## The generic Doppler formula

The Doppler effect expresses that a source emitting in its rest frame at a frequency will appear to emit at a frequency in the observation frame, which depends on the velocity of the source in the observation frame, . The relation between and is given by

(66)
with , where and are respectively the velocity components along and perperdicular to the line of sight. and are often called the radial and transverse velocity components.

For objects far from the observer, because of the universe expansion. For close-by objects, Eq. shows that the dependency on is of second order in , while the dependency on is only of first order in . Hence, for all cases, the Doppler equation is simplified to

(67)
In other words, the frequency change under a change of frame is attributed only to the radial velocity between the rest and observation frame.