The effect of energy amplification variance on the shock-acceleration

TL;DR

This paper investigates how the variance in energy amplification affects the accuracy of two theoretical models predicting particle energy spectra in shock acceleration, especially under different scattering conditions and shock velocities.

Contribution

It compares Peacock's approximation and Vietri's formulation, highlighting the impact of energy-gain variance on their differences across various shock regimes.

Findings

Peacock's approximation is accurate for non-relativistic and highly-relativistic shocks.

Variance in energy-gain influences the difference between models in mildly-relativistic shocks.

Model B shows convergence of the power-law index difference with increasing shock velocity.

Abstract

The shock-acceleration theory predicts a power-law energy spectrum in the test particle approximation, and there are two ways to calculate a power-law index, Peacock's approximation and Vietri's formulation. In Peacock's approximation, it is assumed that particles cross a shock front many times and energy-gains for each step are fully uncorrelated. On the other hand, correlation of the distribution of an energy-gain factor for a particle 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 Peacock's approximation 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 well until now. For demonstration, we consider two cases for the scattering in the upstream: the large-angle scattering (model A) and the regular deflection by large-scale magnetic fields (model B). Especially there is no correlation among the distribution of an energy-gain factor for every step in model A. In this model, we see the power-law index derived from Peacock's approximation differs from the one derived from Vietri's formulation when we consider the 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 model B, we see the difference of the power-law converging along the shock velocity.