Cosmic Ray Air Shower Ground Footprint

Primary Cosmic Ray

Primary energy 10^18.5 eV

Shower age s 1.00

s≈1 at shower maximum; s<1 = still growing (steep core), s>2 = past maximum (flatter, more spread out).

Observatory altitude 0 km a.s.l.

Thinner air raises the Molière radius, spreading the footprint wider.

Ground Array Readout

Shower size N_e

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Molière radius r_M

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S(1000) density at r=1000 m

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How it works

When a high-energy cosmic ray (mostly protons and nuclei) hits the atmosphere it triggers a cascade of secondary particles. By the time that cascade reaches the ground it has spread into a broad, roughly circular footprint a kilometre or more across. Real observatories such as the Pierre Auger Observatory or AGASA don't count every particle — they sample the footprint with an array of ground stations and fit the measured densities to a known shape.

That shape is the Nishimura–Kamata–Greisen (NKG) lateral distribution function, giving the particle density ρ at radius r from the shower core:

ρ(r) = (N_e / r_M²) · C(s) · (r/r_M)^(s-2) · (1 + r/r_M)^(s-4.5)

C(s) = Γ(4.5-s) / [2π · Γ(s) · Γ(4.5-2s)]

Here N_e is the total number of electrons/positrons in the shower (set by the primary energy, roughly N_e ≈ E₀ / 86 MeV for a purely electromagnetic cascade), r_M is the Molière radius (the natural width scale, set by multiple Coulomb scattering — it grows with altitude as the air thins), and s is the "shower age", which tracks how far past its maximum development the cascade is.

Falling particles are sampled directly from this distribution: each one lands at a radius r drawn from the probability ρ(r)·2πr dr, so their spatial pattern on the ground is the real NKG footprint, not a decoration.
Ground pillars show the density every station in the array would measure, colour- and height-coded on a log scale.
S(1000), the density at exactly 1000 m from the core, is highlighted with a ring because real arrays use it as an energy estimator: at that particular radius the reading is almost insensitive to fluctuations in where the shower first started, making it a far more reliable proxy for primary energy than trying to count N_e directly.

High density Low density r = 1000 m ring

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