There is confusion over the velocity definitions in spectral-line astronomy. This is because we have two definitions of radial velocity and neither is particularly meaningful, physically.
The radio velocity definition is
where is the rest frequency of the line, is the observed sky frequency, and c is the speed of light. This expression is often used by Galactic radio astronomers (astronomers from our Galaxy ?) but is now deprecated by the IAU. For reference, this means that the radio velocity increment is
and the frequency in terms of the radio velocity is
The optical velocity definition is
where z is the redshift. This expression is often used in extragalactic astronomy. The optical velocity increment is
and the frequency in terms of the optical velocity is
Neither of these velocity definitions has any physical meaning unless the radial velocity is truly small compared with the speed of light. These expressions are not interchangeable; you must use the correct expression with the correct velocity definition to get a meaningful frequency. Note also that the optical velocity increment depends upon the observing frequency, whereas the radio velocity increment depends on the rest frequency.
We must also define our velocities in some reference frame; the velocity with respect to the observer is not usually of particular interest to anyone (except for solar system work). It is convenient to define the velocity with respect to some astronomical reference frame and the following table lists the standard ones (this table is from the spectral-line chapter of the NRAO Imaging Workshop book by Pjotr Roelfsma).
Velocity Rest Frames | ||
Correct for | Correction (km/s) | Rest Frame Name |
Nothing | 0 | topocentric |
Earth-rotation | <0.5 | |
Earth's motion around | < 0.013 | geocentric |
earth-moon barycentre | ||
Earth's motion around Sun | < 30 | heliocentric |
Sun's motion around solar | <0.012 | barycentric |
system barycentre | ||
Solar motion | < 20 | local standard of rest |
Galactic rotation | 220 | galactocentric |
In Galactic work, the local standard of rest (LSR) frame is generally used. In extragalactic work, the barycentric frame is generally adopted (although it's often misnamed the heliocentric frame).