Theoretical Considerations on the Properties of Accreting Millisecond Pulsars

We examine a number of evolutionary scenarios for the recently discovered class of accretion-powered millisecond X-ray pulsars in ultracompact binaries, including XTE J0929-314 and XTE J1751-305. These systems have very short orbital periods of Porb = 43.6 and 42.4 minutes, respectively, and extremely small mass functions. We focus on a particular scenario that can naturally explain the present-day properties of these systems. This model invokes a donor star that was either very close to the main-sequence turnoff at the onset of mass transfer or had sufficient time to evolve during the mass-transfer phase. We have run a systematic set of binary evolution calculations with a wide range of initial conditions. We find that these two ultracompact binaries can best be fitted by models wherein the donors start to lose mass at orbital periods of ~15 hr. The orbital periods then decrease to a minimum value of ≲40 minutes and finally evolve back up to about 43 minutes. We present the results of our binary evolution calculations for these systems, including interior profiles for the donor stars. We find that the initial properties of the donor star and the exact mode of orbital angular momentum losses do not have to be individually fine-tuned in order to reproduce the observed properties. We also carry out an analysis based on the measured mass functions of XTE J0929-314 and XTE J1751-305 to establish formal probability distributions for the current donor masses, chemical compositions, and thermal bloating factors of these systems. These distributions are evaluated in the context of our binary evolution models. We conclude that the donor masses are likely to be in the range ~0.012-0.025 M☉ and have radii of ~0.042-0.055 R☉. These radii are factors of ~1.1-1.3 times larger than the corresponding radii of zero-temperature stars of the same mass and chemical composition. According to the evolutionary scenario proposed in this paper, the interiors of the donors are largely composed of He, and the surface H abundances are almost certainly less than 10% (by mass). The orbital period derivative of these systems is very likely to be positive, i.e., orb/Porb > 0, with typical values in the range ~3 × 10-10 to 2 × 10-8 yr-1. Long-term average values of the mass-transfer rate could be as high as ~10-11 to 3 × 10-10 M☉ yr-1 but is more likely to be ~10-11 M☉ yr-1. Our study supports the hypothesis that X-ray irradiation has had a minimal effect on enhancing the radius of the low-mass donor. We also show how, in the context of this same basic evolutionary scenario, we can model the properties of the SAX 1808.4-3658 binary millisecond X-ray pulsar (Porb = 2 hr). Finally, we point out that if the proposed scenario to explain the ultracompact systems is correct, then these binaries truly link (as evolutionary cousins) systems that (1) evolve to become wide binary millisecond pulsars containing low-mass He dwarfs and (2) those ordinary low-mass X-ray binaries in which the H-rich donors are slowly reduced to planetary masses.