Study on the Solutions of the Sunyaev-Zeldovich Effect for Clusters of Galaxies

TL;DR

This paper develops formal solutions for the Sunyaev-Zeldovich effect in galaxy clusters, extending to kinematical effects, and provides exact numerical solutions including full-order optical depth terms.

Contribution

It introduces a formalism for the Sunyaev-Zeldovich effect that includes kinematical effects and derives solutions equivalent to the Fokker-Planck approximation, with exact numerical results.

Findings

Kinematical Sunyaev-Zeldovich effect described by pressure-related redistribution function

Derived formal solutions in multiple representations for the effect

Numerical solutions include full-order optical depth terms

Abstract

Based upon the rate equations for the photon distribution function obtained in the previous paper, we study the formal solutions in three different representation forms for the Sunyaev-Zeldovich effect. By expanding the formal solution in the operator representation in powers of both the derivative operator and electron velocity, we derive a formal solution that is equivalent to the Fokker-Planck expansion approximation. We extend the present formalism to the kinematical Sunyaev-Zeldovich effect. The properties of the frequency redistribution functions are studied. We find that the kinematical Sunyaev-Zeldovich effect is described by the redistribution function related to the electron pressure. We also solve the rate equations numerically. We obtain the exact numerical solutions, which include the full-order terms in powers of the optical depth.