ENERGY SPECTRUM OF NONTHERMAL ELECTRONS ACCELERATED AT A PLANE SHOCK

We calculate the energy spectra of cosmic ray (CR) protons and electrons at a plane shock with quasi-parallel magnetic fields, using time-dependent, diffusive shock acceleration (DSA) simulations, including energy losses via synchrotron emission and Inverse Compton (IC) scattering. A thermal leakage injection model and a Bohm type diffusion coefficient are adopted. The electron spectrum at the shock becomes steady after the DSA energy gains balance the synchrotron/IC losses, and it cuts off at the equilibrium momentum $p_{eq}$. In the postshock region the cutoff momentum of the electron spectrum decreases with the distance from the shock due to the energy losses and the thickness of the spatial distribution of electrons scales as $p^{-1}$. Thus the slope of the downstream integrated spectrum steepens by one power of p for $p_{br}$ < p < $p_{eq}$, where the break momentum decreases with the shock age as $p_{br}\;{\infty}\;t^{-1}$. In a CR modified shock, both the proton and electron spectrum exhibit a concave curvature and deviate from the canonical test-particle power-law, and the upstream integrated electron spectrum could dominate over the downstream integrated spectrum near the cutoff momentum. Thus the spectral shape near the cutoff of X-ray synchrotron emission could reveal a signature of nonlinear DSA.