Compton Scattering in Ultrastrong Magnetic Fields: Numerical and Analytical Behavior in the Relativistic Regime

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Abstract
This paper explores the effects of strong magnetic fields on the Compton scattering of relativistic electrons. Recent studies of upscattering and energy loss by relativistic electrons that have used the nonrelativistic, magnetic Thomson cross section for resonant scattering or the Klein-Nishina cross section for nonresonant scattering do not account for the relativistic quantum effects of strong fields (>4 × 1012 G). We have derived a simplified expression for the exact QED scattering cross section for the broadly applicable case in which relativistic electrons move along the magnetic field. To facilitate applications to astrophysical models, we have also developed compact approximate expressions for both the differential and total polarization-dependent cross sections, with the latter representing well the exact total QED cross section even at the high fields believed to be present in environments near the stellar surfaces of soft gamma repeaters and anomalous X-ray pulsars. We find that strong magnetic fields significantly lower the Compton scattering cross section below and at the resonance when the incident photon energy exceeds mec2 in the electron rest frame. The cross section is strongly dependent on the polarization of the final scattered photon. Below the cyclotron fundamental, mostly photons of perpendicular polarization are produced in scatterings, a situation that also arises above this resonance for subcritical fields. However, an interesting discovery is that for supercritical fields, a preponderance of photons of parallel polarization results from scatterings above the cyclotron fundamental. This characteristic is both a relativistic and a magnetic effect not present in the Thomson or Klein-Nishina limits.
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