Dissecting the Compton scattering kernel I: Isotropic media

Compton scattering between electrons and photons plays a crucial role in astrophysical plasmas. Many important aspects of this process can be captured by using the so-called Compton scattering kernel. For isotropic media, exact analytic expressions (valid at all electron and photon energies) do exist but are hampered by numerical issues and often are presented in complicated ways. In this paper, we summarize, simplify and improve existing analytic expressions for the Compton scattering kernel, with an eye on clarity and physical understanding. We provide a detailed overview of important properties of the kernel covering a wide range of energies and highlighting aspects that have not been appreciated as much previously. We discuss analytic expressions for the moments of the kernel, comparing various approximations and demonstrating their precision. We also illustrate the properties of the scattering kernel for thermal electrons at various temperatures and photon energies, introducing new analytic approximations valid to high temperatures. The obtained improved formulae for the kernel and its moments should prove useful in many astrophysical computations, one of them being the evolution of spectral distortions of the cosmic microwave background in the early Universe. A novel code, CSpack, for efficient computations of the Compton scattering kernel and its properties (in the future also including anisotropies in the initial electron and photon distributions) is being developed in a series of papers and will be available within one month.