Shock acceleration of electrons in the presence of synchrotron losses: I. test particle theory

TL;DR

This paper presents a semi-analytical solution for electron acceleration at shocks considering synchrotron losses, analyzing how different diffusion models affect the electron spectrum and its cutoff shape.

Contribution

It provides a novel semi-analytical framework for electron spectra at shocks with synchrotron losses, applicable to various diffusion regimes and including cutoff shape analysis.

Findings

Derived spectra for arbitrary diffusion coefficients

Identified how diffusion models influence spectral cutoff shape

Applicable to shocks with upstream precursor effects

Abstract

We discuss a semi-analytical solution of the transport equation for electrons at a non-relativistic shock in the presence of synchrotron energy losses. We calculate the spectrum of accelerated (test) particles at any point upstream and downstream of the shock for an arbitrary diffusion coefficient and we specialize the results to three cases: 1) diffusion constant in momentum ($D(p)=D_{0}$), 2) Bohm diffusion ($D(p)\propto p$), and 3) Kolmogorov diffusion ($D(p)\propto p^{1/3}$). Of special importance is the determination of the shape of the cutoff in the electron spectrum which depends on the diffusion properties felt by particles in the shock region. The formalism can be generalized to the case of a shock with an upstream precursor induced by the dynamical reaction of accelerated particles.