Global Constraints on Diffusive Particle Acceleration by Strong Nonrelativistic Shocks

Estimating the cosmic-ray acceleration efficiency $\epsilon$ in supernova remnants (SNRs) through observations is a challenging task in general. Based on the Rankine-Hugoniot shock conditions, we find an anticorrelation between $\epsilon$ and the power-law spectral index $\alpha$ of relativistic particle distribution produced via diffusive particle acceleration by nonrelativistic shocks, implying more efficient acceleration in older SNRs with harder radio spectra. Then $\epsilon$ may be estimated from some hard radio spectral index measurements. Assuming the particle distribution in downstream of strong shocks to be a nonrelativistic Maxwellian plus a relativistic power law with a high-energy cutoff, we also find that the injection rate for relativistic particles $\eta$ needs to $\gtrsim 10^{-6}$ for a prominent decrease of the adiabatic index in SNRs, which implies higher compression ratio and lower values of $\alpha$. This threshold of $\eta$ increases with the shock speed $u_1$, which may explain the relatively harder radio spectra of older SNRs with lower $u_1$. We show that $\eta$ and/or the relativistic cutoff momentum $p_\text{m}$ need to be low for old SNRs, and expect a gradual increase of $\epsilon$ as SNR evolves with gradually decreasing $\eta$ and $p_\text{m}$.