Derivation of the Kompaneets equation using the boost operator approach

Abstract

The repeated scattering of photons by thermal electrons at low temperatures is described by the Kompaneets equation and its generalized forms that include anisotropies and higher order temperature corrections. In this work, we use the boost operator approach to derive the related expressions in a transparent way that showcases the generality of the formalism and its application to radiative transfer problems. We consider the simplest form of the Kompaneets equation for the scattering in isotropic media at the leading order in the electron temperature and then include anisotropies in the photon field, reproducing previously obtained expressions for the evolution equations. For this we use expressions for the scattering operator in the electron rest frame up to first order in the electron recoil, O(h nu/m_e c^2), but then work at all orders in the electron momentum, p, as easily obtained with the boost operator approach. This shows how specific transformation rules can be formulated that allow simplification of the otherwise cumbersome and repetitive calculations. We also confirm the expressions for higher order temperature corrections in isotropic media, highlighting the validity of the approach presented here. As part of the derivation, we find expressions for the boost operator in general boost directions which we believe will also be useful in other applications of the formalism.

Figures (2)

• Figure 1: A schematic diagram of the boost operator approach. The lab frame occupation number, $n(\nu_0, {\hat{\boldsymbol\gamma}}_0)$, is boosted into the electron rest frame along the $z$-axis using the boost operator ${}^{0}\hat{\mathcal{B}}^{m}_{\ell' \ell"}(\nu,\beta)$ where the scattering is evaluated. The scattered occupation number, $n'(\nu', {\hat{\boldsymbol\gamma}}')$, is then boosted back into the lab frame using the combination ${}^{-1}\hat{\mathcal{B}}^{m}_{\ell \ell'}(\nu,-\beta)/\gamma$, where the Doppler weight $-1$ and the $1/\gamma$ factor arise from the Lorentz transformation of the scattering optical depth (see section \ref{['sec:Lab frame col term']}).
• Figure 2: Representations of a distribution as seen from two frames boosted relative to each other. The rest frame distribution (left) is isotropic and becomes distorted towards the direction of the boost when viewed from the boosted frame (right). The magnitude of the boost shown here is $\beta=0.8$.