Maximally hard radio spectra from Fermi acceleration in pulsar-wind nebulae

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 literature show an approximately normal, $\alpha=0.2\pm0.2$ distribution. We present $\sim 3\sigma$ evidence for a distinct sub-population PWNe, with spectrum $\alpha=0.01\pm0.06$ near termination shock and significantly softer elsewhere, possibly due recent evacuation surroundings. Such spectra, especially in sub-population, suggest Fermi process, such as diffusive acceleration, at its extreme, $\alpha=0$ limit. that this limit is approached sufficiently anisotropic small-angle scattering, enhanced on either side particles approaching front. In upstream, hardening mostly associated energy gain, driven by same beamed crossing shock. Downstream, main effect diminished escape probability, but lowers acceleration efficiency $\lesssim25\%$ $\alpha=0.3$ $\lesssim1\%$ $\alpha=0.03$.