Numerical studies of diffusive particle acceleration in supernova remnants

We have examined the time-dependent acceleration of cosmic ray particles by the Fermi, first-order diffusive shock process in supernova remnants. The convection-diffusion transport equation for the cosmic ray particles has been numerically solved self-consistently with the gas dynamic equations for the underlying flow. A model SNR characterized by an explosion energy of 10⁵¹ ergs, an ejected mass of 10 M and an ISM gas density of 3 × 10^-3 cm^-3 is followed through the adiabatic, Sedov-Taylor phase. A wide range of diffusion coefficients is considered. From our numerical results, we conclude that the observed spectrum of galactic cosmic rays; namely, a power law with q ∼ 4.2-4.3 extending up to 10¹⁴ eV, may be explained by diffusive acceleration in SNR if the diffusion coefficient is as small as the Bohm limit. At the same time, the cosmic rays can absorb about 30 per cent of the explosion energy. That is apparently enough to replenish the galactic cosmic rays. Although the dynamical feedback of the cosmic ray pressure modifies the shock structure to some degree, the shock is still dominated by the gas pressure at late times, so the temperature of the postshock gas should be large enough to agree with X-ray observations of SNRs. We also briefly discuss the time evolution of cosmic rays and gas flow within plane-parallel shocks.