  Cosmic ray air showers have been known for over 30 years to emit pulsed radio
emission in the frequency range from a few to a few hundred MHz, an effect that
offers great opportunities for the study of extensive air showers with upcoming
fully digital "software radio telescopes" such as LOFAR and the enhancement of
particle detector arrays such as KASCADE Grande or the Pierre Auger
Observatory. However, there are still a lot of open questions regarding the
strength of the emission as well as the underlying emission mechanism.
Accompanying the development of a LOFAR prototype station dedicated to the
observation of radio emission from extensive air showers, LOPES, we therefore
take a new approach to modeling the emission process, interpreting it as
"coherent geosynchrotron emission" from electron-positron pairs gyrating in the
earth's magnetic field. We develop our model in a step-by-step procedure
incorporating increasingly realistic shower geometries in order to disentangle
the coherence effects arising from the different scales present in the air
shower structure and assess their influence on the spectrum and radial
dependence of the emitted radiation. We infer that the air shower "pancake"
thickness directly limits the frequency range of the emitted radiation, while
the radial dependence of the emission is mainly governed by the intrinsic
beaming cone of the synchrotron radiation and the superposition of the emission
over the air shower evolution as a whole. Our model succeeds in reproducing the
qualitative trends in the emission spectrum and radial dependence that were
observed in the past, and is consistent with the absolute level of the emission
within the relatively large systematic errors in the experimental data.
