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Constraining the magnetic field in GRB relativistic collisionless shocks using radio data

Using GRB radio afterglow observations, we calculate the fraction of shocked plasma energy in the magnetic field in relativistic collisionless shocks ($ε_B$). We obtained $ε_B$ for 38 bursts by assuming that the radio afterglow light curve originates in the external forward shock and that its peak at a few to tens of days is due to the passage of the minimum (injection) frequency through the radio band. This allows for the determination of the peak synchrotron flux of the external forward shock, $f_p$, which is $f_p \propto ε_B^{1/2}$. The obtained value of $ε_B$ is conservatively a minimum if the time of the "jet break" is unknown, since after the "jet break" $f_p$ is expected to decay with time faster than before it. Claims of "jet breaks" have been made for a subsample of 23 bursts, for which we can estimate a measurement of $ε_B$. Our results depend on the blast wave total energy, $E$, and the density of the circum-stellar medium (CSM), $n$, as $ε_B \propto E^{-2}n^{-1}$. However, by assuming a CSM magnetic field ($\sim 10$ $μ$G), we can express the lower limits/measurements on $ε_B$ as a density-independent ratio, $B/B_{sc}$, of the magnetic field behind the shock to the CSM shock-compressed magnetic field. We find that the distribution on both the lower limit on and the measurement of $B/B_{sc}$ spans $\sim 3.5$ orders of magnitude and both have a median of $B/B_{sc} \sim 30$. This suggests that some amplification, beyond simple shock-compression, is necessary to explain these radio afterglow observations.