The Doppler Effect and Radial Velocity

When a source of light moves relative to us, the wavelengths of its spectral lines shift away from the values measured in the laboratory — the optical analogue of the acoustic Doppler effect that lifts the pitch of an approaching siren. In astronomy this is not a curiosity but the single most productive measurement in the whole subject. A line of accurately known rest wavelength $\lambda_{\mathrm{rest}}$ — a hydrogen Balmer line, the calcium H and K doublet, an iron absorption feature — appears displaced to $\lambda_{\mathrm{obs}}$ , and the size of that displacement encodes the component of the source's velocity along the line of sight. The crucial restriction is that only line-of-sight motion — the radial velocity $v_r$ — produces the classical shift. Motion purely across the sky, the tangential velocity, betrays itself instead through proper motion, the province of astrometry; spectroscopy and astrometry are thus complementary, together reconstructing a star's full three-dimensional space velocity. And because the shift depends on velocity rather than distance, radial velocities can be read for objects far too remote for any direct distance measurement — a quasar at redshift three yields its shift as readily as the nearest star. The sign convention is universal. A positive $v_r$ (recession) lengthens the wavelength — a redshift — while a negative $v_r$ (approach) shortens it — a blueshift. Astronomers package this in the dimensionless redshift $z \equiv (\lambda_{\mathrm{obs}}-\lambda_{\mathrm{rest}})/\lambda_{\mathrm{rest}}$ , which equals $v_r/c$ at low speed. Push toward $c$ and the relativistic formula takes over, predicting a shift even for motion straight across the sky; push to cosmological distances and the redshift becomes a stretching of space rather than a velocity, taking the same form only at small $z$ . Knowing when a redshift means motion, when gravitation, and when expansion is a recurring subtlety of the field, and the Doppler formula is the foundation on which all three rest.

Key takeaways

The Doppler effect shifts the observed wavelength of a spectral line whenever source and observer separate or approach along the line of sight; only the radial component of velocity produces the shift.
For non-relativistic speeds the fractional shift equals the radial velocity in units of $c$ : $\Delta\lambda/\lambda_{\mathrm{rest}} = v_r/c$ , positive (redshift) for recession, negative (blueshift) for approach.
Measuring the shift of a line of known rest wavelength gives $v_r$ directly — the workhorse of binary orbits, galactic rotation, exoplanet detection, and the expanding Universe.
The full special-relativistic formula $1+z=(1+\beta\cos\theta)/\sqrt{1-\beta^2}$ contains transverse time dilation, so even purely tangential motion produces a second-order redshift; the classical result is its leading term.
The cosmological redshift of distant galaxies is formally distinct from a Doppler shift — it is the stretching of space itself, $1+z=a_{\mathrm{obs}}/a_{\mathrm{emit}}$ — but reduces to $z\simeq v_r/c$ for small $z$ .
Modern spectrographs measure radial velocities to $\sim\mathrm{m\,s^{-1}}$ , enough to reveal a planet tugging on its host star.
Key idea: the fractional wavelength shift of a spectral line measures the line-of-sight velocity of the source, $\Delta\lambda/\lambda_{\mathrm{rest}} = v_r/c$ in the non-relativistic limit — redshifted for recession, blueshifted for approach.

The core insight

The fractional wavelength shift of a spectral line measures the line-of-sight (radial) velocity of the source: $\Delta\lambda/\lambda_{\mathrm{rest}} = v_r/c$ in the non-relativistic limit, redshifted for recession and blueshifted for approach.

Where this lives today

Radial velocity is the beating heart of modern precision astronomy. The 1995 discovery of 51 Pegasi b — a Jupiter-mass planet orbiting its star every 4.2 days — was read entirely from a $\sim59\ \mathrm{m\,s^{-1}}$ periodic Doppler wobble, and the technique has since found thousands of exoplanets. Purpose-built spectrographs now chase far smaller signals: HARPS reaches $\sim1\ \mathrm{m\,s^{-1}}$ and ESPRESSO on the VLT pushes toward $\sim10\ \mathrm{cm\,s^{-1}}$ , the level needed to sense an Earth-mass planet, which tugs the Sun by only $9\ \mathrm{cm\,s^{-1}}$ . Getting there demands ferocious wavelength calibration — laser frequency combs laying down an evenly spaced ruler of reference lines — and careful removal of the observatory's own motion through the heliocentric correction, subtracting Earth's $\sim30\ \mathrm{km\,s^{-1}}$ orbit and spin projected onto the target. On the survey scale, ESA's Gaia mission has measured radial velocities for tens of millions of stars, combining them with micro-arcsecond proper motions and parallaxes into a six-dimensional phase-space map of the Milky Way — worked example 1 carried out for an entire Galaxy. At the far end of the ladder, the redshifts of Type Ia supernovae against their standardized brightnesses trace the expansion history of the Universe; the tension between the locally calibrated Hubble constant ( $H_0\approx73\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ ) and the value inferred from the microwave background ( $\approx67$ ) is one of the sharpest open problems in cosmology, every rung of it anchored in a Doppler or cosmological redshift.

The physical picture

Picture a source emitting the crests of a light wave at a fixed rate. If it recedes, each successive crest is launched from slightly farther away and must travel a little extra distance, so crests arrive spaced farther apart in time — a longer observed wavelength, a redshift. If it approaches, crests pile up and arrive more closely spaced — a blueshift. Nothing about the effect cares how far away the source is; it depends only on whether the line-of-sight separation is growing or shrinking, that is, on the radial velocity. The first figure freezes this into the classic wavefront picture: a source drifting to one side, its crests bunched ahead of it and stretched behind.

Figure

A source moving to the right emits spherical wave crests that bunch ahead of it and spread behind it, so the observer it approaches sees a shorter wavelength blueshift while the observer it recedes from sees a longer wavelength redshift, with the fractional shift Δλ/λ_rest = v_r/c

A useful image for the radial-only rule: hold a pencil as the velocity vector and rotate it. Pointing along your line of sight the shift is maximal; swung across your field of view the line-of-sight separation is momentarily unchanging, so to first order there is no shift at all. The magnitude is intuitively tiny — the Sun's reflex wobble from Jupiter is only about $12.5\ \mathrm{m\,s^{-1}}$ , giving $\Delta\lambda/\lambda\approx4\times10^{-8}$ , a displacement of $\sim3\times10^{-5}\ \mathrm{nm}$ on an optical line — so that such shifts are measured routinely testifies to the stability of modern spectrographs. The second figure shows the everyday act of the trade: reading a velocity off the hydrogen H $\alpha$ line displaced in the spectrum of a receding galaxy.

Figure

The Hα absorption line with rest wavelength 656.3 nm is observed at 662.9 nm in a receding galaxy, so the shift Δλ = 6.6 nm gives z = Δλ/λ_rest = 0.0101 and a radial velocity v_r = cz ≈ +3.0×10³ km s⁻¹ of recession

At relativistic speeds a second effect appears with no classical analogue. The moving source's clock runs slow — time dilation — so its emission is stretched in time by the Lorentz factor $\gamma$ regardless of direction. This transverse Doppler effect redshifts even a source streaking straight across the sky, a purely relativistic prediction that matters for the jets of active galaxies and for tests of special relativity.

The core equations

Astronomers first reduce a measured line to a dimensionless number. The redshift $z$ is defined as the fractional displacement of the line from its rest wavelength; it is positive for a redshift and negative for a blueshift, and it is the quantity actually reported, independent of any velocity interpretation.

z \equiv \frac{\lambda_{\mathrm{obs}} - \lambda_{\mathrm{rest}}}{\lambda_{\mathrm{rest}}} = \frac{\Delta\lambda}{\lambda_{\mathrm{rest}}} \qquad (1)

where: $z$ — dimensionless redshift (positive) or blueshift (negative); $\lambda_{\mathrm{obs}}$ — observed wavelength of the line; $\lambda_{\mathrm{rest}}$ — rest-frame (laboratory) wavelength of the same transition; $\Delta\lambda=\lambda_{\mathrm{obs}}-\lambda_{\mathrm{rest}}$ . Reading: $z$ is a pure number extracted from two wavelengths, one measured and one known — no physics is assumed yet. For sources moving much slower than light — every star and nearby galaxy — the shift is directly proportional to the line-of-sight velocity. This is the equation used a million times a night at telescopes worldwide, derived below from nothing more than counting wave crests.

\frac{\Delta\lambda}{\lambda_{\mathrm{rest}}} = \frac{v_r}{c} \qquad (2)

where: $v_r$ — radial (line-of-sight) velocity, positive for recession; $c$ — speed of light, $2.998\times10^{8}\ \mathrm{m\,s^{-1}}$ . Reading: a fractional shift of $10^{-4}$ means the source moves at $10^{-4}c\approx30\ \mathrm{km\,s^{-1}}$ along the line of sight; the tangential motion contributes nothing. Inverting equation (2) turns the measurement into the answer: a velocity from two wavelengths and a constant. It holds only to first order in $v_r/c$ — for $v_r=300\ \mathrm{km\,s^{-1}}$ an error of one part in a thousand, negligible for stars but not for quasars.

v_r = c\,\frac{\lambda_{\mathrm{obs}} - \lambda_{\mathrm{rest}}}{\lambda_{\mathrm{rest}}} = c\,z \qquad (3)

where: $v_r$ — inferred radial velocity of the source. Reading: this is the low-speed inversion; for $z\gtrsim0.1$ one must switch to the relativistic form, and for cosmological sources abandon the velocity picture entirely. When the speed is a significant fraction of $c$ , the crest-counting argument must be supplemented by the time dilation of the source's clock. The general result depends on both the speed $\beta=v/c$ and the angle $\theta$ between the velocity and the line of sight, through the radial projection $\beta\cos\theta$ .

1 + z = \frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{rest}}} = \frac{1 + \beta\cos\theta}{\sqrt{1-\beta^2}} = \gamma\left(1 + \frac{v_r}{c}\right),\qquad \gamma=\frac{1}{\sqrt{1-\beta^2}} \qquad (4)

where: $\beta=v/c$ — total speed in units of $c$ ; $\theta$ — angle between velocity and line of sight ( $\theta=0$ head-on recession, $\theta=90^\circ$ transverse); $\gamma$ — Lorentz factor; $v_r=v\cos\theta$ . Reading: the numerator is the classical path-lengthening; the denominator $\sqrt{1-\beta^2}$ is the purely relativistic time dilation that survives even when $v_r=0$ . For motion straight along the line of sight the angle drops out and equation (4) collapses to the compact longitudinal form, the one that governs relativistic jets pointed toward or away from us.

1 + z = \sqrt{\frac{1+\beta}{1-\beta}} \qquad (\text{pure radial recession},\ \beta=v_r/c) \qquad (5)

where: $\beta=v_r/c>0$ for recession directly along the line of sight. Reading: inverting gives $\beta=[(1+z)^2-1]/[(1+z)^2+1]$ , which stays below $1$ for any finite $z$ — a relativistic source can never reach an infinite redshift by motion alone. The redshifts of distant galaxies obey the same arithmetic but demand a different interpretation: they arise not from motion through space but from the expansion of space itself, encoded in the ratio of the scale factor $a(t)$ between emission and observation.

1 + z_{\mathrm{cos}} = \frac{a(t_{\mathrm{obs}})}{a(t_{\mathrm{emit}})} \qquad (6)

where: $a(t)$ — scale factor of the Friedmann–Robertson–Walker metric; $t_{\mathrm{emit}},\,t_{\mathrm{obs}}$ — cosmic times of emission and observation. Reading: a galaxy observed at $z=2$ emitted its light when the Universe was one-third its present size; no velocity is implied, and $v=cz$ is a low- $z$ approximation only. Finally, the radial velocity is only one leg of a star's true motion; combining it with the tangential velocity — recovered from proper motion $\mu$ and distance $d$ — reconstructs the full space velocity.

v = \sqrt{v_r^{\,2} + v_t^{\,2}},\qquad v_t = 4.74\,\mu\,d \qquad (7)

where: $v_t$ — tangential velocity in $\mathrm{km\,s^{-1}}$ ; $\mu$ — proper motion in arcseconds per year; $d$ — distance in parsecs; $4.74=\mathrm{AU/yr}$ expressed in $\mathrm{km\,s^{-1}}$ . Reading: spectroscopy gives $v_r$ , astrometry gives $v_t$ ; only together do they yield the three-dimensional velocity through the Galaxy.

Derivation AThe classical Doppler shift from counting wave crests

The goal is to recover the everyday relation $\Delta\lambda/\lambda_{\mathrm{rest}} = v_r/c$ by following two successive wave crests from a source drifting slowly along the line of sight, and to see that the observed wavelength stretches for one reason only: each new crest is launched from a slightly greater distance and so takes slightly longer to arrive. We treat the emission period as fixed, add the extra light-travel time accrued during one period, and convert the stretched arrival period into a stretched wavelength.

Step 1. A source at rest emits crests one rest period apart, $T_{\mathrm{rest}}=\lambda_{\mathrm{rest}}/c$ ; crest 1 leaves at $t=0$ and crest 2 leaves at $t=T_{\mathrm{rest}}$ . Why: the emission period is the clock, and the entire effect is what happens to the interval between crests as they travel to us.

Step 2. During that one period the source recedes by $\Delta d = v_r\,T_{\mathrm{rest}}$ , so crest 2 sets out from farther away than crest 1. Why: only the line-of-sight displacement lengthens the path — a source moving tangentially would launch both crests from the same distance and produce no shift.

Step 3. Crossing that extra gap at speed $c$ costs an added travel time $\Delta t = \Delta d/c = v_r\,T_{\mathrm{rest}}/c$ . Why: light covers the additional distance at $c$ , and this delay is added to the interval between the two arrivals.

Step 4. The observed period is therefore $T_{\mathrm{obs}} = T_{\mathrm{rest}}+\Delta t = T_{\mathrm{rest}}(1+v_r/c)$ ; multiplying by $c$ gives $\lambda_{\mathrm{obs}}=\lambda_{\mathrm{rest}}(1+v_r/c)$ , so $\Delta\lambda/\lambda_{\mathrm{rest}}=v_r/c$ . Why: the interval between arriving crests is the observed period, and wavelength is simply $c$ times period — the shift is the fractional stretch of that interval. *This result is exact to first order in $v_r/c$ ; the fractional error from neglecting higher-order terms is of order $\beta^2$ .*

Go deeper · Derivation B · 1 minThe relativistic Doppler formula and the transverse effect

Derivation BThe relativistic Doppler formula and the transverse effect

The goal is to upgrade the crest-counting argument to special relativity by restoring the one ingredient the classical picture omits — the time dilation of the moving source's clock — and to obtain the general formula $1+z=(1+\beta\cos\theta)/\sqrt{1-\beta^2}$ , valid at any speed $\beta$ and any angle $\theta$ to the line of sight. From it we read off both the longitudinal limit that governs relativistic jets and the purely transverse redshift that has no classical counterpart.

Step 1. Replace the fixed emission period by the dilated one: the source clock ticks at proper period $T_0$ , but in our frame it is seen to run slow at $\gamma T_0$ , with $\gamma=(1-\beta^2)^{-1/2}$ . Why: a clock moving at speed $\beta$ is time-dilated by $\gamma$ regardless of direction — this factor is exactly what the non-relativistic argument leaves out.

Step 2. Reinstate the geometric path-lengthening, now with the radial projection of the velocity, $v_r=v\cos\theta$ , so the arrival period is $T_{\mathrm{obs}}=\gamma T_0\,(1+\beta\cos\theta)$ . Why: only the component along the line of sight adds travel distance between successive crests; $\theta=0$ is head-on motion and $\theta=90^\circ$ is motion straight across the sky.

Step 3. Wavelength ratio equals period ratio, so $1+z=T_{\mathrm{obs}}/T_0=(1+\beta\cos\theta)/\sqrt{1-\beta^2}$ . Why: the observed frequency is inversely proportional to the observed period, and this boxed relation is now exact at all $\beta$ , not just small ones.

Step 4. Set $\theta=0$ : $1+z=\sqrt{(1+\beta)/(1-\beta)}$ , the longitudinal formula for pure recession. Why: this is the case of a source moving directly away, relevant to fast stars and to jets aligned with the line of sight.

Step 5. Set $\theta=90^\circ$ : $1+z=\gamma$ , a redshift even though $v_r=0$ . Why: with no path-lengthening left, only time dilation survives — the transverse Doppler effect, a genuine relativistic redshift for motion with no radial component whatsoever. *The general formula reduces to $z\simeq v_r/c$ for $\beta\ll1$ and to the longitudinal form for $\theta=0$ ; the transverse limit $1+z=\gamma$ is the pure time-dilation redshift.*

Go deeper · Derivation C · 1 minThe cosmological redshift as a stretching of wavelength

Derivation CThe cosmological redshift as a stretching of wavelength

The goal is to show that the redshift of a distant galaxy, though it looks like a Doppler shift and reduces to one nearby, is really a stretching of the photon's wavelength by the expansion of space, set by the ratio of the scale factor between emission and observation. We trace a light ray through the Friedmann–Robertson–Walker metric, extract $1+z=a(t_{\mathrm{obs}})/a(t_{\mathrm{emit}})$ , and recover the Hubble law as its low-redshift limit — which is precisely why $v=cz$ works nearby and breaks down at large $z$ .

Step 1. In the FRW metric $ds^2=-c^2\,dt^2+a^2(t)\,d\chi^2$ a radial light ray obeys $c\,dt=a(t)\,d\chi$ , and the comoving distance $\chi$ between galaxy and observer is fixed. Why: comoving coordinates are carried along by the expansion, so the coordinate separation of two galaxies at rest in the Hubble flow does not change with time.

Step 2. Two successive crests, emitted at $t_{\mathrm{emit}}$ and $t_{\mathrm{emit}}+\delta t_{\mathrm{emit}}$ , travel the same fixed $\chi$ ; equating the comoving path integrals for the two crests gives $\delta t_{\mathrm{obs}}/a(t_{\mathrm{obs}})=\delta t_{\mathrm{emit}}/a(t_{\mathrm{emit}})$ . Why: both crests cover identical comoving distance, so their integrals of $c\,dt/a$ over the flight must match, forcing the arrival interval to scale with the scale factor.

Step 3. Since a wave period is proportional to a wavelength, $\lambda_{\mathrm{obs}}/\lambda_{\mathrm{emit}}=a(t_{\mathrm{obs}})/a(t_{\mathrm{emit}})\equiv 1+z$ . Why: the wavelength is stretched in exact proportion to how much the Universe has expanded during the photon's flight — no velocity appears anywhere in the argument.

Step 4. For a nearby source expand $a(t_{\mathrm{obs}})/a(t_{\mathrm{emit}})\approx 1+H_0(t_{\mathrm{obs}}-t_{\mathrm{emit}})\approx 1+H_0 d/c$ , so $z\simeq H_0 d/c$ and $cz\simeq H_0 d$ — the Hubble law with apparent recession $v=cz$ . Why: at small $z$ the expansion mimics a Doppler recession, which is exactly why the two are so easily conflated.

Step 5. At large $z$ the linear step fails and $1+z$ is the full scale-factor ratio; inserting $v=cz$ literally gives $v>c$ for $z>1$ , which is meaningless. Why: cosmological redshift is not motion through space, and superluminal "velocities" are the tell that the Doppler interpretation has been pushed past its domain of validity. *The result $1+z=a_{\mathrm{obs}}/a_{\mathrm{emit}}$ follows from the FRW null geodesic and recovers $cz\simeq H_0 d$ as $z\to0$ .*

Limiting cases

The general relativistic result $1+z=(1+\beta\cos\theta)/\sqrt{1-\beta^2}$ , with $\beta=v/c$ and $v_r=v\cos\theta$ , contains every regime of interest as a limit:

Case	Result	Meaning
Non-relativistic radial	$z \simeq \dfrac{v_r}{c}$	Leading order for $v\ll c$ ; the everyday astronomical formula. Sign of $v_r$ gives red- or blueshift.
Second-order term	$z \simeq \dfrac{v_r}{c} + \dfrac{1}{2}\dfrac{v^2}{c^2}$	Expansion to $O(\beta^2)$ ; the $\tfrac12\beta^2$ piece is the transverse time-dilation redshift, present even when $v_r=0$ .
Purely transverse	$1+z = \gamma = \dfrac{1}{\sqrt{1-\beta^2}}$	Radial velocity zero, yet a genuine relativistic redshift — the transverse Doppler effect, no classical analogue.
Longitudinal recession	$1+z = \sqrt{\dfrac{1+\beta}{1-\beta}}$	Pure line-of-sight motion; inverts to $\beta<1$ for any finite $z$ .
Ultra-relativistic approach	$1+z = \sqrt{\dfrac{1-\beta}{1+\beta}}\to 0$	Head-on approach ( $v_r\to -c$ ) gives extreme blueshift; relevant to approaching relativistic jets.
Cosmological	$1+z = \dfrac{a_{\mathrm{obs}}}{a_{\mathrm{emit}}}$	Expansion of space, not motion; $z\simeq H_0 d/c$ only at low $z$ .

Common trapA galaxy's redshift is its speed through space, so v = cz always

Two traps hide here. First, the Doppler shift is blind to tangential motion: it measures $v_r$ , not the total space velocity, so a star crossing the sky at $100\ \mathrm{km\,s^{-1}}$ but with zero radial component shows no classical shift at all. Never quote a Doppler velocity as a star's speed without saying "radial". Second, and more damaging, do not read a distant galaxy's redshift as a literal recession velocity through space. The cosmological redshift comes from the expansion of space itself, $1+z=a_{\mathrm{obs}}/a_{\mathrm{emit}}$ ; the formula $v=cz$ is only the low- $z$ approximation of the Hubble law and gives nonsensical " $v>c$ " for $z>1$ . Galaxies at $z=6$ are not outrunning light — their light has simply been stretched by a factor of seven while crossing an expanding Universe. The honest statement is that redshift is an observable; velocity is a model-dependent interpretation of it.

Worked examples

1 · Radial velocity of a galaxy and the space velocity of a star

Part A: a galaxy shows the H $\alpha$ line ( $\lambda_{\mathrm{rest}}=656.3\ \mathrm{nm}$ ) at $\lambda_{\mathrm{obs}}=662.9\ \mathrm{nm}$ . Find its radial velocity. Part B: a nearby star has proper motion $\mu=0.5''\,\mathrm{yr^{-1}}$ at distance $d=10\ \mathrm{pc}$ and radial velocity $v_r=20\ \mathrm{km\,s^{-1}}$ . Find its total space velocity.

Step 1. Reduce the line to a redshift. The shift is $\Delta\lambda=662.9-656.3=6.6\ \mathrm{nm}$ , so $z=\Delta\lambda/\lambda_{\mathrm{rest}}=6.6/656.3=\mathbf{0.0101}$ . Because the line moved to longer wavelength, this is a redshift and the source recedes.

Step 2. Convert to velocity. Since $z\ll1$ the classical inversion applies: $v_r=cz=(2.998\times10^{5}\ \mathrm{km\,s^{-1}})(0.0101)=\mathbf{+3.0\times10^{3}\ km\,s^{-1}}$ , a recession. The relativistic correction here is of order $z\sim1\%$ , comfortably below the measurement precision, so $v=cz$ is justified.

Step 3. Now the star's tangential velocity from astrometry: $v_t=4.74\,\mu\,d=4.74\times0.5\times10=\mathbf{23.7\ km\,s^{-1}}$ , using $\mu$ in arcsec per year and $d$ in parsecs.

Step 4. Combine the two orthogonal legs into the space velocity: $v=\sqrt{v_r^{\,2}+v_t^{\,2}}=\sqrt{20^2+23.7^2}=\sqrt{961.7}=\mathbf{31.0\ km\,s^{-1}}$ . The star's true motion is inclined at $\arctan(v_t/v_r)=\arctan(23.7/20)\approx50^\circ$ to the line of sight — neither purely radial nor purely tangential, which is why both spectroscopy and astrometry were needed.

2 · A quasar at z = 2, and why "v = cz" fails

A quasar shows a line redshifted to $1+z=3$ , i.e. $z=2$ . What velocity does this imply? Contrast the naive Doppler reading with the special-relativistic one, and state the correct interpretation.

Step 1. The naive move is $v=cz=2c$ — a source receding at twice the speed of light. This is already the signal that the low- $z$ formula has been abused: no massive source moves faster than $c$ , so $v=cz$ cannot be a physical velocity here.

Step 2. Apply instead the exact longitudinal formula (5): $1+z=\sqrt{(1+\beta)/(1-\beta)}$ . Squaring, $(1+z)^2=9=(1+\beta)/(1-\beta)$ , so $9(1-\beta)=1+\beta$ , giving $8=10\beta$ and $\beta=\mathbf{0.8}$ . As a special-relativistic Doppler shift, $z=2$ corresponds to $v_r=0.8c<c$ — sensible, and always below $c$ for any finite $z$ .

Step 3. But the correct interpretation of a cosmological quasar is neither. The redshift is $1+z=a_{\mathrm{obs}}/a_{\mathrm{emit}}=3$ : the light was emitted when the Universe was one-third its present size, and the wavelength was stretched threefold by expansion. The proper recession velocity, defined through proper distance in the FRW metric, can legitimately exceed $c$ at high $z$ without violating relativity, because it is the expansion of space, not motion through it. Naive $v=cz$ gives $2c$ (wrong); SR longitudinal gives $\beta=0.8$ ; the physical reading is cosmological, $1+z=3$ .

Physics of the portrait

The radial-velocity method binds together a nineteenth-century wave insight and the whole modern enterprise of measuring the Universe — a chain that runs from a siren on a railway platform to the detection of planets around other stars.

Origins · 1 minChristian Doppler and how this was discovered

Christian Doppler

1803–1853, Austrian mathematician and physicist · the frequency shift of waves from moving sources, and its application to starlight

Christian Doppler was a mathematician at the Prague Polytechnic when, in 1842, he read a paper titled Über das farbige Licht der Doppelsterne — "On the coloured light of double stars" — proposing that the observed frequency of a wave depends on the relative motion of source and observer. His physical reasoning was sound even where his astronomical application was not: he wrongly imagined the effect would visibly colour stars, not realizing that a star's thermal continuum shifts as a whole and that its velocities are far too small to change perceived colour. The acoustic version was confirmed directly in 1845 by Buys Ballot, using trumpeters on a moving railway carriage and listeners with perfect pitch on the platform. The optical effect found its true home in spectroscopy: in 1848 Hippolyte Fizeau independently framed it as the displacement of spectral lines (hence the name Doppler–Fizeau effect), and in 1868 William Huggins measured the first stellar radial velocity, finding Sirius receding at a few tens of kilometres per second from the tiny shift of a hydrogen line. From there the method scaled without limit — Slipher's galaxy velocities, Hubble's redshift–distance law, and, a century and a half later, the $59\ \mathrm{m\,s^{-1}}$ wobble by which Mayor and Queloz found the first planet orbiting a Sun-like star. What Doppler offered to explain the colour of double stars became the ruler with which we measure the cosmos.

History

1842

— Doppler proposes that a wave's observed frequency shifts with the source's motion

1868

— Huggins measures the first stellar radial velocity, the recession of Sirius

1912

— Slipher measures the radial velocities of spiral nebulae, most of them redshifted

1929

— Hubble finds the linear redshift–distance relation, evidence for cosmic expansion

1995

— Mayor and Queloz detect 51 Pegasi b from its 59 m per second radial-velocity wobble

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §4.3 (the Doppler effect, eqs. 4.32–4.39) and §5.2 · Binney & Tremaine, Galactic Dynamics, 2nd ed., §1.1 (space velocities and rotation) · Ryden, Introduction to Cosmology, 2nd ed., Ch. 2 and §5.1 (redshift and the scale factor) · Dodelson, Modern Cosmology, Ch. 1 (redshift in the expanding Universe) · Mo, van den Bosch & White, Galaxy Formation and Evolution, §3.1. Key numerical benchmarks used throughout: the classical shift $\Delta\lambda/\lambda=v_r/c$ ; the general relativistic formula $1+z=(1+\beta\cos\theta)/\sqrt{1-\beta^2}$ with its longitudinal ( $\theta=0$ ) and transverse ( $\theta=90^\circ$ , $1+z=\gamma$ ) limits; the worked H $\alpha$ example ( $z=0.0101$ , $v_r\approx3.0\times10^{3}\ \mathrm{km\,s^{-1}}$ ); the space velocity ( $v_t=23.7$ , $v=31.0\ \mathrm{km\,s^{-1}}$ ); and the quasar inversion ( $z=2\Rightarrow\beta=0.8$ under the SR formula). Standard figures — Jupiter/Earth solar reflex $12.5$ and $0.09\ \mathrm{m\,s^{-1}}$ , $c=2.998\times10^{8}\ \mathrm{m\,s^{-1}}$ , H $\alpha$ rest wavelength $656.3\ \mathrm{nm}$ , 51 Peg b amplitude $\sim59\ \mathrm{m\,s^{-1}}$ — are textbook and catalogue values.

Prerequisites. Luminosity and Radiant Flux · Photometric Systems and Filters · The Equatorial Coordinate System. Leads to. The Cosmic Distance Ladder · Standard Candles · Stellar Parallax and the Parsec.

Ask the tutor: Want to see how a periodic radial-velocity wobble becomes a planet's mass, or how the same redshift is read three different ways from a star, a white dwarf, and a distant galaxy?

Does the Doppler shift measure a star's true speed?

No — it measures only the radial component, the part of the velocity directed along your line of sight. A star could be hurtling across the sky at hundreds of kilometres per second and, if none of that motion is toward or away from you, show essentially no classical Doppler shift. To recover the full three-dimensional space velocity you must add the tangential velocity, which comes from astrometry: the proper motion $\mu$ (how fast the star's position drifts across the sky) times the distance, $v_t=4.74\,\mu d$ . Spectroscopy and astrometry are genuinely complementary tools — one gives $v_r$ , the other $v_t$ — and only together do they pin down where a star is really going.

Which way is a redshift, and how do I remember the sign?

A redshift is a shift to longer wavelength (toward the red end of the spectrum) and corresponds to recession — the source moving away, positive $v_r$ . A blueshift is a shift to shorter wavelength and corresponds to approach, negative $v_r$ . The mnemonic that never fails: receding means the wave has to "reach" farther, stretching it out to the red. The dimensionless redshift $z=(\lambda_{\mathrm{obs}}-\lambda_{\mathrm{rest}})/\lambda_{\mathrm{rest}}$ carries the sign automatically — positive for recession, negative for approach — so if you compute $z$ and get a negative number, you are looking at an approaching, blueshifted object, and everything downstream ( $v_r=cz$ ) inherits the correct sign.

Is the cosmological redshift really different from a Doppler shift?

Yes, conceptually, even though the low-redshift arithmetic is identical. A Doppler shift arises because a source moves through space relative to you; a cosmological redshift arises because space itself expands while the photon is in flight, stretching its wavelength by the factor $1+z=a_{\mathrm{obs}}/a_{\mathrm{emit}}$ . For nearby galaxies the two are indistinguishable — both give $z\simeq v/c$ — which is why Hubble's law is often stated as a recession velocity. But at high $z$ the distinction matters: interpreting $z=6$ as a Doppler velocity would imply faster-than-light motion, whereas the expansion picture has no such problem, because proper recession velocities in an expanding Universe are not motion through space and are not bounded by $c$ .

Can we actually detect the transverse Doppler effect?

Yes, and its confirmation is a classic test of special relativity. Because it is a second-order effect — a redshift of order $\tfrac12\beta^2$ even when $v_r=0$ — it is swamped by the first-order longitudinal shift unless the geometry is carefully controlled. The Ives–Stilwell experiment (1938) isolated it in a beam of fast hydrogen atoms by averaging forward- and backward-emitted light, and modern versions using stored ion beams and optical clocks confirm the $\gamma$ factor to extraordinary precision. In astrophysics it contributes to the spectra of relativistic jets and accretion flows near black holes, where bulk speeds are a large fraction of $c$ and the pure time-dilation redshift is no longer negligible.

How do spectrographs reach metre-per-second precision?

Two ingredients. First, calibration: the instrument must know its own wavelength scale to a part in $10^{9}$ , achieved with a stable reference — historically a thorium–argon lamp or an iodine absorption cell, and now a laser frequency comb that lays down thousands of evenly spaced reference lines like the ticks of an optical ruler. Second, reference-frame correction: the raw velocity is measured at a telescope that is itself moving, so one subtracts Earth's $\sim30\ \mathrm{km\,s^{-1}}$ orbital motion and $\sim0.5\ \mathrm{km\,s^{-1}}$ rotation projected onto the target — the heliocentric (or barycentric) correction, $v_r^{\mathrm{helio}}=v_r^{\mathrm{obs}}+\mathbf{v}_\oplus\cdot\hat{\mathbf{n}}$ . Only after both steps does the residual $\sim1\ \mathrm{m\,s^{-1}}$ wobble of a planet-bearing star emerge from the noise.

How is a Doppler redshift distinguished from a gravitational one?

There are three physically distinct ways a spectral line can redshift. A Doppler redshift comes from relative motion, $z\simeq v_r/c$ . A gravitational redshift comes from light climbing out of a potential well, $z\simeq \Delta\Phi/c^2$ — a line emitted deeper in a star's or black hole's gravity appears redder, independent of any motion. A cosmological redshift comes from the expansion of space, $1+z=a_{\mathrm{obs}}/a_{\mathrm{emit}}$ . All three take the form "line displaced to longer wavelength", so the interpretation is fixed by context: a star's orbit implies Doppler, a white-dwarf or neutron-star surface implies gravitational, and a distant galaxy in the Hubble flow implies cosmological. Precision measurements — such as the gravitational redshift of stars orbiting the Galactic-centre black hole — must model all three at once.