Influence of shower fluctuations and primary composition on studies of the shower longitudinal development

We study the influence of shower fluctuations, and the possible presence of different nuclear species in the primary cosmic ray spectrum, on the experimental determination of both shower energy and the proton air inelastic cross section from studies of the longitudinal development of atmospheric showers in fluorescence experiments. We investigate the potential of the track length integral and shower size at maximum as estimators of shower energy. We find that at very high energy $\left(\sim {10}^{19}-{10}^{20}\mathrm{eV}\right)$ the error of the total energy assignment is dominated by the dependence on the hadronic interaction model, and is of the order of 5%. At lower energy $\left(\sim {10}^{17}-{10}^{18}\mathrm{eV}\right),$ the uncertainty of the energy determination due to the limited knowledge of the primary cosmic ray composition is more important. The distribution of the shower maximum $X_{\max }$ is discussed as a measure of the proton-air cross section. Uncertainties in a possible experimental measurement of this cross section introduced by intrinsic shower fluctuations, the model of hadronic interactions, and the unknown mixture of primary nuclei in the cosmic radiation are numerically evaluated.