Spin-up of low-luminosity, low-mass X-ray binaries

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* less than or similar to 10(8)((M) over dot/10(16) g s(-1)) G, where (M) over dot is the mass accretion rate. However, for (M) over dot < (M) over dot(c) similar to 10(16) 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 similar to 2-10. As a consequence, the spin up of LMXBs produces pulsars with spin periods longer by a factor of similar to 2-10 than those with a Keplerian accretion disk. The observed MSPs could be produced only for B* < 10(7) G even when (M) over dot similar to (M) over dot(c) similar to 10(16) g s(-1). This suggests that the low-luminosity LMXBs with (M) over dot < (M) over 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 less than or similar to 10(30) 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.