On the spin-up of neutron stars to millisecond pulsars in long-period binaries

We study the accretion efficiency of neutron stars in long-period binaries, i.e. with periods longer than about 200d, which accrete from a giant companion. Using alpha-disc models and taking into account the effect of irradiation of the accretion disc by the central accretion light source we derive explicit expressions for the duty cycle and the accretion efficiency in terms of the parameters of the binary system and the disc instability limit cycle. We show that the absence of millisecond pulsars in wide binaries with circular orbits and periods longer than about 200d can be understood as a consequence of the disc instability if the duration of the quiescent phase between two subsequent outbursts is at least a few decades.