Relativistic Corrections to the Sunyaev-Zeldovich Effect for Clusters of Galaxies. V. Effect of the Motion of the Observer

We extend the formalism of the relativistic thermal and kinematical Sunyaev-Zeldovich effects to the observer’s system (the Solar System) moving with a velocity ${\beta }_S$ with respect to the cosmic microwave background radiation. We confirm the results recently obtained by Chluba, Huetsi, and Sunyaev in the lowest order of the observer’s velocity $\beta_{S}. We give a more general analytic expression for the thermal and kinematical Sunyaev-Zeldovich effects corresponding to the observer’s system with the power series expansion approximation in terms of ${\theta }_e$. It is found that the effect of the motion of the observer on the Sunyaev-Zeldovich effect will become important in the future high precision observation projects of the Sunyaev-Zeldovich effect.