Shock dynamics in relativistic jets

We present a formalism of the dynamics of internal shocks in relativistic jets where the source has a time-dependent injection velocity and mass-loss rate. The variation of the injection velocity produces a two-shock wave structure, the working surface, that moves along the jet. This new formalism takes into account the fact that momentum conservation is not valid for relativistic flows where the relativistic mass lost by radiation must be taken into account, in contrast to the classic regime. We find analytic solutions for the working surface velocity and radiated energy for the particular case of a step function variability of the injection parameters. We model two cases: a pulse of fast material and a pulse of slow material (with respect to the mean flow). Applying these models to gamma-ray burst light curves, one can determine the ratio of the Lorentz factors γ₂/γ₁ and the ratio of the mass-loss rates dot{m}_2/dot{m}_1 of the upstream and downstream flows. As an example, we apply this model to the sources GRB 080413B and GRB 070318 and find the values of these ratios. Assuming a Lorentz factor γ₁ = 100, we further estimate jet mass-loss rates between dot{m}_1 ∼ 10^{-5} {and } 1 M_{odot } yr^{-1}. We also calculate the fraction of the injected mass lost by radiation. For GRB 070318 this fraction is ∼7 per cent. In contrast, for GRB 080413B this fraction is larger than 50 per cent; in this case radiation losses clearly affect the dynamics of the internal shocks.