Redshift & Recessional Velocity Calculator
Convert a shifted spectral line into a redshift z and the recessional velocity of a distant galaxy.
nm
nm
Redshift z = 0.06659
- 1
Redshift z
(700 − 656.3) ÷ 656.3 = 0.06659 - 2
Relativistic factor (1+z)²
(1 + 0.06659)² = 1.137604 - 3
Recessional velocity
299,792.458 × (1.1376 − 1) ÷ (1.1376 + 1) = 19,298.60Relativistic Doppler formula — always stays below the speed of light.
How does this calculator work?
Redshift z = (λ_obs − λ_emit) / λ_emit compares an observed spectral line to its rest wavelength. Small redshifts give velocity v = cz; for larger z use the relativistic Doppler form v = c·((1+z)²−1)/((1+z)²+1), with c = 299,792.458 km/s, so the speed always stays below light speed.
Formula
How this is calculated
Redshift z measures how much a spectral line has been stretched toward longer (redder) wavelengths. Enter the observed wavelength λ_obs and the rest (laboratory) wavelength λ_emit of the same emission or absorption line, both in the same units (nanometres here), and z is found from z = (λ_obs − λ_emit) / λ_emit. Alternatively, type a known z directly and it overrides the wavelength inputs.
Velocity follows from z. For small redshifts the simple Doppler approximation v = c·z is accurate, where c = 299,792.458 km/s. As z grows this overestimates the speed and can exceed c, so the relativistic Doppler relation v = c·((1+z)² − 1) / ((1+z)² + 1) is used for the headline result; it always stays below the speed of light. The meter shows v as a fraction of c.
Assumptions: this treats the shift as a special-relativistic Doppler effect, which is the standard textbook conversion. True cosmological recession velocities from the expansion of space depend on a chosen cosmological model and can differ at very high z. Edge cases: λ_emit must be non-zero, and z must be greater than −1 (a blueshift, z < 0, is allowed and yields a negative, approaching, velocity).
Frequently asked questions
The linear v = cz is a good approximation only for small redshifts (roughly z < 0.1). For larger z it overestimates the speed and eventually exceeds c, so use the relativistic Doppler formula, which is reported as the main result.
Yes. If the observed wavelength is shorter than the rest wavelength, z is negative (a blueshift) and the velocity comes out negative, indicating the object is approaching rather than receding.
No. This is the special-relativistic Doppler conversion of redshift to velocity. Cosmological recession velocities from the expansion of the universe require a specific cosmological model and differ from this at high z.
Also known as
TG we-Calculate Editorial Team. (2026). Redshift & Recessional Velocity Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/redshift-velocity-calculator
TG we-Calculate Editorial Team. "Redshift & Recessional Velocity Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/redshift-velocity-calculator.
TG we-Calculate Editorial Team, "Redshift & Recessional Velocity Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/redshift-velocity-calculator
@misc{wecalculate_redshift_velocity_calculator, title = {Redshift & Recessional Velocity Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/redshift-velocity-calculator}}, year = {2026}, note = {TG we-Calculate} }
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