2005ApJ...625..263G

The magnetic field structure in γ-ray burst (GRB) outflows is of great interest, as it can provide valuable clues that might help pin down the mechanism responsible for the acceleration and collimation of GRB jets. The most promising way of probing this magnetic field structure is through polarization measurements of the synchrotron emission from the GRB ejecta, which includes the prompt γ-ray emission and the emission from the reverse shock. Measuring polarization in γ-rays with current instruments is extremely difficult: so far there is only one claim of detection (a very high degree of linear polarization in GRB 021206), which, despite the favorable conditions, remains highly controversial and is probably not real. The emission from the reverse shock that propagates into the ejecta as it is decelerated by the ambient medium peaks in the optical on a timescale of tens of seconds (the so-called optical flash) and dominates the optical emission up to about 10 minutes after the GRB. Unfortunately, no polarization measurements of this optical emission have been made to date. However, after the reverse shock finishes crossing the shell of GRB ejecta, the shocked ejecta cools adiabatically and radiates at lower and lower frequencies. This emission peaks in the radio after about 1 day and is called the ``radio flare.'' We use VLA data of radio flares from GRBs to constrain the polarization of this emission. We find only upper limits for both linear and circular polarization. Our best limits are for GRB 991216, for which we find 3 σ upper limits on the linear and circular polarization of 7% and 9%, respectively. These limits provide interesting constraints on existing GRB models. Specifically, our results are hard to reconcile with a predominantly ordered toroidal magnetic field in the GRB outflow together with a ``structured'' jet, where the energy per solid angle drops as the inverse square of the angle from the jet axis, as is expected in models in which the outflow is Poynting flux dominated.