The evolution of a structured relativistic jet and gamma-ray burst afterglow light curves

We carry out a numerical hydrodynamical modeling for the evolution of a relativistic collimated outflow as it interacts with the surrounding medium and calculate the light curve resulting from synchrotron emission of the shocked fluid. The hydrodynamic equations are reduced to one-dimensional by assuming axial symmetry and integrating over the radial profile of the flow, thus considerably reducing the computation time. We present results for a number of different initial jet structures, including several different power laws and a Gaussian profile for the dependence of the energy per unit solid angle, epsilon, and the Lorentz factor, Gamma, on the angle from the jet symmetry axis. Our choice of parameters for the various calculations is motivated by the current knowledge of relativistic outflows from gamma-ray bursts and the observed afterglow light curves. Comparison of the light curves for different jet profiles with gamma-ray burst afterglow observations provides constraints on the jet structure. One of the main results we find is that the transverse fluid velocity in the comoving frame (v(t)) and the speed of sideways expansion for smooth jet profiles is typically much smaller than the speed of sound (c(s)) throughout much of the evolution of the jet; v(t) approaches c(s) when Gamma along the jet axis becomes of order a few (for a large angular gradient of epsilon, v(t) similar to c(s) while Gamma is still large). This result suggests that the dynamics of relativistic structured jets may be reasonably described by a simple analytic model in which epsilon is independent of time, as long as Gamma along the jet axis is larger than a few.