Relativistic Jet Dynamics and Calorimetry of Gamma-Ray Bursts
  We present numerical solutions of the 2D relativistic hydrodynamics equations describing the deceleration and expansion of highly relativistic conical jets, of opening angles 0.05<theta<0.2, propagating into a medium of uniform density. Jet evolution is followed from a collimated relativistic outflow through to the quasi-spherical non-relativistic phase. We show that relativistic sideways expansion becomes significant beyond the radius R at which the expansion Lorentz factor drops to 1/theta. This is consistent with simple analytic estimates, which predict faster sideways expansion than has been claimed based on earlier numerical modeling. For t>R/c the emission of radiation from the jet blast wave is similar to that of a spherical blast wave carrying the same energy. Thus, the total (calorimetric) energy of GRB blast waves may be estimated with only a small fractional error based on t>R/c observations. 
N. Wygoda, E. Waxman, D. Frail
astro-ph.HE astro-ph.CO