Extensions to the Kompaneets Equation and Sunyaev-Zel'dovich Distortion

Analytical expressions are presented for relativistic corrections to the Kompaneets equation and the Sunyaev-Zel'dovich y-distortion. The latter provides a convenient method of inferring both the temperature and Thomson optical depth of the gas in clusters of galaxies using only observations of the spectrum of CMBR photons passing through the cluster. The relativistic correction gives an additional component of the S-Z effect apart from the y-distortion and kinematic S-Z effect. The perturbative method used is shown to provide a very accurate approximations when kTe~<100keV but not for higher temperatures.