Cosmic-ray Spectrum Steepening in Supernova Remnants. I. Loss-free Self-similar Solution

Abstract

The direct measurements of cosmic rays (CRs), after correction for the propagation effects in the interstellar medium, indicate that their source spectra are likely to be significantly steeper than the canonical E −2 spectrum predicted by the standard Diffusive Shock Acceleration (DSA) mechanism. The DSA has long been held responsible for the production of galactic CRs in supernova remnant (SNR) shocks. The γ-ray "probes" of the acceleration spectra of CRs on-the-spot, inside of the SNRs, lead to the same conclusion. We show that the steep acceleration spectrum can be attributed to the combination of (i) spherical expansion, (ii) tilting of the magnetic field along the shock surface and (iii) shock deceleration. Because of (i) and (ii), the DSA is efficient only on two "polar caps" of a spherical shock where the local magnetic field is within 45 • to its normal. The shock-produced spectrum observed edge-on steepens with the particle energy because the number of freshly accelerated particles with lower energies continually adds up to a growing acceleration region. We demonstrate the steepening effect by obtaining an exact self-similar solution for the particle acceleration at expanding shock surface with an arbitrary energy dependence of particle diffusivity κ. We show that its increase toward higher energy steepens the spectrum, which deeply contrasts with the standard DSA spectrum where κ cancels out.

Cosmic-ray acceleration and escape from post-adiabatic supernova remnants

“…Reacceleration of CRs at slow shocks may arise (Cardillo et al 2016), as does a spectral modification due to an enhanced escape-rate of CRs (Malkov et al 2011). Further, energy transfer from CRs to magnetic turbulence (Bell et al 2019) and deviations from spherical symmetry (Malkov & Aharonian 2019) might lead to softer CR-spectra.…”

The Issue with Diffusive Shock Acceleration

“…Geometry-dependent effects. It has been recently suggested [31] that a spectral steepening may appear also because particles are preferentially injected into DSA at quasi-parallel shocks [32,33]; in this case a steep CR spectrum would originate from the time convolution of different contributions from quasi-parallel and oblique shock regions, each of them with a specific normalization and maximum energy. This effect may disappear when the SNR diameter becomes larger than the coherence length of the background magnetic field, though.…”

Spot-like Proton Acceleration in SNR Shocks: Implications for the Energy Spectra

“…Compared to the case of a fixed acceleration area, freshly injected particles with lower energies are added to the growing acceleration zone at an area-integrated rate that increases with time. This enhanced production of low-energy particles naturally results in a steeper overall spectrum [14].The second steepening mechanism is associated with particle diffusion from the active acceleration zone into the neighboring oblique shock zone. As there is almost no particle confining turbulence there, these particles have a good chance to escape the accelerator.…”

“…To address the problem of steep spectra, different scenarios have been recently proposed [13,14]. The authors of [13] show that the steepened spectrum may be caused by the loss of CR energy due to turbulent magnetic field amplification during CR acceleration when the shock velocity is high.…”

“…The authors of [13] show that the steepened spectrum may be caused by the loss of CR energy due to turbulent magnetic field amplification during CR acceleration when the shock velocity is high. The scenario considered in [14] utilizes a well-known property of collisionless shocks to suppress the proton injection when the magnetic field ahead of the shock makes a large angle to its normal, typically θ Bn π/4. Two different spectrum steepening mechanisms can be associated with the variable magnetic field inclination.…”