We investigate an interesting new class of high-mass X-ray binaries (HMXBs) with long orbital periods (Porb> 30 d) and low eccentricities (e � 0.2). The orbital parameters suggest that the neutron stars in these systems did not receive a large impulse, or “kick, ” at the time of formation. After considering the statistical significance of these new binaries, we develop a self-consistent phenomenological picture wherein the neutron stars born in the observed wide HMXBs receive only a small kick ( � 50 kms −1), while neutron stars born in isolation, in the majority of low-mass X-ray binaries, or in many of the wellknown HMXBs with Porb � 30 d receive the conventional large kicks, with a mean speed of ∼ 300 kms −1. Assuming that this basic scenario is correct, we discuss a physical process that lends support to our hypothesis, whereby the magnitude of the natal kick to a neutron star born in a binary system depends on the rotation rate of its immediate progenitor following mass transfer — the core of the initially more massive star in the binary. Specifically, the model predicts that rapidly rotating pre-collapse cores produce NSs with relatively small kicks, and vice versa for slowly rotating cores. If the envelope of the NS progenitor is removed before it has become deeply convective, then the exposed core is likely to be a rapid rotator. However, if the progenitor becomes highly evolved prior to mass transfer, then a stron
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