Redshift and Recession Velocity Calculator: z, v = cz and the Relativistic Form
Redshift and Recession Velocity Calculator
Rest wavelength and observed wavelength in; the redshift z out, with both recession speeds beside it — the school approximation v ≈ cz and the relativistic Doppler form — and the percentage by which the first one overstates the second. At z = 1 the approximation returns exactly c, which is the clearest sign that it has broken.
Wavelengths to redshift, and redshift to speed
The quasar 3C 273 shows the Hβ line, rest wavelength 486.13 nm, at 563.11 nm. Find its redshift and its recession speed both ways. Use H0 = 70 km/s/Mpc for a rough distance.
One measurement, two interpretations, and where the first one dies
- λ₀
- rest wavelength: the line’s wavelength in a laboratory. Looked up, not measured.
- λ
- observed wavelength, from your spectrograph. Longer than λ₀ for a receding source.
- z
- redshift, a pure ratio with no units. 1 + z is the factor by which every wavelength has been stretched.
- v ≈ cz
- the approximation every school syllabus uses. Good to 1% below z = 0.02 and useless above z = 0.5. At z = 1 it returns exactly c.
- v/c = …
- the special-relativistic Doppler form, which never exceeds c for any z. Better, but still a velocity interpretation of something that is really the expansion of space.
Worked example
The quasar 3C 273 shows the Hβ line, rest wavelength 486.13 nm, at 563.11 nm. Find its redshift and its recession speed both ways. Use H0 = 70 km/s/Mpc for a rough distance.
The shift, then the ratio. Δλ = 563.11 − 486.13 = 76.98 nm. z = 76.98/486.13 = 0.15835. Divide by the rest wavelength, not the observed one — dividing by 563.11 gives 0.1367, an 14% error, and nothing about 0.1367 looks wrong.
The school formula. v ≈ cz = 299 792.458 × 0.15835 = 47,473 km/s, which is 0.158c. Quick, and already wrong by more than the measurement error.
The relativistic formula. (1 + z)² = 1.158353² = 1.341742. Then v/c = (1.341742 − 1)/(1.341742 + 1) = 0.341742/2.341742 = 0.145949, so v = 43,754 km/s.
Compare them. The approximation is high by 3,718 km/s, which is 8.5%. At this redshift that is far larger than any plausible measurement error, so the choice of formula, not the data, is the dominant uncertainty in the answer.
A distance, with heavy caveats. 43,754/70 = 625 Mpc, about 2.04 billion light years. Published distances for 3C 273 are nearer 700 Mpc, because a proper cosmological distance accounts for how the expansion rate has changed over the light’s journey — which v/H0 cannot do. Treat this figure as an order of magnitude.
What the measurement actually says. Independent of every formula above: the light arriving from 3C 273 has every wavelength 1.1584 times longer than it left with. That is the observation. 0.158c and 0.146c are two models of why.
Where the cz approximation stops being usable
| z | v = cz (km/s) | Relativistic v (km/s) | cz is too high by | Verdict |
|---|---|---|---|---|
| 0.001 | 300 | 300 | 0.05% | either formula, freely |
| 0.01 | 2,998 | 2,983 | 0.50% | cz is fine |
| 0.05 | 14,990 | 14,807 | 1.2% | cz is fine for school work |
| 0.158 | 47,473 | 43,754 | 8.5% | the choice of formula now dominates |
| 0.5 | 149,896 | 117,566 | 27.5% | cz should not be used |
| 1.0 | 299,792 | 179,875 | 66.7% | cz returns exactly c |
| 2.0 | 599,585 | 239,834 | 150% | cz returns twice the speed of light |
| 7.0 | 2,098,547 | 289,792 | 624% | cz is meaningless; even v is the wrong question |
Lines worth knowing, and where they land at the default redshift
| Line | Rest λ (nm) | At z = 0.158 | At z = 1 | At z = 3 |
|---|---|---|---|---|
| Ca II K | 393.37 | 455.6 | 786.7 | 1573.5 |
| Hδ | 410.17 | 475.1 | 820.3 | 1640.7 |
| Hγ | 434.05 | 502.8 | 868.1 | 1736.2 |
| Hβ | 486.13 | 563.1 | 972.3 | 1944.5 |
| Mg b | 518.36 | 600.5 | 1036.7 | 2073.4 |
| Na D | 589.29 | 682.6 | 1178.6 | 2357.2 |
| Hα | 656.28 | 760.2 | 1312.6 | 2625.1 |
What a redshift measures, and the three different things people call the speed
The measurement is a ratio of two wavelengths, and it is one of the cleanest measurements in astronomy. You identify a pattern of spectral lines, you note that the whole pattern sits at longer wavelengths than it does in a laboratory, and you measure by how much. The answer, z = Δλ/λ₀, needs no assumption about distance, luminosity or anything else. Everything contentious in this topic happens afterwards, when z is converted into a speed or a distance.
Divide by the rest wavelength. Always the rest wavelength. It is the denominator in every definition of z, and using the observed value instead is the single commonest error here. For the worked example it turns 0.1584 into 0.1367 — a 14% error that produces a perfectly plausible-looking number. One way to never forget it: z is defined so that λ = λ₀(1 + z), and the only way to get that is to divide the shift by λ₀.
Boards write the sign differently, and you should notice which one you are reading. AQA’s specification writes z = Δλ/λ = −v/c for v ≪ c, with the minus sign because their v is measured positive towards the observer. Cambridge International’s 9702 writes Δλ/λ ≈ v/c with no minus, taking v positive for recession. Neither is wrong and both appear in exam papers. This page follows the second: a positive z gives a positive recession speed, and an approaching galaxy comes out with a negative z and a negative speed. Check which convention your paper uses before you worry about a sign.
The approximation v ≈ cz has a condition attached and school papers rarely state it. It is the first term of an expansion valid for z ≪ 1. At z = 0.01 it is right to half a percent. At z = 0.1 it is 5% high. At z = 0.158, the worked example, it is 8.5% high. At z = 1 it returns exactly the speed of light, and above z = 1 it returns more than the speed of light — which tells you nothing about the galaxy and everything about the formula. This page always shows both numbers side by side, with the percentage gap between them, so you can see the approximation failing instead of being told about it.
And now the honest part, which most pages skip: neither speed is really right. Cosmological redshift is not a Doppler shift. The galaxy is not moving through space at some velocity, with its light Doppler-shifted on arrival; the space between you and it has expanded while the light was in transit, and the light’s wavelength was stretched along with it. The factor 1 + z is the ratio by which the universe has expanded since the light was emitted. That framing explains something the Doppler one cannot: galaxies with z > 1.5 or so have recession rates greater than c under the standard model, and this violates nothing, because nothing is moving through space faster than light. The relativistic Doppler figure on this page is the right answer to a different question — how fast would a source have to move past you to produce this shift — and it is shown because it is what a special-relativity course asks for.
Local motion contaminates small redshifts. A galaxy’s measured shift is the expansion plus whatever it is doing within its own cluster, and cluster velocities run to a thousand kilometres per second or more. For a galaxy at 10 Mpc the expansion contributes about 700 km/s, so the peculiar motion can be comparable to the signal; the Andromeda Galaxy is close enough that gravity wins outright and it is blueshifted, approaching us at roughly 300 km/s. This page will return a negative z for that case rather than refusing it, and the note under the headline says what a negative z means. The distance figure, however, goes negative too, and a negative distance is not a distance — Hubble’s law simply does not apply to an approaching galaxy, and the page says so rather than hiding the figure.
What a redshift is actually used for. At the scale where expansion dominates, z is the easiest distance indicator in astronomy: one spectrum gives it, no standard candle needed. That is why galaxy surveys quote redshifts rather than distances, and why a map of the universe is a redshift survey. The conversion from z to a distance in megaparsecs needs a cosmological model, and v/H0 — which is what this page gives — is the crudest version of it: accurate to a few percent at z = 0.05 and out by tens of percent by z = 1, because the expansion rate has not been constant over the light’s journey. Use it for an order of magnitude and nothing finer.
Frequently asked questions
Do I divide by the rest wavelength or the observed one?
The rest wavelength, every time. z = (λ − λ₀)/λ₀, which rearranges to λ = λ₀(1 + z). Dividing by the observed wavelength gives a number that is too small by a factor of (1 + z) — 14% at the default values on this page — and it looks entirely reasonable, so nothing catches it except doing it right.
Why does the calculator give two different speeds?
Because two different formulas are in common use and they disagree once z is more than a few hundredths. v = cz is the first term of a series valid for z ≪ 1 and is what every school syllabus sets; the relativistic Doppler expression is exact for a source genuinely moving through space. Showing one without the other either hides an error or answers a question the exam did not ask, so this page shows both and the percentage gap.
Can a recession speed be faster than light?
The cz approximation will happily return one, above z = 1, and that result is an artefact. The relativistic Doppler formula never does. The deeper answer is that cosmological redshift is not a velocity at all: it is the expansion of space, and under the standard model galaxies beyond roughly z = 1.5 do have recession rates exceeding c without anything moving through space faster than light. Nothing on this page computes that; it is flagged so you know the limit of what is here.
What does a negative z mean?
A blueshift: the observed wavelength is shorter than the rest wavelength, so the source is approaching. It is a normal measurement for nearby galaxies, where local gravity beats the expansion — the Andromeda Galaxy is blueshifted at about 300 km/s. This page returns the negative z and the negative speed, and warns that the Hubble distance figure below is then meaningless, because Hubble’s law describes expansion and not local infall.
Why do all the lines move by different amounts in the diagram?
Because the shift is proportional, not additive. Every wavelength is multiplied by the same (1 + z), so a line at 656 nm moves 1.67 times as far in nanometres as a line at 393 nm. That is exactly why the diagram draws three arrows of different lengths from one redshift. It is also what makes the pattern recognisable: the lines keep their ratios, so the fingerprint survives the stretch.
How accurate is the distance this page gives?
Crude, deliberately. It is v/H0 with the relativistic v, which assumes the expansion rate has been constant for the whole of the light’s journey. At z = 0.05 that is good to a few percent. At the default z = 0.158 it gives 625 Mpc where published figures for 3C 273 are nearer 700 Mpc. By z = 1 it is wrong by tens of percent. A real cosmological distance needs the matter and dark-energy densities as well, which this page does not take.
Related calculators
References
- AQA, A-level Physics 7408 specification, subject content §3.9 “Astrophysics”. §3.9.2.1 gives the apparent magnitude scale with “dimmest visible stars have a magnitude of 6” and “difference of 1 on magnitude scale is equal to an intensity ratio of 2.51”; §3.9.2.2 gives m − M = 5 log(d/10); §3.9.3.1 gives z = Δλ/λ = −v/c for v ≪ c and §3.9.3.2 gives v = Hd.
- Cambridge Assessment International Education, Cambridge International AS & A Level Physics 9702 syllabus for examination in 2025, 2026 and 2027, Topic 25 “Astronomy and cosmology”. 25.1 gives the radiant flux inverse-square law F = L/4πd² and the standard-candle idea; 25.3 gives Δλ/λ ≈ v/c and Hubble’s law v ≈ H0d, marked “SI units only”.
- 3C 273 reference data, University of Nevada Las Vegas physics teaching notes, physics.unlv.edu: redshift z = 0.158 339(67), first recognised as a quasar by Maarten Schmidt in 1963, proper distance about 700 Mpc. A teaching page rather than a catalogue, and cited as such.
- BIPM, The International System of Units (SI), 9th edition (2019). The speed of light in vacuum is exactly 299 792 458 m s−1 by the definition of the metre; the astronomical unit appears in Table 8 as a non-SI unit accepted for use with the SI.
- Hubble Tension or Distance Ladder Crisis?, arXiv:2408.11031. Quotes the Planck CMB value as 67.4 ± 0.5 and the SH0ES Cepheid + Type Ia supernova distance-ladder value (Riess et al. 2022) as 73.04 ± 1.04 km s−1 Mpc−1, and describes the gap as a significant tension within ΛCDM.