2003ApJ...597..998L

Observations by the RHESSI satellite of the large polarization of the prompt γ-ray emission from γ-ray burst (GRB) 021206 imply that the magnetic field coherence scale is larger than the size of the visible emitting region, ∼R/Γ, where R is the radius of the flow and Γ is the associated Lorentz factor. Such fields cannot be generated in a causally disconnected, hydrodynamically dominated outflow. Electromagnetic models of GRBs, in which large-scale, dynamically dominant, magnetic fields are present in the outflow from the very beginning, provide a natural explanation of this large reported linear polarization. We derive the Stokes parameters of the synchrotron emission of a relativistically moving plasma with a given magnetic field configuration and calculate the pulse-averaged polarization fraction of the emission from a relativistically expanding shell carrying a global toroidal magnetic field. For viewing angles larger than 1/Γ, the observed patch of the emitting shell has an almost homogeneous magnetic field, producing a large fractional polarization (56% for a power-law energy distribution of relativistic particles, dn/dε∝ε^–3). The maximum polarization is smaller than the theoretical upper limit for a stationary plasma in a uniform magnetic field because of relativistic kinematic effects.