Formation of the Double Neutron Star System PSR J1930$-$1852

The spin period (185 ms) and period derivative ($1.8\times10^{-17}\,\rm s\,s^{-1}$) of the double neutron star (DNS) system PSR J1930$-$1852 recently discovered indicate that the pulsar was mildly recycled through the process of Roche-lobe overflow. This system has the longest orbital period (45 days) of the known DNS systems, and can be formed from a helium star-NS binary if the initial mass of the helium star was $ \lesssim 4.0M_{\odot} $; otherwise the helium star would never fill its Roche-lobe \citep{t15}. At the moment of the supernova explosion, the mass of the helium star was $ \lesssim3.0M_{\odot} $. We find that the probability distribution of the velocity kick imparted to the new-born neutron star has a maximum at about $30 \,\rm km\,s^{-1}$ (and a tail up to $ 260 \,\rm km\,s^{-1}$), indicating that this NS most probably received a low kick velocity at birth.