Hydrostatic expansion and spin changes during type I X-ray bursts

We present calculations of the spin-down of a neutron star atmosphere due to hydrostatic expansion during a type I X-ray burst. We show that Cumming & Bildsten incorrectly calculated the change in the moment of inertia of the atmosphere during a type I burst, resulting in a factor of 2 overestimation of the magnitude of the spin-down for rigidly rotating atmospheres. We derive the angular momentum conservation law in general relativity, both analytically for the case of slow rotation and numerically for rapidly rotating stars. We show that general relativity has a small effect on the angular momentum conservation law, at the level of 5%-10%. We show how to rescale our fiducial results to different neutron star masses, rotation rates, and equations of state and present some detailed rotational profiles. Comparing our results with recent observations of large frequency shifts in MXB 1658-298 and 4U 1916-053, we find that the spin-down expected if the atmosphere rotates rigidly is a factor of 2-3 less than the observed values. If differential rotation is allowed to persist, we find that the upper layers of the atmosphere do spin down by an amount comparable to or greater than the observed values. However, there is no compelling reason to expect the observed spin frequency to be that of only the outermost layers of the atmosphere. We conclude that hydrostatic expansion and angular momentum conservation alone cannot account for the largest frequency shifts observed during type I bursts.