Quantitative Description of the Sunyaev-Zeldovich Effect: Analytic Approximations

Various aspects of relativistic calculations of the Sunyaev-Zeldovich effect are explored and clarified. We first formally show that the main previous approaches to the calculation of the relativistically generalized thermal component of the effect are equivalent. Our detailed description of the full effect results in a somewhat improved formulation. Analytic approximations to the exact calculation of the change of the photon occupation number in the scattering, $\Delta n$, are extended to powers of the gas temperature and cluster velocity that are higher than in similar published treatments. For the purely thermal and purely kinematic components, we obtain identical terms up to the highest common orders in temperature and cluster velocity, and to second order in the Thomson optical depth, as reported in previous treatments, but we get slightly different expressions for the terms that depend on both the gas temperature and cluster velocity. We also obtain an accurate expression for the crossover frequency.