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We study the four double neutron star systems found in the Galactic disk interms of the orbital characteristics of their immediate progenitors and thenatal kicks imparted to neutron stars. Analysis of the effect of the secondsupernova explosion on the orbital dynamics, combined with recent results fromsimulations of rapid accretion onto neutron stars lead us to conclude that theobserved systems could not have been formed had the explosion been symmetric.Their formation becomes possible if kicks are imparted to the radio-pulsarcompanions at birth. We identify the constraints imposed on the immediateprogenitors of the observed double neutron stars and calculate the rangeswithin which their binary characteristics (orbital separations and masses ofthe exploding stars) are restricted. We also study the dependence of theselimits on the magnitude of the kick velocity and the time elapsed since thesecond explosion. For each of the double neutron stars, we derive a minimumkick magnitude required for their formation, and for the two systems in closeorbits we find it to exceed 200km/s. Lower limits are also set to thecenter-of-mass velocities of double neutron stars, and we find them to beconsistent with the current proper motion observations.Comment: 25 pages, 6 figs (9 parts), 4 tables, AASTeX, Accepted in Ap

Double Neutron Star Systems and Natal Neutron Star Kicks