What redshift is
When a source of light moves away from us, or when the space between us stretches, the light reaches us at longer wavelengths: it is shifted toward the red end of the spectrum. Astronomers measure this redshift with the letter z:
z = (λobserved − λrest) ÷ λrest
A spectral line whose rest wavelength is known, such as the red hydrogen-alpha line at 656.28 nm, shows how much a galaxy’s light has been stretched. A negative z is a blueshift: the object is approaching, as the Andromeda galaxy is.
From redshift to velocity
For small z, the recession velocity is roughly v ≈ c × z. For larger values this simple form would exceed the speed of light, so the calculator uses the relativistic Doppler formula:
v = c × ((1 + z)² − 1) ÷ ((1 + z)² + 1)
For distant galaxies, the redshift comes mostly from the expansion of space rather than from motion through space, so this velocity is a convenient way to express z rather than a literal speed.
From redshift to distance
Hubble’s law says that nearby galaxies recede at a speed proportional to their distance: v = H₀ × d. So:
distance ≈ cz ÷ H₀
Worked example
A galaxy has z = 0.1, with H₀ = 70 km/s/Mpc:
| Result | Value |
|---|---|
| Hydrogen-alpha observed at | 721.91 nm |
| Relativistic Doppler velocity | 28,487 km/s (9.50% of c) |
| Hubble distance | 428 Mpc |
| Same in light-years | about 1.397 billion |
| Scale factor when the light left | 0.9091 |
The scale factor 1 ÷ (1 + z) means the universe was about 91% of its current size when that light was emitted.
Limits
The simple Hubble distance works for z below about 0.1. Beyond that, the expansion history of the universe matters, and astronomers distinguish several kinds of distance (comoving, luminosity, angular diameter). Use a cosmology calculator with a full model for those.
The Hubble constant
The value of H₀ is debated. Measurements from the cosmic microwave background by the Planck satellite give about 67 to 68 km/s/Mpc, while nearby Cepheids and supernovae give about 73. This Hubble tension changes distances by around 8%.
Tips
- In a galaxy’s own frame, every wavelength is multiplied by the same factor (1 + z), so any known line works.
- The most distant galaxies observed by the James Webb Space Telescope have z above 10: their ultraviolet light arrives in the infrared.