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We present a simple model of nonlinear di †usive shock acceleration (also called Ðrst-order Fermi shock acceleration) that determines the shock modiÐcation, spectrum, and efficiency of the process in the plane-wave, steady state approximation as a function of an arbitrary injection parameter, g. The model, which uses a threeÈpower-law form for the accelerated particle spectrum and contains only simple algebraic equations, includes the essential elements of the full nonlinear model and has been tested against Monte Carlo and numerical kinetic shock models. We include both adiabatic and wave heating Alfve n of the upstream precursor. The simplicity and ease of calculation make this model useful for studying the basic properties of nonlinear shock acceleration, as well as providing results accurate enough for many astrophysical applications. It is shown that the shock properties depend upon the shock speed with u 0 respect to a critical value which is a function of the injection rate g and maximum acceleru 0 * P gp max 1@4

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Supernova remnants (SNRs) are widely believed to be the principal source of galactic cosmic rays, produced by diffusive shock acceleration in the environs of the remnant's expanding blast wave. Such energetic particles can produce gamma-rays and lower energy photons via interactions with the ambient plasma. The recently reported observation of TeV gamma-rays from SN1006 by the CANGAROO Collaboration, combined with the fact that several unidentified EGRET sources have been associated with known radio/optical/X-ray-emitting remnants, provides powerful motivation for studying gamma-ray emission from SNRs. In this paper, we present results from a Monte Carlo simulation of non-linear shock structure and acceleration coupled with photon emission in shell-like SNRs. These non-linearities are a by-product of the dynamical influence of the accelerated cosmic rays on the shocked plasma and result in distributions of cosmic rays which deviate from pure power-laws. Such deviations are crucial to acceleration efficiency considerations and impact photon intensities and spectral shapes at all energies, producing GeV/TeV intensity ratios that are quite different from test particle predictions. The Sedov scaling solution for SNR expansions is used to estimate