Cosmological Redshift to Recession Velocity Calculator
Solve the classic low-redshift Hubble relation v ≈ c · z for either of its two unknowns — recession velocity or redshift — with step-by-step math, a live spectrum diagram showing the shifted line, a velocity gauge against the speed of light, and companion cosmological metrics including comoving distance and lookback time.
Redshift is a wavelength ratio
Redshift is defined purely by what a spectrograph measures: the fractional increase in wavelength between the laboratory rest frame and the observed spectrum. z = (λ_obs − λ_rest)/λ_rest. There is nothing in that definition that requires a velocity, a distance, or a cosmological model. It is only when you interpret the stretch as a Doppler shift that the relation v ≈ c·z appears — and that interpretation is valid only in the limit z ≪ 1, where the relativistic Doppler formula linearises.
Cosmological redshift is not a Doppler shift
In an expanding universe, photon wavelengths stretch with the scale factor: 1 + z = anow/athen. The photon does not simply "leave a receding source" — the space through which it travels expands during its journey. For nearby objects the two pictures give the same linear answer, v ≈ cz, because the expansion velocity at small distances is small compared with c. At large z they diverge badly: at z = 1 the classical formula gives 299,792 km/s (exactly the speed of light) while the relativistic recession velocity is only 0.6 c, and the correct cosmological interpretation involves no velocity at all.
The classical formula fails above z ≈ 0.1
v ≈ c·z is the first term of the relativistic expansion, and the error grows quadratically. At z = 0.1 it overstates the velocity by about 5%; at z = 0.5 by about 20%; at z = 1 by about 67%; at z = 3 it is meaningless because the classical value exceeds the speed of light. Any calculation using v ≈ c·z should therefore quote the redshift as the primary, model-independent measurement and treat the velocity as a low-z convenience. This calculator shows both figures side by side so the divergence is always visible.
Peculiar velocities contaminate small z
Below z ≈ 0.01 the cosmological recession velocity is comparable to the peculiar velocity of the galaxy within its local gravitational field — a few hundred km/s in a group, up to 1,000 km/s in a cluster core. The observed redshift is the sum of the cosmological and peculiar components, so the inferred velocity can be badly wrong at small z. For example, the Coma cluster at z ≈ 0.023 has a recessional velocity near 6,900 km/s, but individual cluster members scatter by ±1,000 km/s around that value. Below z ≈ 0.002 the Hubble flow is not even a well-defined concept, because the Local Group is gravitationally bound.
How to Use This Calculator
The redshift–velocity relation links two quantities: v and z. Pick which one you want the tool to solve, then enter the other. The auxiliary inputs drive the spectrum diagram and the companion cosmological metrics.
1. Choose what to solve for
Pick v or z from the toggle. The calculator applies v ≈ c·z or the inverse z = v/c automatically. The active field becomes read-only and shows the SOLVED badge. The speed of light is a fixed physical constant, so it is not a variable.
2. Use realistic presets
Velocity presets run from 300 km/s (Local Group edge) to 150,000 km/s (half the speed of light). Redshift presets run from 0.001 (nearby groups) through 0.03 (Coma-like clusters) to 7.0 (the first galaxies). The slider for z is logarithmic so you can reach both the very low and the very high regime smoothly.
3. Set the auxiliary inputs
The rest wavelength drives the spectrum diagram and the observed-wavelength readout. The Hubble constant and matter density drive the comoving distance, lookback time and age-of-universe figures. None of these affects v or z — the core relation depends only on c.
4. Read the result and verdict
The result card shows the solved value plus the distance, lookback time and v/c fraction, with a colour-coded verdict classifying the regime from the Local Group through nearby Hubble flow, intermediate distances, relativistic regime, and the high-redshift early universe.
5. Check the spectrum and the gauge
The diagram draws a rainbow spectrum with the rest-frame line in amber and the observed line in teal, the wavelength stretch bracketed between them, and a velocity gauge with the marker placed against the speed of light. The observed wavelength is computed from λ_obs = λ_rest(1 + z).
6. Switch to Advanced for depth
Advanced mode adds a classical-versus-relativistic velocity chart, a sensitivity table on a ±1% change in each input, and companion metrics including the relativistic velocity, the divergence between the two, the Hubble-law distance, the exact comoving distance, the lookback time, and the age of the universe at emission.
What This Tool Offers
Built around the low-redshift relation v ≈ c·z and its companion cosmological distance and time integrals.
Two-way solving
Solve for recession velocity in km/s or for the cosmological redshift. The calculator applies v = c·z or z = v/c in whichever direction you need, and flags the point at which the classical relation stops being accurate.
Live spectrum diagram
A rainbow spectrum strip with the rest-frame line in amber, the observed redshifted line in teal, the wavelength stretch bracketed between them, and a velocity gauge showing how close the recessional motion is to the speed of light.
Cosmological-regime verdict
Each result is classified from the Local Group through nearby Hubble flow, intermediate distances, the relativistic regime, and the high-redshift early universe, with a coloured badge and the corresponding accuracy caveat.
Classical vs. relativistic velocity
Both the linear v ≈ c·z value and the exact relativistic recession velocity are computed and compared, so the size of the classical error is always visible in kilometres per second and as a percentage.
Distance and lookback time
The Hubble-law distance c·z/H₀ and the exact flat-ΛCDM comoving distance, lookback time and age of the universe at emission are all computed by numerical integration for any H₀ and Ωm.
Reference tables
A redshift lookup grid with velocity, distance and lookback time, a table of notable objects from the Local Group to the CMB, and a full variable reference table with unit notes.
Reference Tables & Conversions
A redshift explorer, a redshift lookup grid, and notable objects from the nearby universe to the cosmic microwave background.
Redshift explorer
Redshift Lookup: Velocity, Distance and Lookback Time
Computed with v ≈ c·z, the relativistic velocity, the flat ΛCDM comoving distance at H₀ = 70 km/s/Mpc and Ωm = 0.315, and the corresponding lookback time.
| Redshift z | v ≈ c·z (km/s) | Relativistic v (km/s) | v / c | Comoving distance (Mpc) | Lookback time (Gyr) |
|---|
Notable Objects by Redshift
Representative redshifts and the corresponding classical recession velocities. Actual quoted velocities in the literature may use the relativistic formula or the cosmological interpretation.
| Object / class | Redshift z | v ≈ c·z (km/s) | Approx. distance | Notes |
|---|---|---|---|---|
| Andromeda (M31) | −0.001 | −300 | 0.78 Mpc | Blueshifted — approaching the Milky Way |
| Local Group dwarf galaxies | −0.0005 to +0.001 | −150 to +300 | 0.1 – 2 Mpc | Gravitationally bound; no Hubble flow |
| Virgo Cluster | 0.004 | 1,200 | 16.5 Mpc | Nearest rich cluster |
| Coma Cluster | 0.023 | 6,900 | 100 Mpc | Classic Hubble-law calibrator |
| Hercules Cluster | 0.037 | 11,100 | 160 Mpc | Member scatter ±1,000 km/s |
| Great Wall / CfA2 | 0.03 | 9,000 | 130 Mpc | Large-scale structure filament |
| SDSS typical galaxy | 0.1 | 30,000 | 440 Mpc | Classical error ≈ 5% |
| 3C 273 quasar | 0.158 | 47,400 | 750 Mpc | First identified quasar |
| Galaxy cluster at z = 0.5 | 0.5 | 149,900 | 1,880 Mpc | Classical error ≈ 20% |
| Distant supernova at z = 1 | 1.0 | 299,792 (= c) | 3,400 Mpc | Classical formula invalid |
| HUDF galaxies | 1 – 6 | 300,000 – 1,800,000 | 5 – 8 Gpc | Classical value exceeds c |
| GN-z11 (earliest confirmed galaxy) | 10.6 | 3,180,000 | 9.7 Gpc | Age at emission ≈ 0.4 Gyr |
| Reionisation era | 6 – 10 | 1,800,000 – 3,000,000 | 8 – 9.5 Gpc | First stars ionising the IGM |
| Cosmic microwave background | 1,100 | 330,000,000 | 14.0 Gpc | Surface of last scattering |
Common Rest Wavelengths
Laboratory wavelengths of the spectral lines most often used for redshift measurement, in nanometres.
| Line | Rest wavelength | Element / transition | Typical use |
|---|---|---|---|
| Lyα | 121.567 nm | Hydrogen Lyman-alpha | High-redshift galaxies, z > 2 |
| Lyβ | 102.572 nm | Hydrogen Lyman-beta | Very high redshift |
| Ca II K | 393.366 nm | Calcium H and K doublet | Stellar and galaxy kinematics |
| Ca II H | 396.847 nm | Calcium H and K doublet | Stellar and galaxy kinematics |
| Hγ | 434.047 nm | Hydrogen Balmer-gamma | Star-forming galaxies |
| Hβ | 486.133 nm | Hydrogen Balmer-beta | Star-forming galaxies |
| [O III] | 500.684 nm | Doubly ionised oxygen | AGN, emission-line galaxies |
| Mg II | 279.800 nm | Magnesium doublet | Quasar absorption systems |
| Na I D | 589.000 nm | Sodium doublet | Stellar absorption, nearby galaxies |
| Hα | 656.281 nm | Hydrogen Balmer-alpha | Star formation rate, nearby galaxies |
| [N II] | 658.345 nm | Ionised nitrogen | AGN diagnostics, H II regions |
| [S II] | 671.6 / 673.1 nm | Ionised sulphur doublet | Electron density diagnostics |
| 21 cm line | 211,061 nm | Neutral hydrogen spin-flip | Galaxy rotation curves, z < 1 |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| v | Recession velocity (classical) | 0 – 299,792 | km/s |
| z | Cosmological redshift | 0 – 1,100 | dimensionless |
| c | Speed of light in vacuum | 299,792.458 (exact) | km/s |
| λrest | Rest-frame wavelength | 100 – 211,061 | nm |
| λobs | Observed wavelength | — | nm |
| H₀ | Hubble constant | 67.4 – 73.0 | km/s/Mpc |
| Ωm | Present matter density parameter | 0.27 – 0.32 | dimensionless |
| dC | Comoving distance | 0 – 14,000 | Mpc |
| tL | Lookback time | 0 – 13.8 | Gyr |
What Does v ≈ c · z Measure?
The relation v ≈ c·z converts a measured redshift into an apparent recession velocity. It is the low-redshift limit of the relativistic Doppler formula, and it is one of the simplest and most widely used equations in observational cosmology: a redshift of 0.01 corresponds to roughly 3,000 km/s, a redshift of 0.1 to roughly 30,000 km/s. It is the formula behind the original Hubble diagram, it is still the standard back-of-the-envelope conversion, and it is accurate to better than 5% for everything within a few hundred megaparsecs.
Redshift–Velocity Formula
The defining low-redshift relationship is:
Rearranged to solve for redshift:
The redshift itself is defined by the wavelength ratio:
λobs = λrest ( 1 + z )
The exact relativistic recession velocity, valid at all z, is:
The Hubble law relates redshift to distance, and the exact comoving distance in a flat ΛCDM universe is:
dC = ( c / H₀ ) ∫₀z dz′ / E(z′), E(z) = √( Ωm(1+z)³ + ΩΛ )
tL = ( 1 / H₀ ) ∫₀z dz′ / ( (1 + z′) E(z′) )
Redshift–Velocity Variables
| Symbol | Meaning | Notes |
|---|---|---|
| v | Recession velocity | Classical low-z value from v = c·z |
| z | Redshift | Model-independent observable: Δλ/λ |
| c | Speed of light | 299,792.458 km/s exactly |
| λrest | Rest wavelength | Laboratory wavelength of the line |
| λobs | Observed wavelength | Measured wavelength in the spectrum |
| H₀ | Hubble constant | Converts redshift to distance |
| dC | Comoving distance | Proper distance today to the emitting object |
| tL | Lookback time | Time the light has been travelling |
How Each Variable Affects the Result
The classical relation is exactly linear in both directions: doubling the redshift doubles the inferred velocity, and halving the velocity halves the inferred redshift. The slope is the speed of light, 299,792.458 km/s per unit redshift, so a redshift of 0.001 corresponds to 300 km/s and a redshift of 0.1 corresponds to 30,000 km/s. There is no damping, no exponent, and no dependence on any cosmological parameter — the relation depends only on c.
That simplicity is both the strength and the weakness of the formula. It is a strength because it makes redshift-to-velocity conversion trivial and model-independent at low z. It is a weakness because the linearity is an artefact of the small-z approximation, and the formula keeps producing plausible-looking numbers long after it has stopped being valid. At z = 1 the classical formula returns exactly the speed of light, which is physically impossible for a massive galaxy; at z = 3 it returns three times the speed of light, which is nonsense. The relativistic formula caps the velocity asymptotically below c, reaching 0.6 c at z = 1, 0.88 c at z = 3, and 0.98 c at z = 10.
Redshift Calculation Examples
v ≈ c · z = 299,792.458 × 0.03 = 8,993.77 km/s
z = v / c = 9,000 / 299,792.458 = 0.03002
λobs = 656.28 × (1 + 0.03) = 675.97 nm
Δλ = 675.97 − 656.28 = 19.69 nm to the red
Classical: v ≈ c × 1.0 = 299,792 km/s (= c, unphysical)
Relativistic: v = c · (2² − 1)/(2² + 1) = 0.6 c = 179,875 km/s
Classical overstates the velocity by 67%
d ≈ c · z / H₀ = 8,993.77 / 70 = 128.5 Mpc
dC ≈ 130.2 Mpc (numerical integration)
tL ≈ 0.422 Gyr
z = (496.0 − 486.13) / 486.13 = 0.0203
v ≈ c × 0.0203 = 6,086 km/s
Frequently Asked Questions
Not in general. In an expanding universe, photon wavelengths stretch in proportion to the scale factor: 1 + z = anow/athen. The photon is not emitted by a source moving away through fixed space; the space through which it travels expands during the journey, and the wavelength stretches with it. For nearby objects the two pictures coincide, because the recession velocity is small compared with c and the classical Doppler formula reduces to v ≈ cz. But at z = 1 the cosmological redshift is not a velocity at all in any physical sense — the galaxy is not receding through space at 0.6 c; rather, the space between us and it has expanded by a factor of two while the light was in transit.
The error scales as roughly z²/2, so the answer depends on your tolerance. At z = 0.01 the error is about 0.5%; at z = 0.05 it is about 2.5%; at z = 0.1 it is about 5%; at z = 0.5 it is about 20%; at z = 1 it is about 67%; and at z = 3 the classical answer exceeds the speed of light and is meaningless. The practical rule in observational work is to use v ≈ cz for z below about 0.1, switch to the relativistic formula between 0.1 and 1, and use the full cosmological framework above z ≈ 1, where the concept of recession velocity becomes model-dependent anyway.
Not as a physical velocity through space. No object can outrun a light beam in its own local frame, and the relativistic recession velocity computed from the redshift never reaches c. However, in cosmology the rate of change of the proper distance between two comoving observers can exceed c, because the expansion of space is not a velocity through space. Galaxies at z > 1.5 have proper distances growing faster than c, and yet their light still reaches us because the Hubble sphere is not an event horizon. The distinction matters: v ≈ cz produces values above c for z > 1, and those numbers are not velocities — they are artefacts of the small-z approximation.
Identify one or more spectral lines of known laboratory wavelength — Hα, Hβ, [O III], the Ca II doublet, or Lyα — and measure the wavelength at which they appear in the observed spectrum. Then compute z = (λobs − λrest)/λrest for each line and average the results. Using multiple lines is important, both to reduce measurement error and to confirm that the identification is correct. For faint high-redshift objects, the Lyman-alpha break and the Lyα emission line are usually the only features strong enough to detect, and the identification of a single line has to be confirmed by the shape of the overall spectrum.
Peculiar velocity is a galaxy's motion through space relative to the cosmic expansion, driven by the gravitational pull of nearby mass concentrations. It is typically a few hundred km/s in the field, up to 1,000 km/s in a rich cluster core, and can reach 600 km/s for the Local Group's motion relative to the cosmic microwave background. Because the observed redshift is the sum of the cosmological and peculiar contributions, the inferred recession velocity is contaminated at small z. Below z ≈ 0.01 the peculiar velocity can be comparable to the cosmological recession, and below z ≈ 0.002 — within the Local Group — the Hubble flow is not even defined, because the members are gravitationally bound.
Two independent measurement methods give persistently different answers — the "Hubble tension". The cosmic microwave background, interpreted through the standard ΛCDM model, gives H₀ ≈ 67.4 km/s/Mpc. The distance ladder, built from Cepheid variables and Type Ia supernovae in the local universe, gives H₀ ≈ 73.0 km/s/Mpc. The discrepancy is around 9%, far larger than the quoted uncertainties, and it is one of the most important open problems in cosmology. Either one of the measurements has an unrecognised systematic error, or the standard cosmological model is incomplete. This calculator lets you choose either value and see how much the distances change.
Comoving distance is the proper distance between us and a galaxy today, after the expansion has been factored out — it is the distance you would measure with a very long ruler laid out now, at a single instant of cosmic time. Light-travel distance is simply c times the lookback time: the distance light has travelled on its way to us. They are different because the universe expanded while the light was in transit. For a galaxy at z = 1, the light-travel distance is about 3,400 Mly while the comoving distance is about 11,000 Mly. Both are legitimate, but they answer different questions, and mixing them up is a common source of confusion in popular descriptions.
Yes — a negative redshift is called a blueshift, and it occurs when the source is approaching rather than receding. The Andromeda galaxy has z ≈ −0.001, corresponding to an approach velocity of about 300 km/s, because the Local Group is gravitationally bound and M31 is falling toward the Milky Way. The relation v ≈ cz applies to blueshifted objects as well, giving a negative velocity, but the cosmological interpretation breaks down entirely because there is no Hubble flow within a bound system. This calculator restricts itself to z ≥ 0 and does not model blueshifted objects.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v | Recession velocity | m/s |
| c | Speed of light | m/s |
| z | Redshift |
What it means
In cosmology, the redshift z is defined as (λ_observed − λ_rest)/λ_rest. For small redshifts (z << 1), the recession velocity v can be approximated as v ≈ c z. This is used to apply Hubble’s law and to estimate distances to nearby galaxies. For larger redshifts, the special relativistic Doppler formula or cosmological models must be used. Understanding this approximation helps astronomers interpret galaxy spectra and to map large‑scale structure.
Worked example
Cosmological Redshift to Recession Velocity – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| z | 0.1 |
| Parameter | Value |
|---|---|
| z | 0.2 |
Common mistakes
- Cosmological redshift: v ≈ c × z – for small redshifts (z << 1).
- z: Redshift parameter – defined as (λ_obs − λ_emit)/λ_emit.
- c: Speed of light – in km/s or m/s.
- Approximation: For z > 0.1, use the relativistic formula.
- Expansion: v is the recession velocity due to cosmic expansion.
Applications
Cosmological redshift to recession velocity, v ≈ c·z, approximates the velocity of a galaxy from its redshift z, valid for low redshifts. This is the basis for measuring the expansion of the universe and for mapping galaxy distances. Astronomers use it to derive velocities from spectroscopic observations. It is also used in the Hubble diagram and in cosmological studies. Understanding this simple relation is essential for interpreting galaxy spectra and for cosmology.
- Velocity estimation from measured redshift (Hubble flow)
- Determination of distances via Hubble's law
- Study of large‑scale structure and peculiar velocities
- Cosmological parameter estimation
- Educational introduction to redshift and expansion
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