Maximally-hard spectra from diffusive shock-acceleration in pulsar-wind nebulae

TL;DR

This paper investigates the origin of extremely hard radio spectra in pulsar-wind nebulae, proposing that diffusive shock acceleration with anisotropic scattering can produce spectra approaching the theoretical hard limit.

Contribution

It demonstrates that diffusive shock acceleration with small-angle, anisotropic scattering can produce the hardest possible spectra, approaching the alpha=0 limit, explaining the observed spectral features in PWNe.

Findings

Evidence for a sub-population of PWNe with near-zero spectral index.

Hard spectra are linked to anisotropic small-angle scattering at shocks.

Acceleration efficiency decreases significantly for extremely hard spectra.

Abstract

The processes leading to the exceptionally hard radio spectra of pulsar-wind nebulae (PWNe) are not yet understood. Radio photon spectral indices among $29$ PWNe from the literature show an approximately normal, $\alpha=0.2\pm0.2$ distribution. We present $\sim 3\sigma$ evidence for a distinct sub-population of PWNe, with a hard spectrum $\alpha=0.01\pm0.06$ near the termination shock and significantly softer elsewhere, possibly due to a recent evacuation of the shock surroundings. Such spectra, especially in the hard sub-population, suggest a Fermi process, such as diffusive shock acceleration, at its extreme, $\alpha=0$ limit. We show that this limit is approached for sufficiently anisotropic small-angle scattering, enhanced on either side of the shock for particles approaching the shock front. In the upstream, the spectral hardening is mostly associated with an enhanced energy gain, possibly driven by the same beamed particles crossing the shock. Downstream, the main effect is a diminished escape probability, but this lowers the acceleration efficiency to $\lesssim25\%$ for $\alpha=0.3$ and $\lesssim1\%$ for $\alpha=0.03$.