Diffusive propagation of cosmic rays from supernova remnants in the Galaxy. II: anisotropy

In this paper we investigate the effects of stochasticity in the spatial and temporal distribution of supernova remnants on the anisotropy of cosmic rays observed at Earth. The calculations are carried out for different choices of the diffusion coefficient D(E) experienced by cosmic rays during propagation in the Galaxy. The propagation and spallation of nuclei (with charge 1 <= Z <= 26) are taken into account. At high energies (E > 1TeV) we assume that D(E) proportional to (E/Z)(delta), with delta = 1/3 and delta = 0.6 being the reference scenarios. The large scale distribution of supernova remnants in the Galaxy is modeled following the distribution of pulsars with and without accounting for the spiral structure of the Galaxy. Our calculations allow us to determine the contribution to anisotropy resulting from both the large scale distribution of SNRs in the Galaxy and the random distribution of the nearest remnants. The naive expectation that the anisotropy amplitude scales as delta(A) proportional to D(E) is shown to be a wild oversimplification of reality which does not reflect in the predicted anisotropy for any realistic distribution of the sources. The fluctuations in the anisotropy pattern are dominated by nearby sources, so that predicting or explaining the observed anisotropy amplitude and phase becomes close to impossible. Nevertheless, the results of our calculations, when compared to the data, allow us to draw interesting conclusions in terms of the propagation scenario to be preferred both in terms of the energy dependence of the diffusion coefficient and of the size of the halo. We find that the very weak energy dependence of the anisotropy amplitude below 10(5) GeV, as observed by numerous experiments, as well as the rise at higher energies, can best be explained if the diffusion coefficient is D(E) proportional to E-1/3. Faster diffusion, for instance with delta = 0.6, leads in general to an exceedingly large anisotropy amplitude. The spiral structure introduces interesting trends in the energy dependence of the anisotropy pattern, which qualitatively reflect the trend seen in the data. The inhomogeneous spatial distribution of the sources in the Galactic disc induces a large scale anisotropy which is not sensitive to the stochastic nature of nearby SNRs: we find that this additional contribution to delta(A) becomes more important for large values of the size of the halo, H. The two terms are comparable in size for H similar to 2 kpc which corresponds to the scale height of the gradient of the spatial distribution of SNRs in the Galaxy. The dependence on energy of delta(A)(E) is close to monotonic when the large-scale, regular term dominates, and does not seem to reflect the observed anisotropy amplitude. Both contributions to the total anisotropy are illustrated and discussed with the help of semi-analytical results.