| Abstract
| - Context.Models of many astrophysical gamma-ray sources assume they contain a homogeneous distribution of electrons that are injected as a power law in energy and evolve by interacting with radiation fields, magnetic fields, and particles in the source and by escaping. This problem is particularly complicated if the radiation fields have higher energy density than the magnetic field and are sufficiently energetic that inverse Compton scattering is not limited to the Thomson regime. Aims.We present a simple, time-dependent, semi-analytical solution to the electron kinetic equation that treats both continuous and impulsive injection, cooling via synchrotron and inverse Compton radiation (taking Klein-Nishina effects into account), and energy-dependent particle escape. We used this solution to calculate the temporal evolution of the multi-wavelength spectrum of systems where energetic electrons cool in intense photon fields. Methods. The kinetic equation for an arbitrary, time-dependent source function is solved by the method of Laplace transformations. Using an approximate expression for the energy-loss rate that takes synchrotron and inverse Compton losses into account, including Klein-Nishina effects for scattering off an isotropic photon field with either a power-law or black-body distribution, we find explicit expressions for the cooling time and escape probability of individual electrons. This enables the full, time-dependent solution to be reduced to a single quadrature. From the electron distribution, we then construct the time-dependent, multi-wavelength emission spectrum. Results. We compare our solutions with several limiting cases and discuss the general appearance and temporal behaviour of spectral features (i.e., cooling breaks, bumps, etc.). As a specific example, we model the broad-band energy spectrum of the open stellar association Westerlund-2 at different times of its evolution, and compare it with observations. The model calculation matches the observations for a source with an age greater than ≈10 5 yrs. We predict that the GLAST gamma-ray observatory should easily detect this source. Conclusions.The technique we present enables simple, computationally efficient, time-dependent models of homogeneous sources to be constructed and compared with multi-wavelength observations.
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