# The effect of the atmospheric refractive index on the radio signal of extensive air showers

## TL;DR

This study quantifies how natural variations in atmospheric refractive index affect radio measurements of air shower properties, revealing systematic errors in the inferred depth of shower maximum that can be mitigated for improved accuracy.

## Contribution

The paper evaluates the impact of refractive index variations on radio measurements of air showers using simulations, providing correction estimates for more precise $X_{max}$ determination.

## Key findings

- A 4% change in refractivity causes a 3.5-11 g/cm^2 error in $X_{max}$ at 30-80 MHz.
- At 120-250 MHz, the error increases to 10-22 g/cm^2.
- A simple Cherenkov angle model qualitatively agrees with simulation results.

## Abstract

For the interpretation of measurements of radio emission from extensive air showers, an important systematic uncertainty arises from natural variations of the atmospheric refractive index $n$. At a given altitude, the refractivity $N=10^6\, (n-1)$ can have relative variations on the order of $10 \%$ depending on temperature, humidity, and air pressure. Typical corrections to be applied to $N$ are about $4\%$. Using CoREAS simulations of radio emission from air showers, we have evaluated the effect of varying $N$ on measurements of the depth of shower maximum $X_{\rm max}$. For an observation band of 30 to 80 MHz, a difference of $4 \%$ in refractivity gives rise to a systematic error in the inferred $X_{\rm max}$ between 3.5 and 11 $\mathrm{g/cm^2}$, for proton showers with zenith angles ranging from 15 to 50 degrees. At higher frequencies, from 120 to 250 MHz, the offset ranges from 10 to 22 $\mathrm{g/cm^2}$. These offsets were found to be proportional to the geometric distance to $X_{\rm max}$. We have compared the results to a simple model based on the Cherenkov angle. For the 120 to 250 MHz band, the model is in qualitative agreement with the simulations. In typical circumstances, we find a slight decrease in $X_{\rm max}$ compared to the default refractivity treatment in CoREAS. While this is within commonly treated systematic uncertainties, accounting for it explicitly improves the accuracy of $X_{\rm max}$ measurements.