Nonlinear cooling of relativistic particles under equipartition conditions

Aims. In powerful cosmic nonthermal radiation sources with dominant magnetic-field self generation, the plasma physical processes generating these magnetic fields by relativistic plasma instabilities are closely related to the processes energising ultra-high energy radiating electrons in these sources. Then the magnetic field strength becomes time-dependent and adjusts itself to the actual kinetic energy density of the radiating electrons. As a consequence, the synchrotron radiation cooling of individual relativistic electrons exhibits a nonlinear behaviour because of the dependence of the magnetic energy density on the actual time-varying kinetic energy density. Methods. The nonlinear kinetic equation for the intrinsic temperoral evolution of relativistic electrons is solved for the case of instantaneous injection of power-law distributed electrons. Results. The properties of the resulting approximate, nonlinear electron density show significant differences compared to the standard linear solution for constant non-equipartition magnetic-field energy density as, for instance, the different time behaviour of the upper and lower cut-offs of the electron distribution. Also the differential electron fluence as a function of electron energy differs from the linear fluence. For large spectral indices $s > 2$ of the injected power law, the nonlinear fluence exhibits a weaker break at the lower injected electron cut-off $\gamma_1$ than the linear fluence. For small spectral indices $1 < s < 2$, the nonlinear fluence shows no break at all and approaches a ${\propto}\gamma ^{-3}$ power law at all energies below $\gamma_2/2$. Conclusions. For electron radiation processes not subject to equipartition conditions, such as inverse Compton scattering of ambient photon gases and relativistic bremsstrahlung, the energy dependences of the electron number density and the electron fluence can be directly used to infer the frequency dependence of the fluxes and fluences of the generated photons. For steep (spectral index $s > 2$) injected power laws, the nonlinear synchrotron fluence at low frequencies approaches a power law ${\propto}\nu ^{-0.6}$, independent of the value of s, which is identical to the synchrotron fluence behaviour from monoenergetically injected relativistic electrons