Redshift to Velocity Calculator
Turn a redshift z into a recession velocity — relativistic and low-z.
≈ 0.095c. Low-z approximation v ≈ c·z = 29,979 km/s — this overestimates the true velocity, badly so at high z.
A redshift of z = 0.1 gives a recession velocity of about 28,489 km/s — roughly 0.095c — using the relativistic Doppler formula v = c·((1+z)² − 1) ÷ ((1+z)² + 1). The simple low-z estimate v ≈ c·z = 29,979 km/s overestimates it, and that error grows large at high z.
What redshift tells you
When a source moves away from us, its light is stretched to longer, redder wavelengths. Redshift z = Δλ ÷ λ measures that stretch. For small z the recession speed is almost proportional to redshift, v ≈ c·z, but that shortcut fails as z climbs because velocity can never reach the speed of light. The relativistic Doppler formula keeps the answer physical for every z.
c = 299,792.458 km/s · low-z approximation: v ≈ c·z (overestimates at high z)
Worked example
Take a galaxy at redshift z = 0.1 and find its recession velocity relativistically:
- 1 Compute (1 + z)². For z = 0.1, (1 + 0.1)² = 1.1² = 1.21.
- 2 Form the velocity ratio. v ÷ c = (1.21 − 1) ÷ (1.21 + 1) = 0.21 ÷ 2.21 = 0.095.
- 3 Multiply by the speed of light. v = 0.095 × 299,792 ≈ 28,489 km/s, about 0.095c.
- 4 Compare with the shortcut. The low-z estimate v ≈ c·z = 29,979 km/s runs ~5% high here, and much higher for large z.
Redshift → velocity: linear vs relativistic
The linear estimate v ≈ c·z equals or exceeds c once z ≥ 1 — physically impossible. The relativistic value always stays below c.
| Redshift z | Linear (c·z) | Relativistic | Relativistic v ÷ c |
|---|---|---|---|
| 0.01 | 2,998 km/s | 2,983 km/s | 0.010c |
| 0.1 | 29,979 km/s | 28,489 km/s | 0.095c |
| 0.5 | 149,896 km/s | 115,305 km/s | 0.385c |
| 1 | 299,792 km/s (= c) | 179,875 km/s | 0.600c |
| 2 | 599,585 km/s (> c ✗) | 239,834 km/s | 0.800c |
Cosmological redshift, Doppler, and Hubble’s law
Redshift is not always a simple Doppler shift. For nearby objects the redshift really is a velocity effect, and the Doppler formula above applies. For distant galaxies the redshift is mostly cosmological — the space between us and the source has expanded while the light was in transit, stretching the wave. Astronomers still often quote a “recession velocity” from the Doppler formula, but it is an interpretation, not a literal speed through space.
Why v can’t exceed c. The special-relativity Doppler formula is built so that v ÷ c approaches 1 only as z → ∞. No finite redshift ever yields a Doppler velocity at or above the speed of light, which is why the linear shortcut — giving 2c at z = 2 — is clearly wrong at high z.
Hubble’s law is approximate. The rough distance d = v ÷ H₀ (with H₀ ≈ 70 km/s/Mpc) is a first estimate only. The precise distance depends on the expansion history of the universe, and at high z different distance measures diverge, so treat any single figure as an order-of-magnitude guide.