Radial velocity and Doppler conventions

Gas moving away from you shows up at a lower frequency than its rest frequency; gas moving toward you shows up higher. For the hydrogen line, the rest frequency f₀ is 1420.405751768 MHz, and the shift is small enough that one number covers most of what you’ll do: about 4.738 kHz per km/s.

There are two ways to turn a frequency into a velocity, and they give slightly different answers. The radio convention divides by the rest frequency, so velocity is a straight linear function of frequency. That is what HI surveys report and what our Doppler calculator shows first. The optical convention divides by the observed frequency instead, which is the familiar cz used for galaxies. Neither is the relativistic formula; both are approximations that agree at low speed.

Worked example

A peak observed at 1420.000 MHz:

• radio convention: 299,792.458 × (1420.405752 − 1420.000) / 1420.405752 = 85.64 km/s
• optical convention: same numerator divided by 1420.000 instead = 85.66 km/s

At Galactic speeds the difference is a few hundredths of a km/s, well inside a typical channel. It grows with velocity and matters for distant galaxies, so state which convention you used when you publish a number. Going the other way, gas receding at 50 km/s appears at 1420.1689 MHz and gas approaching at 100 km/s at 1420.8795 MHz.

The velocity you get is topocentric

That 85.64 km/s is measured from a telescope that is itself spinning with the Earth, orbiting the Sun, and being carried along by the Sun through the Galaxy. Before it means anything you have to add the observer’s own motion, the local standard of rest correction, which can be 30 km/s or more depending on date and direction. Then you can compare with anyone else’s spectrum or use it to build a rotation curve.

How finely you can measure the velocity depends on your channel width, covered in spectral resolution. LAB-012 has the full procedure on a real spectrum layout.