Origin and Propagation of Cosmic-Ray Electrons

We consider the modulating of the energy spectrum of cosmic-ray electrons as they propagate in interstellar space. The effects of spatial propagation, energy loss, boundary conditions, and the spatial distribution and energy spectrum of the electrons injected into interstellar space by the sources are all taken into account. Two models for spatial propagation are considered: isotropic diffusion and convection diffusion. In the latter, the particle motion parallel to the disk plane is characterized by diffusion, while particles are transported perpendicular to the disk plane by convection, caused by the outward expansion of the field lines. Examples are worked out for several source distributions. It is shown that the high-energy electron distribution (in both energy and space) is determined by the source distribution, rather than the properties of the confinement region (i.e., the propagation model). In order to illustrate how one can use the observable quantities (the electron spectrum and positron fraction at Earth, and the background Galactic radio and $\gamma$ radiation) to determine the properties of the injection spectrum, we have considered the construction of a model of primary cosmic-ray electron sources in some detail. It is found that a combination of a source with a disk distribution and a source concentrated at the Galactic center may be necessary to explain the observed disk $\gamma$ rays.