Cosmic Rays Escaping from Galactic Superbubbles and Application to the Fermi Bubble

We calculate the spectrum of escaping cosmic rays accelerated in the shocks produced by expanding galactic superbubbles powered by supernovae producing a continuous energy outflow in star-forming galaxies. We use the generalized Kompaneets equation solutions adapted to expansion in various external density profiles including exponential and power-law shapes, and assume that the escaping spectrum is dominated by the cosmic rays which have reached their maximum energy. We find that the escaping CR spectrum largely depends on the specific density profiles and power source properties, and the results are compared to and constrained by the observed diffuse cosmic ray spectrum spectrum. As a particular case we then also apply the results to the Milky Way's Fermi bubble, and find that the Fermi bubble cosmic rays could contribute a substantial fraction of the observed cosmic ray flux observed around $10^{17}$ eV.