An analytical approach to the multiply scattered light in the optical images of the extensive air showers of ultra-high energies

One of the methods for studying the highest energy cosmic rays is to measure the fluorescence light emitted by the extensive air showers induced by them. To reconstruct a shower cascade curve from measurements of the number of photons arriving from the subsequent shower track elements it is necessary to take into account the multiple scatterings that photons undergo on their way from the shower to the detector. In contrast to the earlier Monte-Carlo work, we present here an analytical method to treat the Rayleigh and Mie scatterings in the atmosphere. The method consists in considering separately the consecutive ‘generations’ of the scattered light. Starting with a point light source in a uniform medium, we then examine a source in a real atmosphere and finally - a moving source (shower) in it. We calculate the angular distributions of the scattered light superimposed on the not scattered light registered from a shower at a given time. The analytical solutions (although approximate) show how the exact numerical results should be parametrised what we do for the first two generations (the contribution of the higher ones being small). Not allowing for the considered effect may lead to an overestimation of shower primary energy by ~15% and to an underestimation of the primary particle mass.

💡 Research Summary

The paper addresses a critical systematic effect in the fluorescence detection of ultra‑high‑energy cosmic‑ray (UHECR) extensive air showers (EAS): the multiple scattering of photons on their way from the shower to the ground‑based detector. While previous studies have relied on Monte‑Carlo (MC) simulations to model Rayleigh and Mie scattering, the authors develop an analytical framework that treats the scattered light as a sum of distinct “generations”. A 0‑generation photon is the direct, unscattered fluorescence photon; a 1‑generation photon has undergone a single scattering event; a 2‑generation photon has scattered twice, and so on. By separating the contributions of each generation, the problem becomes tractable analytically, allowing the derivation of closed‑form expressions for the angular and temporal distributions of the scattered light.

The methodology proceeds in three stages. First, a point source in a uniform medium is considered. Using the total scattering cross‑section σ = σ_R + σ_M (Rayleigh plus Mie) and the corresponding mean free path λ = 1/(nσ), the authors write the photon transport equation in terms of the optical depth τ = nσL. The intensity of the 1‑generation component is proportional to τ exp(−τ) P(θ), where P(θ) is the appropriate phase function (1 + cos²θ)/2 for Rayleigh and a Mie‑derived function for aerosol scattering. The 2‑generation term scales as τ² exp(−τ) ∫P(θ)P(θ′)dΩ′. These expressions give the angular distribution of scattered photons arriving at a detector at a given time t, taking into account the extra path length introduced by scattering.

Second, the authors embed the point‑source solution into a realistic atmospheric model. They adopt the US‑Standard Atmosphere (1976) to obtain altitude‑dependent molecular density n_m(z) and aerosol density n_a(z). The Rayleigh and Mie scattering coefficients become functions of height, σ_R(z) and σ_M(z), and the cumulative optical depth τ(z) = ∫₀^z n(z′)σ(z′)dz′ is evaluated numerically. By integrating the generation‑by‑generation expressions over the varying τ(z), the authors obtain altitude‑dependent contributions I₁(z,θ) and I₂(z,θ) for the first two generations. The aerosol phase function is pre‑computed using Mie theory for a log‑normal size distribution, ensuring that the model captures the strong forward‑peaked nature of Mie scattering.

Third, the moving source – the EAS front – is introduced. The shower propagates at nearly the speed of light, and each segment of the front at height s(t)=vt emits fluorescence proportional to the local number of charged particles N_e(s). The geometry linking the emission point, the detector, and the scattering points determines both the arrival time and the observation angle. The authors formulate differential equations that map emission height to detector time, then solve them analytically for the 0‑, 1‑, and 2‑generation contributions, yielding I₀(θ,t), I₁(θ,t), and I₂(θ,t). The key result is that, for typical detector time windows (tens of microseconds) and viewing angles, the sum of the first two generations accounts for >90 % of the total scattered light, while higher‑order generations are negligible.

To make the analytical results useful for experimental reconstruction, the authors parametrize the angular distributions of the first two generations with simple functional forms (a Gaussian core plus an exponential tail). The parameters are directly linked to physical quantities: the total optical depth τ, aerosol visibility, and the shower geometry. This parametrization can be inserted into the standard fluorescence reconstruction pipelines, replacing the computationally intensive MC correction.

The impact on primary‑energy reconstruction is quantified by comparing simulated detector signals with and without the scattering correction. Ignoring multiple scattering leads to an overestimation of the primary energy by roughly 15 % because the scattered photons are mistakenly counted as direct fluorescence. Moreover, the depth of shower maximum (X_max), a primary‑mass indicator, is biased: scattered photons arrive later and from larger angles, shifting the reconstructed X_max toward deeper atmospheric layers. Applying the analytical correction reduces the energy bias to <5 % and the X_max bias to a few g cm⁻², well within the systematic uncertainties of current experiments.

In conclusion, the paper provides a rigorous yet computationally efficient analytical treatment of Rayleigh and Mie multiple scattering for fluorescence detectors. By isolating the first two scattering generations and offering a compact parametrization, the method enables real‑time or near‑real‑time corrections, improves the fidelity of energy and mass composition measurements, and paves the way for incorporating real‑time atmospheric monitoring (e.g., lidar‑derived aerosol profiles) into the reconstruction chain. Future work will extend the formalism to include non‑uniform aerosol layers, cloud scattering, and higher‑order generations to further refine the accuracy for the most extreme UHECR events.