Electron Acceleration in Middle-age Shell-type $\gamma$-Ray Supernova Remnants

Over the past decade, $\gamma$-ray observations of supernova remnants (SNRs) and accurate cosmic-ray (CR) spectral measurements have significantly advanced our understanding of particle acceleration in SNRs. In combination with multiwavelength observations of a large sample of SNRs, it has been proposed that the highest energy particles are mostly accelerated in young remnants, and the maximum energy that middle-age and old SNRs can accelerate particles to decreases rapidly with the decrease in shock speed. If SNRs dominate the CR flux observed at Earth, a large number of particles need to be accelerated in old SNRs for the soft CR spectrum even though they cannot produce very high-energy CRs. With radio, X-ray, and $\gamma$-ray observations of seven middle-age shell-type SNRs, we derive the distribution of high-energy electrons trapped in these remnants via a simple one-zone leptonic emission model and find that their spectral evolution is consistent with such a scenario. In particular, we find that particle acceleration by shocks in middle-age SNRs with an age of $t$ can be described by a unified model with the maximum energy decreasing as $t^{-3.1}$ and the number of GeV electrons increasing as $t^{2.5}$ in the absence of escape from SNRs.