Ultrahigh-energy cosmic rays are almost exclusively detected through extensive air showers, which they initiate upon interaction with the atmosphere. The longitudinal development of these air showers can be directly observed using fluorescence detector telescopes, such as those employed at the Pierre Auger Observatory or the Telescope Array. In this article, we discuss the properties of the Greisen function, which was initially derived as an approximate solution to the electromagnetic cascade equations, and its ability to describe the longitudinal shower profiles. We demonstrate that the Greisen function can be used to describe longitudinal air-shower profiles, even for hadronic air showers. Furthermore, we discuss the possibility to discriminate between hadrons and photons from the shape of air-shower profiles using the Greisen function.

• Received 23 March 2023
• Accepted 15 May 2023

DOI:https://doi.org/10.1103/PhysRevD.107.103056