Energy estimation of cosmic rays by air shower radio signals

Technical Summary: Energy Estimation of Cosmic Rays by Air Shower Radio Signals

Problem Statement
Determining the primary energy of cosmic rays is essential for understanding their origin, acceleration, and propagation mechanisms. While Extensive Air Showers (EAS) initiated by cosmic rays destroy the primary particle's direct information, secondary particles and radio emissions offer reconstruction pathways. Previous experiments (e.g., AERA, Tunka-Rex, LOPES) have utilized the Lateral Distribution Function (LDF) of radio signals, correlating signal intensity at specific distances from the shower core with primary energy. However, these methods often require identifying an optimal distance specific to each station's geometry and can be sensitive to the shower core location. The SURA experiment, a four-antenna radio array in Semnan, Iran, has successfully reconstructed arrival directions and core locations but requires a robust method to estimate primary energy that minimizes sensitivity to core position variations.

Methodology
The authors propose a reconstruction method based on comparing experimental radio signals with high-fidelity simulations using the CoREAS extension of CORSIKA. The core of the methodology involves the following steps:

1. Simulation Setup:

   • SURA Configuration: Simulations were performed to match the SURA experiment's specific parameters, including its four Log-Periodic Dipole Arrays (LPDA), local magnetic field components ($B_x = 39.53 \, \mu T$, $B_z = 28.1 \, \mu T$), and altitude (1130 m).
   • Dense Reference Array: A virtual dense array of 12,342 antennas with 1-meter spacing was simulated to generate a high-resolution reference for radio signal intensities. This array serves as a baseline for comparison.
   • Event Generation: Proton-induced showers were simulated with primary energies ranging from $10^{17}$ eV to $7 \times 10^{17}$ eV. The primary arrival direction was set to a zenith angle of $35^\circ$ and azimuth of $40^\circ$ (based on SURA's peak detection rate), with additional simulations for $60^\circ$ and $65^\circ$ zenith angles to test inclination effects.

2. Data Calibration:

   • The study incorporates a two-stage calibration process: real-time FPGA filtering (high-pass, band-pass, notch filters) and post-processing (arrival direction reconstruction, frequency response application, wavelet noise reduction, and signal up-sampling to 0.2 ns resolution).

3. The Scale Factor ($C_{ij}$) Approach:

   • A scale factor, $C_{ij}$, is defined as the ratio of the mean peak intensity of simulated signals from the dense array ($\bar{A}_{sim}$) to the peak intensity of signals from the SURA configuration ($\bar{A}_{exp}$):
     $$C_{ij} = \frac{\bar{A}_{sim}}{\bar{A}_{exp}}$$
   • This factor is derived by minimizing a $\chi^2$ function comparing experimental (simulated SURA) and dense array signals across the antenna array.
   • The method posits that while the absolute signal intensity depends on core location, the ratio (scale factor) between the sparse SURA array and the dense reference array exhibits a specific correlation with primary energy.

Key Contributions

• Core-Location Independence: The paper demonstrates that the scale factor $C_{ij}$ is largely independent of the cosmic ray shower core location. Even for core positions far from the array center (up to $\pm 30$ m and $\pm 125$ m in different tests), the scale factor remains effectively constant for a given energy.
• Energy Correlation: The authors establish a clear inverse correlation between the scale factor and the primary energy. As primary energy increases, the scale factor decreases. This relationship is modeled by an inverse first-order function:
  $$C_{ij} = y_0 + \frac{a}{E}$$
  where $E$ is the primary energy normalized by $10^{17}$ eV, and $y_0$ and $a$ are fitting constants dependent on the arrival direction.
• Robustness to Arrival Direction: The method was tested on inclined showers ($60^\circ$ and $65^\circ$ zenith angles). The inverse correlation between $C_{ij}$ and energy holds across these angles, though the fitting parameters ($a$ and $y_0$) vary slightly with the zenith angle.

Results

• Reconstruction Accuracy: Using the derived scale factor and the fitted inverse function, the primary energy of simulated cosmic rays was reconstructed. The method achieved a maximum reconstruction error of approximately 11% (observed at $5 \times 10^{17}$ eV for the $35^\circ$ zenith angle).
• Error Analysis: The relative error ($\Delta E / E$) was calculated to be as low as 2% for certain energy levels (e.g., $2 \times 10^{17}$ eV) and generally remained within the 1%–11% range across the tested energy spectrum.
• Inclined Showers: For more inclined showers ($60^\circ$ and $65^\circ$), the method remained effective, though the precision varied. The lowest precision was observed at $1 \times 10^{17}$ eV and $5 \times 10^{17}$ eV for the $65^\circ$ angle, while the maximum error for the $60^\circ$ angle occurred at $7 \times 10^{17}$ eV.

Significance and Claims
The paper claims that this technique offers a distinct advantage over previous approaches by reducing sensitivity to the shower core location, a common source of uncertainty in radio-based energy reconstruction. By utilizing a dense reference array to normalize the sparse experimental data, the method provides a reliable estimator for primary energy without requiring the precise determination of the optimal distance from the core for each specific event.

The authors conclude that the method is valid for both vertical and inclined showers and can be applied to the SURA experiment's data once real-world data collection is complete. They note that future work will involve applying this technique to actual SURA experimental data and expanding simulations to include showers with even more distant core locations and varied arrival directions to further validate the method's robustness. The study does not claim to replace existing methods but offers a complementary approach specifically tailored to the geometry and constraints of the SURA array.