Gamma-ray burst early afterglows: Reverse shock emission from an arbitrarily magnetized ejecta

Evidence suggests that gamma-ray burst (GRB) ejecta are likely magnetized, although the degree of magnetization is unknown. When such magnetized ejecta are decelerated by the ambient medium, the characteristics of the reverse shock emission are strongly influenced by the degree of magnetization. We derive a rigorous analytical solution for the relativistic 90 degrees shocks under the ideal MHD condition. The solution is reduced to the Blandford-McKee hydrodynamical solution when the magnetization parameter sigma approaches zero, and to the Kennel-Coroniti solution ( which depends on sigma only) when the shocks upstream and downstream are ultrarelativistic with respect to each other. Our generalized solution can be used to treat the more general cases, e. g., when the shocks upstream and downstream are mildly relativistic with respect to each other. We find that the suppression factor of the shock in the strong magnetic field regime is only mild as long as the shock upstream is relativistic with respect to the downstream, and it saturates in the high-degrees regime. This indicates that generally strong relativistic shocks still exist in the high-sigma limit. This can effectively convert kinetic energy into heat. The overall efficiency of converting ejecta energy into heat, however, decreases with increasing sigma, mainly because the fraction of the kinetic energy in the total energy decreases. We use the theory to study the reverse shock emission properties of arbitrarily magnetized ejecta in the GRB problem assuming a constant density of the circumburst medium. We study the shell-medium interaction in detail and categorize various critical radii for shell evolution. With typical GRB parameters, a reverse shock exists when sigma is less than a few tens or a few hundreds. The shell evolution can still be categorized into the thick and thin shell regimes, but the separation between the two regimes now depends on the sigma-parameter and the thick shell regime greatly shrinks at high sigma. The thin shell regime can also be categorized into two subregions depending on whether the shell starts to spread during the first shock crossing. The early optical afterglow light curves are calculated for GRBs with a wide range of sigma-value, with the main focus on the reverse shock component. We find that as sigma increases from below, the reverse shock emission level increases steadily until reaching a peak at sigma less than or similar to 1, then it decreases steadily when sigma > 1. At large sigma-values, the reverse shock peak is broadened in the thin shell regime because of the separation of the shock crossing radius and the deceleration radius. This novel feature can be regarded as a signature of high sigma. The early afterglow data of GRB 990123 and GRB 021211 could be understood within the theoretical framework developed in this paper, with the inferred sigma-value greater than or similar to 0.1. The case of GRB 021004 and GRB 030418 may be also interpreted with higher sigma-values, although more detailed modeling is needed. Early tight optical upper limits could be interpreted as very high sigma cases, in which a reverse shock does not exist or is very weak. Our model predictions could be further tested against future abundant early afterglow data collected by the Swift UV-optical telescope, so that the magnetic content of GRB fireballs can be diagnosed.