2019MNRAS.482.5268Z

Estimating the cosmic-ray acceleration efficiency ε in supernova remnants (SNRs) through observations is a challenging task in general. Based on the Rankine-Hugoniot shock conditions, we find an anticorrelation between ε and the power-law spectral index α of relativistic particle distribution produced via diffusive particle acceleration by non-relativistic shocks, implying more efficient acceleration in older SNRs with harder radio spectra. Then ε may be estimated from some hard radio spectral index measurements. Assuming the particle distribution in downstream of strong shocks to be a non-relativistic Maxwellian plus a relativistic power law with a high-energy cut-off, we also find that the injection rate for relativistic particles η needs to >=10^–6 for a prominent decrease of the adiabatic index in SNRs, which implies higher compression ratio and lower values of α. This threshold of η increases with the shock speed u₁, which may explain the relatively harder radio spectra of older SNRs with lower u₁. We show that η and/or the relativistic cut-off momentum p_m need to be low for old SNRs, and expect a gradual increase of ε as SNR evolves with gradually decreasing η and p_m.