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Description
It has been a long-standing paradigm that the bulk of Galactic comic-ray gets accelerated at supernova remnant shocks via diffusive shock acceleration. As observations – both direct and indirect – became better, many features appeared in the cosmic-ray spectra at Earth and in the emission spectra of remnants, that require deviations from the expectation of a simple power-law of accelerated particles with a power-law index of s=-2.
Here, we demonstrate how the presence of a density-gradient in the upstream of a shock can lead to a hardening or softening of the resulting particle distribution, depending on the direction of the gradient. We show analytically and with numerical simulations, that the spectral modification is proportional to τacc/τgra, where τacc is the acceleration timescale of particles and τgra the typical timescale over which the upstream-density changes. An increasing density leads to a softening of the spectrum.
A gradient in ambient density in this sense can be created by a change of the abundancy of an element, e.g. in the medium inside a stellar wind-bubble. Here, the burning of hydrogen over time leads to an increasing hydrogen-fraction from the vicinity of the star towards the edge of the wind-bubble. We show that such spatial gradients, which differ in strength between elements, lead to differing spectral indices between different particle species. For a typical wind-bubble around a massive star, the result is a softer proton spectrum compared to helium, as observed for Galactic cosmic-rays.
