The effect of energy amplification variance on shock acceleration

Abstract

Shock-acceleration theory predicts a power-law energy spectrum in the test particle approximation, and there are two ways to calculate the power-law index, namely Peacock's approximation and Vietri's formulation. In Peacock's approximation, it is assumed that particles cross a shock front many times and that the energy gain factors for each step are fully uncorrelated. By contrast, correlation of the distribution of the energy-gain factors is considered in Vietri's formulation. We examine how Peacock's approximation differs from Vietri's formulation. It is useful to know when we can use Peacock's approximation, because in this approximation it is simple to derive the power-law index. In addition, we focus on how the variance of the energy-gain factor has an influence on the difference between Vietri's formulation and Peacock's approximation. The effect of the variance has not been examined in detail until now. As examples, we consider two cases for the scattering in the upstream region: large-angle scattering (model A), and regular deflection by large-scale magnetic fields (model B). In particular, there is no correlation among the distribution of an energy-gain factor for every step in model A. In this model, it can be seen that the power-law index derived from Peacock's approximation differs from that derived from Vietri's formulation when we consider a mildly relativistic shock, and the variance of the energy-gain factor affects this difference. We can use Peacock's approximation for a non-relativistic shock and a highly relativistic shock because the effect of the variance is hidden in these cases. In model B, we see the difference of the power-law index, which converges along the shock velocity.