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Based on astrometric measurements and spectral analysis from $Gaia$ DR3, two quiescent black hole (BH) binaries, $Gaia$ BH1 and BH2, have been identified. Their origins remain controversial, particularly for $Gaia$ BH1. By considering a rapidly rotating ($\omega/\omega_{\rm crit} = 0.8$) and strongly magnetized ($B_{\rm 0} = 5000$ G) merger product, we find that, at typical Galactic metallicity, the merger product can undergo efficient chemically homogeneous evolution (CHE). This results in the merger product having a significantly smaller radius during its evolution compared to that of a normally evolving massive star. Under the condition that the initial triple stability is satisfied, we use the Multiple Stellar Evolution (MSE) code and the MESA code to identify an initial hierarchical triple that can evolve into $Gaia$ BH1. It initially consists of three stars with masses of 9.03 $M_{\odot}$, 3.12 $M_{\odot}$, and 1 $M_{\odot}$, with inner and outer orbital periods of 2.21 days and 121.92 days, and inner and outer eccentricities of 0.41 and 0.45, respectively. This triple initially experiences triple evolution dynamics instability (TEDI) followed by Roche lobe overflow (RLOF). During RLOF, the inner orbit shrinks, and tidal effects gradually suppress the TEDI. Eventually, the inner binary undergoes a merger through contact (or collision). Finally, using models of rapidly rotating and strongly magnetic stars, along with standard core-collapse supernova (SN) or failed supernova (FSN) models, we find that a PMB consisting of an 12.11 $M_{\odot}$ merger product and a 1 $M_{\odot}$ companion star (originally an outer tertiary) can avoid RLOF. After a SN or FSN with a low ejected mass of $\sim$0.22 $M_{\odot}$ and a low kick velocity ($46^{+25}_{-33}$ ${\rm km/s}$ or $9^{+16}_{-8}$ ${\rm km/s}$), the PMB can form $Gaia$ BH1 in the Galactic disk., Comment: 11 pages, 3 figures, Accepted to APJL

Wolf-Rayet stars (WRs) are very important massive stars. However, their origin and the observed binary fraction within the entire WR population are still debated. We investigate some possible merger channels for the formation of WRs, including main sequence (MS)/ Hertzsprung Gap (HG) + MS, He + HG/ Giant Branch (GB). We find that many products produced via binary merger can evolve into WRs, the MS/ HG + MS merger channel can explain WRs with luminosities higher than $\sim 10^{5.4}$\,L$_{\odot}$, while the He + HG/ GB merger channel can explain low-luminosity WRs in the range of $10^{4.7}$\,L$_{\odot}$\,$\sim$\,$10^{5.5}$\,L$_{\odot}$. In the population synthesis analysis of WRs, we assume an initial binary fraction ($f_{\rm ini,bin}$) of 50\% and 100\% for massive stars. We also assume that MS/ HG + MS merger products are non-rotating or rapidly rotating ($\omega/\omega_{\rm crit}=0.8$). In different cases, the calculated single fractions of WRs range from $22.2\%$ to $60.6\%$ in the Milky Way (MW) and from $8.3\%$ to $70.9\%$ in the Large Magellanic Cloud (LMC). The current observations fall within the range of our calculations. When the merger product of MS/HG+MS rotates rapidly, we estimate that there are approximately 1015 to 1396 WRs in the MW and 128 to 204 WRs in the LMC. Our model also roughly reproduces the observed single-peak luminosity distribution of WRs in the MW. However, the weak bimodal luminosity distribution observed in the LMC is not reproduced in our model. We assess that this may be due to the model underestimating the mass-loss rate in the LMC. In conclusion, we consider that the binary merger is significant formation channel for WR formation, and can explain the observed high fraction of the single WRs in the total population., Comment: 14 pages, 8 figures , Accepted to APJ

Observations have shown that the effective temperature of hydrogen-free Wolf-Rayet (WR) stars is considerably lower than that of the standard model, which means that the radius of the observed H-free WR stars is several times larger than that estimated by the standard model. The envelope inflation structure (EIS) caused by the radiation luminosity being close to the Eddington luminosity in the iron opacity peak region of H-free WR stars may be the key to resolve the radius problem of H-free WR stars. We try to explain the H-free WR stars observed in the Milk Way (MW) and the Large Magellanic Cloud (LMC) by the He stars. Using the Modules for Experiments in Stellar Astrophysics code, we compute the evolution of He stars with and without MLT++ prescriptions and discuss their effects on the EIS. We have calculated the evolution of He stars using a new mass-loss rate formula and three different relative rotational velocity and compared our results with observations on Hertzsprung-Russell diagrams. The low luminosity (log$(L/L_{{\odot}})\leq5.2$) H-free WR stars in the MW and the LMC can be explained by the helium giant phase in low-mass He stars, the high $X_{C}$ and $X_{O}$ in WC stars can only evolve through low-mass He stars with a rapid rotation. High-mass He stars with the EIS can explain H-free WR stars with a luminosity exceeding $10^{5.7} L_{{\odot}}$ and an effective temperature above $10^{4.7}$ K in the MW. They can also explain H-free WR stars on the right-hand side of the He zero-age main sequence in the LMC. High-mass stars with the EIS evolve into WO stars at the final evolution stage, and the shorter lifetime fraction is consistent with the small number of observed WO stars., Comment: 9 pages, 7 figures 1 tables, Accepted to A&A

The binary systems consisting of a Be star and a white dwarf (BeWDs) are very interesting.They can originate from the binaries composed of a Be star and a subdwarf O or B star (BesdOBs), and they can merge into red giants via luminous red nova or can evolve into double WD potentially detected by $LISA$ mission. Using the method of population synthesis, we investigate the formation and the destiny of BeWDs,and discuss the effects of the metallicity ($Z$) and the common envelope evolution parameters. We find that BesdOBs are significant progenitors of BeWDs. About 30\% ($Z=0.0001$)-50\% ($Z=0.02$) of BeWDs come from BesdOBs. About 60\% ($Z=0.0001$) -70\% ($Z=0.02$) of BeWDs turn into red giants via a merger between a WD and a non-degenerated star. About 30\% ($Z=0.0001$) -40\% ($Z=0.02$) of BeWDs evolve into double WDs which are potential gravitational waves of $LISA$ mission at a frequency band between about $3\times10^{-3}$ and $3\times10^{-2}$ Hz. The common envelope evolution parameter introduces an uncertainty with a factor of about 1.3 on BeWD populations in our simulations., Comment: 17 pages, 12 figures, 2 table, accepted for publication in RAA

Context. About 0.2-2% of giant stars are Li-rich, whose lithium abundance (A(Li)) is higher than 1.5 dex. Among them, near 6% are super Li-rich with A(Li) exceeding 3.2 dex. Meanwhile, the formation mechanism of these Li-rich and super Li-rich giants is still under debate. Aims. Considering the compact He core of red giants, attention is paid to the effect of element diffusion on A(Li). In particular, when the He core flash occurs, the element diffusion makes the thermohaline mixing zone extend inward and connect to the inner convection region of stars. Then, a large amount of 7Be produced by the He flash can be transferred to stellar surface, finally turning into 7Li. Thus, the goal of this work is to propose the mechanism of A(Li) enrichment and achieve the consistency between the theoretical and observation data. Methods. Using the Modules for Experiments in Stellar Astrophysics (MESA), we simulate the evolution of low-mass stars, with considering the effects of element diffusion on the Li abundances. The timescale ratio of Li-rich giants to normal giants is estimated by population synthesis method. Then we get the theoretical value of A(Li) and make a comparison with observations. Results. Considering the influence of element diffusion in the model results in the increase of lithium abundance up to about 1.8dex, which can reveal Li-rich giants. Simultaneously, introducing high constant diffusive mixing coefficients (Dmix) with the values from 10e11 to 10e15in the model allows A(Li) to increase from 2.4 to 4.5dex, which can explain the most of Li-rich and super Li-rich giant stars. The population synthesis method reveals that the amount of Li-rich giants among giants is about 0.2-2%, which is consistent with observation estimated levels.