2.2. Plausible sources of UHECRs and the Hillas' plot

Following the main ideas behind the concept of Fermi's first order acceleration, when r_L approaches the accelerator size, it becomes very difficult to magnetically confine the CR to the acceleration region, and thus to continue the accelerating process up to higher energies. If one includes the effect of the characteristic velocity c of the magnetic scattering centers ⁽³⁾, the above argument leads to the general condition (sometimes called the "Hillas criterion" [159]),

(14)

for the maximum energy acquired by a particle travelling in a medium with magnetic field B.

In the case of one-shot acceleration scenarios, the maximum reachable energy turns out to have a quite similar expression to the shock acceleration case of Eq. (14). For instance, a dimensional analysis suggests that the maximum energy that can be obtained from a pulsar is [159]

(15)

where is the pulsar angular velocity, B_s the surface magnetic field and R_s the neutron star radius. Therefore, if B_s ~ 10¹² G, R_s ~ 10 km, and ~ 60 s^-1 (as for the Crab pulsar), a circuit connected between pole and equator would see an emf ~ 10¹⁸ V for an aligned or oblique dipole. When realistic models of acceleration are constructed, however, this ideal dimensional limit is not fully realized, because the large potential drop along the magnetic field lines is significantly short-circuited by electron and positrons moving in the opposite directions along the field lines [160].

The dimensional arguments of Eqs. (14) and (15) are conveniently depicted in the "Hillas diagram" [159] shown in Fig. 4. Very few sites can generate particles with energies > 10²⁰ eV: either this occurs on highly condensed objects with huge B or enormously extended objects. Some of these potential astrophysical sources are discussed in what follows. For further details on the electrodynamical limitations of CR sources see, e.g. [164].

Figure 4. The Hillas diagram showing (chain curves) magnetic field versus gyroradius for proton momenta 1015, 1016, ..., 1024 eV / c. The solid curves correspond to different shock-waves velocities: the upper solid curve indicates the maximum attainable energy = 1, the middle and lower solid curves indicate plausible less effective acceleration processes. Typical size and magnetic field of possible acceleration sites are shown for neutron stars (ns), white dwarfs (wd), sunspots (ss), magnetic stars (ms), active galactic nuclei (ag), interstellar space (is), SNRs (sn), radio galaxy lobes (rg), galactic disk (d) and halo (h), clusters of galaxies (cl) and intergalactic medium (ig). Typical jet-frame parameters of the synchrotron proton blazar model [161] and gamma ray burst model [162] are indicated by open squares labeled "bl" and "gb", respectively [163].

³ The size of the accelerating region containing the magnetic field should be as large as 2r_L. Taking into account a characteristic velocity c of the scattering centers this transforms into 2r_L / . Back.