Two-Step Acceleration Model of Cosmic Rays at Middle-Aged SNR

Recent gamma-ray observations of middle-aged supernova remnants revealed a mysterious broken power-law spectrum. Using three-dimensional magnetohydrodynamics simulations, we show that the interaction between a supernova blast wave and interstellar clouds formed by thermal instability generates multiple reflected shocks. The typical Mach numbers of the reflected shocks are shown to be M ~ 2 depending on the density contrast between the diffuse intercloud gas and clouds. These secondary shocks can further energize cosmic-ray particles originally accelerated at the blast-wave shock. This "two-step" acceleration scenario reproduces the observed gamma-ray spectrum and predicts the high-energy spectral index ranging approximately from 3 to 4.