A spectacular radio flare from XRF 050416a at 40 days and implications for the nature of X-ray flashes

We present detailed optical, near-infrared, and radio observations of the X-ray flash XRF 050416a obtained with Palomar and Siding Springs Observatories, as well as the HST and VLA, placing this event among the best-studied X-ray flashes to date. In addition, we present an optical spectrum from Keck LRIS from which we measure the redshift of the burst, z = 0.6528. At this redshift the isotropic-equivalent prompt energy release was about 10(51) ergs, and using a standard afterglow synchrotron model, we find that the blast wave kinetic energy is a factor of 10 larger, EK,(iso) approximate to 10(52) ergs. The lack of an observed jet break to t similar to 20 days indicates that the opening angle is theta(j) greater than or similar to 7 degrees and the total beaming-corrected relativistic energy is greater than or similar to 10(50) ergs. We further show that the burst produced a strong radio flare at t similar to 40 days accompanied by an observed flattening in the X-ray band, which we attribute to an abrupt circumburst density jump or an episode of energy injection (from either a refreshed shock or off-axis ejecta). Late-time observations with HST show evidence for an associated supernova with peak optical luminosity roughly comparable to that of SN 1998bw. Next, we show that the host galaxy of XRF 050416a is actively forming stars at a rate of at least 2 M-circle dot yr(-1) with a luminosity of L-B approximate to 0.5L* and metallicity of Z similar to 0.2-0.8 Z(circle dot). Finally, we discuss the nature of XRF 050416a in the context of short-hard GRBs and under the framework of off-axis and dirty fireball models for X-ray flashes.