Shower center of gravity and hadronic interaction characteristics

The system of equations for the center of gravity of the shower initiated in the atmosphere by high-energy proton is derived and solved under certain assumptions for the case of logarithmically decreasing interaction length of hadrons in the air. The obtained expression splits into two terms. One term represents the center of gravity of the purely electromagnetic cascade at the primary proton energy and another one is a modification of that by hadronic cascading. This expression provides a transparent view of the way in which hadronic interaction characteristics define the longitudinal shower development.