Observational appearance of rapidly rotating neutron stars: X-ray bursts, cooling tail method, and radius determination

Neutron stars (NSs) in low-mass X-ray binaries rotate at frequencies high enough to significantly deviate from sphericity ($\nu_* \sim$ 200--600 Hz). We investigate the effects of rapid rotation on the observational appearance of a NS. We propose analytical formulae relating gravitational mass and equatorial radius of the rapidly rotating NS to the mass $M$ and radius $R$ of a non-rotating NS of the same baryonic mass using accurate fully relativistic computations. We compute spectra from an oblate rotating NS observed at different inclination angles using the modified oblate Schwarzschild (MOS) approximation, where light bending is computed in Schwarzschild metric, but frame dragging and quadrupole moment of a NS are approximately accounted for in the photon redshift calculations. We generalize the cooling tail method to the case of a rapidly rotating NS to obtain the most probable values of $M$ and $R$ of the corresponding non-rotating NS with the same baryonic mass. We approximate the local spectra from the NS surface by a diluted blackbody using previously computed NS atmosphere models. We show that the NS radius could be overestimated by 3--3.5 km for face-on stars of $R\approx 11$ km rotating at $\nu_* =$ 700 Hz if the version of the cooling tail method for a non-rotating NS is used. We apply the method to an X-ray burst observed from the NS rotating at $\nu_* \approx$ 532 Hz in SAX J1810.8$-$2609. The resulting radius of the non-rotating NS (assuming $M=1.5 M_\odot$) becomes $11.8\pm0.5$ km if it is viewed at inclination i=60 deg and $R=11.2\pm0.5$ km for a face-on view, which are smaller by 0.6 and 1.2 km than the radius obtained using standard cooling tail method ignoring rotation. The corresponding equatorial radii of these rapidly rotating NSs are 12.3$\pm 0.6$ km (for i=60 deg) and 11.6$\pm 0.6$\,km (for i=0 deg).