CMB Radiation in the Direction of a Moving Cluster of Galaxies with Hot Gas: Relativistic Corrections

It has been recently realized (Rephaeli 1995) that the relativistic corrections to the spectral distortions of the cosmic microwave background (CMB) measured in the direction of clusters of galaxies containing hot gas are significant and should be detectable with the forthcoming experiments. In the present paper we calculate the correction terms that are proportional to $V_{r}/c\times kT_e/m_ec^2$ and $(V/c)^2$ to the standard formulae describing the spectral distortions caused by the bulk motion of the free electrons (kinematic effect) and due to the presence of the hot gas (thermal effect) for the case of a cluster having a peculiar velocity $V$ ($V_{r}$ is its radial component). The results of our analytical calculations are confirmed by Monte-Carlo simulations (Sazonov & Sunyaev 1998).