The Doppler Effect For Light

You have met the Doppler effect already, in Module 4 (Waves) of the OCR A-Level Physics specification: the change in observed frequency of a wave when source and observer are in relative motion. A fire engine siren sounds high-pitched as it approaches, low-pitched as it recedes. The same effect happens to light — but with cosmological consequences that shaped twentieth-century physics.

This lesson develops the Doppler formula for electromagnetic radiation, distinguishes red shift from blue shift, and prepares the ground for Lessons 8 and 9, where we shall use the Doppler effect to measure the expansion of the universe itself.

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The Doppler Effect, Briefly Recalled

For sound waves, the classical Doppler formula is

f' = f × (v_sound ± v_observer) / (v_sound ∓ v_source)

which you studied in Module 4. The details depend on who is moving and in which direction. For sound waves through air, the medium (the air itself) defines a preferred frame, and the full formula treats source and observer asymmetrically.

For light, the situation is different. There is no medium for light to travel through — no "aether" — as the Michelson–Morley experiment (1887) demonstrated. Light propagates through vacuum at the speed `c) in every inertial frame. The full relativistic Doppler formula is

f' = f × √((1 - β)/(1 + β))     where β = v/c

where v) is the relative velocity along the line of sight (positive for recession). This is the formula that an astronomer uses for objects moving at a significant fraction of c). But for the nonrelativistic case (`v ll c)), which is what OCR A-Level Physics A requires, we can simplify it considerably.

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The Low-Velocity Approximation

Expanding the relativistic formula for small `β):

f' = f × (1 - β + β²/2 - ...) / (1 + β + β²/2 + ...)
   ≈ f × (1 - β)          to first order in β

This gives the change in frequency:

Δf = f' - f = -f × β = -f × v/c

Or, equivalently:

Δf / f = -v/c

The negative sign reflects the convention that v > 0) means the source is *receding* (moving away), and that a receding source appears to have a *lower* frequency (red shift). For an approaching source (v < 0)), the frequency increases (blue shift).

Because frequency and wavelength are related by `c = fλ) (the speed of light is constant in vacuum), an increase in wavelength corresponds to a decrease in frequency, and vice versa. The fractional change in wavelength is equal and opposite to the fractional change in frequency:

Δλ / λ = -Δf / f = v/c

The OCR formula you are expected to memorise and use is:

Δλ / λ ≈ Δf / f ≈ v / c

Here v) is the **radial velocity** of the source (along the line of sight), positive for recession. The approximation is valid when v ll c). For essentially all stellar motions within our Galaxy, and for all but the most distant galaxies, this approximation is excellent.

Exam Tip: OCR gives you the formula `Δλ/λ = Δf/f = v/c) on the data sheet. What you must supply is the physical interpretation, sign, and the algebra to go from observed wavelength shifts to source velocities.

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Red Shift And Blue Shift

When a source moves away from the observer, the emitted waves are "stretched" behind the receding source. The wavelength appears longer — shifted towards the red end of the spectrum. This is called a red shift.

When a source moves towards the observer, the waves are "compressed" in front of it. The wavelength appears shorter — shifted towards the blue end of the spectrum. This is called a blue shift.

graph LR
    subgraph Recession
    S1[Source moving →]
    W1[Wavelength<br/>stretched]
    O1[Observer sees<br/>red shift]
    S1 --> W1 --> O1
    end
    subgraph Approach
    S2[Source moving ←]
    W2[Wavelength<br/>compressed]
    O2[Observer sees<br/>blue shift]
    S2 --> W2 --> O2
    end

The key conventions:

• Red shift → Δλ > 0), v > 0) (receding).
• Blue shift → Δλ < 0), v < 0) (approaching).

By convention, astronomers report red shifts as positive z = Δλ/λ) values and blue shifts as negative. For small velocities, z approx v/c).

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Worked Example 1: Velocity Of A Star From A Spectral Line

A hydrogen absorption line has laboratory wavelength λ = 656.3) nm (the H-α line in the Balmer series). In a particular star's spectrum this line is observed at λ' = 656.8) nm. Find the star's radial velocity and direction.

Step 1. Compute `Δλ):

Δλ = λ' - λ = 656.8 - 656.3 = 0.5 nm

Step 2. Apply the Doppler formula:

v = c × Δλ/λ
  = (3.00 × 10⁸) × (0.5 × 10⁻⁹) / (656.3 × 10⁻⁹)
  = (3.00 × 10⁸) × (0.5 / 656.3)
  = (3.00 × 10⁸) × (7.62 × 10⁻⁴)
  = 2.29 × 10⁵ m s⁻¹

This is about 230 km s⁻¹. Since `Δλ > 0), the line is red-shifted — the star is moving away from us at 230 km/s along the line of sight.

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Worked Example 2: Approaching Galaxy

A particular emission line in the Andromeda galaxy (M31) has laboratory wavelength λ = 486.1) nm (the H-β line). The observed wavelength is λ' = 485.6) nm. What is the radial velocity of Andromeda?

Δλ = 485.6 - 486.1 = -0.5 nm
v = c × Δλ/λ
  = (3.00 × 10⁸) × (-0.5 × 10⁻⁹) / (486.1 × 10⁻⁹)
  = -3.09 × 10⁵ m s⁻¹

The velocity is negative, meaning the galaxy is approaching us. Andromeda is indeed one of the few galaxies that is blue-shifted relative to the Milky Way — it is on a collision course with our own Galaxy, expected to merge in about 4.5 billion years. Most galaxies, at greater distances, are red-shifted because the universe is expanding (see Lesson 8).

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Worked Example 3: Rotation Of A Distant Galaxy

A spiral galaxy is observed along the line of sight. The H-α line (laboratory λ = 656.3) nm) is measured at the two ends of the galaxy's disc. On the western edge it appears at 656.7) nm; on the eastern edge at `656.9) nm. (a) What is the average radial velocity of the galaxy? (b) What is its rotational velocity?

(a) Average λ' = (656.7 + 656.9)/2 = 656.8) nm. Average Δλ = 0.5) nm.

v_avg = c × 0.5/656.3 = 2.29 × 10⁵ m s⁻¹