelope. Although the massive star is complicated by multiple convection zones, its radiative envelope dominates behavior under adiabatic mass loss. Such donor stars are predicted to undergo delayed dynamical instability (Hjellming & Webbink 1987; Ge et al. 2010) if the initial mass ratio q i is larger than q ad . We find that the critical mass ratios q ad and ˜ q ad increase with the radius, as the star evolves from the ZAMS to the BGB (Figure4). The convection envelope dominates once a massive star becomes a red GB or a red supergiant (RSG). However, the thermal timescale becomes very short for convective envelope giants. So the threshold for thermal timescale mass loss through the outer-Lagrangian point q Lout th is likely to become more strict compared to q ad (Ge et al. 2020b). In other words, binary systems, containing such kind of donors, with initial mass ratios q Lout th < q i < q ad may enter a common envelope like phase.  \nt  \nt  \ni  \ni  \nr  \nr  \nc  \nc  \nq  \nq  \n/s8857  \nFigure 5. The evolution of metal-poor ( Z = 0 . 001) stars. Black solid lines indicate massive donor stars, while gray solid lines are low- and intermediate-mass stars. Some of the intermediate and low-mass stars were studied by Ge et al. (2023). We also extend the donor spaces in this study. Open circles are placed at ∆ log 10 R = 0 . 1 from the ZAMS.  \n<!-- image -->  \n/s8857  \nFigure 4. Critical initial mass ratios q ad as functions of stellar radius for 20 M ⊙ models. The solid and dashed lines correspond to Z = 0 . 001 and Z = 0 . 02 stellar models. Red and blue lines correspond to models with standard mixing-length envelopes q ad and artificially isentropic convective envelopes ˜ q ad for fully nonconservative mass transfer. Gray lines show stellar models with standard envelopes q ad for fully conservative mass transfer.  \nAfter the mass of a star, its metallicity is the second most important parameter for its structure and evolution. However, metallicity seems only to shift the critical mass ratios q ad of massive stars (same mass) in radius (Figure 4). The critical mass ratio trend looks quite similar for 20 M ⊙ massive stars of different metallicities. As we described in the previous section, the donor response to adiabatic mass loss is independent of the orbital evolution of a rapidly mass-transferring binary system. However, how much mass and angular momentum is lost from the binary system significantly affects the Roche lobes of two components and the orbital period. So, before entering a delayed dynamical instability, conservative and nonconservative mass transfer alter the initial critical mass ratio q ad dramatically (Figure 4). The critical mass ratios of evlved donors of different metallicities decrease significantly for nonconservative mass transfer. These reduced critical initial mass ratios are caused by a longer thermal timescale mass transfer before entering the delayed unstable process. Surprisingly, the differences between fully nonconservative q ad (red) and ˜ q ad (blue) along to metallicity are not significant (Figure 4).", '4. MODEL GRIDS AND THRESHOLDS FOR UNSTABLE MASS TRANSFER': 'Previously Ge et al. (2020a) studied the adiabatic responses of Population I (Z = 0.02) stars spanning a full range of stellar mass (0 . 10 to 100 M ⊙ ) and all evolu-  \n)  \n)  \n)  \n/s8857  \n/s8857  \n/s8857  \nL  \nL  \nL  \n/  \n/  \n/  \nL  \nL  \nL  \n(  \n(  \n(  \n0  \n0  \n0  \n1  \n1  \n1  \ng  \ng  \ng  \no  \no  \no  \nl  \n<!-- image -->  \nl  \nl  \ntionary stages. Ge et al. (2023) extended the study to metal-poor donor stars. Here we also include nonconservative mass loss. This is essential for binary population synthesis studies of compact binaries with WD, NS, or BH companions.  \nWe cover a mass range from 1 to 100 M ⊙ (Figure 5). We present the initial physical properties of different donor stars and their thresholds for dynamical timescale mass transfer in Appendix A and online. Tables A1 and A2 list interior and global physical properties of metalpoor donor stars. We only provide a 20 M ⊙ stellar sequence in this print version but list all the data in a machine-readable version. Table A3 lists thresholds for conservative and nonconservative dynamical timescale mass transfer for metal-poor stars. Table A4 extends the thresholds for nonconservative dynamical timescale mass transfer in solar-metallicity stars (from 0 . 1 to 100 M ⊙ ).  \nWe plot the critical mass ratios q ad of metal-poor intermediate-mass 5 M ⊙ and massive from 10 to 100 M ⊙ stars in Figures 6 and 7. During core-helium burning, the critical mass ratios increase dramatically because a deep convective envelope is replaced by a thick radiative envelope. Massive donors show similar trendsto the 20 M ⊙ stars. The critical mass ratio q ad increases evolution of the donor until a convective envelope appears (Figure 7).', '5. HIGH MASS X-RAY BINARIES': 'A high-mass X-ray binary (HMXB) is generally defined as a binary star system consisting of a neutron  \nd  \na  \nq  \nFigure 6. Critical mass ratios q ad as functions of stellar radius for 5 M ⊙ stars. The solid (black) and dotted (blue) lines correspond to models of fully conservative and nonconservative mass transfer.  \n<!-- image -->  \nd  \nd  \na  \na  \nq  \nFigure 7. Critical mass ratios q ad as functions of stellar radius for metal-poor massive stars. The masses in units of M ⊙ are labeled for each track. The solid (black) and dotted (blue) lines correspond to models of fully conservative and nonconservative mass transfer.  \n<!-- image -->  \nq  \nstar (NS) or a BH accreting matter from a high-mass ( M ≳ 10 M ⊙ ) stellar companion (van den Heuvel & Heise 1972; Tutukov & Yungelson 1973). Over 400 HMXBs have been found in the Milk Way (Fortin et al. 2023), the Large (Antoniou & Zezas 2016) and Small (Haberl & Sturm 2016) Magellanic Clouds, and nearby galaxies (Lehmer et al. 2021). The donor stars in HMXBs are dominated by supergiant (Sg) O/B stars or Be stars (Fornasini et al. 2023), and there are a few HMXBs with Wolf-Reyet donors or near naked he-  \nstars. Other HMXBs include gamma-ray binaries that are dominated by gamma rays above 1Mev rather than X-rays and ultra-luminous , L X > 1 × 10 39 erg s -1 , sources. HMXBs are important in modern astrophysics, such as first generations of HMXBs may provide X-ray heating to intergalactic gas and impact the onset and duration of Epoch of Reionization (Jeon et al. 2014), understanding the formation and evolution of HMXBs provides important insights into one of the primary formation channels of gravitational wave sources (Fornasini et al. 2023).  \nRecently, updated catalogue of HMXBs have been provided by Neumann et al. (2023) and Fortin et al. (2023). New HMXBs, including ultra-luminous X-ray sources (Misra et al. 2020), have been found since the widely used catalogue of Liu et al. (2006). From the observed HMXBs, we select 82 objects with known orbital information. We gather their orbital periods P orb , compact companion masses M X , donor masses M d , peak X-ray luminosities L X , and eccentricities e along with other useful information in Table 1.  \nWe further select HMXBs undergoing stable mass transfer in circular orbits to compare our theoretical results. Unlike low and intermediate-mass X-ray binaries, most HMXBs are wind mass-transferring systems. So, the mass ratio q = M d /M X of HMXBs can be large, over twenty (Figure 8 and 9). These are much larger than the mass ratios of intermediate-mass X-ray binaries, which reach q max ≈ 6 (Figure 17 and Table 3 of Ge et al. 2023) that are undergoing stable Roche lobe overflow (RLOF). From Figure 8, we see that most of the relatively long orbital-period ( P orb > 10 d) HMXBs have low ratios of donor radius to Roche-lobe radius. On the other hand at relatively short orbital periods ( P orb < 10 d) most HMXBs are close to or undergoing stable RLOF.  \nSimilarly, we find that relatively long orbital-period ( P orb > 10 d) HMXBs can have a moderately ( e > 0 . 35) to highly eccentric orbits (Figure 9). On the contrary, HMXBs in short period ( P orb < 10 d) are generally in low ( e < 0 . 35) eccentricity orbits. When the progenitor of the accreting NS or BH explordes as a supernova, there is likely a natal kick caused by spatially asymmetric mass loss or neutrino emission (Woosley 1987). Such natal kicks can change the eccentricity and the final separation and even unbind otherwise runaway binary (Flannery & van den Heuvel 1975; Hills 1983). Here, we select HMXBs with a low eccentricity ( e < 0 . 35) and a high filling degree ( R/R L > 0 . 95) on the demands that they are closest to stable RLOF.', 'T able 1 . Observ ed High-mass X-ra y Binaries with kno wn masses and orbital p erio ds': '<!-- image -->  \nT able 1 c ont inue d  \nT able 1 (c ontinue d)  \nT able 1 c ont inue d  \n<!-- image -->  \nT able 1 (c ontinue d)  \nBHXBs  \n<!-- image -->  \nNote -* Donor Sp ec tral T yp e  \n<!-- image -->  \nM  \n/  \nM  \n=  \nq  \nFigure 8. The ratio of donor radius R to Roche-lobe radius R L of 82 HMXBs in Table1 on a mass ratio against the orbital period diagram. Symbols represent the types of HMXBs in the Table. Colours (gray means unknown) show the ratio of donor radius to its Roche-lobe radius.  \n<!-- image -->  \nOnly 7 HMXBs composed with a BH accretor have known binary parameters (Table 1). We plot these BHHMXBs in a mass ratio-orbital period diagram (Figure 10). Four have a low eccentricity and high overfilling factor (blue) and are suitable sources to compare with the theoretical mass ratio limits. MWC 656 and LS 5039 (gray) are wind-driven HMXBs. Cyg X-3 contains a WN4 to 6 type donor star which is significantly smaller than a massive main-sequence star. The blue open circles mark the critical mass ratio for massive metal-poor  \n40  \n40  \n40  \n40  \n40  \n40  \n40  \nFigure 10. Seven BH-HMXBs in a mass ratio against the orbital period diagram. Blue symbols mark objects with a high overfilling factor R/R L and low eccentricity e . Gray symbols show wind mass transferring objects or HMXBs with a near naked helium star donor. Blue open circles are the critical mass ratio for the dynamical timescale mass transfer of massive stars in different evolutionary states. The upper boundary of the blue region is a fit of the critical mass ratio as a function of the orbital period.  \n<!-- image -->  \nX  \nM  \n/  \nd  \nM  \n=  \nq  \nFigure 9. The eccentricities of HMXBs on a mass ratio orbital period diagram. Again symbols show the HMXB types.  \n( Z = 0 . 001) donor stars in different evolutionary states. We fit the maximum critical mass ratio q ad envelope as a function of the orbital period. We only show the fitting of Z = 0 . 001 massive stars for simplicity. The maximum critical mass ratios q max ad of Z = 0 . 02 massive stars are slightly larger than Z = 0 . 001 donors. These observed BH-HMXBs in circular orbits and transferring mass through RLOF are located below our theoretically predicated mass ratio limits as required. q  \n8  \n8  \n8  \n8  \n8  \n<!-- image -->  \nX  \nM  \n/  \nd  \nM  \n=  \nq  \nAny systems above the blue envelope must be in a wind mass transfer phase or possibly undergoing thermal RLOF before experiencing a delayed dynamical instability. Indeed LS 5039 has a high eccentricity that is less correlation with RLOF.  \nSimilarly we select 14 HMXB systems with an NS companions and eccentricity e < 0 . 35. We plot these NS-HMXBs in a mass ratio against orbital period diagram (Figure 11). Eight (violet) NS-HMXBs have known donor radii. The other six (grey) do not. We colour the stable mass transfer parameter space for lowmetallicity ( Z = 0 . 001) hydrogen-rich donor stars blue in Figure 11 similarly to Figure 10. So, above the blue envelope binary systems are only possible for wind mass transfer, thermal timescale mass transfer before entering delayed dynamically unstable mass transfer, or if the donor is a naked helium star. Our predicted mass ratio upper limits are for stable mass transfer in a circular  \n40  \n40  \n40  \n40  \n40  \n40  \n40  \n40  \nX  \nM  \nX  \nM  \nX  \nM  \nX  \nM  \nM  \nX  \nM  \nX  \nM  \nX  \nM  \nX  \n/  \n/  \n/  \n/  \n/  \n/  \n/  \n/  \nd  \nM  \nM  \nd  \nM  \nd  \nM  \nd  \nM  \nd  \nM  \nd  \nM  \nd  \nM  \nd  \n=  \n=  \n=  \n=  \n=  \n=  \n=  \n=  \nq  \n30  \n30  \n30  \n30  \n30  \n30  \n30  \n30  \nFigure 11. Fourteen NS-HMXBs on a mass ratio against orbital period diagram. Symbols mark different types of NS-HMXBs. Violet have measured radii while grey do not. The blue and violet regions are theoretically permitted spaces for stable RLOF from the normal and naked helium donors. Above the fitted maximum mass ratio envelope (blue for metal-poor Z = 0 . 001 massive donors and violet for Z = 0 . 02 helium donor stars), systems must be undergoing wind mass transfer, have a highly eccentric orbit, be suffering delayed dynamical instability.  \n<!-- image -->  \nq  \nq  \nq  \nq  \nq  \nq  \nq  \norbit. With relatively long orbital periods ( P orb > 11 d) NS-HMXBs, two eccentric systems are located above the critical mass ratio limit. These are most likely in a wind mass transfer phase. With relatively short orbital periods ( P orb < 11 d), all eight systems located above the critical mass ratio limit. Seven have R/R L larger than 0.96 and so must be in a state of relatively stable RLOF. They may be almost to experience a delayed dynamical instability. However, this stage is expected to last only for a short thermal timescale. A more likely explanation is that their donors are helium stars. We extrapolate the results of Zhang et al. (2024) and fit the maximum critical mass ratios of solar-metallicity helium donor stars. We find these short-period NS-HMXBs are then under the critical mass ratio limit, except for one ready con-  \nstrained eccentric system. We encourage more detailed observations to test this hypothesis.', '6. DISCUSSIONS': 'We have considered the impact of both metallicity and nonconservative mass transfer on the dynamical timescale mass transfer thresholds. Similarly to our previous study (Ge et al. 2023), metal-poor stars on the MS or HG (radiative envelope) tend to be more unstable than metal-rich donors (Figure 4). Metal-poor donors, with deep convective envelopes, are more stable than metal-rich stars (Figure 4). We expect that mass transfer stability thresholds may impact binary systems differently in the Galatic thick and thin disks and the halo.  \nNonconservative mass transfer significantly decreases the critical mass ratio for massive donors that have left the MS compared with the conservative case (Figure 7). Nevertheless, the critical mass ratios of massive MS and HG donors are mostly still greater than 3 or 4. This suggests a moderate increase in the number of binary systems that form from stable mass transfer channels.  \nStellar winds may significantly change the masses of the donor and the orbital separation before Roche-lobe filling. Such winds are included in binary population synthesis studies. Our initial donor stars that fill their Roche lobes are built without wind mass loss (Ge et al. 2010) in order to minimize the impact of the uncertainty of stellar winds. Both observation (Mokiem et al. 2007) and theory (Conti et al. 2008) suggest that stellar wind is metallicity-sensitive. We previously considered the impact of stellar wind (Ge et al. 2020a) and found that the critical mass ratio of a donor depends on the initial mass of the donor and its current core mass, both of which are evaluated in population synthesis models (Hurley et al. 2002) in a different way.  \nRotation is also important in massive star evolution. Rotation and extra mixing (Meynet & Maeder 2003, 2005) can enhance formation of Wolf-Rayet stars. Indeed, Smartt et al. (2004) noticed and Massey et al. (2017) reported a lackation of red supergiants in local group galaxies. So rotation and extra-mixing worthy of future work in mass transfer physics.  \nOur systematic studies of the thresholds for dynamical-timescale mass transfer predict the onset of the CE evolution (Paczynski 1976). After the critical mass ratios for unstable mass transfer, the treatments of CE ejection is the next least understood question in binary star evolutio. The current treatment requires both precise binding energy E bind (Chen & Ivanova 2024) and CE ejection efficiency α CE (Scherbak & Fuller 2023; Ge et al. 2024). Ge et al. (2022, 2024) found evidence for a relation between the initial mass ratio and their common envelope ejection efficiency β CE for low-mass ( M i < 2 M ⊙ ) donor stars. So this work should also be extended to higher mass stars.', '7. SUMMARY': "Previously Ge et al. (2010, 2015, 2020a) systematically presented critical mass ratios, assumed conserved mass and angular momentum, for dynamical-timescale mass  \ntransfer over the entire span of donor star evolutionary states with Z = 0 . 02. Here we extend our analysis to include various metallicities and nonconservative mass transfer. Using 20 M ⊙ donor stars as examples, we analyze the response of stars with metallicities Z = 0 . 02 and Z = 0 . 001 with nonconservative mass transfer. We find the critical mass ratios of donor stars with masses between 1 and 100 M ⊙ with Z = 0 . 001. We further complement the thresholds for nonconservative unstable mass transfer for stars between 0 . 1 and 100 M ⊙ at Z = 0 . 02. Critical mass ratios of metal-poor MS and HG donors with radiative envelopes are smaller than those of solar metallicity stars at the same evolutionary stage. However, critical mass ratios of metal-poor RGB/AGB donors with convective envelope are larger than those of the solar metallicity with the same radii. Nonconservative mass transfer also decreases the critical mass ratios of donors from late MS to the end of the star's evolution. We apply our results to 82 observed HMXBs with measured mass ratios and orbital periods. The observed HMXBs with low eccentricities and overfilled Roche lobes are nicely located under our predicted limits. Our results can be applied to study individual binary objects and binary population synthesis of most binary systems.", '8. ACKNOWLEDGMENTS': 'This project is supported by the National Natural Science Foundation of China (NSFC Nos. 12288102, 12125303, 12090040/3, 12173081), the National Key R&D Program of China (Nos. 2021YFA1600403, 2021YFA1600401), the Key Research Program of Frontier Sciences, CAS (No. ZDBS-LY-7005), CAS-Light of West China Program, Yunnan Fundamental Research Projects (Nos. 202101AV070001, 202201BC070003, 202401BC070007), Yunnan Revitalization Talent Support Program - Science & Technology Champion Project (No. 202305AB350003), and the International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001). HG thanks Professor Ronald Webbink for helpful discussions in building the adiabatic mass-loss model. HG thanks Boyuan Liu, a postdoctral research associate at institute of astronomy at university of cambridge, for his useful comments. CAT thanks Churchill College for his fellowship.  \nAPPENDIX', 'A. TABLES FOR THE INITIAL PHYSICAL PROPERTIES OF DONOR STARS AND THE THRESHOLDS FOR DYNAMICAL TIMESCALE MASS TRANSFER': "We present the interior physical properties of metal-poor stars in Table A1. Only model sequences for 20 M ⊙ donors are listed in this printed table. Further models are available in machine-readable form. The physical parameters are  \n- (1) n -mass-loss sequence number counting from ZAMS,\n- (2) M i -initial mass of the mass-transferring star ( M ⊙ ),\n- (3) log 10 t -age (yr),\n- (4) M He -helium core mass ( M ⊙ ), where the hydrogen abundance X first reaches 0.15 going out from the center,\n- (5) M C -carbon core mass ( M ⊙ ) where the helium abundance Y first reaches 0.25,\n- (6) M deg -mass of the degenerate core ( M ⊙ ),\n- (7) M ce -mass of the convective envelope ( M ⊙ ),\n- (8) ψ c -central electron chemical potential ( µ e , in units of k T ),\n- (9) log 10 ρ c -central density ( g cm -3 ),\n- (10) log 10 T c -central temperature (K),\n- (11) X c -central hydrogen mass fraction,\n- (12) Y c -central helium mass fraction, and\n- (13) X s -surface hydrogen mass fraction.  \nSecondly, we document the global physical properties of metal-poor stars in Table A2. As for Table A1, only model sequences of 20 M ⊙ donors are listed in this printed table. Again the full data are available online in machine-readable form. The variables are listed as follows:  \n- (1) n -mass-loss sequence number,\n- (2) M i -initial mass of the mass-transferring star ( M ⊙ ),\n- (3) log 10 R i -initial radius of the donor star ( R ⊙ ),\n- (4) log 10 T eff -effective temperature (K),\n- (5) log 10 L -stellar luminosity ( L ⊙ ),\n- (6) log 10 t KH -thermal, Kelvin-Helmholtz, timescale (yr),\n- (7) k 2 = I/ ( M i R 2 i )-dimensionless moment of inertia,\n- (8) L H -hydrogen-burning luminosity ( L ⊙ ),\n- (9) L He -helium-burning luminosity ( L ⊙ ),\n- (10) L Z -heavy-element (carbon-, oxygen- etc.) burning luminosity ( L ⊙ ),\n- (11) | L neu | -neutrino luminosity ( L ⊙ ), and\n- (12) L th -gravothermal luminosity ( L ⊙ ).  \nThirdly, we summarize the quantitative results of thresholds for both conservative and nonconservative dynamical-timescale mass transfer in Table A3. The superscripts in Table A3 indicate β ∈ { 1 , 0 . 5 , 0 } . During the mass transfer process, a fraction (1 -β ) of mass is lost from the system and takes away the specific angular momentum of the accretor. Columns (4) to (10) list the dynamical mass transfer thresholds for standard stellar models, and columns (11) to (17) are for models with artificially isentropic convective envelopes (a ∼ on a variable top). Variables are  \n- (1) n -mass-loss sequence number,\n- (2) M i -initial mass of the mass-transferring star ( M ⊙ ),\n- (3) log 10 R i -initial radius of the donor star ( R ⊙ ),\n- (4) ζ 1 ad -critical radius-mass exponent for conservative dynamical-timescale mass transfer,\n- (5) q 1 ad -critical initial mass ratio for conservative dynamical-timescale mass transfer,\n- (6) M 1 KH -mass threshold at which ˙ M = -M i /t KH and the tangent point for the minimum q 1 ad ,\n- (7) q 0 . 5 ad -critical initial mass ratio for semiconservative dynamical-timescale mass transfer,\n- (8) M 0 . 5 KH -mass threshold at which ˙ M = -M i /t KH and the tangent point for the minimum q 0 . 5 ad ,\n- (9) q 0 ad -critical initial mass ratio for fully nonconservative dynamical-timescale mass transfer,\n- (10) M 0 KH -mass threshold at which ˙ M = -M i /t KH and the tangent point for the minimum q 0 ad ,\n- (11) ˜ ζ 1 ad -critical radius-mass exponent for conservative dynamical-timescale mass transfer,\n- (12) ˜ q 1 ad -critical initial mass ratio for conservative dynamical-timescale mass transfer,\n- (13) ˜ M 1 KH -mass threshold at which ˙ M = -M i /t KH and the tangent point for the minimum q 1 ad ,\n- (14) ˜ q 0 . 5 ad -critical initial mass ratio for semiconservative dynamical-timescale mass transfer,\n- (15) ˜ M 0 . 5 KH -mass threshold at which ˙ M = -M i /t KH and the tangent point for the minimum q 0 . 5 ad ,\n- (16) ˜ q 0 ad -critical initial mass ratio for fully nonconservative dynamical-timescale mass transfer, and\n- (17) ˜ M 0 KH -mass threshold at which ˙ M = -M i /t KH and the tangent point for the minimum q 0 ad .  \nFinally, we complement the thresholds for nonconservative unstable mass transfer in Z = 0 . 02 stars in Ta-  \nTable A1.  \nGe et al. Interior physical properties of metal-poor donor stars  \nNote -This table is available entirely in machine-readable form.  \nNote -* The degenerate core mass only makes sense for low- and intermediate-mass stars but not for massive stars.  \nNote -** The age of the ZAMS star is not zero because it contracts slightly to reach a minimum radius after core hydrogen burning.  \nble A4. Variables are the same as in Table A3. The full model grids cover 1567 ( Z = 0 . 02) donors with masses from 0.1 to 100 M ⊙ . Ge et al. (2020a) documented these  \ndonor stars' corresponding initial physical parameters in their tables 1 and 2.", 'REFERENCES': "Amaro-Seoane, P., Audley, H., Babak, S., et al. 2017, arXiv e-prints, arXiv:1702.00786,  \ndoi: 10.48550/arXiv.1702.00786  \nAmaro-Seoane, P., Andrews, J., Arca Sedda, M., et al.  \n2023, Living Reviews in Relativity, 26, 2,  \ndoi: 10.1007/s41114-022-00041-y  \nAntokhin, I. I., Cherepashchuk, A. M., Antokhina, E. A., & Tatarnikov, A. 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