Gamma-ray halos as a measure of intergalactic magnetic fields: A classical moment problem

Abstract

The presence of weak intergalactic magnetic fields can be studied by their effect on electromagnetic cascades induced by multi-TeV $\ensuremath{\gamma}$ rays in the cosmic radiation background. Small deflections of secondary electrons and positrons as the cascade develops extend the apparent size of the emission region of distant TeV $\ensuremath{\gamma}$-ray sources. These $\ensuremath{\gamma}$-ray halos can be resolvable in imaging atmospheric Cherenkov telescopes and serve as a measure of the intergalactic magnetic field strength and coherence length. We present a method of calculating the $\ensuremath{\gamma}$-ray halo for isotropically emitting sources by treating magnetic deflections in the cascade as a diffusion process. With this ansatz the moments of the halo follow from a set of simple diffusion-cascade equations. The reconstruction of the angular distribution is then equivalent to a classical moment problem. We present a simple solution using Pad\'e approximations of the moment's generating function.