General case

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General case

Let's assume that the user wishes to observe a line that appears at the rest frequency, , for a source that has a systemic velocity, , in a given velocity convention #1. The rest frame and observatory frame frequencies are linked by the Doppler effect through

(74)
where is the Doppler function that can be deduced from Eq. , , or , depending on the assumed convention. For instance, the Doppler function of the optical convention is
(75)

Now, let's assume that the analysis exactly uses the same observed frequency, , and systemic velocity, , but it interprets it using another Doppler convention and thus Doppler function, . The spectral axis in the rest frame is then computed as

(76)
with
(77)
This means that the lines will seem to appear at a wrong rest frequency with a wrong full width at half maximum.