Signatures of the transition from galactic to extragalactic cosmic rays

We discuss the signatures of the transition from galactic to extragalactic cosmic rays in different scenarios, giving the most attention to the dip scenario. The dip is a feature in the diffuse spectrum of ultrahigh-energy protons in the energy range $1\times {10}^{18}–4\times {10}^{19}\mathrm{eV}$, which is caused by electron-positron pair production on the cosmic microwave background radiation. The dip scenario provides a simple physical description of the transition from galactic to extragalactic cosmic rays. Here we summarize the signatures of the pair-production dip model for the transition, most notably the spectrum, the anisotropy, and the chemical composition. The main focus of our work is, however, on the description of the features that arise in the elongation rate and in the distribution of the depths of shower maximum $X_{\max }$ in the dip scenario. We find that the curve for $X_{\max }(E)$ shows a sharp increase with energy, which reflects a sharp transition from an iron-dominated flux at low energies to a proton-dominated flux at $E\sim {10}^{18}\mathrm{eV}$. We also discuss in detail the shape of the $X_{\max }$ distributions for cosmic rays of given energy and demonstrate that this represents a powerful tool to discriminate between the dip scenario and other possible models of the transition.

Figure 1

Predicted dip in comparison with the AGASA [48], HiRes [49], Yakutsk [50], and Auger [51] data. The latter are presented as hybrid data, shown by circles, and combined data (surface detector data above 4.5 EeV and fluorescence data below), shown by triangles. The comparison of the dip with Auger data is taken from Ref. 29.

Figure 2

Left panel: the second-knee transition. The extragalactic proton spectrum is shown for the $E^{-2.7}$ generation spectrum and for propagation in a magnetic field with $B_c=1\mathrm{nG}$ and $l_c=1\mathrm{Mpc}$, with the Bohm diffusion at $E\lesssim E_c$. The distance between sources is $d=50\mathrm{Mpc}$. $E_b=E_{\mathrm{cr}}=1\times {10}^{18}\mathrm{eV}$ is the beginning of the transition, $E_{\mathrm{Fe}}$ is the position of the iron knee, and $E_{\mathrm{tr}}$ is the energy where the galactic and extragalactic fluxes are equal. The dash-dot line shows the power-law extrapolation of the KASCADE spectrum to higher energies, which in fact has no physical meaning, because of the steepening of the galactic spectrum at $E_{\mathrm{Fe}}$. Right Panel: the ankle transition, for the injection spectrum of extragalactic protons $E^{-2}$. In both cases the dashed line is obtained as a result of subtracting the extragalactic spectrum from the observed all-particle spectrum.

Figure 3

Average penetration depth ${\overline{X}}_{\max }$ (left panel) and the variance of $X_{\max }$ distribution ${\sigma }_{X_{\max }}$ (right panel) as functions of energy for protons (upper curves) and iron nuclei (lower curves) as calculated using QGSJET, QGSJET-II, and SIBYLL models—solid, dashed, and dotted lines, correspondingly.

Figure 4

Left panel: elongation rate for the dip scenario. Right panel: elongation rate for the ankle scenario. The three lines, which present the calculations, are labeled as in Fig. 3: solid, dashed, and dotted lines correspond to QGSJET, QGSJET-II, and SIBYLL models, respectively. The data points are the measurements of Fly’s Eye (stars) [37], HiRes-Mia (squares) [38], and HiRes (circles) [39] experiments.

Figure 5

Elongation rates for the dip scenario (right panel) and for that of the ankle (left panel) in comparison with the Auger data [40]. The three lines are labeled as in Fig. 3.

Figure 6

Predicted $X_{\max }$ distribution for the dip scenario (left panels) and for the ankle scenario (right panels) in different energy bins in comparison with Fly’s Eye data [45] (points).

Figure 7

Predicted $X_{\max }$ distribution for the dip scenario (left panels) and for the ankle scenario (right panels) in different energy bins in comparison with HiRes data (points).