Formula and Method for Calculating Redshift
Redshift describes how much a light source's wavelength has stretched or compressed compared to the wavelength it was emitted at. It is defined as z = (λobserved − λemitted) / λemitted, a dimensionless ratio. A positive z means the observed light is shifted toward longer (redder) wavelengths — the classic signature of an object receding along the line of sight. A negative z is a blueshift, meaning the source is approaching. This calculator also converts that redshift into an equivalent recession velocity using the special-relativistic Doppler formula, which holds for any redshift from z = −1 up to arbitrarily large positive values.
How the calculation works
Enter the rest (emitted) wavelength — the wavelength the source would show at rest, such as a known spectral line — and the observed wavelength measured by the instrument, both in the same unit. The calculator subtracts the two to get the wavelength shift Δλ, then divides by the rest wavelength to get z. To find the recession velocity, it applies the relativistic Doppler relation 1 + z = √[(1+β)/(1−β)], where β = v/c. Solving for β algebraically gives β = [(1+z)² − 1] / [(1+z)² + 1], and multiplying by the speed of light (c = 299,792.458 km/s) gives the velocity in km/s. For small z, this reduces to the familiar v ≈ cz used for nearby stars.
Common mistakes
- Swapping observed and rest wavelengths: reversing the two flips the sign of z and turns a redshift into a blueshift (or vice versa).
- Mixing units between the two wavelengths: both the rest and observed values must be entered in the same unit (e.g., both in nm) since z is a unitless ratio — mismatched units silently corrupt the result.
- Using v = cz at high redshift: the simple linear approximation only holds for z ≪ 1; for quasars and other high-z sources, the full relativistic Doppler formula (or a proper cosmological model) is required.
- Treating all redshift as Doppler motion: gravitational redshift (light climbing out of a strong gravitational field) and cosmological redshift (space itself expanding) both produce a wavelength shift without the source having any true peculiar velocity through space.
Real-world applications
- The radial velocity method for exoplanet detection tracks the tiny periodic redshift/blueshift of a star's spectral lines caused by the gravitational tug of an orbiting planet.
- Galaxy redshift surveys use z, combined with Hubble's law, to estimate distances and map the large-scale structure of the universe.
- Spectroscopic classification of quasars and active galactic nuclei relies on precise redshift measurements of emission lines like Lyman-alpha and H-alpha.
- Binary star systems and eclipsing binaries are analyzed by tracking the alternating redshift and blueshift of each star as they orbit their common center of mass.