The Rossi X-Ray Timing Explorer observed the soft X-ray transient Aquila X-1 during its outburst in 1997 February and March. We report the discovery of quasi-periodic oscillations (QPOs) in its persistent flux with frequencies in the range of 740-830 Hz, a Q-value of over 100, a fractional rms amplitude of 6.8%±0.6%, and nearly coherent oscillations (NCOs) during a type I burst with a frequency of 549 Hz. The frequency of the QPOs in the persistent flux is correlated with the mass accretion rate on a timescale of hours, but not on a timescale of days. This is most likely the manifestation in a single source of the kilohertz QPO puzzle observed among many sources, i.e., on the one hand, individual sources show a correlation between the QPO frequency and the inferred mass accretion rate, and on the other hand, the dozen or so sources with luminosities spanning two decades have essentially the same QPO frequencies. We propose that this multivalued QPO frequency and mass accretion rate correlation indicates the existence of many similar regimes of the accretion disk. These regimes, with a very similar energy spectrum and QPO frequency, are distinguished from each other by the mass accretion rate or the total X-ray flux. The NCOs during the burst can be made almost perfectly coherent by taking into account a large ν̇. This strongly suggests that this frequency is related to the neutron star spin frequency. The large ν̇ is attributable to the expansion or contraction of the neutron star photosphere during the burst.
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