Caption: Determining the line-of-sight velocity or radial velocity to astronomical objects from the Doppler effect and line spectra.

Features:

1. Continuous spectra are of little use in determining radial velocities since it hard to impossible to know the shape of the continuous spectra in their emission reference frame.

2. On the other hand, line spectra (either emission line spectra or absorption line spectra) are easily used. The observed line spectra are Doppler shifted from their emission reference frame, but you can recognize the line pattern, and so identify the atomic lines or molecular lines. The known rest frame wavelengths of the atomic lines or molecular lines are, in fact, their emission reference frame wavelengths---and so these are known too.

3. Once you have an emission reference frame wavelength, one can determine the radial velocity from the relativistic Doppler effect formula for wavelength shift

         λ_2 = λ_1 * sqrt[(1+v/c)/(1-v/c)] ,

   where λ_2 is the observed line wavelength, λ_1 is the emitted line wavelength, the vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns, and v is the radial velocity to the astronomical object. Note v is positive/negative for recession/approach.

4. The inverse function for v/c is

         v/c = [(λ_2/ λ_1)**2 - 1]/[(λ_2/ λ_1)**2 + 1] ,

5. If |(λ_2/ λ_1)-1| << 1, you can use 1st order Doppler effect formula

         Δλ/λ = v/c ,

   to determine v/c. This is done in the example calculation in the image.

6. Note that extragalactic sources outside of the Local Group of Galaxies have a cosmological redshift as well as a Doppler shift and the cosmological redshift grows with distance and becomes dominant beyond ∼ 5 Mpc (see Wikimedia Commons: File:Hubble constant.JPG). Beyond ∼ 100 Mpc, the Doppler shift may become negligible depending on your needed accuracy/precision.

   Note the cosmological redshift is NOT a Doppler shift in the opinion of yours truly though they are related effects. They have different formulae, except to 1st order in small v/c. Some people call the cosmological redshift a Doppler shift, but yours truly thinks this is wrong.