Atmospheric multiple scattering of fluorescence light from extensive air showers and effect of the aerosol size on the reconstruction of energy and depth of maximum

The reconstruction of the energy and the depth of maximum $X_{\rm max}$ of an extensive air shower depends on the multiple scattering of fluorescence photons in the atmosphere. In this work, we explain how atmospheric aerosols, and especially their size, scatter the fluorescence photons during their propagation. Using a Monte Carlo simulation for the scattering of light, the dependence on the aerosol conditions of the multiple scattered light contribution to the recorded signal is fully parameterised. A clear dependence on the aerosol size is proposed for the first time. Finally, using this new parameterisation, the effect of atmospheric aerosols on the energy and on the $X_{\rm max}$ reconstructions is presented for a vertical extensive air shower observed by a ground-based detector at $30~$km: for typical aerosol conditions, multiple scattering leads to a systematic over-estimation of $5\pm1.5\%$ for the energy and $4.0\pm 1.5~$g/cm$^2$ for the $X_{\rm max}$, where the uncertainties refer to a variation of the aerosol size.