Redshift to Velocity Calculator

A spectral line's redshift is defined directly from the two wavelengths: z = (λ_observed − λ_emitted) ÷ λ_emitted. A positive z means the observed wavelength is stretched longer than it was emitted (redshifted); a negative z means it's compressed shorter (blueshifted).

For motion purely along the line of sight, special relativity ties z to the source's recession velocity through the relativistic Doppler formula, 1 + z = √((1+β)/(1−β)), where β = v/c. Solving that for β gives β = ((1+z)² − 1) ÷ ((1+z)² + 1), which this calculator evaluates directly from z. Because (1+z) is always a ratio of two positive wavelengths, this expression for β always lands strictly between −1 and 1 — the formula can never imply a velocity at or beyond the speed of light.

This is the purely radial, special-relativistic case (no cosmological expansion or gravitational redshift); it's the same formula used for a receding star or galaxy's line-of-sight velocity at speeds where the simpler non-relativistic v ≈ cz approximation starts to break down.