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The single most important fact about every galaxy beyond our own Local Group is that its light is shifted to longer wavelengths. Spectral lines that emerge from hydrogen, calcium and iron in distant galaxies arrive at our telescopes red-shifted, sometimes by enormous fractions. The shift is not a property of the galaxy's chemistry — every galaxy's hydrogen is the same hydrogen — but of its motion relative to us, and at very large distances, of the expansion of space itself. This lesson establishes the Doppler effect for light, derives the non-relativistic redshift relation Δλ/λ = v/c, introduces the cosmological redshift as a distinct (and ultimately more accurate) picture, and shows how spectral line shifts are used in everything from measuring the rotation of distant galaxies to detecting exoplanets.
Spec mapping. This lesson covers AQA 7408 section 3.9.3 on cosmology: the Doppler effect for electromagnetic waves; redshift z = Δλ/λ ≈ v/c at non-relativistic speeds; the distinction between kinematic Doppler and cosmological redshift; applications including galaxy radial velocities, binary spectroscopic stars, and exoplanet radial-velocity detection. (Refer to the official AQA specification document for exact wording.)
Synoptic links. This lesson builds on wave properties (AQA 3.3): the same frequency-period-velocity relations apply to electromagnetic waves in vacuum. It connects forward to Hubble's law and the Big Bang (next lesson) and feeds into the exoplanet detection content of the final lesson. The relativistic Doppler correction sits beyond the AQA specification but is mentioned for completeness.
The Doppler effect — named for Christian Doppler, who explained the colour shift of moving binary stars in 1842 — is the change in observed frequency or wavelength of a wave when the source moves relative to the observer.
For sound travelling through a medium (the air), the source's speed relative to the air matters, and the formula depends on whether the source or the observer is moving. For electromagnetic waves in vacuum there is no medium and only the relative velocity between source and observer matters. At speeds v ≪ c the relation is
Δλ / λ = Δf / f = v / c
where v is the speed of the source's recession (taken as positive when moving away), λ is the rest-frame wavelength, and Δλ = λ_observed − λ_emitted.
Convention. A positive Δλ means the line has shifted to longer wavelengths ("redshift", source moving away). A negative Δλ means the line has shifted to shorter wavelengths ("blueshift", source moving towards).
The redshift z is defined as
z = Δλ / λ_emitted = (λ_observed − λ_emitted) / λ_emitted
so at non-relativistic speeds z ≈ v/c. Redshift is dimensionless and is the universal parameter astronomers use to characterise the motion (or cosmological location) of distant objects.
The Hα line of hydrogen has a rest wavelength λ₀ = 656.28 nm. In the spectrum of a nearby galaxy, the line is observed at 657.40 nm. Calculate the redshift z and the recession velocity v.
Solution. Δλ = 657.40 − 656.28 = 1.12 nm. z = Δλ/λ = 1.12 / 656.28 = 1.71 × 10⁻³. v = cz = 3.00 × 10⁸ × 1.71 × 10⁻³ = 5.12 × 10⁵ m s⁻¹ = 512 km s⁻¹. The galaxy is receding at about 0.17% of the speed of light.
A spectral line at rest wavelength 486.13 nm (Hβ) appears at 485.30 nm in the spectrum of the Andromeda galaxy. Calculate the velocity of approach.
Solution. Δλ = 485.30 − 486.13 = −0.83 nm. z = −0.83 / 486.13 = −1.71 × 10⁻³. v = cz = −5.12 × 10⁵ m s⁻¹. The minus sign means the galaxy is approaching at about 512 km s⁻¹. (This is approximately the real radial velocity of M31 relative to the Milky Way.)
A quasar shows the Lyα line of hydrogen (rest wavelength 121.6 nm) at observed wavelength 729.6 nm. Calculate z.
Solution. z = (729.6 − 121.6) / 121.6 = 608 / 121.6 = 5.00. The observed wavelength is six times the rest wavelength. Calling cz the "velocity" gives v = 5c — which is impossible. At z this large, the non-relativistic formula breaks down.
For sources moving at a significant fraction of c, the correct special-relativistic formula is
1 + z = √((1 + β) / (1 − β)) with β = v/c
This always gives 0 < (v/c) < 1 for any positive z, no matter how large. At z = 5, β ≈ 0.946 — the quasar is receding at 95% of the speed of light, if its redshift is a kinematic Doppler shift (which, at cosmological distances, it is not — see below).
AQA only requires the non-relativistic Δλ/λ ≈ v/c at z ≪ 1. The relativistic version is mentioned here for completeness and would not be examined.
For galaxies in the Hubble flow (well outside our Local Group), the redshift is not a Doppler shift in the conventional sense. The galaxies are not moving through space; rather, space itself is expanding, stretching the wavelengths of photons as they traverse it.
Imagine a wave drawn on a sheet of rubber. As the rubber stretches, the wave's wavelength grows in proportion. If the rubber doubles in size between emission and reception, the wavelength doubles, and the observed wavelength is twice the emitted wavelength — z = 1. This is the cosmological redshift.
The mathematical relation is
1 + z = a(t_observe) / a(t_emit)
where a(t) is the scale factor of the universe at time t. At the time the photon was emitted the universe was smaller; by the time it arrives, the universe has expanded by a factor 1 + z. For nearby galaxies (low z) the Doppler-shift picture is a good approximation. For distant galaxies (z > 0.1) the cosmological picture is the correct one.
The distinction matters but is rarely tested at A-Level — you only need to know:
By measuring the redshifts or blueshifts of spectral lines, astronomers determine the radial velocity (line-of-sight motion) of any luminous object. Galactic radial velocities have been catalogued for tens of millions of galaxies, providing the data underpinning Hubble's law (next lesson).
If two stars orbit each other, each one alternately approaches and recedes from Earth over one orbital period. The spectral lines therefore oscillate in wavelength sinusoidally. By measuring the period and amplitude of the line shifts, astronomers determine the orbital parameters of the binary even when the two stars cannot be resolved spatially — this is spectroscopic binary detection.
For a single-lined spectroscopic binary (one star much brighter than the other), only one set of lines shifts. For a double-lined system, both sets shift in antiphase.
A spiral galaxy seen edge-on shows different parts of itself moving at different radial velocities — one side approaches, the other recedes. By measuring the velocity of the hydrogen 21 cm line across the galaxy's image, astronomers build a rotation curve v(r). For most spiral galaxies the rotation curve flattens at large r — direct evidence of the dark matter halo that surrounds the visible disc.
A star with an orbiting planet wobbles slightly because of the planet's gravitational pull (both bodies orbit their common centre of mass). The stellar wobble produces a tiny Doppler shift in the stellar spectral lines — typically of order m s⁻¹ to 100 m s⁻¹ depending on the planet's mass and orbital distance.
The required spectroscopic precision is staggering. A Jupiter-mass planet in a Jupiter-like orbit produces a stellar wobble of about 12.5 m s⁻¹. The corresponding Δλ/λ at 500 nm is 12.5/3×10⁸ = 4.2 × 10⁻⁸ — a wavelength shift of about 0.00002 nm. Modern spectrographs (HARPS, ESPRESSO) routinely achieve 1 m s⁻¹ precision, with future instruments aiming for the 10 cm s⁻¹ needed to detect Earth-mass planets around Sun-like stars.
A spectral line of a binary star system oscillates between λ_max = 656.50 nm and λ_min = 656.06 nm with a period of 5.40 days. The rest wavelength is λ₀ = 656.28 nm. Estimate (a) the orbital speed of the visible component along the line of sight, and (b) the orbital radius if the orbit is circular and edge-on.
Solution. (a) The wavelength oscillates with semi-amplitude Δλ = (656.50 − 656.06)/2 = 0.22 nm. Velocity semi-amplitude v = c × Δλ/λ₀ = 3.00 × 10⁸ × 0.22 / 656.28 = 1.00 × 10⁵ m s⁻¹ = 100 km s⁻¹. (b) For circular motion at speed v over period T, the orbital circumference is 2πr = vT. T = 5.40 × 86 400 s = 4.66 × 10⁵ s. r = vT / (2π) = (1.00 × 10⁵)(4.66 × 10⁵) / 6.28 = 7.41 × 10⁹ m. This is about 5% of the Earth-Sun distance — a tight binary.
A star is observed to have a sinusoidal radial velocity variation of semi-amplitude K = 56 m s⁻¹ and period T = 4.23 days. Estimate the minimum mass of the orbiting planet, given the star has mass M_⋆ = 1.0 M_⊙ and the orbit is assumed circular and edge-on. (For an edge-on orbit, m_p × v_p = M_⋆ × K from momentum conservation. The planet's orbital speed v_p follows from Kepler's third law.)
Solution. Kepler's third law: T² = 4π²a³/(GM_⋆) gives a = (GM_⋆T²/4π²)^(1/3) = (6.67 × 10⁻¹¹ × 1.99 × 10³⁰ × (4.23 × 86 400)²/4π²)^(1/3) = (6.67 × 10⁻¹¹ × 1.99 × 10³⁰ × 1.34 × 10¹¹ / 39.5)^(1/3) = (4.50 × 10⁸ × ... ) = 7.70 × 10⁹ m. v_p = 2πa/T = 2π × 7.70 × 10⁹ / 3.65 × 10⁵ = 1.32 × 10⁵ m s⁻¹. Mass: m_p = M_⋆ × K / v_p = 1.99 × 10³⁰ × 56 / 1.32 × 10⁵ = 8.4 × 10²⁶ kg ≈ 0.44 M_J. About half a Jupiter mass — a "hot Jupiter".
Each star's lines shift in antiphase as they orbit. By measuring the period and the amplitude of the wavelength oscillations, astronomers obtain the orbital speeds — and combined with Kepler's laws, the masses.
| Source | Typical v / c | Typical Δλ at 500 nm | What is measured |
|---|---|---|---|
| Earth's orbital motion | 10⁻⁴ | 0.05 nm | Heliocentric correction (subtract) |
| Stellar radial velocity | 10⁻⁵ – 10⁻³ | 0.005 – 0.5 nm | Stellar motion, binary orbits |
| Galaxy peculiar velocity | 10⁻³ | 0.5 nm | Local group dynamics |
| Hubble flow (nearby) | 10⁻³ – 10⁻¹ | 0.5 – 50 nm | Hubble's law |
| Quasars / high-z galaxies | 10⁻¹ – ~1 | 50 – 500 nm | Cosmological expansion |
| Exoplanet host stars | ~10⁻⁸ | ~2 × 10⁻⁵ nm | Radial-velocity planet detection |
The required precision spans fifteen orders of magnitude. Modern spectrographs use simultaneous calibration against stable reference sources (iodine cells, laser frequency combs) to push down to the level needed for Earth analogues.
Specimen question modelled on the AQA paper format.
(a) State the equation linking the redshift z, the change in wavelength Δλ and the rest wavelength λ₀ for a source moving away at non-relativistic speed v. (1 mark)
(b) The hydrogen Hα line is observed in the spectrum of a galaxy at 6620 Å. Its rest wavelength is 6562.8 Å. Calculate the redshift z and the recession velocity v in km s⁻¹. (3 marks)
(c) A star is observed to show a sinusoidal radial-velocity variation of semi-amplitude 75 m s⁻¹ with a period of 21.0 days, as part of an exoplanet search programme. (i) Calculate the maximum and minimum observed wavelength of a spectral line of rest wavelength 5500 Å, ignoring the host star's overall recession velocity. (3 marks) (ii) Comment briefly on whether the wavelength shifts of this magnitude are detectable with modern instruments. (2 marks)
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