Redshift Calculator — z, Recession Velocity & Distance
Enter the emitted (rest-frame) and observed wavelengths of a spectral line to compute redshift z, the relativistic recession velocity and the approximate Hubble distance.
nm
nm
km/s/Mpc
Positive z — source is receding (redshift)
- 1
Wavelength shift Δλ
700 − 656.3 nm = 43.7 - 2
Redshift z
43.7 ÷ 656.3 = 0.06659Positive z means the source is receding (redshift); negative means approaching (blueshift).
How does this calculator work?
Redshift z = (λ_observed − λ_emitted) / λ_emitted. Recession velocity (relativistic) = c × [(z+1)² − 1] / [(z+1)² + 1]. Hubble distance ≈ zc/H₀ (valid for z < 0.1; H₀ ≈ 70 km/s/Mpc, 2025 estimate, editable). Enter the emitted and observed wavelengths of a known spectral line.
Formula
How this is calculated
Redshift z quantifies how much a spectral line has shifted between emission and observation. For the standard definition: z = (λ_observed − λ_emitted) / λ_emitted. A positive z means the observed wavelength is longer than the emitted one — the source is receding and light is stretched toward red. A negative z is a blueshift, indicating approach.
The recession velocity uses the relativistic Doppler formula v = c × [(z+1)² − 1] / [(z+1)² + 1], which is always correct and prevents v from exceeding c. For small z this simplifies to v ≈ z × c, accurate to within 1% for z < 0.1. Note that for large-z galaxies (z > ~1.5) the "recession velocity" derived this way can exceed c under general relativity — this is a coordinate effect allowed by GR (expansion of space), not a violation of special relativity.
The Hubble distance d ≈ zc / H₀ is only a valid estimate for z < 0.1, where the universe can be treated as locally flat and the expansion rate constant. At larger z a proper comoving distance requires numerical integration of the Friedmann equations. The Hubble constant H₀ is approximately 70 km/s/Mpc from CMB-based measurements (2025 estimates), but remains under active discussion due to the "Hubble tension" between early- and late-universe measurement methods.
Frequently asked questions
Doppler redshift occurs when a source moves through space, compressing or stretching wave crests during transit. Cosmological redshift occurs because the universe itself is expanding, stretching the wavelength of photons while they travel. The formula z = (λ − λ₀)/λ₀ describes the net observed shift in both cases, but the physical interpretation and the velocity formula differ for large z.
Special relativity limits the speed of objects moving through space to below c. Cosmological recession is the expansion of space itself between two points — not motion through space. Very distant galaxies (z > ~1.5) recede faster than light in this coordinate sense, yet their photons still reach us because the expansion rate was slower earlier in cosmic history.
Nearby galaxies: z < 0.01. The most distant confirmed galaxies: z > 10 (light emitted when the universe was < 5% of its current age). The cosmic microwave background: z ≈ 1100. Stars in our galaxy: |z| < 0.001 (due to stellar motion and gravitational redshift).
Also known as
TG we-Calculate Editorial Team. (2026). Redshift Calculator — z, Recession Velocity & Distance [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/redshift-calculator
TG we-Calculate Editorial Team. "Redshift Calculator — z, Recession Velocity & Distance." TG we-Calculate. 2026. https://we-calculate.com/calculator/redshift-calculator.
TG we-Calculate Editorial Team, "Redshift Calculator — z, Recession Velocity & Distance," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/redshift-calculator
@misc{wecalculate_redshift_calculator, title = {Redshift Calculator — z, Recession Velocity & Distance}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/redshift-calculator}}, year = {2026}, note = {TG we-Calculate} }
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