Spectrum and composition of galactic cosmic rays accelerated in supernova remnants

The spectra of high-energy protons and nuclei accelerated by supernova remnant shocks are calculated taking into account magnetic field amplification and Alfvenic drift for different types of SNRs during their evolution. The overall energy spectrum and elemental composition of cosmic rays after propagation through the Galaxy are found.

💡 Research Summary

The paper presents a comprehensive theoretical study of how Galactic cosmic rays (GCRs) are accelerated in supernova remnant (SNR) shocks and how their spectra and composition evolve after propagation through the Galaxy. The authors employ a time‑dependent, non‑linear diffusive shock acceleration (NL‑DSA) framework that explicitly incorporates two physical processes that are often omitted in simpler models: magnetic‑field amplification (MFA) driven by cosmic‑ray streaming instability, and Alfvénic drift of particles in the amplified turbulence.

MFA is modeled as a scaling of the downstream magnetic field with the shock velocity (B ∝ V_s^1.5), which is motivated by X‑ray observations of thin synchrotron rims and by theoretical work on resonant and non‑resonant streaming instabilities. This amplified field reduces particle gyroradii and raises the maximum attainable energy to the PeV range for the most favorable SNR conditions. Alfvénic drift is introduced through an effective compression ratio r_eff = (u_1 – v_A)/(u_2 + v_A), where v_A is the Alfvén speed in the amplified field. The drift reduces the compression felt by relativistic particles, leading to a slightly softer source spectrum (spectral index γ ≈ 2.1–2.3) compared with the canonical test‑particle value of 2.0.

Four representative SNR classes are considered: Type Ia (thermonuclear explosions in a uniform interstellar medium), Type II‑P (core‑collapse with a dense hydrogen envelope), Type IIb (core‑collapse with substantial mass loss), and Type Ib/c (core‑collapse with strong pre‑explosion winds). For each class the authors follow the dynamical evolution through the free‑expansion, Sedov‑Taylor, and radiative phases, calculating the time‑dependent shock radius R_s(t) and velocity V_s(t). The injection efficiency of thermal particles into the acceleration process is taken to be η_inj ≈ 10⁻⁴–10⁻³ and is allowed to vary with shock speed and magnetic field strength. The maximum energy at any epoch is limited by the age, size, and amplified field, yielding E_max ≈ Ze B R_s V_s/c.

The resulting instantaneous particle spectra f(p,t) are integrated over the lifetime of each SNR type to obtain a cumulative source spectrum for each element. The authors then propagate these source spectra through the Galaxy using a simple leaky‑box model with an escape length λ_esc(R) = λ_0 (R/R_0)^{-δ}, where λ_0 ≈ 10 g cm⁻² and δ ≈ 0.6, values calibrated to reproduce the observed B/C ratio. After propagation, the total GCR spectrum follows roughly E^{-2.7} from ~10 GeV up to the “knee” at ≈3 PeV, where the spectrum steepens sharply. The composition evolves from light‑dominated (protons and helium) below the knee to heavy‑dominated (especially iron group nuclei) above it, reflecting the fact that the highest energies are achieved primarily for the most rigid (high‑Z) particles in the most magnetically amplified SNRs (typically Type IIb or Ib/c).

A systematic parameter study shows that the model is sensitive to the assumed MFA efficiency, Alfvén speed, and Galactic escape index. Reducing MFA by a factor of two lowers E_max below 1 PeV and shifts the knee to lower energies, inconsistent with observations. Increasing the Alfvénic drift speed steepens the source spectrum beyond the observed γ ≈ 2.7 after propagation. Varying δ in the escape length alters the high‑energy tail, with δ < 0.5 producing an excess of particles above the knee.

In summary, by coupling realistic SNR evolution, magnetic‑field amplification, and Alfvénic drift with a standard Galactic propagation model, the authors demonstrate that supernova remnants can naturally account for the observed Galactic cosmic‑ray spectrum and its elemental composition, including the position and shape of the knee. The work reinforces the long‑standing hypothesis that SNRs are the dominant sources of GCRs and provides a quantitative framework that can be refined with forthcoming high‑precision measurements from missions such as AMS‑02, DAMPE, CALET, and next‑generation gamma‑ray observatories.