Orbital Contraction of Post-Common-Envelope Binaries with a Circumbinary Disk

Tight and compact binary systems, such as double neutron star binaries, are believed to undergo a common envelope evolution phase, resulting in strongly bound orbits. During this phase, the outer layers of the primary star are expelled, resulting in orbital shrinkage. However, a part of the expelled material may remain as a circumbinary disk, which can further influence subsequent orbital evolution. In this study, we investigated orbital evolution in the presence of a circumbinary disk within a simplified framework by assuming that orbital contraction and disk dissipation occur over the viscous timescale. The results showed that the orbit of the binary system after the common envelope evolution phase was further contracted by up to $\sim 17 \%$ due to the presence of the circumbinary disk, irrespective of the disk's mass and structure. This additional orbital contraction following the common envelope evolution phase may have significant implications for the formation rate of double neutron star binaries that merge within a cosmic timescale.