Redshift ↔ Observed Wavelength Calculator

Give any two of redshift, rest wavelength, and observed wavelength, and get the third exactly — plus the classical and relativistic recession velocities implied by that redshift.

Redshift / wavelength calculator

1 + z = λobs / λrest — positive z is a redshift (receding source, or expanding space), negative is a blueshift (approaching). Give any two of redshift, rest wavelength, and observed wavelength; the third follows exactly.

+0.158
visibleΔλ = +76.804 nmrest 486.1 nmobserved 562.9 nm

A line's rest position vs. where it's actually observed, against the visible-light band — some of these presets land inside it, some fall well outside what an optical telescope alone can see.

Recession velocity implied by this redshift

Classical (v = cz)+47367.33 km/s (+15.80% c)
Relativistic Doppler+43665.21 km/s (+14.57% c)

Even the relativistic value is a special-relativistic Doppler velocity — strictly, the velocity of a source moving through space. At genuinely cosmological distances, redshift instead comes from the expansion of space itself, and there isn't a single well-defined "recession velocity" the way special relativity defines one — see below for the full explanation.

In 1963, Maarten Schmidt noticed that the emission lines in the spectrum of a faint radio source, 3C 273, matched the hydrogen Balmer series — just shifted 16% to the red. That single number, a redshift of 0.158, revealed 3C 273 to be a quasar billions of light-years away, one of the most luminous objects ever observed, hiding in plain sight because nobody had tried multiplying its wavelengths by 1.158. Redshift is that direct: one number, read straight off a spectrum, that unlocks distance, velocity, and sometimes the discovery itself.

The definition

Redshift compares an observed wavelength to the wavelength a source emits in its own rest frame:

1+z=λobsλrest1 + z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{rest}}}

Rearranged, this gives whichever quantity is missing — λ_obs = λ_rest(1+z), or λ_rest = λ_obs/(1+z). z is dimensionless: positive means redshifted (wavelengths stretched), negative means blueshifted (compressed) — the Andromeda Galaxy is one of the rare blueshifted objects, approaching us at around 300 km/s, which is why it’s on a collision course over the next few billion years.

Two velocities, and why they disagree

A redshift is often translated into a velocity, but there are two different formulas in circulation, and they only agree when z is small.

The naive version simply says v = cz — a first-order, non-relativistic reading. It’s the formula most people learn first, and it’s an excellent approximation for nearby galaxies. It also breaks completely for large z: push it far enough and it predicts velocities faster than light, which isn’t a real result, just a warning sign the formula’s been used outside where it works.

v=czv = cz

The special-relativistic Doppler formula stays below c at every redshift, however large:

v=c (1+z)2−1(1+z)2+1v = c\,\frac{(1+z)^2 - 1}{(1+z)^2 + 1}

The calculator computes both, and flags it explicitly whenever the classical value has crossed into the physically meaningless faster-than-light regime.

Doppler redshift vs. cosmological redshift

Both formulas above describe a source moving through space — genuine special-relativistic Doppler shift. But most of the large redshifts astronomers actually measure, for distant galaxies and quasars, aren’t that at all. They come from cosmological redshift: space itself expanding while the light is in transit, stretching its wavelength along the way, described by general relativity’s Friedmann-Robertson-Walker metric rather than special relativity’s Doppler formula.

The two phenomena use the same definition of z and can look identical for a single measurement, but they mean different things, and at high redshift there’s no longer a single well-defined “recession velocity” the way special relativity defines one — the relativistic Doppler value above is best read as an illustrative special-relativistic analogy, not a literal cosmological velocity, once z gets large. Working out an actual cosmological distance or lookback time from z requires a full cosmological model (matter density, dark energy density, and the expansion rate today) — genuinely a different calculation from anything on this page.