On the Efficiency of Internal Shocks in Gamma-Ray Bursts

The fraction of a fireball kinetic energy that is radiated by internal shocks is sensitive to the amplitude of initial fluctuations in the fireball. We give a simple analytical description for the dissipation of modest-amplitude fluctuations and confirm it with direct numerical simulations. At high amplitudes, the dissipation occurs in a nonlinear regime with efficiency approaching 100%. Most of the fireball energy can then be radiated away by the prompt gamma-ray burst, and only a fraction remains for the afterglow.