Filamentary Diffusion of Cosmic Rays on Small Scales

We investigate the diffusion of cosmic rays (CRs) close to their sources. Propagating individual CRs in purely isotropic turbulent magnetic fields with maximal scale of spatial variations $l_{\max }$, we find that CRs diffuse anisotropically at distances $r\lesssim l_{\max }$ from their sources. As a result, the CR densities around the sources are strongly irregular and show filamentary structures. We determine the transition time $t_{*}$ to standard diffusion as $t_{*}\sim {10}^4\mathrm{yr}(l_{\max }/150\mathrm{pc}{)}^{\beta }(E/\mathrm{PeV}{)}^{-\gamma }(B_{\mathrm{rms}}/4\mu \mathrm{G}{)}^{\gamma }$, with $\beta \simeq 2$ and $\gamma =0.25–0.5$ for a turbulent field with a Kolmogorov power spectrum. We calculate the photon emission due to CR interactions with gas and the resulting irregular source images.

Figure 1

Eigenvalues $d_i$ (solid lines) of the diffusion tensor $D_{ij}=⟨x_i{x}_j⟩/(2t)$ together with the average diffusion coefficient $D$ (dashed line) as function of time $t$. For $B_{\mathrm{rms}}=4\mu \mathrm{G}$, $l_{\max }=150\mathrm{pc}$, $\alpha =5/3$, and CR energy $E={10}^{15}\mathrm{eV}$.

Figure 2

Relative cosmic ray densities around their source projected in the panel planes, for energies $E=100\mathrm{TeV}$ (upper row), 1 PeV (middle row), 10 PeV (lower row), and times $t=500\mathrm{yr}$ (left column), 2 kyr (middle column), and 7 kyr (right column). Same field realization in each panel. Each panel corresponds to a $600\mathrm{pc}\times 400\mathrm{pc}$ field of view, with the source located in the center.

Figure 3

Eigenvalues $d_i^{(b)}$ of the diffusion tensor $D_{ij}$ as a function of time $t$ for energies $E=100\mathrm{TeV}$ (left), 1 PeV (middle), and 10 PeV (right). Same field realization as in Fig. 2. Dashed lines for the average diffusion coefficient $D^{(b)}\simeq D$.

Figure 4

Left panel: Relative cosmic ray densities along the lines of sight as seen from a specific observer located 500 pc away from the source. $E=1\mathrm{PeV}$ and $t=1\mathrm{kyr}$ for a given magnetic field realization. Box size is $20°\times 20°$, with the source at (0,0). Right panel: Corresponding relative surface brightness in $\gamma$ rays with energies $E_{\gamma }\ge 100\mathrm{GeV}$.