ConverterHub

Redshift Velocity Calculator

Convert a galaxy's redshift into how fast it recedes.

How it is calculated

Redshift z is the fractional stretch of a spectral line: z = (λobserved − λemitted) ÷ λemitted. The tool turns z (zero or more, no unit) into a velocity in two ways, using c = 299,792.458 km/s. The classical estimate is v = c × z. The relativistic Doppler estimate is v = c × ((1 + z)² − 1) ÷ ((1 + z)² + 1), which always stays below c. It also shows the relativistic speed as a percentage of c. Velocities are rounded to whole km/s. For small z the two agree closely, while at z of about 1 or more the classical figure becomes misleading and can even exceed c. A negative z, which would be a blueshift, gives no result. The velocity is a convention for presenting redshift, since for distant galaxies cosmologists attribute redshift to the expansion of space.

Worked example

A galaxy shows z = 0.5. Classical: 299,792.458 × 0.5 = 149,896 km/s. Relativistic: (1 + 0.5)² = 2.25, then (2.25 − 1) ÷ (2.25 + 1) = 1.25 ÷ 3.25 = 0.3846, and 0.3846 × 299,792.458 = 115,305 km/s. That is 38.46% of the speed of light. The simple formula overstates the speed here by about 34,600 km/s. For z = 0.01 the two results are 2,998 and 2,983 km/s, nearly the same.

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Common mistakes

  • Using v = c × z for a large redshift. At z = 3 it gives 899,377 km/s, three times the speed of light, whereas the relativistic result is 264,523 km/s.
  • Entering the observed wavelength or a percentage in the z field. Redshift is a plain ratio, so a 50% stretch in wavelength is z = 0.5.
  • Treating the result as the exact speed of a distant galaxy through space. It is a Doppler-style conversion of z, and the actual cosmological interpretation differs at large distances.
  • Entering a negative z for an approaching object. This tool only accepts zero and above.

About this calculator

This calculator converts a galaxy's cosmological redshift (z) into its recession velocity, offering both the classical (non-relativistic) approximation v = cz — valid for small z — and the full relativistic Doppler formula, which is needed for high-redshift objects where v approaches or would otherwise exceed the speed of light under the simple formula. Redshift is measured as z = (λ_observed − λ_emitted) / λ_emitted, the fractional stretch in a spectral line's wavelength caused by the expansion of space.

Astronomers use redshift as their primary tool for measuring how fast distant galaxies are receding from us due to the expansion of the universe (per Hubble's law), and combined with distance measurements, it underpins our understanding of cosmic expansion and the age of the universe. Enter a redshift value to get the corresponding recession velocity.

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