Last page was the idea; this page is the number. One compact equation turns a tiny shift in a star’s light into the star’s speed — which is how astronomers clock objects trillions of kilometres away without ever leaving the lab.
📘 What you need to know
For light from a non-relativistic source, the Doppler shift is: Δf/f = Δλ/λ ≈ v/c
The ≈ means “approximately equals” — the equation only works when v ≪ c
Δλ = λ0 − λ: the observed wavelength minus the reference (laboratory) wavelength
Each side of the equation is a ratio, so the Doppler shift has no units — wavelengths can stay in nm
We usually treat the observer as stationary, so the relative speed Δv is just the source’s speed v
In practice the shift is measured from atomic spectral lines — the whole line pattern slides towards red (receding) or blue (approaching)
The Doppler Equation for Light
For a light-emitting source that isn’t moving anywhere near the speed of light, the fractional shift in frequency (or wavelength) equals the source’s speed as a fraction of c:
Doppler shift for light (v ≪ c)
Δf ÷ f = Δλ ÷ λ ≈ v ÷ c
Taking the symbols one at a time: Δf is the change in frequency (Hz) and f is the reference (original) frequency; Δλ is the change in wavelength (m) and λ the reference wavelength; v is the relative velocity of source and observer (m s⁻¹); and c is the speed of light — which lives in your data booklet, so no memorising needed.
Two small-print points that earn real marks:
The ≈ sign matters: this is an approximation that only holds when v is much smaller than c
Both sides are ratios of matching quantities, so the units cancel — the Doppler shift itself has no units, and you can keep wavelengths in nanometres without converting
What exactly is Δλ?
It’s the observed wavelength minus the one the source “really” emits — the reference value you’d measure from the same atoms in a laboratory:
Change in wavelength
Δλ = λ0 − λ
where λ0 is the observed wavelength and λ the reference. The sign is a free gift: a positive Δλ means the light arrived stretched (red-shifted → source receding), a negative Δλ means it arrived squashed (blue-shifted → source approaching).
Whose speed is v?
Strictly, the equation uses the relative speed along the line joining source and observer, Δv = vs − vo. But in almost every question we take the observer (us, on Earth) to be stationary, so vo = 0 and Δv is simply the speed v of the source.
Spectral Lines: the Fingerprint Trick
How do we know what wavelength a star’s light “should” have? Atomic spectral lines. Every element absorbs light at its own fixed set of wavelengths, printing a barcode of dark lines across the spectrum — and that barcode is identical whether the atoms are in a lab on Earth or in a galaxy far away.
So compare the two. If the pattern from a distant galaxy matches the lab pattern but the whole barcode has slid towards the red end, the galaxy is moving away — and the size of the slide, Δλ, hands you its speed.
The dark absorption lines are an element’s fingerprint. The distant galaxy shows the same fingerprint, but every line has slid the same distance towards the red (long-wavelength) end — so the galaxy is receding, and the slide Δλ gives its speed.
🧭 Solving a Doppler-for-light problem
Identify the two wavelengths: the reference λ (laboratory) and the observed λ0 (from the star or galaxy)
Find the shift: Δλ = λ0 − λ — keep the sign, and feel free to stay in nm
Rearrange and solve:v = cΔλ ÷ λ (dividing by the reference wavelength)
Quick recap: Δf/f = Δλ/λ ≈ v/c, valid for v ≪ c. Measure Δλ from shifted spectral lines, divide by the reference wavelength, multiply by c — the sign tells you towards or away.
WE 1
A spectral line measured from a stationary source in the laboratory has a wavelength of 442 nm. The same line in the light from a distant star, moving directly away from Earth, is measured at 597 nm. Calculate the speed at which the star is receding.
List the quantities
λ = 442 nm, λ₀ = 597 nm, c = 3.0 × 10⁸ m s⁻¹
Δλ = 597 − 442 = 155 nmRearrange the Doppler equationΔλ/λ ≈ v/c ⇒ v = cΔλ/λSubstitutev = (3.0 × 10⁸ × 155) ÷ 442v ≈ 1.1 × 10⁸ m s⁻¹No need to convert nm to m — the nanometres cancel in the ratio Δλ/λ.
WE 2
A distant galaxy is viewed edge-on from Earth, so its stars orbit the galactic centre towards us on one side and away on the other. A spectral line with laboratory wavelength 486.13 nm is measured at 486.35 nm from the left-hand edge of the galaxy and 485.91 nm from the right-hand edge.
(a) State and explain which side of the galaxy is moving towards the Earth.
(b) Calculate the rotational speed of the galaxy.
Part (a)
The right-hand side (485.91 nm) is observed at a shorter wavelength than the reference (486.13 nm) — it has been blue-shifted
the right-hand side is moving towards EarthPart (b) — average the two shiftsΔλ = (486.35 − 485.91) ÷ 2 = 0.22 nmApply the Doppler equationv = cΔλ/λ = (3.0 × 10⁸ × 0.22) ÷ 486.13v ≈ 1.4 × 10⁵ m s⁻¹ (about 136 km s⁻¹)One edge is red-shifted, the other blue-shifted by the same amount — averaging the two removes any overall motion of the galaxy and leaves pure rotation.
💡 Top tips
Because the equation is a ratio, wavelengths can stay in nm throughout — converting to metres just invites power-of-ten slips
Keep the minus signs: the sign of Δλ (or Δf) is the direction — positive Δλ means receding, negative means approaching
c is given in the data booklet; the Doppler equation is the thing to remember
Anchor your colours: red light has the longest wavelength and lowest frequency of the visible spectrum; blue the shortest wavelength and highest frequency
Rotating object seen edge-on? One side red-shifts, the other blue-shifts — take half the difference between the two observed wavelengths
⚠ Common mistakes
Using the equation when v is close to c — it’s an approximation for v ≪ c only
Computing Δλ = λ − λ0 (backwards) and then declaring the wrong direction of motion
Dividing by the observed wavelength instead of the reference wavelength λ
Converting one wavelength to metres but not the other — stay consistent (or just stay in nm)
Forgetting to halve the wavelength difference for an edge-on rotating galaxy
Up next: galactic redshift — what it means that almost every galaxy’s barcode is slid towards the red, and how that points to an expanding universe.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
