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.

Baryon Loading Efficiency and Particle Acceleration Efficiency of Relativistic Jets: Cases for Low Luminosity Bl Lacs

“…Largeangle scattering in strong turbulent fields would allow the index harder as p 1 ∼ 1 (e.g. Stecker 2007;Aoi et al 2008;Summerlin & Baring 2012), although our results indicate η g ∼ 10 4.5 implying weak turbulence.…”

Linking high-energy cosmic particles by black-hole jets embedded in large-scale structures

“…References [77,81] find a main minimum and a second minimum in their fitting. The main minimum requires a hard injection spectral index s = 0.9 (or 1.61 with the EGMF), which is different from typical expectations from the diffusive shock acceleration mechanism (although such a hard spectrum could be reached by the shock acceleration at ultrarelativistic shocks [91,92] or other mechanisms such as the shear acceleration [93]). The second minimum suggests s = 2.0 (or 2.3 with the EGMF) and is similar to the mixed composition scenario shown here, but with a σ(X max ) lighter than the Auger measurements between 10 18.8 eV and 10 19.4 eV.…”

High-energy cosmic ray nuclei from tidal disruption events: Origin, survival, and implications

“…However, the spectral index can be affected by the deflection angle, the ratio of the scattering mean free path to the particle gyroradius and the orientation of the magnetic field to the shock normal. In the large-angle scattering case, where magnetic fluctuations are sufficiently large, particles can gain significant energy in the single scattering and may lead to a harder spectrum [75][76][77]. Such a possibility has been discussed to explain a hard spectrum of blazars [77,78].…”