Where Does Tracing of Cosmic Ray in Real Atmosphere Terminate?

TL;DR

This paper investigates the physical processes that terminate cosmic-ray trajectories in Earth's atmosphere, moving beyond simplified models by considering energy loss and scattering, and provides criteria for realistic boundary altitude estimations.

Contribution

It introduces a detailed analysis of energy loss and scattering effects on cosmic-ray propagation, offering improved criteria for atmospheric boundary modeling in simulations.

Findings

Bethe-Bloch energy loss dominates at low rigidities.

Hard scattering interactions dominate at higher rigidities.

Recommended boundary altitude is at least 50 km for protons, more for heavy nuclei.

Abstract

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 ($\Delta\mathfrak{R}/\mathfrak{R}$), 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 $\Delta\mathfrak{R}/\mathfrak{R}+\langle N\rangle\lesssim1$ for protons, increasing by more than $15$ km for heavy nuclei such as iron.