Gamma-rays from molecular clouds illuminated by cosmic rays escaping from interacting supernova remnants

Recently, the gamma-ray telescopes AGILE and Fermi observed several middle-aged supernova remnants (SNRs) interacting with molecular clouds. It is likely that their gamma-rays arise from the decay of neutral pions produced by the inelastic collision between cosmic rays (CRs) and nucleons, which suggests that SNRs make the bulk of Galactic CRs. In this paper, we provide the analytical solution of the distribution of CRs that have escaped from a finite-size region, which naturally explains observed broken power-law spectra of themiddle-aged SNRs. In addition, the typical value of the break energy of the gamma-ray spectrum, 1-10 GeV, is naturally explained from the fact that the stellar wind dynamics shows a separation between the molecular clouds and the explosion centre of about 10 pc. We find that the runaway-CR spectrum of the four middle-aged SNRs (W51C, W28, W44 and IC 443) interacting with molecular clouds could be the same, even though it leads to different gamma-ray spectra. This result is consistent with that of recent studies of Galactic CR propagation, and supports that SNRs are indeed the sources of Galactic CRs.