The Fokker-Planck Expansion for Compton Scattering

Since individual Compton scatterings of low-energy photons by non-relativistic electrons produces only a small fractional change in the photon energy, it is convenient to describe the evolution of the photon spectrum under the action of Compton scattering by hot electrons in terms of a Fokker-Planck equation. The Kompaneets equation is such an equation and describes the evolution accurately for electron temperatures below ~10 keV. For higher electron temperatures the standard technique is to abandon the Fokker-Planck approach and simply integrate the collision integral in the Boltzmann equation. It turns out that by doing a higher order expansion for the Fokker-Planck equation one obtains fairly accurate corrections to the Kompaneets equation for moderate temperatures (< 100 keV). The advantage of this approach over the collision integral is that it leads to analytical results for the corrections to the Kompaneets equation, and from this one can obtain specific spectral shapes to describe corrections to the Sunyaev-Zel'dovich (S-Z) y-distortion.

The first corrections to the Kompaneets equation were first presented in a conference proceedings and more recently submitted as a Letter to The Astrophysical Journal. In the meantime Challinor and Lasenby has also obtained the same result. Several people have tried this approach in the past, but did not performed the expansion in a completely self-consistent manner.

To supplement the aforementioned paper we give some more details of the derivations in a set of TeX notes (TeX dvi or PostScript) as well as a Mathematica notebook (but see below).

Greek Letters and Cutoffs

The computation was done under Mathematica v2.2 which had no greek letters, except in the comments. You can figure it out: b for beta, d for delta, etc. This notebook was then converted to Mathematica v3.0 which has a Save As HTML option. Unfortunately this option writes isolated greek letters, not as imbedded images, but rather using an unimplemented HTML notation: &agr; for \alpha, &bgr; for \beta, &Dgr; for \Delta, &thgr; for \theta, etc. This makes it hard to read. The problem of cut-off equations has been fixed by a patch available from Wolfram Research.
Last modified: Tue Sep 25 15:08:57 CDT