The Small Contribution of Molecular Bremsstrahlung Radiation to the Air-Fluorescence Yield of Cosmic Ray Shower Particles

A small contribution of molecular Bremsstrahlung radiation to the air-fluorescence yield in the UV range is estimated based on an approach previously developed in the framework of the radio-detection of showers in the gigahertz frequency range. First, this approach is shown to provide an estimate of the main contribution of the fluorescence yield due to the de-excitation of the C $^3\Pi_{\mathrm{u}}$ electronic level of nitrogen molecules to the B $^3\Pi_{\mathrm{g}}$ one amounting to $Y_{[337]}=(6.05\pm 1.50)~$ MeV$^{-1}$ at 800 hPa pressure and 293 K temperature conditions, which compares well to previous dedicated works and to experimental results. Then, under the same pressure and temperature conditions, the fluorescence yield induced by molecular Bremsstrahlung radiation is found to be $Y_{[330-400]}^{\mathrm{MBR}}=0.10~$ MeV$^{-1}$ in the wavelength range of interest for the air-fluorescence detectors used to detect extensive air showers induced in the atmosphere by ultra-high energy cosmic rays. This means that out of $\simeq 175~$ photons with wavelength between 330 and 400 nm detected by fluorescence detectors, one of them has been produced by molecular Bremsstrahlung radiation. Although small, this contribution is not negligible in regards to the total budget of systematic uncertainties when considering the absolute energy scale of fluorescence detectors.