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Electron-capture supernovae (EC-SNe) provide an alternative channel for producing neutron stars (NSs). They play an important role in the formation of double NS (DNS) systems and the chemical evolution of galaxies, and contribute to the NS mass distribution in observations. It is generally believed that EC-SNe originate from $e$-captures on $\rm^{24}Mg$ and $\rm^{20}Ne$ in the massive degenerate oxygen-neon (ONe) cores with masses close to the Chandrasekhar limit ($M_{\rm Ch}$). However, the origin of EC-SNe is still uncertain. In this paper, we systematically studied the EC-SNe in NS+He star systems by considering the explosive oxygen burning that may occur in the near-$M_{\rm Ch}$ ONe core. We provided the initial parameter spaces for producing EC-SNe in the initial orbital period $-$ initial He star mass (log$P_{\rm orb}^{\rm i}-M_{\rm He}^{\rm i}$) diagram, and found that both $M_{\rm He}^{\rm i}$ and minimum $P_{\rm orb}^{\rm i}$ for EC-SNe increase with metallicity. Then, by considering NS kicks added to the newborn NS, we investigated the properties of the formed DNS systems after the He star companions collapse into NSs, such as the orbital periods, eccentricities and spin periods of recycle pulsars ($P_{\rm spin}$), etc. The results show that most of the observed DNS systems can be produced by NS kicks of $\lesssim50\rm\,km\,s^{-1}$. In addition, we found that NSs could accrete more material if the residual H envelope on the He star companions is considered, which can form the mildly recycled pulsars ($P_{\rm spin}\sim20\,$ms) in DNS systems., Comment: 13 pages, 14 figures, 3 tables, accepted for publication in MNRAS, comments welcome!

Type Ia supernovae (SNe Ia) are successful cosmological distance indicators and important element factories in the chemical evolution of galaxies. They are generally thought to originate from thermonuclear explosions of carbon-oxygen white dwarfs in close binaries. However, the observed diversity among SNe Ia implies that they have different progenitor models. In this article, we performed the long-term evolution of NS+He star binaries with different initial He star masses ($M_{\rm He}^{\rm i}$) and orbital periods ($P_{\rm orb}^{\rm i}$) for the first time, in which the He star companions can explode as SNe Ia eventually. Our simulations indicate that after the He stars develop highly degenerate oxygen-neon (ONe) cores with masses near the Chandrasekhar limit, explosive oxygen burning can be triggered due to the convective Urca process. According to these calculations, we obtained an initial parameter space for the production of SNe Ia in the $\rm log\,$$P^{\rm i}_{\rm orb}-M^{\rm i}_{\rm He}$ plane. Meanwhile, we found that isolated mildly recycled pulsars can be formed after He stars explode as SNe Ia in NS+He star binaries, in which the isolated pulsars have minimum spin periods ($P_{\rm spin}^{\rm min}$) of $\sim 30-110\rm\,ms$ and final orbital velocities of $\sim \rm 60-360\,km\,s^{-1}$, corresponding to initial orbital periods of $0.07-10\rm\,d$. Our work suggests that the NS+He star channel may contribute to the formation of isolated mildly recycled pulsars with velocity $\rm \lesssim 360\,km\,s^{-1}$ in observations, and such isolated pulsars should locate in the region of pulsars with massive WD companions in the $P_{\rm spin}-\dot P_{\rm spin}$ diagram., Comment: 10 pages, 12 figures, 1 table, accepted for publication in MNRAS, comments welcome!