How this instrument works
Redshift z is a ratio, not a distance and not a speed. Take the wavelength you actually observe, subtract the wavelength that same spectral line has in a stationary lab source, and divide by that lab value: z = (λ_obs − λ_rest) ⁄ λ_rest. Dividing by λ_rest is what makes z dimensionless and portable — a redshift of 0.0666 means the same thing whether you measured it on the hydrogen H-alpha line, a sodium doublet, or a whole absorption spectrum, because every wavelength in that spectrum stretches by the same fraction.
The formula is shaped that way because a receding source drags every crest of the wave out a little further before the next one leaves, so the whole waveform gets stretched proportionally — the mechanism is the same whether you attribute it to classical Doppler motion or to space itself expanding while the light is in transit. At low z the two pictures give almost the same number, which is why the simple v ≈ zc conversion works. That agreement fails as z grows: by the time z approaches 1, zc would already exceed the speed of light, so astronomers switch to the relativistic Doppler formula or a cosmological redshift-distance relation instead of this linear shortcut.
Sign carries information here too. A positive z means the observed wavelength is longer than the rest wavelength — redshifted, receding — which is the ordinary case for galaxies outside the Local Group. A negative z means the wavelength came in shorter than the lab value: a blueshift, meaning the source is closing the distance, as with the Andromeda galaxy, which is falling toward the Milky Way rather than fleeing it.
- Look up or measure the rest wavelength of a known spectral line and enter it in "Rest (lab) wavelength" — for hydrogen H-alpha that is 656.3 nm.
- Enter the wavelength you actually recorded for that same line in "Observed wavelength", in the same unit.
- Read "Redshift z" — the dimensionless fractional shift between the two.
- Read "Implied recession velocity (non-relativistic)" — z times the speed of light, reliable only while z stays well under 1.
Worked example — the hydrogen H-alpha line at 700 nm
A galaxy's spectrum shows the hydrogen H-alpha line, normally fixed at 656.3 nm in a lab source, arriving instead at 700 nm. Enter 7.00 × 10⁻⁷ m as the observed wavelength and 6.563 × 10⁻⁷ m as the rest wavelength. The formula gives z = (7.00 × 10⁻⁷ − 6.563 × 10⁻⁷) ⁄ 6.563 × 10⁻⁷ = 0.0665854, meaning the line has stretched by about 6.66 percent of its rest value.
Multiplying that redshift by the speed of light gives the implied recession velocity: v = 0.0665854 × 299,792,458 m/s ≈ 19,961,802 m/s, or roughly 19,962 km/s — nearly 6.66 percent of light speed itself. This is exactly the comparison Vesto Slipher was making with spiral nebulae before 1915, and the correlation Edwin Hubble later plotted against distance to establish that the universe is expanding; the same ratio is still how redshift surveys assign a recession speed to a galaxy today.
Questions
Is the recession velocity from this calculator exact?
No, it is the non-relativistic approximation v ≈ zc, which treats redshift as if it converted directly into speed. That holds well for z under roughly 0.1, drifts noticeably by z ≈ 0.3–0.5, and breaks outright as z nears 1, where zc would exceed the speed of light. High-redshift work uses the relativistic Doppler formula or a cosmological redshift-distance relation instead.
Why does the rest wavelength have to be so precise?
Because z is measured relative to it — a small error in the rest value shifts every downstream number. The rest wavelength must come from the identical transition measured in a stationary lab source; pairing an observed line with the rest wavelength of a different transition, or a rounded textbook figure, produces a redshift that looks plausible but is simply wrong.
What does a negative redshift mean?
A blueshift: the observed wavelength is shorter than the rest wavelength, so z comes out negative and the implied velocity is negative too. Physically the source is closing the gap rather than receding. The Andromeda galaxy is the textbook case — its spectral lines are blueshifted because it is falling toward the Milky Way, on a course that will bring the two galaxies together.
Does this only apply to galaxies?
No. The same ratio applies to any moving source with a known spectral feature — a star's orbital wobble in exoplanet spectroscopy, a laboratory ion beam, or sound waves in the analogous acoustic Doppler shift. Astronomy is simply where redshift became a standard measurement, because a galaxy's whole spectrum shifts together and any recognisable line will do.
Is cosmological redshift really the same thing as a Doppler shift?
Not quite, though the arithmetic is identical at low z. A nearby star's redshift is genuine Doppler motion through space; a distant galaxy's redshift is mostly the wavelength stretching while light travels through expanding space over billions of years. Both are computed with the same z = (λ_obs − λ_rest) ⁄ λ_rest formula, which is why one calculator serves both cases at modest z.
Can I use this for a quasar at z greater than 2?
You can compute z itself accurately, since that part of the formula has no speed limit built in. The velocity readout is the part to distrust: v ≈ zc was never meant for z above roughly 0.5, and at quasar redshifts it should be treated only as a rough sense of scale, not a physical speed.