Radiation formation length in astrophysical high brightness sources

The radiation formation length for relativistic particles, $l_c \sim γ^2 λ$ ($γ$ is the Lorentz factor, $λ$ is the emitted wavelength), is much lager than the inter-particle distances in many astrophysical applications. This leads to the importance of plasma effects even for the high energy emission. The consequences are nontrivial: (i) averaging of the phases of the emitting particles reduces the power (a.k.a., a circle current does not emit); (ii) density fluctuations may lead to the sporadic production of coherent emission; (iii) plasma effects during assembly of a photon may lead to the suppression of the emission (Razin-Tsytovich effect for the superluminal modes), or, in the opposite limit of subluminous normal modes, to the newly discussed synchrotron super-radiance. For synchrotron emission the radiation formation length is the same for all emitted waves, $\sim c/ω_B$ (non-relativistic Larmor length); for curvature emission it is $R/γ$ - macroscopically long in pulsar magnetospheres (e.g., kilometers for radio). The popular model of coherent curvature emission by bunches", with kilometers-long radiation formation length, particles swinging-out in a rotating magnetospheres before they finish emitting a wave, extreme requirements on the momentum spreads, and demands on the electric energy needed to keep the electrostatically repulsing charges together, all make that model internally inconsistent. Long radiation formation lengths affect how emission from PIC simulations should be interpreted: phases of the emitted wave should be added over the radiation formation length, not just the powers from the instantaneous acceleration of each particle.