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 their magnetic field B < 10^{8}({\dot M}/10^{16}g/s)^{1/2}G, where {\dot M} is the mass accretion rate. However, for {\dot M} < {\dot M}_c ~ 10^{16}g/s 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 ~2-10 than those with a Keplerian accretion disk. The observed MSPs could be produced only for B < 10^7G even when {\dot M}\sim {\dot M}_c ~ 10^{16}g/s. This suggests that the low luminosity LMXBs with {\dot M} < {\dot M}_c would not be able to spin-up to the observed MSPs. This rules out any undetected populations of persistent, low luminosity LMXBs and potentially a 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 <~ 10^{30} erg/s. 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. 
Insu Yi (1) and Jonathan E. Grindlay (2) ((1) Institute for Advanced   Study, (2) Harvard-Smithsonian Center for Astrophysics)