Imprint of a 2 Myr old source on the cosmic ray anisotropy

We study numerically the anisotropy of the cosmic ray (CR) flux emitted by a single source calculating the trajectories of individual CRs. We show that the contribution of a single source to the observed anisotropy is instead determined solely by the fraction the source contributes to the total CR intensity, its age and its distance,and does not depend on the CR energy at late times. Therefore the observation of a constant dipole anisotropy indicates that a single source dominates the CR flux in the corresponding energy range. A natural explanation for the plateau between 2--20 TeV observed in the CR anisotropy is thus the presence of a single, nearby source. For the source age of 2 Myr, as suggested by the explanation of the antiproton and positron data from PAMELA and AMS-02 through a local source [arXiv:astro-ph/1504.06472], we determine the source distance as $\sim 200$ pc. Combined with the contribution of the global CR sea calculated in the escape model, we can explain qualitatively the data for the dipole anisotropy. Our results suggest that the assumption of a smooth CR source distribution should be abandoned between 200 GeV and 1 PeV.Comment: 4 pages, 4 eps figures; v2: minor changes, to appear in ApJ

On (Sub)stochastic and Transient Weightings of Infinite Strong Digraphs

In the present paper, for a given (possibly, infinite) strongly connected digraph $\cal{D},$ we consider the class $\cal{S}_{<}({\cal D})$ of all truthly substochastic weightings of ${\cal D}$ (here, the word "truthly" means that there exists a vertex whose out-weight is strictly less than $1$). For a finite subdigraph $\cal{F}$ of $\cal{D}$ weighted by $S\in {\cal S}_{<}({\cal D}),$ let $\ell_{max}(\cal{F})$ be the length of its longest directed cycle and $\lambda_{S}(\cal{F})$ be the Perron root (spectral radius) of its weighted adjacency matrix. We prove that the infimum of $\ell_{max}(\cal{F})\bigl(1-\lambda_{S}(\cal{F})\bigr)$ taken over all $\cal{F}$ is positive for every $S\in \cal{S}_{<}({\cal D})$ if and only if $\cal{D}$ admits a finite cycle transversal. The result obtained provides general theorems on the set ${\cal T}({\cal D})$ of transient weightings of ${\cal D}.$ In particular, we present a theorem of alternatives for finite approximations to elements of ${\cal T}({\cal D})$ and simply reprove V. Cyr's criterion for ${\cal T}({\cal D})$ to be empty