Acceleration of X-ray Emitting Electrons in the Crab Nebula

TL;DR

This paper models how electrons are accelerated to high energies at the Crab Nebula's termination shock, reproducing observed X-ray spectra through a turbulence-dependent power-law distribution.

Contribution

It introduces a detailed particle acceleration model involving drift motion and turbulence effects, explaining observed X-ray spectra in the Crab Nebula.

Findings

Power-law electron spectrum with index -1.8 to -2.4 depending on turbulence

Reproduces NuSTAR X-ray flux and photon index

Explains Chandra's hard photon index in the nebula core

Abstract

We study particle acceleration at the termination shock of a striped pulsar wind by integrating trajectories in a prescribed model of the magnetic field and flow pattern. Drift motion on the shock surface maintains either electrons or positrons on "Speiser" orbits in a ring-shaped region close to the equatorial plane of the pulsar, enabling them to be accelerated to very high energy by the first-order Fermi mechanism. A power-law spectrum results: $dN_{\rm e}/d\gamma\propto\gamma^{\alpha_{\rm e}}$, where $\alpha_{\rm e}$ lies in the range $-1.8$ to $-2.4$ and depends on the downstream turbulence level. For sufficiently strong turbulence, we find $\alpha_{\rm e} \simeq -2.2$, and both the photon index and the flux of $1-100$ keV X-rays from the Crab Nebula, as measured by NuSTAR, can be reproduced. The particle spectrum hardens to $\alpha_{\rm e} \simeq -1.8$ at lower turbulence levels, which may explain the hard photon index observed by the Chandra X-ray Observatory in the central regions of the Nebula.