Sunyaev-zel’dovich Effect by Multiple Scattering : Numerical Solution of the Transfer Equations

The radiative transfer equations for multiple inverse Compton scattering of the Cosmic Microwave Background Radiation (CMBR) by the hot intra-cluster electrons are solved numerically. The spherical isothermal and inhomogeneous β model has been considered for the electron distribution. The anisotropy of the CMBR caused by scattering, known as thermal Sunyaev-Zel’dovich effect, along the radial axis of the medium is compared with the analytical solution of Kompaneets equation. The X-ray data of several clusters of galaxies at low redshifts provide an estimation of the central electron density n0 to be of the order 10. It is found that for this value of n0 the effect of multiple scattering is negligible. The numerically calculated anisotropy along the radial axis matches well with the analytical solution that describes single scattering. The result incorporating multiple scattering is fitted with the recent observation of Sunyaev-Zel’dovich effect in the cluster Abell 2163. It is shown that if n0 is greater by an order of magnitude, which could be possible for cluster of galaxies at comparatively higher redshift, multiple scattering would play a significant role at the Wien region of the anisotropy spectrum. A fitting formula for the correction to the Sunyaev-Zel’dovich effect due to multiple scattering is provided.