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The shape of the longitudinal development of the showers generated in theatmosphere by very high energy cosmic ray particles encodes information aboutthe mass composition of the flux, and about the properties of hadronicinteractions that control the shower development. Studies of the energydependence of the average and width of the depth of maximum distribution ofshowers with $E \gtrsim 10^{17.3}$ eV measured by the Pierre Auger Observatory,suggest, on the basis of a comparison with current models, that the compositionof the cosmic ray flux undergoes a very important evolution, first becominglighter and then rapidly heavier. These conclusions, if confirmed, would haveprofound and very surprising implications for our understanding of the highenergy astrophysical sources. Studies of the shape of the depth of maximumdistribution in the same energy range have been used by Auger and by theTelescope Array Collaboration to measure the interaction length of protons inair, a quantity that allows to estimate the $pp$ cross sections for values of$\sqrt{s}$ well above the LHC range. In this paper we argue that it isdesirable to combine the studies of the cosmic ray composition with those aimedat the measurement of the $p$--air cross section. The latter allow to obtainestimates for the fraction of protons in the flux that can be of great help indecoding the composition and its energy dependence. Studies that considermultiple parameters to characterize the depth of maximum distributions alsooffer the possibility to perform more sensitive tests of the validity of themodels used to describe high energy showers.

The spectra and composition of Ultra High Energy Cosmic Rays and the measurement of the proton-air cross section