Electromagnetic Radiation in the Tamm Problem

The 'Tamm Problem' is the calculation of the electromagnetic fields from a single particle travelling a finite distance at superluminal velocities in a medium. It was first addressed by Frank Tamm in 1939 as an extension to the Frank-Tamm theory of Vavilov-Cherenkov radiation from a particle moving an infinite distance in a medium. It is exactly the problem which must be solved in order to calculate the radio-emission from high-energy particle cascades simulated by numerical (Monte Carlo) methods, as is performed by the codes REAS, COREAS, and ZHAireS in the case of extensive air showers, and ZHS for cascades in a dense medium such as ice or the Lunar regolith. Despite its importance, the commonly-used solutions to the radiated fields in the Tamm problem - the 'ZHS' approach and 'endpoints' formalism - are not exact solutions, and are only known to be correct in the far-field and away from the Cherenkov angle. In this contribution, an exact solution to the Tanun problem is presented in the form of a numerically-evaluable integral. Using this exact expression, the regimes of applicability of the ZHS and endpoints approach are evaluated.