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Context: An analytical solution of the generalized diffusive and convectivetransport equation is derived to explain the transport of cosmic ray protonswithin elliptical galaxies. Aims: Cosmic ray transport within ellipticalgalaxies is an interesting element in understanding the origin of highenergetic particles measured on Earth. As probable sources of those highenergetic particles, elliptical galaxies show a dense interstellar medium as aconsequence of activity in the galactic nucleus or merging events betweengalaxies. Thus it is necessary for an appropriate description of cosmic raytransport to take the diffusive and convective processes in a denseinterstellar environment into account. Here we show that the transportequations can be solved analytically with respect to the given geometry andboundary conditions in position space, as well as in momentum space. 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 presentthe solution in momentum space with respect to an escape term for cosmic rayprotons depending on the spatial shape of the galaxy. For a delta-shapeinjection function, the momentum solution is obtained analytically. We findthat the spectral index measured on Earth can be obtained by appropriatelychoosing of the strength of Fermi I and Fermi II processes. From these resultswe calculate the gamma-ray flux from pion decay due to proton-protoninteraction to give connection to observations. Additionally we determine theescape-spectrum of cosmic rays. The results show that both spectra are harderthan the intrinsic power-law spectrum for cosmic rays in elliptical galaxies.

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