Constraints on the Extremely High-Energy Cosmic Ray Accelerators from Classical Electrodynamics

We find the general requirements, set by classical electrodynamics, to 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 field 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 accelerator's size and magnetic field strength results in the ultimate lower limit to the overall source energy, which scales as the fifth power of attainable particle energy. Hard gamma-rays accompanying generation of EHECRs can be used as a probe for potential acceleration sites. We apply the results to several populations of astrophysical objects -- potential EHECR sources -- and discuss their ability to accelerate protons to $10^{20}$ eV and beyond. A possibility to gain from ultrarelativistic bulk flows is emphasized, with Active Galactic Nuclei and Gamma-Ray Bursts being the examples.