Application of afxy-code for parameterization of ionization yield function Y in the atmosphere for primary cosmic ray protons

In this work is obtained new approximation for the yield function Y for cosmic ray induced ionization in the Earth atmosphere on the basis of simulated data. The parameterization is obtained using inverse nonlinear problem solution with afxy(analyze fx=y)-code. Short description of the methods is given. The found approximation is for primary proton nuclei, inclined up to 70 degrees zenith angle. This permits to estimate the direct ionization by primary cosmic rays explicitly. The parameterization is applicable to the entire atmosphere, from ground level to upper atmosphere. Several implications of the found parameterization are discussed.

💡 Research Summary

The paper presents a new analytical approximation for the ionization yield function Y, which quantifies the number of ion pairs produced per gram of air by a single primary cosmic‑ray particle of a given energy. Using previously generated Monte‑Carlo simulation data for primary protons with energies ranging from 500 MeV to 1 TeV, the authors formulate the problem as an inverse nonlinear system f(x)=y, where the unknown vector x contains the parameters of the chosen functional form and y represents the (log‑transformed) simulated Y values.

To solve this ill‑conditioned inverse problem they employ the afxy‑code (analyze f(x)=y), a robust iterative scheme based on a regularized Gauss‑Newton update (equation 5). Two normed spaces are used: the L∞ norm for process regularization (equation 6) and the L2 norm for problem regularization (equations 7‑8). The automatic regularization parameters ε_k and α_k adapt during the iteration, ensuring convergence even when the Jacobian matrix is near singular.

The chosen functional form is a simple rational function of atmospheric depth h (in g cm⁻²):

 Y(h) = (a h² + b h + c) / (d h² + e h + f)  (10)

where a–f are fitted coefficients. By taking logarithms of both sides and scaling the argument, the authors achieve rapid convergence: only about 100 iterations are required to reduce the normalized χ² to values between 0.13 (500 MeV) and 0.001 (1 TeV). The fitted parameters for each energy are listed in Table 1, and the resulting curves are plotted against the original simulation points in Figures 1‑8. The fits reproduce the position and magnitude of the Pfotzer maximum (the peak ionization around 15 km altitude) and also match the low‑altitude behavior, with only minor over‑estimation at the maximum for the 50 GeV and 100 GeV cases.

Because the rational function is analytically integrable, the total atmospheric ionization can be obtained directly from the parameterization. Moreover, the coefficients vary smoothly with energy and can be interpolated using exponential or power‑law functions, allowing the calculation of Y for any proton energy within the studied range without re‑running Monte‑Carlo simulations.

The authors discuss the practical implications of this work. The new parameterization enables fast computation of ionization rates q(h,λ_m) (equation 3) for any geographic location (through the geomagnetic latitude λ_m) and atmospheric profile, bypassing the need for extensive detector networks or repeated heavy simulations. This facilitates studies of how solar activity, which modulates the primary cosmic‑ray spectrum, influences atmospheric ionization, cloud condensation nuclei formation, and potentially climate. The paper also highlights the broader utility of the afxy‑code for solving other nonlinear inverse problems in physics and applied mathematics.

In conclusion, the study delivers a compact, accurate, and analytically tractable representation of the proton‑induced ionization yield function across the entire atmosphere. It demonstrates that the afxy‑code is an effective tool for tackling ill‑conditioned nonlinear systems, and it opens the way for more comprehensive modeling that includes other primary particles, varied incident angles, and different atmospheric conditions.