Constraints on the extremely high-energy cosmic ray accelerators from classical electrodynamics

We formulate the general requirements, set by classical electrodynamics, on the sources of extremely high-energy cosmic rays (EHECRs). It is shown that the parameters of EHECR accelerators are strongly limited not only by the particle confinement in large-scale magnetic fields or by the difference in electric potentials (generalized Hillas criterion) but also by the synchrotron radiation, the electro-bremsstrahlung, or the curvature radiation of accelerated particles. Optimization of these requirements in terms of an accelerator's size and magnetic field strength results in the ultimate lower limit to the overall source energy budget, which scales as the fifth power of attainable particle energy. Hard \ensuremath{\gamma} rays accompanying generation of EHECRs can be used to probe potential acceleration sites. We apply the results to several populations of astrophysical objects---potential EHECR sources---and discuss their ability to accelerate protons to ${10}^{20} \mathrm{eV}$ and beyond. The possibility of gain from ultrarelativistic bulk flows is addressed, with active galactic nuclei and gamma-ray bursts being the examples.