Interpretation 2: At fixed

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Interpretation 2: At fixed

Let's assume that the measured spectrum is made of a line centered around in the observatory frame. The associated rest frame frequency, i.e., the frequency of the line at rest, will be . The observation is naturally set up so that the source frame is also the rest frame. This implies that , and are linked through

(6)

The modeller will use the Doppler effect to interpret the observed frequency difference as a local velocity difference (projected along the line of sight direction) around the systemic velocity of the source. Let's write ( ) the velocity associated to ( ). The velocity difference will be noted

(7)
In order to derive this velocity difference, we associate the velocity to each observed frequency through the Doppler effect, i.e.,
(8)
It is easy to deduce that
(9)
i.e., the local velocity difference is proportional to the observed frequency difference. We can define a linear velocity axis associated to the set of brightnesses, as
(10)
This velocity scale is only meaningful locally, i.e., as long as the brightnesses can be associated to the line rest frequency under consideration.