Time-dependent Evolution of Proton Spectra in Supernova Remnants and Their Contribution to Galactic Cosmic Rays

Abstract: Recent $γ$-ray observations indicate that the proton spectra of supernova remnants (SNRs) are well described by broken power laws, with both the spectral break energy, $E_{\mathrm{br}}$, and the low-energy spectral index, $α$, exhibiting systematic evolution with SNR age. The physical origin of these evolutionary trends and their implications for the Galactic cosmic-ray (CR) population remain poorly understood. In this work, we develop the temporal evolution model for protons in SNRs by extending the semi-analytical framework of Zhang & Fang, in which both the maximum acceleration energy and the injection spectral index evolve with the dynamical evolution of the remnant. The calculated proton spectra reproduce the age-dependent trends of both $E_{\mathrm{br}}$ and $α$ inferred from observations. We adopt the proton spectrum at the onset of the radiative phase as the source spectrum for Galactic CR propagation and incorporate the intrinsic dispersion of source spectral indices among SNRs. The resulting cumulative Galactic proton spectrum is then calculated within a diffusion model. The propagated spectrum agrees well with the observed CR proton flux over a broad energy range, particularly above several tens of GeV. Our results provide a self-consistent framework linking the time-dependent evolution of proton acceleration in individual SNRs to the Galactic CR proton spectrum observed at Earth, and further support the long-standing hypothesis that SNRs are the dominant sources of Galactic CR protons below the knee.