[Paper Review] Calculations of CR energy spectra within the NoRD model

V. V. Uchaikin, Renat T. Sibatov|arXiv (Cornell University)|Mar 19, 2017

Numerical methods in inverse problems18 references3 citations

TL;DR

This paper presents calculations of galactic cosmic ray (CR) energy spectra within the Nonlocal Relativistic Diffusion (NoRD) model, which incorporates relativistic particle speed limits and nonlocal diffusion in a turbulent interstellar medium. The key finding is that the 'knee' in the CR spectrum arises naturally from the relativistic speed limit, without requiring an artificial injection cutoff, and the NoRD model produces a power-law spectrum above the knee—consistent with observations—unlike models with exponential cutoffs.

ABSTRACT

Energy spectra of galactic cosmic rays calculated within the framework of the NoRD (nonlocal relativistic diffusion) model are presented. The model accounts for the turbulent character of the interstellar medium and the relativistic speed limit requirement. Calculations account for spiral distribution of sources, boundedness of halo, spallation of nuclei, energy dependence of the diffusion coefficient, tempered power law injection spectrum. We show that the knee in the background spectra from the ensemble of supernovae can arise due to relativistic speed limit requirement. We compare our calculations with the results obtained in frames of the local models (LoD and LoRD) and nonlocal nonrelativistic model (NoD).

Motivation & Objective

• To model galactic cosmic ray (CR) energy spectra using the Nonlocal Relativistic Diffusion (NoRD) model, which accounts for relativistic particle speeds and turbulent interstellar medium.
• To investigate how the relativistic speed limit affects CR propagation and spectral features, particularly the 'knee' in the energy spectrum.
• To compare the NoRD model with local (LoD, LoRD) and nonlocal nonrelativistic (NoD) models, assessing spectral steepening and source contributions.
• To determine whether the NoRD model can reproduce the observed power-law spectrum above the knee without requiring an artificial exponential cutoff in injection spectra.
• To validate the model using realistic source distributions, including discrete supernovae and a spiral-structured halo with bounded diffusion.

Proposed method

• The NoRD model uses a nonlocal, relativistic diffusion propagator with a Lévy-stable form, replacing the local Laplacian with a fractional operator Δ^{α/2} for superdiffusive transport (α ∈ (1,2]).
• The model incorporates a relativistic speed limit by restricting the propagator to the domain |r - rs| ≤ ct, introducing a ballistic front at the light-cone boundary.
• Spatial and temporal source distributions are modeled using 15 known young supernovae and 30,000 simulated sources with radial and vertical distributions based on pulsar data and spiral structure algorithms.
• Diffusion coefficient is energy-dependent: D(E) = D₀(R/3 GV)^δ, with δ = 0.6 and D₀ = 0.0729 pc²/yr, consistent with observational constraints.
• Boundary conditions are applied via the image method to model the finite halo thickness (2H = 8 kpc), with zero flux at z = ±H.
• Spallation effects are included using the formalism of Blasi and Amato, accounting for nuclear fragmentation in the interstellar medium.

Experimental results

Research questions

• RQ1 Does the inclusion of a relativistic speed limit in the NoRD model naturally produce the 'knee' in the cosmic ray energy spectrum without requiring an artificial injection cutoff?
• RQ2 How does the NoRD model's spectral shape above the knee compare to observational data, particularly in contrast to models with exponential injection cutoffs?
• RQ3 To what extent does the nonlocal, superdiffusive nature of the NoRD model alter the spectral steepening compared to local diffusion models (LoD, LoRD)?
• RQ4 Can the NoRD model reproduce observed CR spectra for protons and CNO-group nuclei using a physically motivated injection spectrum with γ ≈ 2.07?
• RQ5 How do the contributions of distant and nearby sources differ in shaping the CR spectrum under the NoRD framework?

Key findings

• The 'knee' in the CR spectrum emerges naturally from the relativistic speed limit in the NoRD model, without requiring an artificial cutoff in the injection spectrum.
• The NoRD model produces a power-law spectrum above the knee (E > E_knee), in agreement with observations, whereas models with exponential injection cutoffs predict a steeply falling tail.
• The energy of the knee (E_knee) in the NoRD model is in close agreement with that obtained in Blasi & Amato (2012) using an exponential injection cutoff, but the NoRD spectrum remains power-law above E_knee.
• The model successfully accounts for the observed spectral hardening and steepening using a physically consistent injection index γ ≈ 2.07, avoiding the need for unphysically large γ values (e.g., γ > 2.7) required in previous nonrelativistic NoD models.
• The NoRD model shows that sources beyond the relativistic reach (|r - rs| > ct) do not contribute to the spectrum at high energies, effectively excluding them and causing spectral steepening.
• Numerical results for protons and CNO-group nuclei across four models (LoD, LoRD, NoD, NoRD) confirm that the NoRD model best reproduces the observed power-law behavior above the knee.