A Minimal Model for Understanding Secondary Cosmic Rays

We take a phenomenological approach in a minimal model to understand the spectral intensity of secondary cosmic-ray particles like positrons, antiprotons, Lithium, Beryllium and Boron. Our analysis shows that cosmic rays at $\sim$ GeV energies pass through a significant amount of matter in regions surrounding the sources. This grammage decreases with increasing cosmic-ray energy and becomes negligible beyond $\sim 100$ GeV. During the subsequent propagation in the interstellar medium cosmic rays of all energies up to $\sim 10^5$ GeV/$n$ pass through about 1-2 g cm$^{-2}$ of matter before leaking into the intergalactic medium. It is in the interstellar medium that the bulk of the positrons and antiprotons are generated. Also cosmic-ray nuclei like C, N, and O at all energies generate additional amounts of Li, Be and B nuclei with a spectrum similar to those of C, O etc. The implications of these findings of the minimal model to the observations of gamma rays and also the importance of spatial and temporal discreteness of cosmic-ray sources for modeling cosmic-ray propagation are briefly pointed out.