The motion of the earth with respect to the distant matter frame produces a spectral distortion in the observed background radiation given by 1

The second-order approximation in β gives (de Bernardis et al., 1989) 2 where α≅(dlnI/dln v) = (v/I)(dI/dv). The first term in cosθ of equation (2) is the usual dipole anisotropy, the second constant term is the equivalent of the transverse Doppler effect in special relativity, and the third term is the quadrupole anisotropy. It is interesting to note that the quadrupole term is zero in the Rayleigh-Jeans region of the blackbody spectrum: Q ≈ 1/2(3–α)(2–α)β2cos2θ.