Redshift & Hubble Distance Calculator

Frequently Asked Questions

What is redshift?

The stretching of light to longer wavelengths, z = (λ_observed − λ_emitted)/λ_emitted, as a source moves away from us.

How is recession velocity found?

For small z, v ≈ c·z; at large z a relativistic formula is needed because v cannot exceed the speed of light.

What is Hubble's law?

v = H₀·d, so dividing recession velocity by the Hubble constant gives an approximate distance to the galaxy.

Why is the distance only approximate?

It depends on the assumed Hubble constant and ignores cosmological model details, so it is an order-of-magnitude estimate.

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How This Calculator Works

When light from a distant galaxy is stretched to longer (redder) wavelengths by cosmic expansion, the fractional shift is the redshift z = Δλ ÷ λₘₑₛₜ = (λₒ₋ₛ − λₘₑₛₜ) ÷ λₘₑₛₜ. For small z the recession velocity is well approximated by v ≈ c z, where c = 299,792 km/s. Hubble's law then converts velocity to distance: d = v ÷ H₀, with the Hubble constant H₀ ≈ 70 km/s per Mpc (editable here, since measured values range ≈67-74). This tool returns z, recession velocity in km/s and as a fraction of c, and distance in megaparsecs and millions of light-years.

Worked Example: A Galaxy's Hydrogen Line

Hydrogen's Hα line is emitted at λₘₑₛₜ = 656.3 nm in the lab. A galaxy's spectrum shows that line at λₒ₋ₛ = 670.0 nm. Then z = (670.0 − 656.3) ÷ 656.3 = 13.7 ÷ 656.3 ≈ 0.0209. Recession velocity v ≈ c z = 299,792 × 0.0209 ≈ 6,260 km/s. Distance d = v ÷ H₀ = 6,260 ÷ 70 ≈ 89 Mpc, roughly 292 million light-years away. Because z < 0.1, the simple linear approximation is accurate to a few percent here.

Real-World Uses

Cosmic distance ladder: estimate distances to galaxies far beyond parallax or standard-candle reach.

Mapping large-scale structure: redshift surveys turn 2-D sky positions into 3-D maps.

Measuring H₀: comparing independent distances and redshifts constrains the expansion rate.

Identifying quasars and high-z objects: extreme redshift flags the most distant, oldest light.

Common Mistakes

Redshift is not a Doppler shift through space - at cosmological scales it is the expansion of space itself stretching the wavelength, so "velocity" is a convenient interpretation, not a literal speed through a medium. Mixing up rest and observed wavelength flips the sign: a larger observed wavelength means positive z (receding); a smaller one is a blueshift (approaching). Using mismatched wavelength units cancels out in the ratio, but a typo in either wavelength propagates directly into the distance.

Limitations & Related

The v ≈ cz relation is a low-z approximation; above z ≈ 0.1 special-relativistic and cosmological corrections matter, and at z > 1 the naive formula would imply faster-than-light speeds, which it must not be read as. A full treatment requires the relativistic Doppler formula or an FLRW cosmological model with matter and dark-energy density parameters. Hubble's law also assumes the smooth cosmic expansion (Hubble flow) dominates over local peculiar motions, which is only true beyond ≈10-20 Mpc. Related tools include the Doppler-shift and escape-velocity calculators.

Redshift & Hubble Distance Calculator

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