Donaldson R F Redshift & Doppler Velocity Calculator
Convert between redshift (z), a wavelength shift, and the radial velocity it implies.
How this is worked out
Spectroscopic redshift is defined as z = (λobserved − λrest) / λrest: a positive z means the light has been stretched to longer (redder) wavelengths. At everyday astronomical speeds (far below the speed of light), the classical approximation v = cz works well. For large redshifts — distant galaxies and quasars, where z can exceed 1 — this tool also shows the exact special-relativistic Doppler formula, 1 + z = √((1+v/c)/(1−v/c)), which never lets the implied velocity reach or exceed the speed of light no matter how large z gets.
Frequently asked questions
When does the classical formula stop being good enough?
The two formulas agree closely while z is small (roughly z < 0.1, corresponding to speeds under about 10% of light speed) but diverge increasingly after that; for cosmological redshifts, the relativistic figure is the physically meaningful one, and for the most distant objects (z > a few), even the simple relativistic Doppler picture used here is really a simplification of cosmological expansion, not literal motion through space.
Can z be negative?
Yes — a negative z is a blueshift, meaning the object is approaching (or, in a cosmological context, that space between us and it is contracting relative to now), such as the Andromeda Galaxy, which is blueshifted because it's falling toward the Milky Way.
Related tools: Stellar Magnitude & Distance Calculator, Astronomical Distance Converter, Light Travel Time Calculator.
Source / Last verified: the redshift definition, classical Doppler approximation (v = cz) and exact special-relativistic Doppler formula are standard astrophysics and special relativity, built directly on the speed of light already cited in this site's other astronomy tools (NIST, exact by the SI definition of the metre). Read/checked 1 October 2026: the relativistic formula correctly reproduces the textbook identity v = 0.6c at z = 1, and never exceeds the speed of light at any z, however large.