Spectrum and acceleration of cosmic rays

The differential spectrum of CR is described by a power law of the form

dN/dE ∝E−α (1.37)

where the spectral index α varies slightly in the three main regions of the spectrum, which are separated by two peculiar structures, the so called knee at about 1015 _{eV and}

the ankle at some 1018 _{eV (e.g. Nagano & Watson, 2000). The spectrum has initially a}

spectral indexα=2.7, it then steepens after the knee withα=3.0 between the knee and the ankle, and finally it hardens again at energies above the ankle; the spectral index for the ultra high energy CR (UHECR) is not well defined due to the lack of data (Cronin, 1999). Between the knee and the ankle, there is increasing evidence for a third minor structure, the so called second knee, where the spectrum softens from an index of α=3.0 to α=3.3 (e.g. Gaisser, 2005).

A possible explanation for the nature of the knee is that this structure is associated with the upper limit for the energy that acceleration from shocks in SN explosions can supply to the particles. If this is the case, the knee represents a separation between particles of Galactic origin and the ones coming from the extragalactic universe (Peters, 1960). The relatively smooth behavior of the cutoff is due to the different cutoff energies of different nuclei, depending on the magnetic rigidity of the particle, i.e. R(Z)=Pc/Ze, which provides the scaling relation between the energy of the particle (E=Pc) and its charge (Ze) (Gaisser, 2010). In the scenario where the cutoff of the spectrum occurs at the characteristic rigidity

1.2 Cosmic rays 31

Figure 1.12: Left: size versus magnetic field of the possible sites of CR acceleration (Hillas, 1984; Cronin, 1999). Protons cannot be accelerated to energies above 1020 _{eV in sites}

under the dashed line. Right: Proton energy as a function of the propagation distance through the CMB; beyond 100 Mpc, all the three lines tent to an energy lower than 1020 eV (Cronin, 1999).

Rc, protons are the first to bend at Ec=eRc, and then come He nuclei at Ec=2eRc and so

on up to the heavier nuclei; the energy range occupied by this smooth cutoff is a factor of 30 (Gaisser, 2010). In the original idea by Peters, at this energy another component of extragalactic nature, with a harder spectrum, should dominate. In reality, what is shown by the observations is a softening of the spectrum, and a hardening only above the ankle (e.g. Gaisser, 2005). While CR above the ankle are known to have an extragalactic origin (region E-G in the right panel of Fig.1.11), the region between the knee and the ankle is the most controversial. The total power required to fill this region (region B in the right panel of Fig.1.11) can be calculated assuming a spectrum Q(E)∝E−2 _{and a different energy}

dependence of the propagation in this energy range, i.e. τesc ∝E−0.33. The result is a power

requirement for the filling of the B region of about 2×1039erg s−1, less than 10% of the total power requirement for all the galactic CR (Gaisser, 2005); the source of these CRs is still unknown.

Particles are not easy to accelerate up to 1020 eV. A basic condition for the acceleration of a proton to such a high energy is that the product of the magnetic field B and the size of the acceleration region R is higher than 3×1017 _{Gauss cm (Cronin, 1999). Hillas (1984)}

presented an illustrative plot which shows the difficulty of accelerating CR to 1020 eV. We present his original plot, revised by Nagano & Watson (2000) in the left panel of Fig.1.12.

The upper limit on the energy E can be estimated to be (Cronin, 1999)

where E18 is the maximum energy (in units of 1018 eV), Z the nuclear charge of the

particle, B the magnetic field of the acceleration source and L its size, and β=v/c is the velocity of the shock wave. The lines plotted in the left panel of Fig.1.12 correspond to a proton with E=1020 _{eV and different β factors. Efficient accelerators should lie above the}

solid β=1 line (Nagano & Watson, 2000). As seen in the plot, the only objects which can accelerate protons to these high energies are AGN, colliding galaxies and galaxy clusters. Another interesting effect which influences the energetics of the UHECR is their interaction with the CMB. At energies higher than 5×1019 eV, the cross section for the interaction of CR with CMB photons is high (Greisen, 1966; Zatsepin & Kuz’min, 1966). The right panel of Fig.1.12 shows the propagation of protons through the CMB; regardless of the initial energy of the proton, after a propagation of 100 Mpc (or longer) the proton kinetic energy will be lower than 1020 _{eV. This distance corresponds to a redshift of 0.025, and it}

is indeed very small when compared with cosmological distances. As a consequence, any CR proton recorded to have an energy higher than 1020 eV must have originated within 100 Mpc distance from the Earth, and there are not many sources in this distance range to satisfy the Hillas’ criteria. The authors of the original papers (Greisen, 1966; Zatsepin & Kuz’min, 1966) indeed claimed discovery of an upper limit for the spectrum of CR.