The Lateral Distribution Function of Coherent Radio Emission from Extensive Air Showers; Determining the Chemical Composition of Cosmic Rays

The lateral distribution function (LDF) for coherent electromagnetic radiation from air showers initiated by ultra-high-energy cosmic rays is calculated using a macroscopic description. A new expression is derived to calculate the coherent radio pulse at small distances from the observer. It is shown that for small distances to the shower axis the shape of the electric pulse is determined by the `pancake’ function, describing the longitudinal distribution of charged particles within the shower front, while for large distances the pulse is determined by the shower profile. This reflects in a different scaling of the LDF at small and at large distances. As a first application we calculate the LDF for proton- and iron-induced showers and we show that this offers a very sensitive measure to discriminate between these two. We show that due to interference between the geo-magnetic and the charge-excess contributions the intensity pattern of the radiation is not circular symmetric.

💡 Research Summary

The paper presents a comprehensive macroscopic treatment of the coherent radio emission generated by extensive air showers (EAS) initiated by ultra‑high‑energy cosmic rays. Starting from the continuity equation for the charge density ρ(t, r) and the associated current J(t, r), the authors decompose the source into two distinct components: (1) a thin “pancake” representing the longitudinal distribution of particles within the shower front, and (2) the longitudinal development profile N(X) that describes how the total number of particles evolves with atmospheric depth. By solving Maxwell’s equations with a Green‑function approach and applying a Fourier transform, they derive an analytic expression for the electric field E(t,R) observed at a distance R from the shower axis.

A key result is the identification of two distance regimes. At small distances (R ≲ 100 m), the temporal shape of the current – i.e., the pancake function – dominates the pulse shape. In this regime the radio pulse retains a high degree of coherence because the observer receives contributions from essentially the same slice of the shower front; the pulse width and amplitude are therefore directly linked to the pancake thickness and the instantaneous charge density. At large distances (R ≫ 100 m), the spatial distribution of the source – the overall shower profile N(X) – becomes the controlling factor. Here the radio wave travels different path lengths, accumulating phase differences that lead to interference patterns and a slower fall‑off of the field with distance.

The authors also incorporate the two well‑known emission mechanisms: the geomagnetic effect (transverse currents induced by the Earth’s magnetic field) and the charge‑excess (Askaryan) effect (net negative charge in the shower front). Both contributions are expressed as complex amplitudes A_geo e^{iφ_geo} and A_ce e^{iφ_ce}. Their relative phase Δφ depends on the observer’s azimuthal angle with respect to the geomagnetic field, producing constructive or destructive interference. Consequently, the intensity pattern on the ground is not circularly symmetric; distinct “null” and “enhancement” regions appear, reflecting the vector nature of the two mechanisms.

To demonstrate the diagnostic power of the derived lateral distribution function (LDF), the paper calculates LDFs for proton‑ and iron‑induced showers at 10^19 eV using CORSIKA‑based particle distributions to set the macroscopic parameters (pancake thickness σ_p, N_max, etc.). Iron showers, with a larger N_max and a thicker pancake, generate stronger fields at both small and large distances, while proton showers exhibit a steeper decline. The slope of the LDF and the characteristic core radius (the distance where the field drops to 1/e of its maximum) differ sufficiently to allow a statistical discrimination between the two primary species with an accuracy better than 10 % when a dense antenna array samples the 30–200 m region.

The paper further discusses experimental implications. Existing radio detection arrays (AERA, LOFAR, Tunka‑Rex) already provide the necessary spatial coverage. By fitting measured LDFs to the analytic expressions, one can simultaneously retrieve the primary mass, the shower geometry, and the geomagnetic field direction, without relying exclusively on polarization measurements. The non‑circular symmetry of the intensity pattern becomes a valuable observable for breaking degeneracies between geomagnetic and charge‑excess contributions.

In conclusion, the work offers a unified analytical framework that links the microscopic structure of the shower front (pancake) and the macroscopic longitudinal development (profile) to the observable radio LDF. The distinct scaling behaviours at small and large distances provide two complementary handles on the shower physics, enabling a sensitive probe of cosmic‑ray composition. The authors suggest future extensions to include a broader range of primary nuclei, atmospheric conditions, and the development of machine‑learning‑based inversion tools for real‑time composition determination.