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We present analytic approximations to the optically thin synchrotron andsynchrotron self-Compton (SSC) spectra when Klein-Nishina (KN) effects areimportant and pair production and external radiation fields can be neglected.This theory is useful for analytical treatment of radiation from astrophysicalsources, such as gamma-ray bursts (GRBs), active galactic nuclei and pulsarwind nebula, where KN effects may be important. We consider a source with acontinuous injection of relativistic electrons with a power-law energydistribution above some typical injection energy. We find that thesynchrotron-SSC spectra can be described by a broken power-law, and provideanalytic estimates for the break frequencies and power-law indices. In general,we show that the dependence of the KN cross-section on the energy of theupscattering electron results in a hardening of the energy distribution of fastcooling electrons and therefore in a hardening of the observed synchrotronspectrum. As a result the synchrotron spectrum of fast cooling electrons, belowthe typical injection energy, can be as hard as $F_\nu \propto \nu^0$, insteadof the classical $\nu^{-1/2}$ when KN effects are neglected. The synchrotronenergy output can be dominated by electrons with energy above the typicalinjection energy. We solve self-consistently for the cooling frequency and findthat the transition between synchrotron and SSC cooling can result in adiscontinuous variations of the cooling frequency and the synchrotron and SSCspectra. We demonstrate the application of our results to theory by applyingthem to prompt and afterglow emission models of GRBs.

Klein-Nishina Effects on Optically Thin Synchrotron and Synchrotron Self-Compton Spectrum