Cascades in the Earth's magnetosphere initiated by photons with the parameters of the highest energy AGASA events

We investigate the cascading effects of extremely high energy (EHE) photons in the Earth's magnetosphere assuming that these photons arrive with the parameters of the highest energy AGASA events (energies, arrival directions). For the location of the AGASA Observatory, we determine the directions in the sky from which photons can cascade with a high (low) probability. In the case of the primary photons with the parameters of the events with the energies $> 10^{20}$ eV, we compute the average cascade spectra of secondary photons entering the Earth's atmosphere, and estimate their fluctuations around these average values by selecting the events with the largest and smallest number of secondary cascade photons. It is shown that most photons with the parameters of the highest energy AGASA events should initiate cascades in the Earth's magnetosphere with a high probability even though they tend to arrive from directions in the sky for which the perpendicular component of the magnetic field is weaker. On the other hand, if these events are caused by the photons with lower energies, then the fluctuations in their shower development in the magnetosphere and the atmosphere should be higher than in the case of photons with the energies estimated by the AGASA experiment.