Linear Polarization and Proper Motion in The Afterglow of Beamed GRBs

We investigate the polarization and proper motion expected in a beamed Gamma-Ray Burst's ejecta. We find that even if the magnetic field has well defined orientation relative to the direction of motion of the shock, the polarization is not likely to exceed 20%. Taking into account the dynamics of beamed ejecta we find that the polarization rises and decays with peak around the jet break time (when the Lorentz factor of the flow is comparable to the initial opening angle of the jet). Interestingly, we find that when the offset of the observer from the center of the beam is large enough, the polarization as function of time has three peaks, and the polarization direction of the middle peak is rotated by 90 degrees relative to the two other peaks. We also show that some proper motion is expected, peaking around the jet break time. Detection of both proper motion and the direction of polarization can determine which component of the magnetic field is the dominant.