Relativistic jet dynamics and calorimetry of gamma-ray bursts

We present numerical solutions of the two-dimensional relativistic hydrodynamics equations describing the deceleration and expansion of highly relativistic conical jets, of opening angles 0.05 ≤ θ₀ ≤ 0.2, propagating into a medium of uniform density. Jet evolution is followed from a collimated relativistic outflow to the quasi-spherical non-relativistic phase. We show that relativistic sideways expansion becomes significant beyond the radius r _θ at which the expansion Lorentz factor drops to θ^-1 ₀. This is consistent with simple analytic estimates, which predict faster sideways expansion than has been claimed based on earlier numerical modeling. For t > t_s = r _θ/c the emission of radiation from the jet blast wave is similar to that of a spherical blast wave carrying the same energy (significant deviations at t t_s occur only for well off-axis observers, θ_obs 1 ≫ θ₀). Thus, the total (calorimetric) energy of gamma-ray burst blast waves may be estimated with only a small fractional error based on t > t_s observations.