Analytical view of diffusive and convective cosmic ray transport in elliptical galaxies

Context. An analytical solution of the generalized diffusive and convective transport equation is derived to explain the transport of cosmic ray protons within elliptical galaxies. Aims. Cosmic ray transport within elliptical galaxies is an interesting element in understanding the origin of high energetic particles measured on Earth. As probable sources of those high energetic particles, elliptical galaxies show a dense interstellar medium as a consequence of activity in the galactic nucleus or merging events between galaxies. Thus it is necessary for an appropriate description of cosmic ray transport to take the diffusive and convective processes in a dense interstellar environment into account. Here we show that the transport equations can be solved analytically with respect to the given geometry and boundary conditions in position space, as well as in momentum space. Methods. From the relativistic Vlasov equation, which is the most fundamental equation for a kinetic description of charged particles within the interstellar medium in galaxies, one finds a generalized diffusion-convection equation in quasilinear theory. This has the form of a ‘leaky box’ equation, meaning particles are able to escape the confinement region by diffusing out of the galaxy. We apply here the ‘diffusion approximation’, meaning that diffusion in gyrophase and pitch angle are the fastest particle-wave interaction processes. An analytical solution can be obtained using the ‘scattering time method’, i.e. separation of the spatial and momentum problems. Results. The spatial solution is shown using a generalized source of cosmic rays. Additionally, the special case of a jet-like source is illustrated. We present the solution in momentum space with respect to an escape term for cosmic ray protons depending on the spatial shape of the galaxy. For a delta-shape injection function, the momentum solution is obtained analytically. We find that the spectral index measured on Earth can be obtained by appropriately choosing of the strength of Fermi I and Fermi II processes. From these results we calculate the gamma-ray flux from pion decay due to proton-proton interaction to give connection to observations. Additionally we determine the escape-spectrum of cosmic rays. The results show that both spectra are harder than the intrinsic power-law spectrum for cosmic rays in elliptical galaxies.