We have calculated the evolution of 60 model binary systems consisting of helium stars in the mass range of M _ { He } = 2.5 - 6 \mbox { M$ { } _ { \odot } $ } with a 1.4 M _ { \odot } neutron star companion to investigate the formation of double neutron star systems .
Orbital periods ranging from 0.09 to 2 days are considered , corresponding to Roche lobe overflow starting from the helium main sequence to after the ignition of carbon burning in the core .
We have also examined the evolution into a common envelope phase via secular instability , delayed dynamical instability , and the consequence of matter filling the neutron star ’ s Roche lobe .
The survival of some close He-star neutron-star binaries through the last mass transfer episode ( either dynamically stable or unstable mass transfer phase ) leads to the formation of extremely short-period double neutron star systems ( with P \lesssim 0.1 days ) .
In addition , we find that systems throughout the entire calculated mass range can evolve into a common envelope phase , depending on the orbital period at the onset of mass transfer .
The critical orbital period below which common envelope evolution occurs generally increases with M _ { He } .
In addition , a common envelope phase may occur during a short time for systems characterized by orbital periods of 0.1 - 0.5 days at low He-star masses ( \sim 2.6 - 3.3 \mbox { M$ { } _ { \odot } $ } ) .

The existence of a short-period population of double neutron stars increases the predicted detection rate of inspiral events by ground-based gravitational-wave detectors and impacts their merger location in host galaxies and their possible role as \gamma -ray burst progenitors .
We use a set of population synthesis calculations and investigate the implications of the mass-transfer results for the orbital properties of DNS populations .