Where Does Tracing of Cosmic Ray in Real Atmosphere Terminate?

In backtracing simulations, which are widely employed to determine cosmic-ray particle trajectories in the geomagnetic field, the atmosphere is typically approximated as an artificial sharp boundary at some low altitude where the traced trajectory terminates. In this paper, we extend beyond this simplified assumption and investigate two realistic physical processes that terminate cosmic-ray particle propagation in the atmosphere: Bethe-Bloch energy loss mechanisms and hard scattering interactions with atmospheric atoms using total cross sections based on the Glauber-Gribov formalism. The former mechanism dominates at low rigidities (for protons below $\sim0.57$~GV), while the latter becomes dominant at higher rigidities. Consequently, we introduce two dimensionless variables up to detailed numerical criteria: the relative rigidity shift due to Bethe-Bloch effects ($Δ\mathfrak{R}/\mathfrak{R}$), and the expected number of hard scattering events ($\langle N\rangle$). Using the corrected US Standard Atmosphere 1976 model, we demonstrate that the altitude dependence can be factorized as approximately $\exp(-0.14h/\textrm{km})$. Additionally, the effect of the local curvature radius of the trajectory near perigee can be similarly factorized. Our calculations indicate that the simplified sharp-boundary altitude should be at least $50$ km with $Δ\mathfrak{R}/\mathfrak{R}+\langle N\rangle\lesssim1$ for protons, increasing by more than $15$ km for heavy nuclei such as iron.

💡 Research Summary

The paper addresses a fundamental limitation in the widely used back‑tracing technique for classifying cosmic‑ray particles observed in low‑Earth orbit (LEO). Traditionally, a sharp atmospheric boundary (e.g., 20 km, 40 km, or 100 km) is imposed where the backward integration of the particle’s trajectory is stopped. This ad‑hoc approach ignores the actual microscopic interactions that terminate a particle’s motion once it penetrates the dense lower atmosphere.
The authors identify two physical processes that can end a back‑traced trajectory: (1) continuous energy loss described by the Bethe‑Bloch formula, and (2) hard scattering events with atmospheric nuclei, for which they compute total cross sections using the Glauber‑Gribov formalism. The Bethe‑Bloch mechanism dominates at low rigidities (below ≈ 0.57 GV for protons), while hard scattering becomes dominant at higher rigidities.
To quantify the impact of each mechanism, the paper introduces two dimensionless variables: the relative rigidity shift Δℛ/ℛ caused by Bethe‑Bloch losses, and the expected number of hard‑scattering events ⟨N⟩. Both quantities are evaluated by integrating over the atmospheric density profile taken from the corrected US Standard Atmosphere 1976 model (updated CO₂ concentration to 2025 levels). Remarkably, the altitude dependence of both Δℛ/ℛ and ⟨N⟩ can be factorized as an exponential, exp(−0.14 h / km), reflecting the near‑perfect exponential decay of atmospheric density below 86 km.
The authors also consider the curvature of the particle’s path near perigee, characterized by the local radius of curvature r = ℛ/B (ℛ is rigidity, B the magnetic field). By replacing the straight‑line tangent path with an upward‑bending circular arc of radius r, they derive a curvature factor g(r) that multiplies the exponential altitude term. A compact fit, g(r) ≈ 0.47 · (r km)^{0.72}/