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Redshift to Velocity Calculator

Enter a redshift z and pick a model to find the recession velocity.

Result

115,304.8 km/s
38.46% c
Relativistic

How it is calculated

Redshift z is the fractional stretch of a wavelength, z = (observed − emitted) ÷ emitted, and has no unit. The tool offers two conversions with c = 299,792.458 km/s. The simple model is v = c · z. The relativistic Doppler model is v = c · ((1+z)² − 1) ÷ ((1+z)² + 1). It also reports the speed as a percentage of c, v ÷ c × 100. The velocity is shown in km/s with one decimal. Only positive z is accepted; zero and negative values (blueshift) give no result. Relativistic is the default, and any model value other than simple also uses it. The simple model is not capped, so a z above 1 returns speeds above 100% of c.

Worked example

A galaxy with z = 0.5, relativistic model. Square the stretch factor: (1 + 0.5)² = 2.25. Numerator: 2.25 − 1 = 1.25. Denominator: 2.25 + 1 = 3.25. Ratio: 1.25 ÷ 3.25 = 0.3846. Multiply by c: 0.3846 × 299,792.458 = 115,304.8 km/s, which is 38.46% c. With the simple model the same z gives 0.5 × 299,792.458 = 149,896.2 km/s (50.00% c), a noticeably larger figure.

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Common mistakes

  • Using the simple model for large redshifts. At z = 2 it reports 200% of the speed of light, while the relativistic model gives 80% c.
  • Entering the observed wavelength or the ratio observed ÷ emitted instead of z. A ratio of 1.5 corresponds to z = 0.5.
  • Typing a negative z for an approaching object. The tool only accepts z above zero.
  • Reading the result as a speed through space for very distant galaxies. The calculation is a Doppler conversion of z, nothing more.
  • Entering redshift as a percentage, for example 50 for z = 0.5. The field expects the plain decimal.

About this calculator

When light from a distant galaxy is stretched to longer, redder wavelengths, astronomers measure that shift as redshift z. At low z, the recession velocity is simply v = c·z, where c is the speed of light — a good approximation for nearby galaxies. But as z approaches or exceeds 1, this simple relation breaks down and can even suggest speeds faster than light, so astronomers switch to the relativistic Doppler formula, v = c·[(1+z)² − 1] / [(1+z)² + 1], which stays correctly bounded below c.

This conversion underlies Hubble's law and the expansion of the universe: since Edwin Hubble's 1929 observations, redshift-to-velocity calculations have let cosmologists estimate how fast galaxies are receding and, combined with distance measurements, how fast the universe is expanding.

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Comments (1)

  • Editorial team

    Pick the model by the size of z. At z = 0.1 the simple product gives 29,979.2 km/s and the relativistic Doppler formula gives 28,487.1 km/s, about 5% apart, while at z = 0.01 they differ by only a fraction of a percent. Neither is the cosmological expansion velocity that astronomers quote for distant galaxies, which needs a different model. Which redshift range do you work with: nearby galaxies or high-z quasars?

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