Comptonisation of radiation below the cyclotron frequency in a strong magnetic field

We present new equations describing the Thomson scattering of the o-mode and z-mode in a strongly magnetised plasma, valid below the fundamental cyclotron frequency. Scattering by nonrelativistic thermal electrons leads to a frequency diffusion equation for the more strongly scattered o-mode and this equation is effectively the Kompaneets equation with cross section 2/15th of the Thomson cross section. Transfer of the photons tends to be dominated by the less strongly scattered z-mode; an o-mode photon is scattered occasionally into a z-mode photon, which then diffuses rapidly due to its large mean free path before being scattered back into an o-mode photon. Our results should have applications in X-ray pulsars and γ-ray burst sources, as well as magnetic white dwarfs occurring in cataclysmic variables