From spectral lines to distance: reading a redshift
How astronomers turn a shifted spectral line into z, then into a recession velocity and a rough distance, and where the simple formulas break down.
Spectral lines are the fingerprints
Atoms emit and absorb light at fixed, known wavelengths, which appear as sharp lines in a spectrum. Hydrogen-alpha sits at 656.3 nm in a laboratory, and the [OIII] oxygen line sits at 500.7 nm. When we see those same lines from a galaxy at longer wavelengths, the amount of shift tells us how much the light was stretched on its way to us. Identifying which line we are looking at is the essential first step, because z is meaningless without a known rest wavelength.
Turning a shift into z
Redshift z is just the fractional change in wavelength, so a line moved from 656.3 nm to 660 nm has been stretched by 3.7 nm, giving z of about 0.00564. Because it is a ratio, z is dimensionless and independent of the units you measure in. A larger shift means a larger z, and the same formula works whether the shift is tiny or the wavelength has nearly doubled.
From z to velocity and distance
The simplest interpretation treats the shift as a Doppler effect and multiplies z by the speed of light to get a recession velocity. Dividing that velocity by the Hubble constant then gives a distance through Hubble's law. For the [OIII] example at z near 0.06, this chain yields roughly 17962 km/s and about 257 megaparsecs at H0 of 70. Both steps are approximations that are best at modest redshift.
Why the estimates are only a starting point
The v = cz shortcut ignores special relativity, so it inflates velocity as z climbs and would nonsensically exceed light speed past z of one. Real cosmological distances also depend on the full expansion history of the universe, including dark energy and matter density, not just a single Hubble constant. For precise work astronomers use a cosmology model, but for a quick sense of scale the simple formulas in this tool are a reasonable first pass.