Doppler Effect

Full definition

The classic Doppler experience, everyone has lived it: an ambulance tears past you. The siren emits exactly the same note throughout. Yet you hear it high-pitched as the vehicle approaches, then low-pitched as it recedes. At the precise moment it crosses, the pitch jumps abruptly. That's the acoustic Doppler effect.

The explanation is intuitive. Sound waves leave the siren at regular intervals. When the vehicle approaches, each new wave is emitted slightly closer to you than the previous: wave crests bunch, the perceived frequency rises, the sound grows high. When it recedes, the opposite: crests spread out, frequency drops, pitch falls. The emitted frequency hasn't changed at all.

The same phenomenon applies to every kind of wave: sound, light, radio, gravitational waves. For light, blue (short wavelength) corresponds to high-pitched sound, red (long wavelength) to low-pitched. Hence the terms blueshift (approaching source) and Doppler redshift (receding source). Christian Doppler sensed this as early as 1842 contemplating binary stars: the star orbiting toward us should appear bluer, the one swinging away, redder.

It is a titanic tool in astrophysics. Without leaving Earth, we measure the radial velocity of billions of stars and galaxies. We thereby discover:

• Spectroscopic binary stars (lines oscillating symmetrically around a mean position). • The rotation of the Sun and planets (receding limb redshifts, approaching limb blueshifts). • Exoplanets by radial velocity (star wobbles slightly around the common centre of mass). • Galaxy rotation curves (which led to the discovery of dark matter). • Recession of distant galaxies (Hubble-Lemaître law — but beware: at very large distances, redshift is cosmological, not Doppler).

Formulas and scale

The non-relativistic case (v ≪ c) is disarmingly simple:

Δf / f = Δλ / λ = v_r / c

with v_r the radial velocity (the line-of-sight component; transverse motion is ignored at first order). Positive for a receding source (reddening), negative for an approaching one (blueing). At v = 1 km/s, Δλ/λ = 3.3 × 10⁻⁶ — phenomenal spectroscopic precision is needed to detect it.

In the relativistic regime, the correct formula (longitudinal relativistic Doppler) becomes:

f_obs / f_em = √((1 - β)/(1 + β))

with β = v/c. It reduces to Δf/f ≈ -v/c at first order, but now incorporates relativistic time dilation.

Transverse Doppler. A purely relativistic case, with no acoustic analogue: even without radial velocity, a source moving laterally at high speed undergoes redshift due to time dilation. Experimentally verified by Ives-Stilwell (1938) on canal rays in a tube, matching special relativity to within 1 %.

Astrophysical orders of magnitude.

• Solar equatorial rotation: ~2 km/s — easily measurable. • Stellar wobble due to a Jupiter-type exoplanet: ~50 m/s — the 1990s limit, now routine. • Wobble from a super-Earth in the habitable zone: ~0.1-1 m/s — the realm of ESPRESSO (VLT, 2018). • Earth-type planet around a Sun-like star: 9 cm/s — the ultimate goal, at current technical limits.

Regimes and variants

Several variants of the phenomenon must be distinguished.

Acoustic Doppler. In air or a material medium. Depends on absolute velocity relative to the medium (source and observer roles are not symmetric). The classical formula:

f_obs = f_em · (v_sound + v_obs) / (v_sound - v_source)

with signs depending on motion direction. Example: siren at 1,000 Hz, train at 30 m/s approaching, v_sound ≈ 340 m/s. You perceive f ≈ 1,097 Hz approaching, f ≈ 919 Hz after passing. Musically trained ears clearly hear a semitone; the Doppler jump greatly exceeds a full tone.

Optical (classical) Doppler. For light sources at velocities small compared to c. Unlike acoustic, light needs no medium; only the relative source-observer velocity matters (relativity principle). Governs all classical stellar spectroscopy.

Relativistic Doppler. At high velocity, incorporates time dilation. Includes transverse Doppler. Needed as soon as β > ~0.1.

Doppler and gravitational waves. Gravitational waves also undergo Doppler shift. During a compact-binary merger, the frequency seen from Earth is shifted by the system's radial velocity relative to us. LIGO corrects data for this shift.

Mössbauer effect — precision measurement. For the iron-57 γ line (14.4 keV), Mössbauer spectroscopy reaches a fractional precision of 10⁻¹², enabling measurement of velocities as small as mm/s. This tool allowed Pound and Rebka to verify Einstein's gravitational redshift in 1959, over 22.5 metres in a Harvard tower.

How is it used in astronomy?

The Doppler effect is the magic wand that lets us measure velocities without leaving Earth. Headline applications.

Radial-velocity exoplanets. A planet makes its star wobble around the common centre of mass. Watching stellar lines, we detect the periodic Doppler oscillation. The historic method that yielded 51 Pegasi b, the first exoplanet around a solar-type star, discovered by Michel Mayor and Didier Queloz at OHP in October 1995 — 2019 Nobel Prize in Physics. The ESPRESSO spectrograph (VLT, 2018) reaches 10 cm/s, on the threshold of detecting an Earth in the habitable zone.

Spectroscopic binary stars. When two stars too close to be visually resolved orbit each other, their spectral lines oscillate symmetrically around a mean. Amplitude → mass ratio, period → orbital distance. This is how the majority of stars are weighed.

Galactic rotation curves. Measuring Doppler along a galactic diameter gives the rotation velocity at each radius. Vera Rubin and Kent Ford in the 1970s revealed that these curves don't fall off at the edges — a key piece of evidence for dark matter.

Planetary atmospheric velocimetry. On Mars, Venus, Titan: measuring atmospheric winds via submillimetre molecular-line Doppler (ALMA). On the Sun: Doppler helioseismology (SDO), probing the solar interior as seismic waves probe Earth.

And for amateurs? With a high-resolution spectroscope (LHIRES III, Alpy 600), you can measure solar equatorial rotation (~2 km/s on each side of the disk) in a morning. Push further: binary stars Algol and β Lyrae show lines oscillating with their periods (2.87 d and 12.9 d respectively). Measurable in a seasoned amateur observatory.

Not to be confused with

Several related phenomena are close but distinct.

Cosmological redshift. Doppler redshift is due to source motion through space. Cosmological redshift is due to stretching of space between source and observer during the journey. At small redshifts (z ≪ 0.1), both yield the same numerical result — but for a galaxy at z = 5, interpreting the shift as a Doppler velocity via v = c·z makes no physical sense: it's not motion through space, it's space stretching.

Gravitational redshift. Predicted by general relativity, it occurs when light climbs out of a gravitational potential well. Nothing to do with velocity. Verified by the Pound-Rebka experiment in 1959. Contributes significantly to white-dwarf lines (~100 km/s apparent) and neutron-star lines.

Sagnac effect. Frequency shift observed in a rotating optical ring. Not a classical Doppler effect (it persists even when source and detector co-rotate). Basis of the fibre-optic gyroscope.

Compton scattering. A photon bouncing off a moving electron changes wavelength. This is an interaction, not a Doppler effect (though they are distant cousins).

Chromatic dispersion. A prism separates colours of a white beam because refractive index depends on wavelength. Nothing to do with motion. A prism creates no Doppler shift — it reveals the spectrum already present in the incoming light.