Redshift Velocity Calculator
Compute the recession velocity of a nearby galaxy from its redshift z using v = c·z, where c is the speed of light. Part of the Hubble-Lemaître law — the empirical relation that established the expanding universe.
Last updated: May 2026
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About this calculator
For nearby galaxies (z << 1), the recession velocity is approximately v = c·z, where c is the speed of light (299,792.458 km/s) and z is the redshift (the fractional shift of spectral lines toward longer wavelengths: z = Δλ/λ₀). Combined with the Hubble-Lemaître law v = H₀·d, this gives distance d = c·z/H₀, where H₀ is the Hubble constant (km/s/Mpc) — currently 67.4 from Planck CMB or 73.0 from the local distance ladder, the so-called "Hubble tension." Variables: redshift z is the dimensionless spectroscopic redshift (Andromeda has z ≈ -0.001 — blueshift; nearby galaxies typically z = 0.001 to 0.1; Hubble Deep Field reaches z = 8+). Edge cases: at z > 0.3 the simple v = c·z approximation breaks down — relativistic Doppler and cosmological-expansion treatments diverge from the linear formula, and the right interpretation requires choosing a cosmological model and a distance definition (luminosity, comoving, angular-diameter — they all differ at high z). Blueshift (negative z) corresponds to motion toward us, observed for the Andromeda galaxy and other Local Group members. For galaxies in the Hubble flow (beyond ~30 Mpc), peculiar velocities (deviations from pure expansion) are 100–500 km/s, dominating distance estimates from redshift alone for nearby objects.
How to use
Example 1 — A nearby galaxy. A galaxy is observed with redshift z = 0.1. Enter Redshift = 0.1. Velocity = 0.1 × 299,792.458 ≈ 29,979 km/s — about 10% of the speed of light. Using a Hubble constant H₀ = 70 km/s/Mpc, the corresponding distance is d = v/H₀ ≈ 428 Mpc (~1.4 billion light-years). Example 2 — A high-redshift quasar. Observed redshift z = 3. Enter 3. Calculator output = 3 × 299,792.458 ≈ 899,377 km/s, which exceeds c and is unphysical — the simple v = c·z formula is only valid for z ≲ 0.1. At high z the correct relativistic Doppler formula v/c = ((1+z)² − 1) / ((1+z)² + 1) gives v ≈ 0.882c for z = 3. For cosmological distances, the right framework is comoving distance in an expanding-universe metric rather than Doppler velocity at all — galaxies at z = 3 are receding faster than light in the comoving sense, which does not violate special relativity because no information travels through space faster than c.
Frequently asked questions
What is the Hubble constant and why does it matter?
The Hubble constant H₀ is the current expansion rate of the universe, with conventional units of km/s/Mpc — for every megaparsec of distance, galaxies recede an additional H₀ km/s on average. It sets the timescale of the universe (Hubble time = 1/H₀ ≈ 14 billion years, close to the actual age), the size of the observable universe (Hubble radius = c/H₀ ≈ 14 billion light-years), and the critical density needed for a flat geometry (ρ_c = 3H₀²/(8πG)). Best modern values: Planck satellite CMB measurement gives H₀ = 67.4 ± 0.5 km/s/Mpc; the SH0ES local distance ladder gives H₀ = 73.0 ± 1.0 km/s/Mpc. The disagreement (~5 sigma) is the "Hubble tension" — one of the major open questions in cosmology. Possible resolutions include systematic errors in either measurement, new physics in the early universe (early dark energy, modified neutrino sector), or modifications to the standard cosmological model.
Why does v = c·z fail at high redshift?
The simple Doppler interpretation v = c·z is a low-redshift approximation. At higher z, two effects matter: (1) relativistic Doppler — if the recession were truly a Doppler shift through space (which it isn't, but the analogy helps), v/c = ((1+z)² − 1) / ((1+z)² + 1), saturating below c regardless of how high z gets. A galaxy at z = 5 has Doppler v ≈ 0.94c, not 5c. (2) Cosmological expansion — in an expanding universe, the redshift isn't really Doppler at all; it's the stretching of space itself between emission and detection. The "recession velocity" defined as v = dD_comoving/dt can exceed c at z > 1.5 without violating special relativity (no information travels faster than light through space). For interpreting high-z observations, use cosmological distance measures (comoving, luminosity, angular diameter), not Doppler velocity. The simple v = c·z works only for z < 0.1 or so.
What is peculiar velocity and how does it affect distance estimates?
Peculiar velocity is a galaxy's motion relative to the Hubble flow — not from cosmic expansion, but from local gravitational interactions with neighbouring galaxies and large-scale structures. Typical peculiar velocities for nearby galaxies are 200–500 km/s; for galaxies in dense clusters, up to 1000–2000 km/s. This adds noise to distance estimates from pure-redshift methods: a galaxy 50 Mpc away at recession 3500 km/s might also have a peculiar 500 km/s toward us, giving observed v = 3000 km/s and inferred distance 43 Mpc — a 14% error. For very nearby galaxies (Local Group, Local Volume), peculiar velocity can exceed Hubble flow entirely; Andromeda is approaching us at ~110 km/s despite cosmic expansion. The standard method to separate the effects: use independent distance measurements (Cepheids, SN Ia) along with redshifts, fit the Hubble-Lemaître relation in the regime where peculiar velocities are small relative to Hubble flow (typically > 30 Mpc), and apply statistical corrections for known velocity fields like the Local Group infall toward Virgo.
What are the most common mistakes interpreting redshift?
The first is using v = c·z at high redshift where it breaks down — galaxies at z > 1 have non-Doppler interpretations of their redshift in expanding-universe cosmology, and naive application of c·z gives velocities exceeding c. The second is confusing cosmological redshift (from expansion) with Doppler redshift (from peculiar motion); both produce wavelength shifts but have different distance interpretations. The third is forgetting peculiar velocities: for nearby galaxies, observed redshift includes both Hubble recession and peculiar motion, which can have either sign. The fourth is using the wrong Hubble constant value — the persistent ~10% discrepancy between Planck (67.4) and SH0ES (73.0) means quoted distances depend on which is used; for headline numbers, always state which H₀ is assumed. The fifth is confusing redshift z with the "scale factor" a = 1/(1+z); at z = 1 the universe was half its current size (a = 0.5), not just slightly smaller. Cosmological distance, lookback time, and age at emission are all separate quantities related to but distinct from z.
When should I not use this calculator?
Skip it for high-redshift objects (z > 0.3) — the linear v = z·H₀ approximation severely underestimates the relevant velocity-equivalent at those scales; use cosmological distance calculators with proper Friedmann-Robertson-Walker metric integration. Don't use it for nearby galaxies where peculiar velocity dominates over Hubble flow (within ~10 Mpc) — observed redshift doesn't reliably indicate distance. It's the wrong tool for blueshifted objects (negative z, like Andromeda); they're not in the Hubble flow regime at all. Avoid it for objects with very anomalous redshifts (gravitational redshift near compact objects, peculiar high-velocity stars in our own galaxy); the formula assumes pure cosmological recession. Finally, don't use it as the only distance estimate for any specific galaxy when you need precision; combine redshift with peculiar-velocity surveys, Tully-Fisher or Fundamental Plane distance estimators, or supernova standard candles for accuracy beyond a few percent.