Orbital Characteristics of Binary Systems after Asymmetric Supernova Explosions

We present an analytical method for studying the changes of the orbital characteristics of binary systems with circular orbits due to a kick velocity imparted to the newborn neutron star during a supernova (SN) explosion. Assuming a Maxwellian distribution of kick velocities we derive analytical expressions for the distribution functions of orbital separations and eccentricities immediately after the explosion, of orbital separations after circularization of the post-SN orbits, and of systemic velocities of binaries that remain bound after the explosion. These distributions of binary characteristics can be used to perform analytical population synthesis calculations of various types of binaries, the formation of which involves a supernova explosion. We study in detail the dependence of the derived distributions on the kick velocity and the pre-SN characteristics, we identify all the limits imposed on the post-SN orbital characteristics, and we discuss their implications for the population of X-ray binaries and double neutron star systems. We show that large kick velocities do not necessarily result in large systemic velocities; for typical X-ray binary progenitors the maximum post-SN systemic velocity is comparable to the relative orbital velocity prior to the explosion. We also find that, unless accretion-induced collapse is a viable formation channel, X-ray binaries in globular clusters have most probably been formed by stellar dynamical interactions only and not directly from primordial binaries.