A single high-energy particle striking the upper atmosphere triggers a cascading chain of collisions, multiplying into millions of secondary particles by the time it nears the ground.
N(X) ~ exp(a*X) (Heitler cascade model)
shower max ~ log(E_primary)
- Entry angle — angle the primary particle enters the atmosphere at, changing its path length.
- Descent speed — playback speed of the cascade as it develops through the atmosphere.
- Primary particle energy — energy of the initiating cosmic ray, from ordinary to ultra-high-energy.
- Shower particle lifetime — how long secondary particles remain part of the visible cascade before decaying.
Ultra-high-energy cosmic-ray air showers detected by observatories like Pierre Auger can spread secondary particles across tens of square kilometres from a single primary particle.