Universality of Electron Distributions in Extensive Air Showers

Based on extensive air shower simulations, it is shown that electron distributions with respect to two angles determining the electron direction at a given shower age, for a fixed electron energy and lateral distance, are universal. This means that the distributions do not depend on the primary particle energy or mass (thus, neither on the interaction model), shower zenith angle, or shower to shower fluctuations, if they are taken at the same shower age. Together with previous work showing the universality of the distributions of the electron energy, lateral distance (integrated over angles), and angle (integrated over lateral distance) for fixed electron energy, this paper completes a full universal description of the electron states at various shower ages. Analytical parametrizations of the full electron states are given. It is also shown that some distributions can be described by a number of variables smaller than five, with the new ones being products of old ones raised to some power. The accuracy of the present parametrization is sufficiently good to apply to showers with a primary energy uncertainty of 14% (as is the case at the Pierre Auger Observatory). The shower fluctuations in the chosen bins of the multidimensional variable space are about 6%, determining the minimum uncertainty needed for the parametrization of the universal distributions. An analytical way of estimating the effect of the geomagnetic field is given. Thanks to the universality of the electron distribution in any shower, a new method of shower reconstruction can be worked out from the data from observatories using the fluorescence technique. The light fluxes (both fluorescence and Cherenkov) for any shower age can be exactly predicted for a shower with any primary energy and shower maximum depth, so that the two quantities can be obtained by best fitting the predictions to the measurements.