The Spectral Index of Accelerated Particles in Shock Waves with Oblique Magnetic Field

On the basis of Fermi acceleration in oblique shock waves, the spectral indices of accelerated particles are obtained. We adopt the test particle approximation and assume that particle distribution is isotropic in the local fluid frames. We calculate the spectral index from the average energy gain and the escape probability per one cycle incorporating magnetic mirror effects based on the adiabatic approximation. Our results apply to any value of shock speed and any obliqueness as far as the shock is subluminal. For a given obliqueness and the compression ratio of 4, the index decreases first with an increase of the shock speed, reaches a minimum value below 2.0, and then increase to infinity as the shock speed approaches the subluminal limit. As for the dependence on the obliqueness, the index decreases with an increase of the obliqueness reaching a minimum value around 1.84 in a highly oblique region and it increases to infinity at the subluminal limit.