Evaluation of the Primary Energy of UHE Photon-induced Atmospheric Showers from Ground Array Measurements

A photon induced shower at $E_{prim}\ge 10^{18}$ eV exhibits very specific features and is different from a hadronic one. At such energies, the LPM effect delays in average the first interactions of the photon in the atmosphere and hence slows down the whole shower development. They also have a smaller muonic content than hadronic ones. The response of a surface detector such as that of the Auger Observatory to these specific showers is thus different and has to be accounted for in order to enable potential photon candidates reconstruction correctly. The energy reconstruction in particular has to be adapted to the late development of photon showers. We propose in this article a method for the reconstruction of the energy of photon showers with a surface detector. The key feature of this method is to rely explicitly on the development stage of the shower. This approach leads to very satisfactory results ($\simeq 20%$). At even higher energies ($5.10^{19}$ eV and above) the probability for the photon to convert into a pair of e$^+$e$^-$ in the geomagnetic field becomes non negligible and requires a different function to evaluate the energy with the proposed method. We propose several approaches to deal with this issue in the scope of the establishment of an upper bound on the photon fraction in UHECR.