Relativistic Shock Acceleration: A Hartree-Fock Approach

TL;DR

This paper introduces a Hartree-Fock approach to analyze particle acceleration at relativistic shocks, deriving a simple equation for the power-law index and providing detailed distributions, validated against numerical solutions.

Contribution

It presents a novel Hartree-Fock method to approximate eigenfunctions in relativistic shock acceleration, simplifying the calculation of the power-law index and distributions.

Findings

Derived a transcendental equation for the power-law index

Provided angular and spatial distributions upstream of the shock

Validated results against numerical eigenfunction solutions

Abstract

We examine the problem of particle acceleration at a relativistic shocks assuming pitch-angle scattering and using a Hartree-Fock method to approximate the associated eigenfunctions. This leads to a simple transcendental equation determining the power-law index, $s$, given the up and downstream velocities. We compare our results with accurate numerical solutions obtained using the eigenfunction method. In addition to the power-law index this method yields the angular and spatial distributions upstream of the shock.