We report on Hubble Space Telescope (HST) observations of the late-time afterglow and host galaxy of GRB 021004 (z = 2.33). Although this gamma-ray burst (GRB) is one of the best observed so far in terms of sampling in the time domain, multiwavelength coverage, and polarimetric observations, there is substantial disagreement between different interpretations of data sets on this burst in the literature. We have observed the field of GRB 021004 with the HST at multiple epochs from 3 days until almost 10 months after the burst. With the STIS PRISM and G430L spectroscopy, we cover the spectral region from about 2000 to 5700 ?, corresponding to 600-1700 ? in the rest frame. From the limit on the flux recovery blueward of the Lyman limit, we constrain the H I column density to be above 1 × 1018 cm-2 (5 σ). On the basis of ACS and NICMOS imaging, we find that the afterglow evolved achromatically within the errors (any variation must be less than 5%) during the period of the HST observations. The color changes observed by other authors during the first 4 days must be related to a stochastic phenomenon superimposed on an afterglow component with a constant spectral shape. This achromaticity implies that the cooling break has remained on the blue side of the optical part of the spectrum for at least 2 weeks after the explosion. The optical-to-X-ray slope βOX is consistent with being the same at 1.4 and 52.4 days after the burst. This indicates that the cooling frequency is constant and, hence, according to fireball models, that the circumburst medium has a constant density profile. The late-time slope of the light curve (α2, Fν ∝ t) is in the range α2 = 1.8-1.9 and is inconsistent with a single power law. This could be due to a late-time flattening caused by the transition to nonrelativistic expansion or could be due to excess emission (a "bump" in the light curve) about 7 days after the burst. The host galaxy is, like most previously studied GRB hosts, a (very) blue starburst galaxy with no evidence for dust and with strong Lyα emission. The star formation rate of the host is about 10 M☉ yr-1 on the basis of both the strength of the UV continuum and the Lyα luminosity. The spectral energy distribution of the host implies an age in the range 30-100 Myr for the dominant stellar population. The afterglow was located very close (~100 pc) to the center of the host, implying that the progenitor was possibly associated with a circumnuclear starburst.
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