Temporal evolution of the energy spectrum of ultrarelativistic electrons under the action of synchrotron losses and turbulent acceleration

A solution of the kinetic equation for the isotropic part of the distribution function of ultrarelativistic electrons is obtained, which describes the temporal evolution of the spectrum under the action of synchroton losses and turbulent acceleration. Initial distribution is assigned in the formation of a power-law spectrum, and the properties of the turbulence are assigned by the coefficient of turbulent acceleration, which is consistent with the coefficient of synchrotron losses to assure the isotropization of the distribution function without significant particle acceleration. The solution preserves the initial spectrum up to certain values of the time and energy connected with the inequality, and is transformed into a power-law spectrum with a different exponent. Possibilities of applying the solution to the description of processes in supernova shells and radio galaxies are discussed, and it is found that synchrotron losses also play an important role in hypothetical turbulent plasma reactions.