We present an accurate numerical table for the relativistic corrections to the thermal Sunyaev-Zeldovich 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 numerical table is provided for the ranges 0.002 ≤ θ e ≤ 0.100 and 0 ≤ X ≤ 20, where θ e ≡ k B T e /m e c 2 , X ≡ ω/k B T 0 , T e is the electron temperature, ω is the angular frequency of the photon, and T 0 is the temperature of the cosmic microwave background radiation. We also present an accurate analytic fitting formula that reproduces the numerical results with high precision.
We extend the formalism of the relativistic thermal Sunyaev-Zel'dovich effect to the system moving with a velocity β ≡ v/c with respect to the cosmic microwave background radiation. In the present formalism, the kinematic Sunyaev-Zel'dovich effect for the cluster of galaxies with a peculiar velocity β is derived in a straightforward manner by the Lorentz boost of the generalized Kompaneets equation. We give an analytic expression for the kinematic Sunyaev-Zel'dovich effect which is valid up to O(β 2 ) with the power series expansion approximation in terms of θ e ≡ k B T e /mc 2 , where T e and m are the electron temperature and the electron mass, respectively. It is found that the relativistic corrections to the kinematic Sunyaev-Zel'dovich effect are significant. For a typical electron temperature k B T e = 10keV, one obtains −8.2% and +1.3% corrections from the O(βθ e ) and O(βθ 2 e ) contributions, respectively. The O(β 2 ) correction is extremely small, +0.2% for β = 1/300 at k B T e = 10keV. Therefore it can be safely neglected. These relativistic corrections are directly reflected on the determination of the peculiar velocity β of the cluster of galaxies with the observation of the kinematic Sunyaev-Zel'dovich effect.
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