Radio pulses from cosmic ray air showers - Boosted Coulomb and Čerenkov fields

High-energy cosmic rays passing through the Earth's atmosphere produce extensive showers whose charges emit radio frequency pulses. Despite the low density of the Earth's atmosphere, this emission should be affected by the air refractive index because the bulk of the shower particles move roughly at the speed of radio waves, so that the retarded altitude of emission, the relativistic boost, and the emission pattern are modified. In this paper we consider the contribution of the boosted Coulomb and the Cerenkov fields and calculate analytically the spectrum using a very simplified model in order to highlight the main properties. We find that typically the lower half of the shower charge energy distribution produces a boosted Coulomb field, of amplitude comparable to the levels measured and to those calculated previously for synchrotron emission. Higher energy particles instead produce a Cerenkov-like field, whose amplitude may be smaller because both the negative charge excess and the separation between charges of opposite signs are small at these energies.