PAMELA and AMS-02 $e^{+}$ and $e^{-}$ spectra are reproduced by three-dimensional cosmic-ray modeling

The PAMELA collaboration recently released the $e^{+}$ absolute spectrum between 1 and 300 GeV in addition to the positron fraction and the $e^{-}$ spectrum previously measured in the same period. We use the newly developed three-dimensional upgrade of the dragon package to model those data. This code allows us to consider a realistic spiral-arm source distribution in the Galaxy, which impacts the high-energy shape of the propagated spectra. At low energy we treat solar modulation with the HelioProp code and compare its results with those obtained using the usual force-field approximation. We show that all PAMELA data sets can be consistently, and accurately, described in terms of a standard background on top of which a charge symmetric $e^{+}+e^{-}$ extra component with harder injection spectrum is added; this extra contribution is peaked at $\sim 1–10\mathrm{TeV}$ and may originate from a diffuse population of sources located in the Galactic arms. For the first time, we compute the energy required to sustain such a relevant positron flux in the Galaxy, finding that it is naturally compatible with an astrophysical origin. We considered several reference propagation setups; we find that models with a low (or null) reacceleration—tuned against light nuclei data—nicely describe both PAMELA leptonic and hadronic data with no need to introduce a low-energy break in the proton and Helium spectra, as it would be required for high reacceleration models. We also compare our models with the preliminary $e^{-}$ and $e^{+}$ absolute spectra recently measured by AMS-02. We find that those data, differently from what is inferred from the positron fraction alone, favor a high energy cutoff $\sim 10\mathrm{TeV}$ of the extra component if this is uniquely generated in the Galactic arms. A lower cutoff may be allowed if a relevant contribution from powerful $e^{-}+e^{+}$ nearby accelerators (e.g., one or few pulsars) is invoked.

Figure 1

The $\mathrm{B}/\mathrm{C}$ ratio computed with dragon for the three propagation setups considered in this paper and modulated in the force-field approximation ($\mathrm{\Phi }=0.5\mathrm{GV}$) are compared with PAMELA [19] and AMS-02 [20] preliminary experimental data.

Figure 2

The proton and helium spectra computed with dragon for the three propagation setups considered in this paper and modulated in the force-field approximation are compared with PAMELA and AMS-02 experimental data. The local interstellar spectrum is also shown. The PAMELA data reported in this, and the following figures of this paper are extracted from the Cosmic-Ray Database (http://lpsc.in2p3.fr/cosmic-rays-db/) [21].

Figure 3

The face-on view of the primary electron density at 100 GeV on the Galactic plane as computed with dragon is represented (arbitrary units).

Figure 4

The $e^{-}$ (purple solid) and $e^{+}$ (red solid) spectra propagated with the KRA setup, $E_{\text{cut}}=10\mathrm{TeV}$, and modulated with HelioProp are compared with PAMELA experimental data. The residuals of the model with respect to the data are shown in the lower panel.

Figure 5

Red (orange) and purple (blue) lines, respectively, represent the $e^{+}$, $e^{-}$ spectra computed for the extra component cutoff energy $E_{\text{cut}}=10(1)\text{TeV}$. Solid, dotted-dashed, and dashed lines represent charge-dependent modulated, force-field modulated, and interstellar (unmodulated) spectra, respectively. The solid grey lines have been obtained with a modulation setup suited for the AMS-02 data taking period. Red and black dotted lines represent the modulated $e^{+}$ background and extra component, respectively. From top to bottom, the three panels refer to PF, KRA, and KOL models, respectively.

Figure 6

The positron fraction corresponding to the models described in Sec. 2 is compared with PAMELA and AMS-02 data. From top to bottom, the three panels have been obtained for the PF, KRA, and KOL models, respectively. The line notation is the same as that of Fig. 5.

Figure 7

The $e^{-}$ and $e^{+}$ spectra (upper panel) and the positron fraction (lower panel) obtained by adding to the KRA model components the $e^{-}$ contribution due to the Vela supernova remnant (dashed line) are shown. The line notation is the same as that of Fig. 5. Only modulated spectra, under the condition suited for the PAMELA data taking period, are shown.

Figure 8

The $e^{-}$ and $e^{+}$ positron spectra (upper panel) and the positron fraction (lower panel) obtained by adding to the component shown in Fig. 7 the contribution of Monogem and Geminga pulsars (triple-dotted-dashed lines). The cutoffs $E_{\text{cut}}=1\mathrm{TeV}$ for the Galactic extra component and $E_{\text{PSR}}=1.2\mathrm{TeV}$ for those pulsars are adopted. The line notation is the same as that of Fig. 5.