Anisotropic Diffusion Modeling of Cosmic-Ray Lepton Propagation

TL;DR

The work addresses how fully anisotropic, spatially varying diffusion in a realistic Galactic magnetic field shapes the propagation of cosmic-ray leptons and their observed spectra. It computes the diffusion-tensor $\hat{D}_{ij}(\mathbf{r},E)$ by local test-particle integrations in a prescribed GMF, rotates to Galactic coordinates, and solves the stationary transport equation with energy losses on a 4D grid, explicitly incorporating $D_{\parallel}$ and $D_{\perp}$. Compared to isotropic diffusion, the fully anisotropic model (Model A) fits the DAMPE/H.E.S.S. and AMS-02 data with an injection index $\Gamma \approx 2.169$ and shows a reduced need for hard injection spectra, while producing a propagation horizon elongated along the regular field. This approach demonstrates that spatially varying anisotropic diffusion is crucial for consistent interpretation of lepton spectra and diffuse emission, and provides a framework tied to GMF models and energy-loss physics, with data and code availability on request.

Abstract

We analyze DAMPE and H.E.S.S. measurements of the total cosmic-ray electron-positron spectrum, together with the AMS-02 positron fraction, using an anisotropic, spatially varying diffusion framework. The diffusion-tensor components are computed via numerical integration of test-particle trajectories in a prescribed Galactic magnetic-field model. We show that accounting simultaneously for the spatial dependence and anisotropy of the diffusion tensor yields an accurate description of the local electron and positron data up to TeV energies. The inferred injection spectral index, $Γ=-2.169$, is fully consistent with expectations from diffusive shock-acceleration theory. Within this framework, the observed spectral softening arises naturally from enhanced energy losses experienced by leptons propagating over larger distances along the regular magnetic field.

Figures (4)

• Figure 1: Energy-loss rate as a function of lepton energy at the Solar position. The magnetic field is taken from model Unger_2024, ISRF energy densities from Ref. Delahaye2010, and radiative and Coulomb loss formulae from Ref. BlumenthalGould1970.
• Figure 2: Fit to the combined $e^-+e^+$ spectrum using DAMPE dampe2017, H.E.S.S.hess2024 data for the total lepton flux and AMS-02 ams02electrons2014 data for the positron fraction over the energy interval (0.04–60) TeV. The upper panel shows the best–fit spectrum, and the lower panel shows the residuals normalized to $\sigma$. DAMPE data points overlapping with the H.E.S.S. energy range, as well as H.E.S.S. points with only upper energy bounds, were excluded from the fit.
• Figure 3: Spatial distribution of CR leptons in Model A (anisotropic diffusion) at 15 GeV (left) and 400 GeV (right). The model parameters correspond to the best fit in Table \ref{['tab:table2']}. The Solar position is marked with a star at $(x,y,z)=(-8.2,0.0,0.2)$ kpc. The ellipses indicated by gray dashed lines represent the approximate shape of the lepton propagation horizon.
• Figure 4: Comparison of $e^-+e^+$ distributions in Model A (anisotropic diffusion $\hat{D}$) and Model B (isotropic diffusion $D_0$) at 15 GeV (left) and 400 GeV (right). The profiles are shown along $x$ at locations a $(y=0.0,z=0.2)$ and b $(y=0.0,z=1.0)$, and along $z$ at locations c $(x=0.0,y=0.0)$ and d $(x=-8.2,y=0.0)$ kpc. The shaded regions indicate the characteristic lepton horizon. The Solar position is marked with a star at $(x,y,z)=(-8.2,0.0,0.2)$ kpc.