2015ApJ...800..103T

Recent gamma-ray observations show that middle-aged supernova remnants (SNRs) interacting with molecular clouds can be sources of both GeV and TeV emission. Models involving reacceleration of preexisting cosmic rays (CRs) in the ambient medium and direct interaction between SNR and molecular clouds have been proposed to explain the observed gamma-ray emission. For the reacceleration process, standard diffusive shock acceleration (DSA) theory in the test particle limit produces a steady-state particle spectrum that is too flat compared to observations, which suggests that the high-energy part of the observed spectrum has not yet reached a steady state. We derive a time-dependent DSA solution in the test particle limit for situations involving reacceleration of preexisting CRs in the preshock medium. Simple estimates with our time-dependent DSA solution plus a molecular cloud interaction model can reproduce the overall shape of the spectra of IC 443 and W44 from GeV to TeV energies through pure π⁰-decay emission. We allow for a power-law momentum dependence of the diffusion coefficient, finding that a power-law index of 0.5 is favored.