The Distribution of Cosmic Ray Electrons in Star-Forming Galaxies

We derive explicit, algebraic expressions for the steady-state number density of cosmic ray electrons as a function of position and energy using Green's function of the diffusion equation with energy losses for an axisymmetric distributions of the particle sources in the galactocentric radius $r$ and distance to the mid-plane $z$. The solution is obtained for a Gaussian distribution of the particle sources in $r$ and $z$ but we show that it can be used for an arbitrary spatial distribution of the sources. The accuracy of our results is about 10\% or better in wide ranges of $r$, $z$ and particle energies. These solutions can be used in the interpretation of radio astronomical observations of galaxies, particularly in the studies of the radio luminosities for large galaxy samples, and represent a physically justifiable and efficient alternative to the assumption of the energy equipartition between cosmic rays and interstellar magnetic fields.