Redshift and Recession Velocity Calculator

Redshift occurs when light from a distant object is stretched to longer (redder) wavelengths by the object’s motion away from the observer or by the expansion of spacetime itself. The redshift value (z) can be converted to a recession velocity.

The Formulas:

Redshift: z = (λ_observed − λ_rest) / λ_rest

For low velocities (z « 1), the classical Doppler formula applies:

v ≈ z × c

Where c = speed of light = 299,792 km/s

For high redshifts (relativistic, z approaching or exceeding 1):

v = c × [(z+1)² − 1] / [(z+1)² + 1]

Worked Example:

A galaxy is observed with the Hydrogen-alpha line at 780 nm instead of its rest wavelength of 656.3 nm.

z = (780 − 656.3) / 656.3 = 123.7 / 656.3 = 0.188481

The simple approximation v ≈ zc gives 0.188481 × 299,792.458 = 56,505 km/s. The relativistic form gives something noticeably smaller:

v = c × [(1+z)² − 1] ÷ [(1+z)² + 1] = 51,258 km/s

That’s about 17.1% the speed of light. The classical figure overshoots by 10.2%, which is exactly why the relativistic form matters at these redshifts.

This calculator reports both numbers on every run and takes the relativistic one as the headline. There is no threshold and no switching. The “z > 0.1” line in the result panel is telling you whether the simple v = zc shortcut is still trustworthy, not which formula was used.

Cosmological Distance Estimate:

Using Hubble’s Law: v = H_0 × d

Where H_0 ≈ 70 km/s/Mpc (Hubble constant)

Distance = 51,258 / 70 = 732 Megaparsecs ≈ 2.4 billion light-years

Famous Redshift Reference Values:

Why the calculator disagrees with that table, and which one to believe.

Type 0.003 in and this page returns 12.8 Mpc for Virgo, not 16.5. Type 0.158 and it returns 624 Mpc for 3C 273, not 749. Neither is a bug, and the gap is the interesting part.

For Virgo the culprit is peculiar velocity. The cluster is close enough that its galaxies are still falling around inside it at several hundred km/s, and our own group is falling toward it. Redshift measures the total line-of-sight speed, expansion and local tumbling together, and at 16 Mpc the tumbling is a large fraction of the whole. Nearby distances come from Cepheids and surface-brightness fluctuations instead, which is where 16.5 Mpc comes from.

For 3C 273 the culprit is the definition of “distance”. Dividing v by H₀ answers a different question than the comoving distance quoted in catalogues, and the two separate as z climbs. At z = 0.158 they differ by about 20%. Past z ≈ 0.5 a naive v/H₀ stops being useful at all, because the expansion rate was not constant over the light’s journey.

Rule of thumb: below z ≈ 0.01 distrust the redshift, above z ≈ 0.5 distrust the division. In between, v/H₀ is within a few percent.

Practical Tips:

• Negative z (blueshift) means the object is approaching. That is rare, and only happens for galaxies close enough for gravity to beat expansion
• Cosmological redshift is not a Doppler shift. Nothing is moving; the wavelength stretched because the space carrying it stretched