Escape-limited model of cosmic-ray acceleration revisited

Context. The spectrum of cosmic rays (CRs) is affected by their escape from an acceleration site. This may be observed not only in the gamma-ray spectrum of young supernova remnants (SNRs) such as RX J1713.7-3946, but also in the spectrum of CRs showering the Earth. Aims. The escape-limited model of cosmic-ray acceleration is studied in general. We discuss the spectrum of runaway CRs from the acceleration site. The model will also be able to constrain the spectral index at the acceleration site and the ansatz with respect to the unknown injection process into the particle acceleration. Methods. Our methods are analytical derivations. We apply our model to CR acceleration in SNRs and in active galactic nuclei (AGN), which are plausible candidates of Galactic and extragalactic CRs, respectively. In particular we take account into the shock evolution with cooling of escaping CRs in the Sedov phase for young SNRs. Results. The spectrum of escaping CRs generally depends on the physical quantities at the acceleration site like the spectral index, the evolution of the maximum energy of CRs and the evolution of the normalization factor of the spectrum. It is found that the spectrum of runaway particles can be both softer and harder than that of the acceleration site. Conclusions. The model explains spectral indices of both Galactic and extragalactic CRs produced by SNRs and AGNs, respectively, suggesting the unified picture of CR acceleration.