Previously, observations with the Rossi X-ray Timing Explorer showed that millisecond oscillations occur preferentially in thermonuclear X-ray bursts with photospheric radius expansion from sources rotating near 600 Hz, while they occur with equal likelihood in X-ray bursts with and without radius expansion for sources rotating near 300 Hz. With a larger sample of data than in previous studies, we find that the detectability of the oscillations is not directly determined by the properties of the X-ray bursts. Instead, we find that (1) the oscillations are observed almost exclusively when the accretion rate onto the neutron star is high, but that (2) radius expansion is only observed at high accretion rates from the 600 Hz sources, whereas it occurs only at low accretion rates in the 300 Hz sources. The persistent millisecond pulsars provide the only apparent exceptions to these trends. The first result might be explained if the oscillation amplitudes are attenuated at low accretion rates by an extended electron corona. The second result indicates that the rotation period of the neutron star determines how the burst properties vary with accretion rate, possibly through the differences in the effective surface gravity or the strength of the Coriolis force.
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