Depth development of extensive air showers from muon time distributions

We develop an algorithm that has the potential to relate the depth development of ultra high energy extensive air showers and the time delay for individual muons. The time distributions sampled at different positions at ground level by a large air shower array are converted into distributions of production distances using an approximate relation between production distance, transverse distance and time delay. The method is naturally restricted to inclined showers where muons dominate the signal at ground level but could be extended to vertical showers provided that the detectors used can separate the muon signal from electrons and photons. We explore the accuracy and practical uncertainties involved in the proposed method. For practical purposes only the muons that fall outside the central region of the shower can be used, and we establish cuts in transverse distance. The method is tested using simulated showers by comparing the production distance distributions obtained using the method with the actual distances in the simulated showers. It could be applied in the search for neutrinos to increase the acceptance to highly penetrating particles, as well as for unraveling the relative compositions of protons and heavy nuclei. We also illustrate that the obtained depth distributions have minimum width when both the arrival direction and the core position are well reconstructed.