Measurement of the proton-air cross-section at $\sqrt{s}=57$ TeV with the Pierre Auger Observatory

We report a measurement of the proton-air cross-section for particle production at the center-of-mass energy per nucleon of 57 TeV. This is derived from the distribution of the depths of shower maxima observed with the Pierre Auger Observatory: systematic uncertainties are studied in detail. Analysing the tail of the distribution of the shower maxima, a proton-air cross-section of $[505\pm22(stat)^{+28}_{-36}(sys)]$ mb is found.