1998ApJ...505..828Y

We examine the spin-up of low-luminosity, low-mass X-ray binaries (LMXBs) to millisecond pulsars (MSPs). In the conventional spin-up model of the Ghosh & Lamb type, where the stellar magnetic field interacts with the Keplerian accretion disk, MSPs could be produced from LMXBs if the magnetic field B_*≲10⁸M{dot}/10¹⁶ g.s^–1) G, where M{dot} is the mass accretion rate. However, for M{dot}<M{dot}_c∼10¹⁶ g.s^–1, accretion is likely to occur via a quasi-spherical flow with a sub-Keplerian rotation. The sub-Keplerian rotation rate is smaller than the Keplerian rate by a factor ∼2-10. As a consequence, the spin up of LMXBs produces pulsars with spin periods longer by a factor of ∼2-10 than those with a Keplerian accretion disk. The observed MSPs could be produced only for B_* < 10⁷ G even when M{dot}∼M{dot}_c∼10¹⁶ g.s^–1. This suggests that the low-luminosity LMXBs with M{dot}<M{dot}_c would not be able to spin-up to the observed MSPs. This rules out any undetected populations of persistent, low-luminosity LMXBs and a potentially significant fraction of the atoll sources as a possible source population of the observed MSPs. If a large number of undetected, persistent, low-luminosity LMXBs do exist, they could produce MSPs near the pulsar death line with intrinsic electromagnetic luminosity ≲10³⁰ ergs.s^–1. The observed MSPs could possibly arise from a population of soft X-ray transients (SXTs) containing neutron stars, although this is not supported by current estimates of the outburst rate or numbers of such systems. Accretion-induced collapse of low magnetic field white dwarfs remains a possible channel for MSP formation.