Scheme of the upper atmosphere used for the analytical solution. The upper horizontal blue line denotes the top of the atmosphere, where a particle with energy E enters at the zenith angle θ. Energy deposition at the atmospheric depth h is considered in a thin layer of dh thickness where the particle has energy ε. The pathlength of the particle in the air between the top of the atmosphere and the depth h is denoted as X = h/cos(θ).