We obtained four Chandra/ACIS-S observations beginning two weeks after the end of the November 2000 outburst of the neutron star (NS) transient Aql X-1. Over the five month span in quiescence, the X-ray spectra are consistent with thermal emission from a NS with a pure hydrogen photosphere and R_{infty}=15.9+{0.8}-{2.9} (d/5 kpc) km at the optically implied X-ray column density. We also detect a hard power-law tail during two of the four observations. The intensity of Aql X-1 first decreased by 50+/-4% over three months, then increased by 35+/-5% in one month, and then remained constant (<6% change) over the last month. These variations in the first two observations cannot be explained by a change in the power-law spectral component, nor in the X-ray column density. Presuming that R_{infty} is not variable and a pure hydrogen atmosphere, the long-term changes can only be explained by variations in the NS effective temperature, from kT_{eff, infty}=130+3-5 eV, down to 113+3-4 eV, finally increasing to 118+9-4 eV for the final two observations. During one of these observations, we observe two phenomena which were previously suggested as indicators of quiescent accretion onto the NS: short-timescale (<1e4 sec) variability (at 32+8-6% rms), and a possible absorption feature near 0.5 keV. The possible absorption feature can potentially be explained as due to a time-variable response in the ACIS detector. Even so, such a feature has not been detected previously from a NS, and if confirmed and identified, can be exploited for simultaneous measurements of the photospheric redshift and NS radius.
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