  The formation and evolution of double neutron stars (DNS) have traditionally
been studied using binary population synthesis. In this work, we take an
alternative approach by focusing only on the second supernova (SN) forming the
DNS and the subsequent orbital decay and merger due to gravitational wave
radiation. Using analytic and numerical methods, we explore how different NS
natal kick velocity distributions, pre-SN orbital separations, and progenitor
He-star masses affect the post-SN orbital periods, eccentricities, merger
times, systemic velocities, and distances traveled by the system before
merging. Comparison with the set of 17 known DNSs in the Milky Way shows that
DNSs have pre-SN orbital separations ranging between 1 and 44 $R_{\odot}$.
Those DNSs with pre-SN separations $\sim$1 $R_{\odot}$ have merger time
distributions that peak $\sim$10-100 Myr after formation, regardless of the
kick velocity received by the NS. These DNSs are typically formed with systemic
velocities $\sim$10$^2$ km s$^{-1}$ and may travel $\sim$1-10 kpc before
merging. Depending on progenitor mass of the second-born NS, the short merger
time can account for the $r$-process enrichment observed in compact stellar
systems such as ultra-faint dwarf galaxies. For Milky Way-mass galaxies only
DNSs 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 far
as $\sim$Mpc before merging, which as previous studies have pointed out has
implications for identifying the host galaxies to short gamma ray bursts and
gravitational wave events.
