A shift in a line is a speedometer, and it works at any distance

Almost everything astronomy measures gets worse with distance. Brightness falls as the inverse square, angular sizes shrink, parallax runs out, and resolving anything requires an instrument that scales with the range.

One measurement does not. A source’s velocity along the line of sight is read from the position of a spectral line, and the position of a line does not depend on how far away it is. A galaxy a billion parsecs off has its recession speed measured to a few kilometres per second — the same precision available for a star next door — provided only that enough photons arrive to make the line visible.

That is the reason the Doppler shift is the most productive single measurement in the subject. It supplies stellar masses, the rotation of galaxies, the existence of unseen planets, the expansion of the universe, and the temperature of a gas cloud, and it does all of them by measuring where a line is rather than how bright anything looks.

Why it is a ratio, and never a displacement #

The relation is

$$\frac{\Delta\lambda}{\lambda_0} = \frac{v}{c},$$

and the form matters. What a source’s motion changes is not “the wavelength, by so many nanometres” but the wavelength by a fixed fraction. A line at 656 nm and one at 393 nm, in the same source moving at the same speed, shift by 1.97 nm and 1.18 nm respectively.

The first figure shows exactly that. The shift is not a rigid translation of the whole spectrum; it is a stretch, with everything moving proportionally away from zero.

Two practical consequences follow immediately. First, the measurement improves with the number of lines: fitting a whole forest of them for a single common value of $v/c$ is enormously better than measuring one, and a modern radial-velocity determination cross-correlates a stellar spectrum against a template containing thousands of lines at once. The largest use the technique has been put to is finding planets by the wobble they impose on their stars, where the shift to be measured is four parts in $10^8$. Second, the redder part of the spectrum gives a larger displacement for the same velocity, which is why infrared spectrographs have an advantage that partly offsets their poorer detectors.

What the measurement cannot see #

The technique measures one component of the velocity and is blind to the other two, and the blindness is total rather than partial.

Only motion along the line of sight shifts a wavelength. A star moving at 300 km/s directly across the sky produces exactly no Doppler shift. So every velocity obtained this way is $v\cos\theta$ for an unknown $\theta$, and it is a lower bound on the true speed.

The consequence that shaped two decades of exoplanet work is the $\sin i$ problem. A planet’s mass inferred from the wobble it imposes on its star comes out as $M\sin i$, with $i$ the unknown inclination of the orbit. A face-on system produces no signal at all; an edge-on one produces the full amplitude. A reported “half a Jupiter mass” could be exactly that seen edge-on, or a brown dwarf seen nearly face-on, and nothing in the spectrum distinguishes them.

The complementary measurement is proper motion, which sees the transverse component and is blind to the radial one — and which does degrade with distance, since it is an angular rate. Combining the two gives a full space velocity, and it can only be done for stars near enough for astrometry.

The asymmetry between the two is worth a number, because it explains the shape of the field. Converting a proper motion into a transverse speed requires multiplying by the distance, so its error inherits the distance’s error in full; a star with a 10% distance has a 10% transverse velocity at best. A radial velocity carries no distance at all. So for nearby stars the two components are known to comparable precision and a genuine three-dimensional velocity exists, while for anything past a few kiloparsecs one component is known to metres per second and the other two are not known at all. Galactic dynamics was, until Gaia, a subject conducted almost entirely on one coordinate out of three.

What was actually measured #

The chain from a photograph to a velocity has one link that is not obvious, and it is worth setting out because it is where the technique’s real difficulty lives: the rest wavelength has to come from somewhere.

A shift is measured against a reference, and the reference is a laboratory. The wavelengths of the hydrogen Balmer lines, the sodium D doublet, the calcium H and K lines and several thousand others are measured in discharge tubes and furnaces on Earth, to precisions of parts in $10^{9}$. Astronomy borrows those numbers wholesale. Every stellar velocity in the literature is, at bottom, a comparison between a star and a laboratory light source, and the assumption that atomic physics is the same in both places is doing load-bearing work.

That assumption is testable and has been tested. Comparing the ratios of different lines’ wavelengths in distant quasars against laboratory values is a test of whether the fine-structure constant has changed over cosmic time; the answer, to a few parts in $10^{6}$, is that it has not.

The instrumental chain is the other half, and it is discussed at length in the essay on stellar wobbles: iodine cells, thorium–argon lamps, vacuum-enclosed spectrographs and laser frequency combs, taking the achievable precision from kilometres per second in the 1980s to about 25 centimetres per second now.

What is worth adding here is the correction that must be removed before any of it means anything. The observer is on a rotating planet orbiting the Sun, so the raw measurement contains the Earth’s rotation (up to 0.46 km/s), its orbital motion (29.8 km/s), and the Earth–Moon barycentric wobble (about 0.013 km/s). All three are far larger than the signal being sought, and all three are subtracted using a solar-system ephemeris. A measurement at 25 cm/s therefore requires knowing the observatory’s own velocity to better than that — which requires the Earth’s orbit, its rotation, and the observatory’s position on it, to a precision nobody needed before radial velocities got this good.

The width of a line is a thermometer too #

A line has a position and it also has a width, and the width is a Doppler measurement of a different kind.

The atoms in a gas are in thermal motion, moving in all directions with a distribution of speeds set by the temperature. Each atom’s emission or absorption is shifted by its own velocity component along the line of sight, so the ensemble produces a line smeared over a range of wavelengths — a Gaussian, whose width is

$$\frac{\Delta\lambda}{\lambda_0} = \sqrt{\frac{2kT}{mc^2}}.$$

That is thermal Doppler broadening, and it turns a line profile into a temperature, independently of the continuum’s colour. The mass in the denominator makes it species-dependent: at the same temperature, hydrogen lines are broadened four times more than oxygen lines, which is a check that the broadening is thermal rather than something else.

Several other things broaden lines and each is a separate measurement. Pressure broadening depends on density, which separates a giant’s tenuous atmosphere from a dwarf’s compressed one and is how luminosity class is assigned from a spectrum alone. Rotational broadening comes from one limb of a star approaching while the other recedes, and it gives $v\sin i$ for the star’s own spin. Turbulence adds its own. Disentangling them requires comparing lines of different species and different strengths, and the result is a set of independent numbers — temperature, density, rotation, turbulence — from the shapes of features that are, individually, tiny dips in a curve.

Where the classical formula stops #

The relation $\Delta\lambda/\lambda = v/c$ is a first-order approximation, and the corrections matter at speeds astronomy routinely deals with.

The exact relativistic expression for motion directly along the line of sight is

$$\frac{\lambda_{\text{obs}}}{\lambda_0} = \sqrt{\frac{1 + \beta}{1 - \beta}}, \qquad \beta = v/c,$$

which reduces to the classical form when $\beta$ is small. At $\beta = 0.1$ the classical formula is wrong by 5%; for the jets of active galaxies, where $\beta$ exceeds 0.99, it is wrong by orders of magnitude.

There is also a transverse Doppler shift with no classical counterpart. A source moving purely across the line of sight is redshifted, by a factor $1/\sqrt{1-\beta^2}$, purely from time dilation — its clock runs slow, so its emitted frequency is lower. It is second order in $\beta$ and therefore tiny, and it was measured in 1938 by Ives and Stilwell using a beam of hydrogen ions, as one of the first direct confirmations of time dilation.

And there is a shift that is not a Doppler shift at all, which is the most important distinction in the whole subject. Cosmological redshift is not caused by motion through space; it is caused by the expansion of space during the light’s journey, stretching every wavelength by the same factor the universe grew. The formula looks similar and the physics is different, and treating a redshift of 7 as a velocity gives a nonsensical answer of several times the speed of light. What $z = 7$ means is that the universe was one-eighth its present size when the light left.

The third case is gravitational redshift, in which a photon climbing out of a gravitational well loses energy. It is measured in white dwarf spectra — Sirius B’s lines are shifted by about 80 km/s equivalent, which is not a velocity but a mass-to-radius ratio — and it was measured on Earth in 1959 by Pound and Rebka over a 22-metre tower, to 1%.

Four different phenomena move a spectral line, and separating them is a question about the source rather than about the measurement. The line does not know which one shifted it.

The distinction has an operational form worth stating. A Doppler shift and a cosmological one produce identical spectra; nothing measured at the telescope tells them apart. What separates them is the pattern — a Doppler field is as often blue as red and shows structure on the sky, while cosmological redshift is positive for everything and increases monotonically with distance. The classification is made from the population, never from the individual measurement.

The same pair of spectra at two more speeds shows the range over which the ratio is the whole content of the measurement.

The generalisation: the same measurement, from radio to gamma rays #

The technique’s reach is what makes it the workhorse, and the range is worth laying out because it is unusually wide even for astronomy.

21 cm. Neutral hydrogen emits a line at 1420 MHz from a hyperfine transition, and it is bright, ubiquitous and at a wavelength dust does not absorb. Mapping its Doppler shift across a galaxy gives the rotation curve, and it was such maps that established flat rotation curves and made dark matter unavoidable.

Molecular lines. Carbon monoxide’s rotational transitions trace cold molecular gas where star formation happens; their shifts give the kinematics of collapsing clouds and the outflows that stop the collapse.

X-ray lines. Iron’s K-alpha line near 6.4 keV, emitted from the inner accretion disc of a black hole, is shifted and skewed by orbital motion at a substantial fraction of $c$ combined with gravitational redshift. Fitting the resulting profile measures the black hole’s spin — a parameter with no other observational route.

The unifying statement is that any transition of known rest frequency is a velocity probe, and the choice of transition is dictated by what the source emits and what the intervening medium transmits. A single physical relation, applied over eleven decades of wavelength.

The measurement that found the expansion #

The most consequential use of the technique was made before anyone understood what it meant, and it is worth recording because it shows how far a measurement can run ahead of its interpretation.

Between 1912 and 1925 Vesto Slipher, at Lowell Observatory, measured radial velocities for about forty spiral nebulae — objects whose nature was then entirely unsettled, and which many astronomers took to be clouds inside the Milky Way. Each spectrum took upwards of twenty hours of exposure on a photographic plate.

The velocities were extraordinary. Most were positive, several exceeded 1,000 km/s, and the largest was three times anything measured for a star. Nothing inside the galaxy could move that fast and stay bound, which was the first strong argument that the spirals were external systems.

What Slipher did not have was distances. Supplying them took Hubble’s Cepheid work at Mount Wilson, and plotting Slipher’s velocities against Hubble’s distances gave the linear relation published in 1929. The velocities in that paper are almost entirely Slipher’s, measured over a decade before the relation existed and for a completely different purpose.

A velocity that is not the source’s #

There is a class of measurement in which the shift is real, the arithmetic is right, and the number is not a velocity of anything — and it is common enough to be worth a warning.

The clearest case is a spectral line formed in a moving medium between the observer and the source. An interstellar cloud imprints absorption lines on a distant star’s spectrum, and those lines carry the cloud’s velocity rather than the star’s. A spectrum containing both therefore contains two velocities, and separating them is a matter of recognising which lines belong to which — usually easy, because the interstellar lines are narrow and the stellar ones are broad, and occasionally not.

A subtler case is a line whose formation depth moves. In a stellar atmosphere with a systematic flow — rising granules and sinking lanes — the strong lines form higher and the weak lines deeper, and the two therefore report different velocities for the same star. That is not an error; it is a measurement of the atmosphere, and the difference between them is what the convective blueshift is.

And a line can be shifted by a change in what is emitting rather than by motion at all. A spot rotating across a stellar disc removes light from one side of the profile and displaces the fitted centroid, at the same amplitude a small planet would produce, without the star having moved.

The measurement is of a line, and a line’s position is changed by anything that changes the line. Attributing it to motion is an interpretation, and it is the interpretation that fails first.

Where the model stops #

Line of sight only. The blindness to transverse motion, discussed above, and it does not improve with better instruments.

A known rest wavelength. For an unidentified line there is no reference, and identification and velocity have to be solved together — which for high-redshift objects with few lines is genuinely ambiguous.

A single velocity. A real source has a distribution of velocities within it, so what is measured is a weighted average over whatever the beam contains. An unresolved galaxy gives one number for a hundred billion stars moving in every direction.

Stellar surfaces are not still. The convective granulation and starspots that limit exoplanet detection are Doppler signals from a source that is not moving as a whole, and no amount of instrumental precision distinguishes them from real motion.

The figures have a limitation that is worth being blunt about. The first two draw the shift at a magnitude chosen to be visible, and no real spectrum looks like that. A shift of 240 km/s is eight parts in $10^{4}$ — for a line drawn 2.4 pixels wide, a displacement of a small fraction of a pixel. Every published diagram of the Doppler shift exaggerates by two or three orders of magnitude, and the honest statement is that the effect this whole essay describes has never been seen by eye in a stellar spectrum, only fitted.

Two more readings of the same pair, at the same speeds through different windows.

The ladder from here #

Later rungs on this anchor: the classical and relativistic formulae derived. Transverse Doppler shift and the Ives–Stilwell experiment. Barycentric correction, and the ephemeris it needs. Cross-correlation velocities, and the CCF. Thermal, pressure, rotational and turbulent broadening, separated. The curve of growth. $v\sin i$ and stellar rotation. The 21 cm line and galactic rotation curves. Gravitational redshift, and Sirius B. Cosmological redshift, and why it is not a velocity. And the relativistic iron line, which measures a black hole’s spin from a profile.

Christian Doppler proposed the effect in 1842 and applied it, incorrectly, to explain the colours of binary stars — he thought a star’s colour was its Doppler shift, which would require speeds a substantial fraction of $c$. The effect is real, the application was wrong, and the name stuck to the effect.