An analytical approach to the multiply scattered light in the optical images of the extensive air showers of ultra-high energies

Abstract: One of the methods for studying the highest energy cosmic rays is to measure the fluorescence light emitted by the extensive air showers induced by them. To reconstruct a shower cascade curve from measurements of the number of photons arriving from the subsequent shower track elements it is necessary to take into account the multiple scatterings that photons undergo on their way from the shower to the detector. In contrast to the earlier Monte-Carlo work, we present here an analytical method to treat the Rayleigh and Mie scatterings in the atmosphere. The method consists in considering separately the consecutive 'generations' of the scattered light. Starting with a point light source in a uniform medium, we then examine a source in a real atmosphere and finally - a moving source (shower) in it. We calculate the angular distributions of the scattered light superimposed on the not scattered light registered from a shower at a given time. The analytical solutions (although approximate) show how the exact numerical results should be parametrised what we do for the first two generations (the contribution of the higher ones being small). Not allowing for the considered effect may lead to an overestimation of shower primary energy by ~15% and to an underestimation of the primary particle mass.