Brightness temperature constraints on coherent processes in magnetospheres of neutron stars

We discuss constraints that the observed brightness temperatures impose on coherent processes in pulsars and Fast Radio Bursts (FRBs), and in particular on the hypothesis of coherent curvature emission by bunches. We estimate the peak brightness temperature that a bunch of charge $Ze$ can produce via synchrotron and/or curvature emission as $k_B T \sim (Z e)^2/\lambda$, where $\lambda$ is the typical emitted wavelength. We demonstrate that the bunch's electrostatic energy required to produce observed brightness temperature is prohibitively high, of the order of the total {\it bulk } energy. We compare corresponding requirements for the Free Electron Laser mechanism (Lyutikov 2021) and find that in that case the constraints are much easier satisfied.