Energy Spectrum Of Nonthermal Electrons Accelerated At A Plane Shock

TL;DR

This paper models the energy spectra of cosmic ray electrons and protons accelerated at a plane shock, incorporating energy losses and nonlinear effects, revealing spectral features that could inform observations of X-ray synchrotron emission.

Contribution

It provides a detailed simulation of electron and proton spectra at shocks, including energy losses and nonlinear DSA effects, highlighting spectral signatures near the cutoff.

Findings

Electron spectrum reaches a steady state with a cutoff at equilibrium momentum.

Downstream spectrum steepens by one power of p between break and cutoff.

Spectral shape near cutoff can indicate nonlinear DSA effects.

Abstract

We calculate the energy spectra of cosmic ray (CR) protons and electrons at a plane shock with quasi-parallel magnetic fields, using time-dependent, diffusive shock acceleration (DSA) simulations, including energy losses via synchrotron emission and Inverse Compton (IC) scattering. A thermal leakage injection model and a Bohm type diffusion coefficient are adopted. The electron spectrum at the shock becomes steady after the DSA energy gains balance the synchrotron/IC losses, and it cuts off at the equilibrium momentum p_{eq}. In the postshock region the cutoff momentum of the electron spectrum decreases with the distance from the shock due to the energy losses and the thickness of the spatial distribution of electrons scales as p^{-1}. Thus the slope of the downstream integrated spectrum steepens by one power of p for p_{br}<p<p_{eq}, where the break momentum decrease with the shock age as p_{br}\propto t^{-1}. In a CR modified shock, both the proton and electron spectrum exhibit a concave curvature and deviate from the canonical test-particle power-law, and the upstream integrated electron spectrum could dominate over the downstream integrated spectrum near the cutoff momentum. Thus the spectral shape near the cutoff of X-ray synchrotron emission could reveal a signature of nonlinear DSA.