Polarization with a 3-dimensional Mixed Magnetic Field and Its Application to GRB 170817A

A large-scale ordered magnetic field plays a very important role in the formation and acceleration of a gamma-ray burst (GRB) jet. During the GRB prompt phase, some dissipation processes may happen and disturb the magnetic field, making the field become random to some extent. Therefore, a mixed magnetic field consisting of an ordered component and a random component is plausible for the early afterglow phase. Here we investigate the polarization evolution and light curve of an afterglow under such a 3-dimensional mixed magnetic field. Three kinds of ordered component (i.e., aligned, toroidal and radial) are discussed. We find that the three cases are distinguishable through jet polarization evolution. The polarization angle for a 3D mixed magnetic field with an aligned ordered component can evolve gradually but only changes abruptly by $90^\circ$ in the toroidal and radial cases. Furthermore, during the reverse shock crossing time, the polarization degree (PD) can be non-zero for the toroidal case but roughly zero for the radial case. Since an aligned component in a jet corresponds to a magnetar central engine and a toroidal component corresponds to a black hole, GRB central engines are distinguishable through polarization observations even if the magnetic field is mixed in a jet. In addition, our polarization calculation can be applied to GRB 170817A associated with GW170817. Using the recently-observed PD upper limit $12\%$ of GRB 170817A at $t=244$ days, the magnetic field strength ratio of the ordered to random components in this afterglow is constrained to be $\lesssim0.9$.