**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

(5) |

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
(*k*_{B} *T*_{e} / *m*_{e}
*c*^{2})(_{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
(*k*_{B} *T*_{e} / *m*_{e}
*c*^{2})^{2}(_{pec} / *c*) term is only about 1% of
the nonrelativistic kinetic SZE, and the
(_{pec} /
*c*)^{2} term is only 0.2%.