Proton Synchrotron Origin of the Very High Energy Emission of GRB 190114C

We consider here a proton-synchrotron model to explain the MAGIC observation of GRB 190114C afterglow in the energy band $0.2 - 1$~TeV, while the X-ray spectra are explained by electron-synchrotron emission. Given the uncertainty of the particle acceleration process, we consider several variations of the model, and show that they all match the data very well. We find that the values of the uncertain model parameters are reasonable: {isotropic} explosion energy $\sim 10^{54.5}$~erg, ambient density $\sim 10-100~{\rm cm^{-3}}$, and fraction of electrons/protons accelerated to a high energy power law of a few per-cents. All these values are directly derived from the observed TeV and X-ray fluxes. The model also requires that protons be accelerated to observed energies as high as a few $10^{20}$~eV. Further, assuming that the jet break takes place after $10^6$~s gives the beaming-corrected energy of the burst to be $\approx 10^{53}$~erg, which is one to two orders of magnitude higher than usually inferred, because of the small fraction of electrons accelerated. Our modeling is consistent with both late time data at all bands, from optical to X-rays, and with numerical models of particle acceleration. Our results thus demonstrate the relevance of proton-synchrotron emission to the high energy observations of GRBs during their afterglow phase.