Electron acceleration and synchrotron radiation in decelerating plasmoids

An equation is derived to calculate the dynamics of relativistic magnetized plasma which decelerates by sweeping up matter from the ISM. Reduction to the non-radiative and radiative regimes is demonstrated for the general case of a collimated plasmoid whose area increases as a power of distance x from the explosion, and in the specific case of a blast-wave geometry where the area increases quadratically with x. An equation for the evolution of the electron momentum distribution function in the comoving fluid frame is used to calculate the observed synchrotron radiation spectrum. The central uncertainty involves the mechanism for transfering energy from the non-thermal protons to the electrons. The simplest prescription is to assume that a fixed fraction of the comoving proton power is instantaneously transformed into a power-law, "shock"-like electron distribution function. This permits an analytic solution for the case of a relativistic plasmoid with a constant internal magnetic field. Effects of parameter changes are presented. Breaks in the temporal behavior of the primary burst emission and long wavelength afterglows occur on two time scales for external ISM density distributions n(ext) proportional to x(-eta). The first, as emphasized by Meszaros & Rees, represents the time when the outflow sweeps up approximate to M(th)/Gamma(0) of material from the ISM, where M(th) is the mass in the outflow ejecta and Gamma(0) is the initial Lorentz factor of the plasmoid. The second represents the time when synchrotron cooling begins strongly to regulate the number of nonthermal electrons that are producing radiation which is observed at a given energy. Results are applied to GRB and blazar variability. The slopes of the time profiles of the X-ray and optical light curves of the afterglows of GRB 970228 and GRB 970508 are explained by injection of hard electron spectra with indices s<2 in a deceleration regime near the radiative limit. We also propose that blazar flares are due to the transformation of the directed kinetic energy of the plasmoid when interacting with the external medium, as would occur if relativistic outflows pass through clouds orbiting the central supermassive black hole. Published by Elsevier Science B.V.