# Electron and positron spectra in the three dimensional spatial-dependent propagation model

## TL;DR

This paper extends the spatial-dependent propagation model to electrons and positrons, explaining spectral hardening, positron excess, and anisotropy with spiral source distribution, providing a more consistent cosmic-ray description.

## Contribution

It introduces a spiral distribution of sources into the spatial-dependent propagation model, improving the explanation of cosmic-ray spectra and positron excess.

## Key findings

- Electron spectrum hardening above tens of GeV explained
- Positron excess above 10 GeV addressed with local source
- Electron anisotropy below observational limits

## Abstract

The spatial-dependent propagation model has been successfully used to explain diverse observational phenomena, including the spectral hardening of cosmic-ray nuclei above $200$ GV, the large-scale dipole anisotropy and the diffusive gamma distribution. In this work, we further apply the spatial-dependent propagation model to both electrons and positrons. To account for the excess of positrons above $10$ GeV, an additional local source is introduced. And we also consider a more realistic spiral distribution of background sources. We find that due to the gradual hardening above $10$ GeV, the hardening of electron spectrum above tens of GeV can be explained in the SDP model and both positron and electron spectra less than TeV energies could be naturally described. The spatial-dependent propagation with spiral-distributed sources could conforms with the total electron spectrum in the whole energy. Meanwhile compared with the conventional model, the spatial-dependent propagation with spiral-distributed sources could produce larger background positron flux, so that the multiplier of background positron flux is $1.42$, which is much smaller than the required value by the conventional model. Thus the shortage of background positron flux could be solved. Furthermore we compute the anisotropy of electron under spatial-dependent propagation model, which is well below the observational limit of Fermi-LAT experiment.