Leading relativistic corrections to the Kompaneets equation

Abstract

We calculate the first relativistic corrections to the Kompaneets equation for the evolution of the photon frequency distribution brought about by Compton scattering. The Lorentz invariant Boltzmann equation for electron-photon scattering is first specialized to isotropic electron and photon distributions, the squared scattering amplitude and the energy-momentum conserving delta function are each expanded to order v 4 /c 4 , averages over the directions of the electron and photon momenta are then carried out, and finally an integration over the photon energy yields our FokkerPlanck equation. The Kompaneets equation, which involves only first-and second-order derivatives with respect to the photon energy, results from the order v 2 /c 2 terms, while the first relativistic corrections of order v 4 /c 4 introduce third-and fourth-order derivatives. We emphasize that our result holds when neither the electrons nor the photons are in thermal equilibrium; two effective temperatures characterize a general, non-thermal electron distribution. When the electrons are in thermal equilibrium our relativistic Fokker-Planck equation is in complete agreement with the most recent published results, but we both disagree with older work.

Spectral and temporal properties of Compton scattering by mildly relativistic thermal electrons

“…Then, treated Compton scattering using either integral or differential form depending on the value of the fractional photon energy change. Challinor & Lasenby (1998), Itoh et al (1998), Brown & Preston (2012) and Nozawa & Kohyama (2015) also applied their results to the Sunyaev-Zeldovich effect. Without relativistic corrections, the Kompaneets equation (including the stimulated scattering term) only allows us to determine the Compton parameter, y ≡ 4θτ, of the medium (hot gas with kT e ∼ 10 keV in a cluster of galaxies) upscattering the cosmic microwave background radiation (Zeldovich & Sunyaev 1969).…”

Relativistic corrections to the Kompaneets equation

“…In this section, we first review the prescription used in Brown & Preston [16]. Let us start with the following identity relation:…”

“…On the other hand, Brown & Preston [16] studied the leading order relativistic corrections to the Kompaneets Email address: snozawa@josai.ac.jp (Satoshi Nozawa) equation by calculating the Boltzmann equation in the delta function expansion approximation. Although their expansion scheme is quite different from the previous schemes [9,10], their leading order corrections on the SZ effect reproduced the previous results [9,10].…”

“…In this paper, we study the SZ effect for clusters of galaxies. We explore the relativistic corrections to the Kompaneets equation in terms of two different expansion approximation schemes, namely, the Fokker-Planck expansion approximation [17] and delta function expansion approximation [16]. We show that two expansion approximation formalisms are equivalent under the Thomson approximation.…”

Double Compton and Cyclo-Synchrotron in Super-Eddington Discs, Magnetized Coronae, and Jets

“…Other extensions to Kompaneets can be found elsewhere that account for relativistic effects (Sazonov & Sunyaev 1998;Challinor & Lasenby 1998;Nagirner et al 1997;Challinor & Lasenby 1998;Brown & Preston 2012;Nozawa & Kohyama 2015), high temperatures (Sampson 1959;Xie et al 2010;Garain & …”