Redshift Calculator

Background.

Redshift, denoted z, measures how much a spectral line's wavelength has shifted between when light was emitted by a distant source and when it was observed here: z = (λ_observed − λ_emitted) / λ_emitted. This calculator computes z directly from those two wavelengths, then reports recession velocity two different ways — the simple classical (Doppler) approximation v = cz, and the correct relativistic Doppler formula v = c·[(1+z)² − 1]/[(1+z)² + 1] — because most competing redshift calculators only offer the classical formula, and that formula silently gives wrong, sometimes physically impossible, answers once z grows past roughly 0.1.

The honesty gap is dramatic and easy to demonstrate: for a quasar's Lyman-alpha line redshifted from 121.6 nm to 486.4 nm — a real, observationally realistic redshift of exactly z = 3 — the classical formula v = cz predicts a recession velocity of about 3.00 times the speed of light, an outright physical impossibility. The relativistic formula, applied to the identical z = 3 input, correctly returns about 0.882c, a value that respects special relativity's speed limit at every redshift, no matter how large. This calculator computes both values side by side and adds a plain-language recommendation — 'classical is fine here' or 'use the relativistic figure' — based on whether |z| has crossed the roughly 0.1 threshold past which the two formulas meaningfully diverge, so you never have to remember the threshold yourself or wonder which number is trustworthy for your specific input.

A note on scope, in the same spirit of honesty: the relativistic-Doppler formula implemented here describes a source receding directly away from an observer through flat spacetime under special relativity. That is an excellent approximation for many practical purposes and is exactly the formula astronomy references present as the 'more accurate' alternative to v = cz. It is not, however, identical to a full cosmological-redshift calculation for very distant galaxies, which properly requires general relativity's FRW (Friedmann–Robertson–Walker) spacetime metric and a specific cosmological model — the expansion of space itself, not an object's motion through pre-existing space, is what drives redshift at truly cosmological distances. Swinburne University's astronomy encyclopedia makes exactly this distinction, and this calculator's content follows it rather than blurring the two physically distinct pictures together.

Redshift shows up everywhere in astrophysics: it is how astronomers estimate how fast (and, via Hubble's law, how far away) galaxies are receding from us, how they detect exoplanets through the tiny periodic Doppler wobble a planet induces in its host star's spectrum, and how they measure orbital velocities within our own galaxy. Positive z (redshift, longer observed wavelength) means the source is receding; negative z (blueshift, shorter observed wavelength) means it is approaching — the Andromeda Galaxy, famously, is blueshifted, because it is one of the relatively few galaxies moving toward the Milky Way rather than away from it under cosmic expansion.

What is redshift calculator?

Redshift is defined purely from an observed change in wavelength: z = (λ_observed − λ_emitted) / λ_emitted, where λ_emitted is the wavelength a spectral line has in the source's own rest frame (a value physicists and chemists have measured precisely in laboratories for essentially every atomic transition) and λ_observed is the wavelength actually measured by a telescope or spectrograph here. Because z is built from a ratio of two wavelengths, it is dimensionless — a pure number, with no units — which makes it convenient to compare across observations made at completely different parts of the electromagnetic spectrum, from radio to gamma rays, as long as the same spectral line's rest wavelength is known.

Astronomers distinguish two physically different causes of redshift, and this calculator's content is careful not to conflate them. Doppler redshift arises from an object's actual motion through space relative to the observer — the same physical effect that makes a passing ambulance's siren drop in pitch as it moves away. Cosmological redshift, by contrast, arises from the expansion of space itself stretching a photon's wavelength during its long journey to us, even for a source that is not, in any meaningful sense, 'moving' through the space immediately around it. At low redshift the two effects are numerically almost indistinguishable, which is why the classical v = cz Doppler formula works fine for nearby galaxies; at high redshift they genuinely diverge, and the relativistic Doppler formula this calculator uses is the correct special-relativistic treatment of an object's motion — a substantial improvement over v = cz, but still distinct from a full general-relativistic cosmological treatment at the very largest observed redshifts.

How to use this calculator.

1. 01Enter the rest-frame (emitted) wavelength of the spectral line you're measuring, in nanometers.
2. 02Enter the observed wavelength actually measured.
3. 03Read the redshift z — positive means the source is receding (redshifted), negative means approaching (blueshifted).
4. 04Compare the classical and relativistic recession-velocity estimates, and follow the plain-language recommendation for which one to trust at your specific z.

The formula.

Redshift is computed directly from the two wavelengths: z = (λ_obs − λ_emit)/λ_emit. A positive result means the observed wavelength is longer than the rest wavelength (redshift, consistent with a receding source); a negative result means the observed wavelength is shorter (blueshift, consistent with an approaching source).

The classical (low-velocity) Doppler approximation treats redshift as directly proportional to velocity as a fraction of light speed: z ≈ v/c, which rearranges to the familiar v = cz. This is an excellent approximation when v is much smaller than c — equivalently, when |z| is small — but it is only an approximation, and it has no built-in mechanism to prevent v from exceeding c as z grows, which is exactly the failure this calculator demonstrates at z = 3.

The relativistic Doppler formula instead accounts properly for special relativity's velocity-addition rules and time dilation, giving v = c·[(1+z)² − 1]/[(1+z)² + 1]. As z grows without bound, this expression approaches c but mathematically never reaches or exceeds it — the (1+z)² term grows in both the numerator and denominator, and the ratio [(1+z)² − 1]/[(1+z)² + 1] is bounded strictly below 1 for every finite z, which is exactly the behavior a correct relativistic formula must have. Both wavelength inputs must be strictly positive — a wavelength of zero or less has no physical meaning — and that single guard is sufficient to guarantee both formulas stay well-defined and finite for every valid input, including blueshifted (negative-z) sources.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Swinburne University of Technology. COSMOS — The SAO Encyclopedia of Astronomy. "Cosmological redshift." z = (λobs − λrest)/λrest; classical z≈v/c at low velocity; distinguishes cosmological expansion from Doppler motion.
2. [2]NASA Science. "Hubble Cosmological Redshift." Light is stretched to longer, redder wavelengths as the universe expands and galaxies recede.
3. [3]NASA Goddard Space Flight Center. StarChild. "Redshift and Hubble's Law."
4. [4]National Institute of Standards and Technology. CODATA recommended value: speed of light in vacuum, c = 299,792,458 m/s, exact.