Modeling the broadest spectral band of the Crab nebula and constraining the ions acceleration efficiency

Although it is widely accepted that the electromagnetic spectrum from radio to very-high-energy $\gamma$-rays of pulsar wind nebulae (PWNe) originates from leptons, there is still an open question that protons (or more generally, ions) may exist in pulsar wind and are further accelerated in PWN. The broadband spectrum of the prototype PWN Crab, extended recently by the detection of the Tibet AS$\gamma$ and HAWC experiments above 100 TeV, may be helpful in constraining the acceleration efficiency of ions. Here, we model the broadest energy spectrum of Crab and find that the broadband spectrum can be explained by the one-zone leptonic model in which the electrons/positrons produce the emission from radio to soft $\gamma$-rays via the synchrotron process, and simultaneously generate the GeV-TeV $\gamma$-rays through inverse Compton scattering including the synchrotron self-Compton process. In the framework of this leptonic model, the fraction of energy converted into the energetic protons is constrained to be below $0.5\ (n_{\rm t}/10\ {\rm cm}^{-3})^{-1}$ per cent, where $n_{\rm t}$ is the target gas density in the Crab. However, this fraction can be up to $7\ (n_{\rm t}/10\ {\rm cm}^{-3})^{-1}$ per cent if only the $\gamma$-rays are used.