Cosmic Ray Air Shower — Ground Footprint & LDF Plot (2D)

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

This 2D companion shows the same shower footprint from directly overhead, plus the actual analysis tool real observatories use: a log–log lateral distribution plot. When a high-energy cosmic ray hits the atmosphere it triggers a cascade that spreads into a broad circular footprint by the time it reaches the ground. Observatories such as the Pierre Auger Observatory sample that footprint with an array of ground stations and fit the measured densities to a known curve.

That curve 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)]

N_e is the total shower size (set by primary energy, N_e ≈ E₀ / 86 MeV for an EM cascade), r_M is the Molière radius (grows with altitude as the air thins), and s is the shower age.

Top panel — top-down view of the ground array: each square is a station, colour-coded by the local particle density on a log scale. Faint rings expanding from the core are individual secondary particles landing at radii drawn from the real ρ(r)·2πr probability, so their spatial spread is the NKG footprint, not a decoration.
Bottom panel — the log–log LDF plot: the smooth curve is the theoretical ρ(r), the dots are what each station in the array would actually read. This is exactly the plot a cosmic-ray physicist fits to real data.
S(1000), density at r = 1000 m, is marked because at that radius the reading is almost insensitive to where the shower first started — making it a far more reliable energy proxy than counting N_e directly.

High density Low density r = 1000 m ring