2.2. Kinetic Sunyaev-Zel'dovich Effect
If the cluster is moving with respect to the CMB rest frame, there will be an additional spectral distortion due to the Doppler effect of the cluster bulk velocity on the scattered CMB photons. If a component of the cluster velocity, pec, is projected along the line of sight to the cluster, then the Doppler effect will lead to an observed distortion of the CMB spectrum, referred to as the kinetic SZE. In the nonrelativistic limit, the spectral signature of the kinetic SZE is a pure thermal distortion of magnitude
where pec is along the line of sight; that is, the emergent spectrum is still described completely by a Planck spectrum, but at a slightly different temperature, lower (higher) for positive (negative) peculiar velocities (Sunyaev & Zel'dovich, 1972; Phillips, 1995; Birkinshaw, 1999). Figure 2 illustrates the kinetic SZE (dashed) for a typical galaxy cluster with a peculiar velocity of 500 km s-1. Figure 3 shows the SZE spectrum of the galaxy cluster A2163 along with the best-fit model consisting of thermal (dashed) and kinetic (dotted) SZE components.
Relativistic perturbations to the kinetic SZE are due to the Lorentz boost to the electrons provided by the bulk velocity (Nozawa et al., 1998a; Sazonov & Sunyaev, 1998a). The leading term is of order (kB Te / me c2)(pec / c), and for a 10 keV cluster moving at 1000 km s-1 the effect is about an 8% correction to the nonrelativistic term. The (kB Te / me c2)2(pec / c) term is only about 1% of the nonrelativistic kinetic SZE, and the (pec / c)2 term is only 0.2%.