The redshift calculator above turns a pair of wavelengths into the redshift parameter z, and z into a velocity. Redshift is the single most-used number in observational cosmology, because it is what a spectrograph actually measures: a known spectral line appears at the wrong wavelength, and how wrong it is encodes how fast the source is separating from us. The tool works from wavelengths, from frequencies, from a published z, or backwards from a velocity.
Arb Digital builds free tools that separate what is measured from what is inferred. This page stops at z and the velocity that corresponds to it. It does not convert either into a distance, because that requires a Hubble constant and a set of cosmological assumptions, and the site's Hubble law calculator handles that step explicitly rather than smuggling it in.
What This Redshift Calculator Does
Every chemical element absorbs and emits light at wavelengths fixed by atomic physics, and those wavelengths are the same everywhere. Hydrogen-alpha is at 656.28 nm in a laboratory on Earth and at 656.28 nm in a galaxy a billion light years away. So when a spectrum arrives with that line sitting at 700 nm instead, the discrepancy is not a property of the hydrogen. It is a property of the journey.
The redshift parameter z packages that discrepancy as a fraction: the change in wavelength divided by the rest wavelength. A z of 0 means no shift. A z of 1 means the light arrived at twice the wavelength it left with. A z of 7, which the most distant confirmed galaxies exceed, means the wavelength was stretched eightfold, which is why light emitted as ultraviolet arrives as infrared and why infrared telescopes are the ones that see the early universe.
Converting z to a velocity is where the tool earns its place. The obvious approach, multiplying z by the speed of light, is a low-speed approximation that fails badly once z is more than a few tenths, and taken literally it gives velocities above c for z greater than 1. The relativistic Doppler relation does not, and the calculator reports both figures side by side precisely so the divergence between them is visible.
How to Use It
- Choose your starting point. Wavelengths if you are working from a spectrum, frequencies if your source is quoted in the radio convention, z if you took it from a catalogue, velocity if you are going the other way.
- Enter the rest wavelength carefully. It must be the laboratory value for the specific line you identified. Misidentifying which line you are looking at is the dominant error in amateur redshift work, and no arithmetic can rescue it.
- Use any consistent unit for the pair. Only the ratio matters, so nanometres, angstroms and metres all give the same z as long as both values use the same one.
- Compare the two velocity figures. The relativistic one and the naive cz one agree at small z and diverge sharply at large z. That gap is the whole reason this page reports both.
- Read the scale factor. It tells you how much smaller the universe was when the light set out, which for cosmological redshifts is a more physically honest reading than any velocity.
The Formulas: How Redshift and Velocity Are Calculated
The definition is a fraction. As set out in the Swinburne University COSMOS astronomy encyclopedia entry on redshift, z = (λobs − λrest)/λrest, so 1 + z is simply the ratio of the observed wavelength to the rest wavelength. Because frequency and wavelength are inversely related, the frequency form is 1 + z = frest/fobs, with the ratio the other way up.
The velocity step uses the relativistic Doppler relation for light. OpenStax University Physics Volume 3, section 5.7 on the Doppler effect for light, gives the receding case as λobs = λs√((1 + v/c)/(1 − v/c)). Substituting 1 + z for the wavelength ratio and rearranging for the velocity fraction β = v/c gives β = ((1 + z)² − 1)/((1 + z)² + 1). The speed of light itself is the exact defined value of 299,792,458 m/s published in the NIST CODATA value for the speed of light in vacuum.
Work the defaults. Hydrogen-alpha at a rest wavelength of 656.28 nm observed at 700 nm gives z = (700 − 656.28)/656.28 = 0.06662. Then (1 + z)² = 1.13767, so β = 0.13767/2.13767 = 0.064404, and the recession velocity is 0.064404 × 299,792 = 19,308 km/s. The naive figure, cz, would be 0.06662 × 299,792 = 19,972 km/s — about 3.4 per cent higher. At this modest redshift the two are close. At z = 1 the naive figure is the full speed of light while the correct one is 60 per cent of it, and beyond that the naive formula is simply wrong rather than approximate.
Three Different Things Called Redshift
The same measured z can arise from three physically distinct causes, and the page cannot tell which one applies to your spectrum. A Doppler redshift comes from genuine motion through space, which is what a star orbiting in a binary system shows. A cosmological redshift comes from the expansion of space itself while the light is in transit, stretching the wave along with the space it crosses. A gravitational redshift comes from light climbing out of a gravitational well and losing energy without any relative motion at all.
For nearby objects the distinction is academic and the Doppler interpretation the calculator uses is the right one. For distant galaxies it is not academic at all. A cosmological redshift is not really a velocity, and the relativistic Doppler formula applied to it gives a number that is best read as a convenient label rather than as a speed anything is travelling. This is why the scale factor is reported: for a cosmological redshift, saying that the universe was 1/(1 + z) of its present size when the light departed is a statement that survives scrutiny in a way that a velocity does not.
Real spectra usually contain more than one contribution. A galaxy's measured redshift mixes the cosmological expansion with the galaxy's own peculiar motion through its cluster, and for nearby galaxies the peculiar motion can dominate entirely. Andromeda is blueshifted, approaching the Milky Way, because at that distance local gravity beats the expansion comfortably.
Why the Naive Formula Fails and Where
Multiplying z by the speed of light is the first-order approximation to the relativistic relation, and like every first-order approximation it is excellent until it is not. At z = 0.01 the two agree to within one per cent. At z = 0.1 the gap is about five per cent. At z = 0.5 the naive answer is over-large by roughly a third, and at z = 1 it returns exactly c, which no massive object can reach.
The relativistic version approaches c asymptotically instead. At z = 3 it gives about 0.882c, at z = 7 about 0.969c, and no finite redshift ever reaches or exceeds the speed of light. That is the correct behaviour, and it is the reason the two figures sit next to each other in the results grid rather than one replacing the other.
There is a further subtlety the grid cannot show. In an expanding universe the recession of very distant galaxies is not motion through space and is not bounded by c at all; galaxies beyond a certain distance recede faster than light without violating anything, because the expansion is of space rather than in it. That regime needs a cosmological model rather than a Doppler formula, and it is well outside what this page claims to do.
How This Differs From the Adjacent Astronomy Tools
The boundary in one sentence: this page converts between wavelengths, z and a relativistic recession velocity for light, while the Doppler effect calculator handles the classical moving-source and moving-observer problem for sound and light, where the source and the observer both have velocities relative to a medium or to each other.
The Hubble law calculator takes the velocity this page produces and converts it into a distance using a Hubble constant, which is the next step in the chain and a step that carries real cosmological assumptions. For the relativistic quantities behind the velocity, see the time dilation calculator, the length contraction calculator and the relativistic kinetic energy calculator. The wavelength calculator converts between wavelength, frequency and wave speed for the underlying wave relationship.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using cz as the velocity at large redshift — it is a small-z approximation, and past z = 1 it returns speeds at or above the speed of light, which is a sign the formula has left its range rather than a discovery.
- Misidentifying the spectral line — z depends entirely on knowing which rest wavelength you are comparing against, and picking the wrong line produces a confident, precise, wrong answer.
- Inverting the frequency ratio — wavelength grows with redshift while frequency falls, so the frequency form of 1 + z is the rest value over the observed one, not the other way round.
- Reading a cosmological redshift as a real speed — expansion stretches the light in transit rather than the source moving away through space, and the scale factor is the more meaningful reading.
- Forgetting peculiar motion — a nearby galaxy's own movement within its cluster can swamp the expansion entirely, which is why some nearby galaxies are blueshifted.
Related Free Tools From Arb Digital
For distance from a recession velocity, use the Hubble law calculator, and for the classical moving-source problem the Doppler effect calculator. The time dilation calculator, the length contraction calculator and the relativistic kinetic energy calculator cover the rest of special relativity, and the escape velocity calculator covers the gravitational side. Convert wave quantities with the wavelength calculator and cosmic scales with the astronomical distance converter, and browse the full free online tools hub for everything else.
Frequently Asked Questions
That the light arrived at twice the wavelength it left with, since 1 + z is the ratio of observed to rest wavelength. It does not mean the source is receding at the speed of light, which is what the naive cz formula would wrongly suggest.
Because that is a first-order approximation valid only at small z. It is within a per cent at z = 0.01, roughly a third too large at z = 0.5, and returns exactly the speed of light at z = 1, which no massive object can reach. The relativistic relation approaches c asymptotically instead.
Yes. A negative z is a blueshift, meaning the observed wavelength is shorter than the rest wavelength and the source is approaching. The Andromeda galaxy is blueshifted because its motion towards the Milky Way outweighs cosmic expansion at that distance. The limit is z = −1, which would mean zero observed wavelength.
Not in the ordinary sense. Cosmological redshift comes from space itself expanding while the light is in transit, stretching the wave rather than the source moving away through space. The velocity a Doppler formula returns for it is a useful label rather than a speed anything travels at.
It is 1/(1 + z), the size of the universe when the light was emitted relative to its size now. At z = 1 the universe was half its current scale; at z = 7 it was one eighth. For cosmological redshifts this is a more physically honest reading than any velocity figure.
No, deliberately. Converting a redshift into a distance requires a Hubble constant and a set of cosmological assumptions, so that step belongs on the Hubble law calculator where those assumptions are stated explicitly rather than hidden inside this page.
Any units, provided both use the same one. Redshift is a ratio, so nanometres, angstroms, micrometres and metres all give an identical z. Mixing two different units is the one thing that breaks it.
In an expanding universe, yes, and without violating relativity, because sufficiently distant galaxies are carried apart by the expansion of space rather than moving through it. That regime requires a cosmological model rather than the Doppler relation used here, and it is outside what this page calculates.
This tool is provided for educational use. It applies the relativistic Doppler relation for light and does not model cosmological expansion, gravitational redshift or peculiar motion, all of which contribute to real astronomical measurements. Results should not be treated as a cosmological distance or age determination.
