The Most Probable Cause for the High Gamma-Ray Polarization in GRB 021206

The exciting detection of a very high degree of linear polarization, $P=(80\pm 20)%$, in the prompt gamma-ray emission of GRB 021206, supports synchrotron emission as being the dominant radiation mechanism. There were also claims that this implies a magnetic field ordered on large scales within the ejecta, that must therefore be produced at the source, which in turn was used as an argument in favor magnetic fields playing an active role in the production of GRB jets. However, an alternative explanation was suggested which also works with a magnetic field that is generated in the internal shocks and does not originate at the source: a very narrow jet, of opening angle $\theta_j\sim 1/\gamma$, where $\gamma\gtrsim 100$ is the Lorentz factor during the GRB, viewed at $\theta_j<\theta_{obs}\lesssim\theta_j+1/\gamma$. We calculate $P$ for these two scenarios, and find that it is significantly easier to produce $P\gtrsim 50%$ with an ordered field. More specifically, we obtain $P\sim(43-61)%$ for an ordered transverse magnetic field, $B_{ord}$, whereas a random field within the plane of the shock, $B_\perp$, produces $P\lesssim(38-54)%$ for a single pulse in the GRB light curve, but the integrated emission over many pulses (as measured in GRB 021206) is a factor of $\sim 2$ lower. A magnetic field normal to the shock front, $B_\parallel$, produces $P\sim(35-62)%$ for the emission integrated over many pulses. However, polarization measurements from GRB afterglows suggest a more isotropic shock-produced field configuration that would reduce $P$ by a factor $\sim 2-3$. Therefore, an ordered magnetic field, $B_{ord}$, that originates at the source, can produce the observed polarization most naturally, while $B_\parallel$ is less likely, and $B_\perp$ is the least likely of the above.