4 Maximum energy of accelerated particles

The energy attainable by Fermi acceleration at the external shock is limited by the requirement that the time to deflect the particle upstream must be shorter than the age of the fireball. This yields a maximum value  

$$E \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\h... ...-6}\: \eta_3^{1/3} {\cal E}_{52}^{1/3} n_0^{-1/3}\: \mbox{eV},$$ (2)

where Z is the particle's charge, ${\cal E}\equiv {\cal E}_{52}\, 10^{52} \mbox{erg}$ is the fireball energy, and $B \equiv B_{-6}\, 10^{-6} \,\mbox{G}$and $n \equiv n_0\, \mbox{cm}^{-3}$ are the surrounding medium magnetic field and density (Gallant & Achterberg 1998, 1999). For electrons, the deflection time must also be shorter than the radiative loss time, but the age limitation turns out to be more restrictive.

Given Galactic magnetic fields, Eq. (2) rules out the acceleration of UHECRs by repeated shock crossings at the external blast waves of GRB fireballs. Nonetheless, the initial boost by a factor $\sim\! \Gamma_{\rm sh}^2$ could yield UHECRs if there were relativistic particles of sufficient energy upstream, as it requires only the time for these particles to be deflected downstream, where the magnetic field could be amplified by turbulence. If the seed relativistic particles are cosmic rays typical of the Galactic ISM, however, this process is much too inefficient to account for the observed UHECRs.

Gallant & Achterberg (1998, 1999) instead proposed that in the context of the neutron star binary merger scenario for GRBs (e.g. Narayan et al. 1992), the fireball expands into a pulsar wind bubble blown in the ISM by the progenitor system. Because the energy density of the surrounding medium is then predominantly in the form of relativistic particles, these can be boosted by the blast wave with very high efficiency.

Gallant & Achterberg (1999) also showed that for parameters typical of the millisecond pulsars in the neutron star binaries observed in our Galaxy, the GRB blast wave would decelerate within the pulsar wind bubble, yielding a spectrum ${\rm d}N / {\rm d}E \propto E^{-2}$ for the boosted particles. Moreover, this spectrum would typically extend over the energy region , which is precisely where the UHECR component is observed.

Acknowledgements

This work was supported by ASTRON project 781-71-050 and by the European Commission under TMR programme contract FMRX-CT98-0168.