Redshift Calculator

A galaxy’s light arrives stretched — its spectral lines shifted toward the red by the factor (1 + z). Enter the measured redshift, and this calculator returns what it implies: the recession velocity in its classical and relativistic forms, and the Hubble-law distance the velocity suggests.

Stretched Light, Receding Sources

Redshift is defined with austere simplicity: z = (λ_observed − λ_emitted) ÷ λ_emitted — the fractional stretching of light's wavelength between emission and reception. For slow recession the Doppler effect explains it (z ≈ v ÷ c), and the calculator's classical figure, v = zc, is that approximation. But galaxies' redshifts are not really Doppler shifts from motion through space — they are the expansion of space accumulating along the light's journey — and the relativistic Doppler formula, v = c × ((1 + z)² − 1) ÷ ((1 + z)² + 1), is the better bridge to a velocity language, which is why the calculator prints both and labels the relativistic one as the headline. The distinction the two figures embody — a 5% disagreement at z = 0.1, wildly more at z = 2 — is the difference between a universe that explodes outward and a universe that grows.

Worked Example: z = 0.1, a Respectable Galaxy

The defaults run a mid-range spiral: z = 0.1. Classically: 0.1 × 299,792 = 29,979 km/s. Relativistically: c × (1.21 − 1) ÷ (1.21 + 1) = 28,487 km/s — the honest braking of the exact formula against the linear guess. Hubble's law, v = H₀d with H₀ = 70 km/s/Mpc, then puts the galaxy at about 407 Mpc, roughly 1,330 million light-years of lookback-flavoured distance (cosmologists distinguish several distances at such range; the calculator's is the Hubble-law estimate and says so). The hydrogen-alpha line that left at 656.3 nm arrives at 721.9 nm — deep red, at the visible's edge — which is the literal redness the quantity is named for, priced by the (1 + z) factor the widget prints.

Hubble's Law and the Expanding Universe

Slipher measured the redshifts (from 1912), Hubble plotted them against distances (1929), and the proportionality — farther galaxies recede faster — became the first evidence that the universe expands. The calculator's H₀ = 70 sits inside the modern argument about its exact value (early-universe methods give ~67, local distance ladders ~73, the "Hubble tension" being cosmology's favourite current quarrel), and the honest page prints its assumed constant rather than smuggling one. At small z the law is well-tested and the calculator's distance is serviceable; past z ≈ 1 the expansion's acceleration history bends the simple line, and proper distances need the cosmological model integrated — the calculator's warning line is the boundary of its licence, and the numbers beyond it are orientation, not surveying.

The Redshift Desert and the Gunn–Peterson Trough

Between us and the quasars at z ≈ 6 lies a feature this calculator frames: at high enough redshift, the hydrogen along the line of sight — its Lyman-alpha line stretched by exactly the (1 + z) factor priced here, for every cloud at every intermediate z — absorbs so completely that the spectrum shows a black trough, the signature of the universe before reionisation finished clearing it. Astronomers date the end of the cosmic dark ages by where the trough lifts, near z ≈ 6. The calculator's z is thus also a depth gauge into the universe's own fog: type 6 and imagine the light fighting through a younger, thicker cosmos than the one the z = 0.1 example sails through in clear air.

The Fingers of God

Plot galaxies by redshift-derived distance and the map wrinkles: inside clusters, galaxies' random orbital velocities (hundreds of km/s) add their own Doppler shifts to the Hubble flow, smearing each cluster along the line of sight into a radial spike — astronomers call the pattern "Fingers of God," pointing, with unintended piety, at the observer. The artifact is this calculator's caveat made visible: redshift is not purely distance until the peculiar velocities are averaged or modelled away, and the great surveys (2dF, SDSS, DESI) spend enormous care on exactly that separation. The map of the universe is drawn in z and cleaned by hand, one finger at a time.

Gravitational Redshift: The Other Stretch

z has a second source that this calculator's Doppler framing should not swallow: light climbing out of a gravity well loses energy and reddens — measured for the Sun's light (a tiny z ≈ 2 × 10⁻⁶), confirmed in white dwarfs' spectra, and engineered into GPS, whose satellite clocks' rates are corrected for both special-relativistic and gravitational shifts lest positions drift by kilometres daily. The effect is Einstein's equivalence principle at the workbench, and its presence in the toolkit matters for the reader's interpretation: a measured z is evidence with more than one possible author, and cosmology's reading (expansion) wins for distant galaxies by the law's pattern across the sky, not by any single spectrum's testimony.

Related Tools

More from the Physics Calculators hub: the Wave Velocity Calculator — λ is the thing being stretched — and the Parallax Calculator, the distance method that anchors the ladder Hubble's law extends.

Frequently Asked Questions

Is the galaxy really moving at that speed?

Not through its local space, no — the modern reading is that space itself expanded while the light travelled, stretching the wavelength en route; the "velocity" is a bookkeeping translation, useful for intuition and for the Hubble diagram, misleading if imagined as a rocket. A corollary the calculator's classical column dramatises: at z > 1, zc exceeds c, which no motion can do — while the expansion reading has no such problem (distant galaxies' recession "rates" exceed c routinely and legally). The relativistic figure, capped below c by construction, is the compromise language the page headlines, with this FAQ as its footnote.

What about blueshifts?

They exist and are local: Andromeda approaches at about 110 km/s (z ≈ −0.00037), its light compressed by real Doppler motion within the Local Group, where gravity overrules the Hubble flow. The calculator takes z ≥ 0 because its subject is the cosmological population; a negative z is simply the same Doppler arithmetic with the sign of approach, and nearby galaxies' velocities are dominated by such local "peculiar motions" — a few hundred km/s of noise atop the Hubble signal, which is why the law only emerges cleanly beyond tens of megaparsecs.

How is z measured in practice?

By spectroscopy: spread the galaxy's light (the diffraction grating page's art), identify known lines — hydrogen's Balmer series, calcium's H and K, oxygen's forbidden green — and divide their observed displacement by their laboratory wavelengths. The lines' pattern is unmistakable even when stretched beyond recognition, which is how redshifts past 6 are read from the infrared where visible light has been stretched into it. The calculator begins where the spectrograph ends: with the single number z in hand, asking what universe it implies.

What does z = 1 mean physically?

That the universe was half its present size when the light was emitted — the (1 + z) factor is literally the ratio of the cosmos's scale between reception and emission, the deepest reading of the number: z = 1, half size; z = 3, a quarter; the cosmic microwave background at z ≈ 1,089 comes from a universe a thousand times smaller and hotter. The calculator's "light stretched by a factor of" line prints this ratio deliberately — of every quantity on the page, it is the one that is exactly true regardless of cosmological model, and it turns a spectrum into a time machine's dial.

How far can redshift surveys see?

Routine galaxy surveys catalogue to z ≈ 1–2; quasars and gamma-ray bursts reach z ≈ 7–9; the current record-holding galaxies from JWST sit near z ≈ 14, light from within 300 million years of the beginning — where the calculator's relativistic velocity hugs c (0.996c) and its Hubble distance has become pure orientation. The interesting truth is that "distance" at such depth needs several definitions (light-travel, comoving, luminosity distances all diverge), and the honest treatise is a cosmology text; this page's job is the velocity translation and the first-order estimate, clearly fenced.

What is the cosmic microwave background's redshift?

z ≈ 1,089 — the radiation was emitted when the universe was 380,000 years old and some 1,090 times smaller, its temperature then about 3,000 K (visible light, glowing like a lamp filament) and now, stretched by that factor, a 2.725 K microwave whisper. Enter 1089 in the calculator and the relativistic velocity returns about 0.999998 of c while the classical column cheerfully reports 1,089 times light speed — the cleanest demonstration on the page of why the classical formula is a local approximation and the (1 + z) factor the durable truth.

Who discovered redshift as a cosmic phenomenon?

Vesto Slipher, at Lowell Observatory, measured the spectra (from 1912, heroically slow photographic work) and found nearly all spiral "nebulae" redshifted; Edwin Hubble supplied the distances and the velocity–distance relation in 1929, with Milton Humason extending the measurements; Georges Lemaître had, in 1927, already derived the expansion theoretically and estimated its constant from the same data. The calculator's line descends from all of them — Slipher's z values were the facts, Hubble's law the pattern, Lemaître's the interpretation — and the modern quarrel over H₀'s third significant figure is their inheritance, still unsettled, still printed on this page with the assumed constant declared.