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The formation and evolution of double neutron stars (DNS) have traditionallybeen studied using binary population synthesis. In this work, we take analternative approach by focusing only on the second supernova (SN) forming theDNS and the subsequent orbital decay and merger due to gravitational waveradiation. Using analytic and numerical methods, we explore how different NSnatal kick velocity distributions, pre-SN orbital separations, and progenitorHe-star masses affect the post-SN orbital periods, eccentricities, mergertimes, systemic velocities, and distances traveled by the system beforemerging. Comparison with the set of 17 known DNSs in the Milky Way shows thatDNSs have pre-SN orbital separations ranging between 1 and 44 $R_{\odot}$.Those DNSs with pre-SN separations $\sim$1 $R_{\odot}$ have merger timedistributions that peak $\sim$10-100 Myr after formation, regardless of thekick velocity received by the NS. These DNSs are typically formed with systemicvelocities $\sim$10$^2$ km s$^{-1}$ and may travel $\sim$1-10 kpc beforemerging. Depending on progenitor mass of the second-born NS, the short mergertime can account for the $r$-process enrichment observed in compact stellarsystems such as ultra-faint dwarf galaxies. For Milky Way-mass galaxies onlyDNSs with the tightest pre-SN orbits and large kick velocities ($\gtrsim$10$^2$km s$^{-1}$) can escape. However, those DNSs that do escape may travel as faras $\sim$Mpc before merging, which as previous studies have pointed out hasimplications for identifying the host galaxies to short gamma ray bursts andgravitational wave events.

Double Neutron Star Formation: Merger Times, Systemic Velocities, and Travel Distances