A new analytical model of the cosmic-ray energy flux for Galactic diffuse radio emission

Low-frequency radio observations of diffuse synchrotron radiation offer a unique vantage point for investigating the intricate relationship between gas and magnetic fields in the formation of structures within the Galaxy, spanning from the diffuse interstellar medium (ISM) to star-forming regions. Achieving this pivotal objective hinges on a comprehensive understanding of cosmic-ray properties, which dictate the effective energy distribution of relativistic electrons, primarily responsible for the observable synchrotron radiation. Notably, cosmic-ray electrons (CRe) with energies between 100 MeV and 10 GeV play a crucial role in determining the majority of the sky brightness below the GHz range. However, their energy flux ($j_e$) remains elusive due to solar modulation. We propose deriving observational constraints on this energy gap of interstellar CRe through the brightness temperature spectral index of low-frequency radio emission, here denoted as $\beta_{\rm obs}$. We introduce a new parametric analytical model that fits available data of $j_e$ in accordance with the $\beta_{\rm obs}$ values measured in the literature between 50 MHz to 1 GHz for diffuse emission in the Milky Way. Our model allows to account for multiple observations considering magnetic-field strengths consistent with existing measurements below 10 $\mu$G. We present a first all-sky map of the average component of the magnetic field perpendicular to the line of sight and validate our methodology against state-of-the art numerical simulations of the diffuse ISM. This research makes headway in modeling Galactic diffuse emission with a practical parametric form. It provides essential insights in preparation for the imminent arrival of the Square Kilometre Array.