Doppler Shift & Radial Velocity Calculator

Convert a spectral line's rest and observed wavelengths into radial velocity (or the reverse), using either the classical v/c approximation or the exact relativistic Doppler relation — with a velocity gauge and a chart showing exactly where the two formulas diverge.

Doppler shift / radial velocity calculator

A spectral line's known rest wavelength λ₀, compared to where it's actually observed, gives the radial velocity toward or away from us: v_r/c ≈ (λ_obs − λ₀)/λ₀ at low speed, or the exact relativistic relation λ_obs/λ₀ = √[(1+β)/(1−β)] at any speed. Solve either direction, and switch modes to see exactly when the approximation starts to matter.

Rest wavelength (λ₀)

Observed wavelength (λ_obs)

v_r = +100.5 km/s

Redshifted — receding from the observer · Δλ = 0.22 nm · β = v/c = +3.352 × 10⁻⁴

Classical: +100.5 km/s · Relativistic: +100.48 km/s

At this speed, relativistic corrections are far below 0.5% — the classical approximation is essentially exact.

Dashed = classical (1+β), solid = exact relativistic. They agree closely near β=0 and diverge visibly as |v| becomes a real fraction of c.

A star’s spectral lines sit at precisely known wavelengths in the lab — Hα always at 656.28 nm, the sodium D line always at 589.0 nm. When a star is moving toward or away from us, those lines shift: measure how far, and you have the star’s radial velocity, no other measurement required. This is the working principle behind everything from Vesto Slipher’s first galaxy velocity measurements in the 1910s to today’s exoplanet searches, which detect worlds by the meters-per-second wobble they induce in their host star’s spectrum.

The classical approximation

For velocities much smaller than light, the relation is a simple ratio:

\frac{v_r}{c} \approx \frac{\lambda_{\rm obs} - \lambda_0}{\lambda_0}

A positive shift (λ_obs > λ0, redshifted) means the source is receding; negative (blueshifted) means it’s approaching. Take the worked example this calculator opens with: Hα’s rest wavelength is 656.28 nm; observed at 656.50 nm, that’s a radial velocity of about +100.5 km/s — receding.

Where the exact relativistic relation is needed

The classical formula is a first-order approximation. For purely radial motion, the exact relativistic Doppler relation is

\frac{\lambda_{\rm obs}}{\lambda_0} = \sqrt{\frac{1+\beta}{1-\beta}}, \qquad \beta = \frac{v_r}{c}

which already folds in relativistic time dilation, not just the classical light-travel-time effect. At everyday stellar velocities (tens to hundreds of km/s, β ~ 10⁻⁴ to 10⁻³) the two formulas are indistinguishable. They start to diverge visibly only once a source is moving at a real fraction of the speed of light — relativistic jets, some supernova ejecta, and (going the other way) precision tests of special relativity itself are where the distinction actually earns its keep.

Running it in reverse

Enter a rest wavelength and a radial velocity instead, and the calculator predicts the observed wavelength — useful for planning an observation (where should this line actually land, given the target’s known velocity?) or for building intuition about how large a shift a given velocity really produces before you go looking for it in real data.

Reading the two charts

The radial velocity gauge places the current velocity on a scale that auto-fits to whatever you enter, colored blue for approaching and red for receding. A shaded zone appears only when |v| exceeds 0.1c — the point past which relativistic corrections exceed about 0.5% — so its absence is itself informative: for ordinary stellar and exoplanet work, you’ll essentially never see it.
The classical-vs-relativistic curve plots both formulas across the full range of β, dashed for classical and solid for relativistic. They sit on top of each other near β = 0 and peel apart as |v| grows — a direct visual answer to “how much does the approximation actually cost me here,” rather than a rule of thumb taken on faith.