Dynamical Properties of Internal Shocks Revisited

Abstract: Internal shocks between propagating plasma shells, originally ejected at different times with different velocities are believed to play a major role in dissipating the kinetic energy, thereby explaining the observed lightcurve and spectra in a large range of transient objects. Even if initially the colliding plasmas are cold, following the first collision the plasma shells are substantially heated, implying that in a scenario of multiple collisions, most collisions take place between plasmas of non-zero temperatures. Here, we calculate the dynamical properties of plasmas resulting from collision between arbitrarily hot plasma shells, moving at arbitrary speeds. We provide simple analytical expressions valid for both the ultra-relativistic and Newtonian velocities, for both hot and cold plasmas. We derive the minimum criteria required for the formation of the two-shock wave system, and show that in the relativistic limit, the minimum Lorentz factor is proportional to the square root of the ratio of the initial plasmas enthalpies. We provide basic scaling laws of synchrotron emission from both the forward and reverse shock waves, and show how these can be used to deduce the properties of the colliding shells. Finally, we discuss the implications of these results in the study of several astronomical transients, such as x-ray binaries, radio load quasars and gamma-ray bursts.