Eclipsing binary millisecond pulsars (MSPs; the so-called black widows and redbacks) can provide important information about accretion history, pulsar irradiation of their companion stars, and the evolutionary link between accreting X-ray pulsars and isolated MSPs. However, the formation of such systems is not well understood, nor the difference in progenitor evolution between the two populations of black widows and redbacks. Whereas both populations have orbital periods between 0.1 and 1.0 days, their companion masses differ by an order of magnitude. In this paper, we investigate the formation of these systems via the evolution of converging low-mass X-ray binaries by employing the MESA stellar evolution code. Our results confirm that one can explain the formation of most of these eclipsing binary MSPs using this scenario. More notably, we find that the determining factor for producing either black widows or redbacks is the efficiency of the irradiation process, such that the redbacks absorb a larger fraction of the emitted spin-down energy of the radio pulsar (resulting in more efficient mass loss via evaporation) compared to that of the black widow systems. We argue that geometric effects (beaming) are responsible for the strong bimodality of these two populations. Finally, we conclude that redback systems do not evolve into black widow systems with time.
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