On the gamma ray burst origin of extremely energetic cosmic rays

Air shower experiments have detected cosmic ray events of energies up to 300 EeV. Most likely these cosmic rays have originated from compact objects. Their exact sources are yet to be identified. It has been suggested before that gamma ray bursts are possible sources of ultra-high energy cosmic rays. The two models of gamma ray burst emissions most often discussed are the internal and external shock models. We have calculated the proton spectrum above 60 EeV from all gamma ray bursts distributed up to a redshift of 0.02 in the internal shock model assuming redshift and luminosity distributions consistent with observations, log normal distributions for their values of Lorentz factors, variability times and duration of bursts. Within the external shock model we have calculated the proton flux above 60 EeV from all nearby gamma ray bursts assuming log normal distributions in their values of total energies, Lorentz factors at the deceleration epoch and compared with the observed data. We find that gamma ray bursts can produce cosmic ray proton flux comparable to the flux observed by the Pierre Auger experiment both within the internal and external shock models. We have also studied the dependence of the maximum proton energies and the cooling breaks in the proton spectrum on the various parameters like Lorentz factor, energy of the GRB fireball, variability time (in case of internal shocks), ambient particle density (in case of external shocks). Our results are important to understand how the various observable parameters determine which mechanism e.g.  pγpγ interactions, synchrotron cooling of protons will dominate over one another inside these sources.