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Supernova explosion in a binary is treated as an instantaneous mass loss of the exploding star. We assume that an additional kick velocity is imparted to young neutron stars due to possible asymmetry of the collapse (see Section 5 for further discussion). In this case, the eccentricity and semi-major axis of the binary after the explosion can be straightforwardly calculated (Boersma, 1961). We do this using the following scheme.
During the CE stage, an effective spiral-in of the binary components occurs. This complicated process (first introduced by Paczynsky, 1976) is not fully understood as yet, so we use the conventional energy consideration to find the binary system parameters after the CE by introducing a parameter measuring what fraction of the system's orbital energy (between the beginning and the end of the spiralling-in process) is transformed into the binding energy (gravitational minus thermal) of the ejected envelope. Thus
where is the mass of the core of the mass losing star of initial mass and radius (which is simply a function of the initial separation and the initial mass ratio ), and no substantial mass growth is assumed for the accretor (see, however, Chevalier, 1993). The lower is, the closer the binary becomes after the CE stage. We will show below that the evolutionary scenario allows values of over a wide range, -10.