Cosmological Redshift Calculator

Doppler redshift: z = (λ_obs − λ_emit)/λ_emit, v ≈ c·z

From an emitted and an observed wavelength, this tool computes the redshift z = (λ_obs − λ_emit)/λ_emit and the non-relativistic radial velocity v ≈ c·z, with c = 299,792,458 m/s. For |z| ≳ 0.1, the relativistic Doppler formula v = c·((1+z)² − 1)/((1+z)² + 1) should be used instead — at z = 0.5, c·z overestimates v by ~20%. Example: Hα emitted at λ = 656.3 nm, observed at 700 nm → z ≈ 0.0666, v ≈ 19,963 km/s (galaxy receding). A negative z is a blueshift (source approaching).

Applications

Stellar radial-velocity measurements; exoplanet detection via the host-star Doppler wobble (HARPS, HARPS-N, ESPRESSO — sub-m/s precision); spectroscopy of binary stars and pulsating variables; observational cosmology (galaxy redshift surveys); and laboratory recoil-free Doppler shifts via the Mössbauer effect (Δλ/λ ~ 10⁻¹⁵, used to confirm gravitational redshift in the Pound–Rebka experiment, 1959).

FAQ

When does v = c·z fail? When v is a non-trivial fraction of c. The rule of thumb: keep c·z if z < 0.1 (error < 5%); above that, use the relativistic formula. Cosmological redshifts (z > 1) are not velocities at all — they come from spacetime expansion.

Which spectral lines are used? Hydrogen Balmer (Hα 656.3 nm, Hβ 486.1 nm), Ca II H&K (393.4 / 396.8 nm), Na D doublet (589.0 / 589.6 nm) and Mg b (518 nm). High-resolution spectrographs cross-correlate hundreds of lines at once for m/s precision.

Is the velocity here radial? Yes — only the line-of-sight component. Transverse motion gives a much smaller second-order Doppler effect (∝ v²/c²) and does not appear in z to first order.