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Formation of black widows through ultra-compact X-ray binaries with He star companions

Black widows (BWs) are a type of eclipsing millisecond pulsars (MSPs) with companion masses $M_2$ $\lesssim 0.05\,\rm M_\odot$, which can be used to study the accretion history and the radiation of pulsars, as well as the origin of isolated MSPs. Recent observations indicate that there are two sub-types of BWs. One is the BWs with $M_2$ $\sim$ $0.01-0.05\,\rm M_\odot$, whereas another with $M_2$ $\lesssim 0.01\,\rm M_\odot$. However, the origin of the latter is still highly uncertain. In this paper, we investigated the formation of BWs with $M_2$ $\lesssim 0.01\,\rm M_\odot$ through ultra-compact X-ray binaries (UCXBs) with He star companions, in which a neutron star (NS) accretes material from a He star through Roche-lobe overflow. By considering different He star masses and evaporation efficiencies with the stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA), we evolved a series of NS+He star systems that can undergo UCXB stage. We found that this channel can explain the formation of BWs with $M_2$ $\lesssim 0.01\,\rm M_\odot$ within the Hubble time, especially three widely studied BWs, i.e. PSRs J1719-1438, J2322-2650 and J1311-3430. We also found that X-ray irradiation feedback does not affect the evolutionary tracks of evaporation process. Our simulations indicate that the UCXB channel with He star companions are the potential progenitors of isolated MSPs, and that the origin of BWs with $M_2$ $\lesssim 0.01\,\rm M_\odot$ is different with another sub-type of BWs. In addition, the present work suggests that the BWs with $M_2$ $\lesssim 0.01\,\rm M_\odot$ may not be produced by redback systems.