Bayesian parameter constraints for neutron star masses and radii using X-ray timing observations of accretion-powered millisecond pulsars

We present a Bayesian method to constrain the masses and radii of neutron stars (NSs) using the information encoded in the X-ray pulse profiles of accreting millisecond pulsars. We model the shape of the pulses using "oblate Schwarzschild" approximation, which takes into account the deformed shape of the star together with the special and general relativistic corrections to the photon trajectories and angles. The spectrum of the radiation is obtained from an empirical model of Comptonization in a hot slab in which a fraction of seed black-body photons is scattered into a power-law component. By using an affine-invariant Markov chain Monte Carlo ensemble sampling method, we obtain posterior probability distributions for the different model parameters, especially for the mass and the radius. To test the robustness of our method, we first analysed self-generated synthetic data with known model parameters. Similar analysis was then applied for the observations of SAX J1808.4-3658 by the Rossi X-ray Timing Explorer (RXTE). The results show that our method can reproduce the model parameters of the synthetic data, and that accurate constraints for the radius can be obtained using the RXTE pulse profile observations if the mass is a priori known. For a mass in the range 1.5-1.8 M_⊙, the radius of the NS in SAX J1808.4-3658 is constrained between 9 and 13 km. If the mass is accurately known, the radius can be determined with an accuracy of 5% (68% credibility). For example, for the mass of 1.7 M_⊙ the equatorial radius is R_eq = 11.9^+0.5_-0.4 km. Finally, we show that further improvements can be obtained when the X-ray polarization data from the Imaging X-ray Polarimeter Explorer will become available.