Ambient magnetic field amplification in shock fronts of relativistic jets: an application to GRB afterglows

Abstract

Strong downstream magnetic fields of order of $\sim 1$G, with large correlation lengths, are believed to cause the large synchrotron emission at the afterglow phase of gamma ray bursts (GRBs). Despite of the recent theoretical efforts, models have failed to fully explain the amplification of the magnetic field, particularly in a matter dominated scenario. We revisit the problem by considering the synchrotron emission to occur at the expanding shock front of a weakly magnetized relativistic jet over a magnetized surrounding medium. Analytical estimates and a number of high resolution 2D relativistic magneto-hydrodynamical (RMHD) simulations are provided. Jet opening angles of $\theta = 0^{\circ} - 20^{\circ}$, and ambient to jet density ratios of $10^{-4} - 10^2$ were considered. We found that most of the amplification is due to compression of the ambient magnetic field at the contact discontinuity between the reverse and forward shocks at the jet head, with substantial pile-up of the magnetic field lines as the jet propagates sweeping the ambient field lines. The pile-up is maximum for $\theta \rightarrow 0$, decreasing with $\theta$, but larger than in the spherical blast problem. Values obtained for certain models are able to explain the observed intensities. The maximum correlation lengths found for such strong fields is of $l_{\rm corr} \leq 10^{14}$ cm, $2 - 6$ orders of magnitude larger than the found in previous works.