We develop a simple one-zone model of the steady-state Crab nebula spectrum encompassing both the radio/soft $X$-ray and the GeV/multi-TeV observations. By solving the transport equation for GeV-TeV electrons injected at the wind termination shock as a log-parabola momentum distribution and evolved via energy losses, we determine analytically the resulting differential energy spectrum of photons. We find an impressive agreement with the observed spectrum of synchrotron emission, and the synchrotron self-Compton component reproduces the previously unexplained broad $200$-GeV peak that matches the Fermi/LAT data beyond $1$ GeV with the MAGIC data. We determine the parameters of the single log-parabola electron injection distribution, in contrast with multiple broken power-law electron spectra proposed in the literature. The resulting photon differential spectrum provides a natural interpretation of the deviation from power-law customarily fit with empirical multiple broken power-laws. Our model can be applied to the radio-to-multi-TeV spectrum of a variety of astrophysical outflows, including pulsar wind nebulae and supernova remnants, as well as to interplanetary shocks.

9 pages, 7 figures. MNRAS, in press