Blazar synchrotron emission of instantaneously power-law injected electrons under linear synchrotron, non-linear SSC, and combined synchrotron-SSC cooling

Context. The broadband spectral energy distributions (SED) of blazars show two distinct components which in leptonic models are associated with synchrotron and synchrotron self-Compton (SSC) emission of highly relativistic electrons. In some sources the SSC component dominates the synchrotron peak by one or more orders of magnitude implying that the electrons mainly cool by inverse Compton collisions with their self-made synchrotron photons. Therefore, the linear synchrotron loss of electrons, which is normally invoked in emission models, has to be replaced by a nonlinear loss rate depending on an energy integral of the electron distribution. This modified electron cooling changes significantly the emerging radiation spectra.
Aims: It is the purpose of this work to apply this new cooling scenario to relativistic power-law distributed electrons, which are injected instantaneously into the jet.
Methods: We assume a spherical, uniform, nonthermal source, where the distribution of the electrons is spatially and temporally isotropic throughout the source. We will first solve the differential equation of the volume-averaged differential number density of the electrons, and then discuss their temporal evolution. Since any non-linear cooling will turn into linear cooling after some time, we also calculated the electron number density for a combined cooling scenario consisting of both the linear and non-linear cooling. For all cases, we will also calculate analytically the emerging optically thin time-integrated synchrotron intensity spectrum, also named the fluence, and compare it to a numerical solution.
Results: The first result is that the combined cooling scenario depends critically on the value of the injection parameter α₀. For values α₀ ≪ 1 the electrons cool mainly linear, while in the opposite case the cooling begins non-linear and becomes linear for later times. Secondly, in all cased we find that for small normalized frequencies f < 1 the fluence spectra F(f) exhibit power-laws with constant spectral indices F(f) f^-\vartheta. We find for purely linear cooling \vartheta_SYN = 1/2, and for purely non-linear cooling \vartheta_SSC = 3/2. In the combined cooling scenario we obtain for the small injection parameter \vartheta_1 = 1/2, and for the large injection parameter \vartheta_2 = 3/2, which becomes \vartheta_1 = 1/2 for very small frequencies. These identical behaviors, as compared to the existing calculations for monoenergetically injected electrons, prove that the spectral behavior of the total synchrotron fluence is independent from the functional form of the energy injection spectrum.

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