Leading relativistic corrections to the Kompaneets equation

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• 作者：Lowell S. Brown ; ^{brownl@lanl.gov} ; Dean L. Preston
• 关键词：Kompaneets ; Relativistic corrections ; Sunyaev&ndash ; Zel&rsquo ; dovich effect
• 刊名：Astroparticle Physics
• 出版年：2012
• 期刊代码：4_09276505
• 类别：ph
• 出版时间：June, 2012
• 卷：35
• 期：11
• 页码：742-748
• 文件大小：245 K

摘要

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 v⁴/c⁴, averages over the directions of the electron and photon momenta are then carried out, and finally an integration over the photon energy yields our Fokker-Planck equation. The Kompaneets equation, which involves only first- and second-order derivatives with respect to the photon energy, results from the order v²/c² terms, while the first relativistic corrections of order v⁴/c⁴ 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.