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 050416a obtained with Palomar and Siding Springs Observatories as well as the Hubble Space Telescope and Very Large Array, placing this event among the best-studied X-ray flashes to date. In addition, we present an optical spectrum from the Low Resolution Spectrograph on Keck 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 erg, and using a standard afterglow synchrotron model we find that the blastwave kinetic energy is a factor of 10 larger, E(K,iso)~10^52 erg. The lack of an observed jet break to t~20 days indicates that the opening angle is larger than 7 degrees and the total beaming-corrected relativistic energy is greater than 10^50 erg. We further show that the burst produced a strong radio flare at t~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 (either from a refreshed shock or off-axis ejecta). Late-time observations with the Hubble Space Telescope 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 50416a is actively forming stars at a rate of at least 2 solar masses per year with a luminosity of 0.5 L* and metallicity of 0.2-0.8 Z_solar. Finally, we discuss the nature of XRF 050416a in the context of short-hard gamma-ray bursts and under the framework of off-axis and dirty fireball models for X-ray flashes.