Stellar Radial Velocity (Doppler Spectroscopy) Calculator

Doppler spectroscopy measures how fast a star is moving toward or away from us by detecting tiny shifts in the wavelengths of its spectral lines.

The Doppler formula:

v_r = (Δλ / λ₀) × c = ((λ_obs - λ_rest) / λ_rest) × c

Where:

• v_r = radial velocity (positive = receding, negative = approaching)
• λ_rest = rest (lab) wavelength
• λ_obs = observed wavelength
• c = 299,792.458 km/s

Blueshift vs redshift:

• λ_obs < λ_rest → star is approaching (blueshift, Δλ negative)
• λ_obs > λ_rest → star is receding (redshift, Δλ positive)

Common spectral lines used:

• H-alpha: λ₀ = 656.28 nm (hydrogen)
• H-beta: λ₀ = 486.13 nm (hydrogen)
• Ca II K: λ₀ = 393.37 nm (calcium)
• Ca II H: λ₀ = 396.85 nm (calcium)
• Na D doublet: λ₀ ≈ 589 nm (sodium)

Exoplanet detection: This technique can detect exoplanets by the wobble they induce in their host star. Jupiter causes the Sun to wobble at about 12.5 m/s. Earth’s effect on the Sun is only about 0.09 m/s, which is below what any current instrument can pull out of a real star. The HARPS spectrograph (High Accuracy Radial velocity Planet Searcher) can measure velocities to ~1 m/s precision, and ESPRESSO pushes that toward 0.1 m/s.

The gap between 0.1 m/s of instrument precision and 0.09 m/s of Earth-like signal is not the whole story, and this is the part popular articles skip. Stars are not smooth. Granulation, spots and magnetic cycles move the apparent line centre by a few m/s on their own, so the noise floor for finding an Earth twin is set by the star rather than by the spectrograph. That is why the radial velocity method has found thousands of hot Jupiters and almost no Earth analogues.

Worth calibrating your expectations on the size of these shifts before using the calculator. Stars in our neighbourhood move at tens of km/s relative to us, which on the H-alpha line at 656.28 nm is a shift of only about 0.1 nm. A whole nanometre of shift would mean 457 km/s, faster than all but the rarest hypervelocity stars. If you enter a shift of several nanometres you are no longer doing stellar spectroscopy, you are measuring a distant galaxy. The radial velocity method was used to discover the first confirmed exoplanet around a Sun-like star in 1995 (51 Pegasi b).