VLBA Astrometry of the Galactic Double Neutron Stars PSR J0509+3801 and PSR J1930-1852: A Preliminary Transverse Velocity Distribution of Double Neutron Stars and Its Implications

The mergers of double neutron stars (DNSs) systems are believed to drive the majority of short $\gamma$-ray bursts (SGRBs), while also serving as production sites of heavy r-process elements. Despite being key to i) confirming the nature of the extragalactic SGRBs, ii) addressing the poorly-understood r-process enrichment in the ultra-faint dwarf galaxies (UFDGs), and iii) probing the formation process of DNS systems, the space velocity distribution of DNSs is still poorly constrained due to the small number of DNSs with well-determined astrometry. In this work, we determine new proper motions and parallaxes of two Galactic DNSs -- PSR J0509+3801 and PSR J1930-1852, using the Very Long Baseline Array, and estimate the transverse velocities $v_\perp$ of all the 11 isolated Galactic DNSs having proper motion measurements in a consistent manner. Our correlation analysis reveals that the DNS $v_\perp$ is tentatively correlated with three parameters: spin period, orbital eccentricity, and companion mass. With the preliminary $v_\perp$ distribution, we obtain the following findings. Firstly, the refined $v_\perp$ distribution is confirmed to agree with the observed displacements of the localized SGRBs from their host galaxy birth sites. Secondly, we estimate that around 11% and 25% of DNSs remain gravitationally bound to UFDGs with escape velocities of 15$\mathrm{~km~s^{-1}}$ and 25$\mathrm{~km~s^{-1}}$, respectively. Hence, the retained DNSs might indeed be responsible for the r-process enrichment confirmed so far in a few UFDGs. Finally, we discuss how a future ensemble of astrometrically determined DNSs may probe the multimodality of the $v_\perp$ distribution.