Relativistic Corrections to the Sunyaev-Zel'dovich Effect for Clusters of Galaxies. IV. Analytic fitting formula for the Numerical Results

Abstract

We present an accurate analytic fitting formula for the numerical results for the relativistic corrections to the thermal Sunyaev-Zel&#39;dovich effect for clusters of galaxies. The numerical results for the relativistic corrections have been obtained by numerical integration of the collision term of the Boltzmann equation. The fitting is carried out for the ranges 0.02 < theta_{e} < 0.05 and 0 < X < 20, where theta_{e} = k_{B}T_{e}/m_{e}c^{2}, X = omega/k_{B}T_{0}, T_{e} is the electron temperature, omega is the angular frequency of the photon, and T_{0} is the temperature of the cosmic microwave background radiation. The accuracy of the fitting is generally better than 0.1%. The present analytic fitting formula will be useful for the analyses of the thermal Sunyaev-Zel&#39;dovich effect for high-temperature galaxy clusters.