Redshift calculator
Enter the rest wavelength of a spectral line and the wavelength you observe to get its redshift z, the line-of-sight velocity by the relativistic Doppler formula and the scale factor. Or enter a velocity to see the redshift and where the line moves. Negative values mean a blueshift, an approaching source.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose whether to start from wavelengths or from a velocity.
- Enter the rest wavelength of the line in nm (656.28 nm for hydrogen-alpha) and the observed wavelength.
- Or enter the radial velocity in km/s, negative if the source approaches.
- Redshift: z = (λ − λ₀) ÷ λ₀, so λ = λ₀ × (1 + z).
- Relativistic Doppler: β = ((1 + z)² − 1) ÷ ((1 + z)² + 1), and back 1 + z = √((1 + β) ÷ (1 − β)).
- Classical estimate: v ≈ c × z, good for z ≪ 1. Scale factor: a = 1 ÷ (1 + z).
Worked example
Hydrogen-alpha, 656.28 nm, observed at 700 nm: z = 43.72 ÷ 656.28 = 0.066618. (1 + z)² = 1.137673, so β = 0.137673 ÷ 2.137673 = 0.064404 and the radial velocity is 19,307.7 km/s; c × z would give 19,971.55 km/s. The scale factor is 0.937543.
A source receding at 0.1 c (29,979.2458 km/s): 1 + z = √(1.1 ÷ 0.9), so z = 0.105542, and the line moves to 725.545 nm.
Tips
- Use the same unit for both wavelengths; z has no unit, so nm, Å or µm all work.
- Air or vacuum wavelengths: keep both from the same convention.
- For distances, combine the velocity with the Hubble’s law calculator (valid for small z).
Frequently asked questions
How do you calculate redshift?
z = (λ_observed − λ_emitted) ÷ λ_emitted. The hydrogen-alpha line at 656.28 nm seen at 700 nm has z = 43.72 ÷ 656.28 = 0.066618. A negative z is a blueshift.
How do I get a velocity from z?
For a source moving along the line of sight, special relativity gives v ÷ c = ((1 + z)² − 1) ÷ ((1 + z)² + 1). For small z this is close to v = c × z; at z = 1 the exact speed is 0.6 c, while c × z would give the impossible value c.
What is the scale factor?
For cosmological redshift, 1 + z tells how much the universe has expanded since the light left its source. The scale factor 1 ÷ (1 + z) is the size of the universe then compared with now; at z = 1 it was half as big.
Is a galaxy's redshift a Doppler shift?
For nearby galaxies the Doppler picture works well. For distant ones the redshift comes from the expansion of space, and the "velocity" from the Doppler formula is only a way of expressing z, not a speed through space.