Universality of electron distributions in extensive air showers

Based on extensive air shower simulations it is shown that the electron distributions with respect to the two angles, determining electron direction at a given shower age, for a fixed electron energy and lateral distance, are universal. It 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, of the lateral distance (integrated over angles) and of the 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 smaller than five numbers of variables, the new ones being products of the old ones raised to some powers. The accuracy of the present parametrization is sufficiently good for applying to showers with the primary energy uncertainty of 14% (as it is 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 parametrization of the universal distributions. An analytical way of estimation of the effect of the geomagnetic field is given. Thanks to the universality of the electron distributions in any shower a new method of shower reconstruction can be worked out from data of the 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