COSMIC RAY SPECTRUM IN SUPERNOVA REMNANT SHOCKS

We perform kinetic simulations of diffusive shock acceleration (DSA) in Type Ia supernova remnants (SNRs) expanding into a uniform interstellar medium (ISM). Bohm-like diffusion due to self-excited $Alfv\acute{e}n$ waves is assumed, and simple models for $Alfv\acute{e}nic$ drift and dissipation are adopted. Phenomenological models for thermal leakage injection are considered as well. We find that the preshock gas temperature is the primary parameter that governs the cosmic ray (CR) acceleration efficiency and energy spectrum, while the CR injection rate is a secondary parameter. For SNRs in the warm ISM of $T_0\lesssim10^5K$, if the injection fraction is $\xi\gtrsim10^{-4}K$, the DSA is efficient enough to convert more than 20% of the SN explosion energy into CRs and the accelerated CR spectrum exhibits a concave curvature flattening to $E^{-1.6}$, which is characteristic of CR modified shocks. Such a flat source spectrum near the knee energy, however, may not be reconciled with the CR spectrum observed at Earth. On the other hand, SNRs in the hot ISM of$T_{0}\approx10^{6}K$ with a small injection fraction, $\xi$<$10^{-4}$, are inefficient accelerators with less than 10% of the explosion energy getting converted to CRs. Also the shock structure is almost test-particle like and the ensuing CR spectrum can be steeper than $E^{-2}$. With amplified magnetic field strength of order of $30{\mu}G$ $Alfv\acute{e}n$ waves generated by the streaming instability may drift upstream fast enough to make the modified test-particle power-law as steep as $E^{-2.3}$, which is more consistent with the observed CR spectrum.