Cosmology » The Red-shift Equation
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 8.16P
- Use the red-shift equation to link change in wavelength to the velocity of a galaxy
- Find the change in wavelength, then the velocity of a galaxy
- Rearrange the equation to find the observed wavelength
- Compare how fast two galaxies are moving away
This lesson is examined in Paper 2 only. The red-shift equation is given on the formulae page of the exam, so you do not have to memorise it, but you must know how to use it.
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Unlock This CourseFrom red-shift to a speed
In the lesson Red-shift of Light from Galaxies you saw that the dark lines in the spectrum of a galaxy are shifted towards the red end. A bigger shift means the galaxy is moving away faster. This lesson turns that idea into numbers. If you measure how far a line has moved, you can work out the velocity of the galaxy.
Key terms:
- Reference wavelength (λ0): the wavelength of a dark line when it is measured on Earth.
- Observed wavelength (λ): the wavelength of the same line in the light that arrives from the galaxy.
- Change in wavelength (Δλ): the observed wavelength minus the reference wavelength.
The red-shift equation
The equation (given on the formulae page)
change in wavelength ÷ reference wavelength = velocity of a galaxy ÷ speed of light
(λ − λ0) ÷ λ0 = Δλ ÷ λ0 = v ÷ c
The symbols mean:
- λ0 is the reference wavelength, measured on Earth.
- λ is the observed wavelength, from the galaxy.
- Δλ = λ − λ0 is the change in wavelength.
- v is the velocity of the galaxy, in m/s.
- c is the speed of light, 3 × 108 m/s.
The left side compares the shift to the original wavelength. It tells you what fraction of the reference wavelength the line has moved. The right side says the galaxy is moving away at the same fraction of the speed of light. So a line that has moved by 1% means a galaxy moving at 1% of the speed of light.
Units and standard form
Wavelengths of light are tiny, so they are often given in nanometres (nm). Because the equation divides one wavelength by another, the wavelength units cancel. You can leave both in nm. Just make sure Δλ and λ0 use the same unit.
The velocity v comes out in m/s because c is in m/s. Galaxy speeds are very large, so answers are usually in standard form, such as 3 × 106 m/s.
Method for finding the velocity
- Find Δλ = λ − λ0.
- Divide Δλ by λ0. This number has no unit.
- Multiply by c to get v = (Δλ ÷ λ0) × c.
Worked examples
Worked example 1: finding the velocity
A dark line has a reference wavelength of 400 nm. In the light from a galaxy it is at 404 nm. Calculate the velocity of the galaxy.
Step 1: Δλ = 404 − 400 = 4 nm
Step 2: Δλ ÷ λ0 = 4 ÷ 400 = 0.01
Step 3: v = 0.01 × 3 × 108
Answer: v = 3 × 106 m/s
Worked example 2: a bigger shift
A line with a reference wavelength of 600 nm is seen at 618 nm. Calculate the velocity of the galaxy.
Step 1: Δλ = 618 − 600 = 18 nm
Step 2: Δλ ÷ λ0 = 18 ÷ 600 = 0.03
Step 3: v = 0.03 × 3 × 108
Answer: v = 9 × 106 m/s
Worked example 3: units in metres
A line has a reference wavelength of 4.0 × 10−7 m. The change in wavelength is 2.0 × 10−8 m. Calculate the velocity of the galaxy.
Step 1: Δλ ÷ λ0 = (2.0 × 10−8) ÷ (4.0 × 10−7) = 0.05 (the metres cancel)
Step 2: v = 0.05 × 3 × 108
Answer: v = 1.5 × 107 m/s
Rearranging to find the observed wavelength
Sometimes you are given the velocity and asked for the wavelength you would see. Work in the other direction.
Method for finding the observed wavelength
- Work out v ÷ c.
- Multiply by λ0 to get Δλ = (v ÷ c) × λ0.
- Add Δλ to λ0 to get λ.
Worked example 4: finding the observed wavelength
A galaxy is moving away at 1.5 × 107 m/s. A dark line has a reference wavelength of 450 nm. Calculate the wavelength at which the line is observed.
Step 1: v ÷ c = (1.5 × 107) ÷ (3 × 108) = 0.05
Step 2: Δλ = 0.05 × 450 = 22.5 nm
Step 3: λ = 450 + 22.5 = 472.5 nm
Answer: the line is observed at 472.5 nm.
Comparing two galaxies
Because v = (Δλ ÷ λ0) × c, a galaxy with a bigger shift has a bigger velocity. If the reference wavelength is the same, doubling Δλ doubles v.
Worked example 5: two galaxies
The same dark line has a reference wavelength of 500 nm. In galaxy A it is seen at 502 nm. In galaxy B it is seen at 510 nm. Which galaxy is moving away faster, and how many times faster?
Galaxy A: Δλ = 2 nm, so Δλ ÷ λ0 = 2 ÷ 500 = 0.004, and v = 0.004 × 3 × 108 = 1.2 × 106 m/s
Galaxy B: Δλ = 10 nm, so Δλ ÷ λ0 = 10 ÷ 500 = 0.02, and v = 0.02 × 3 × 108 = 6 × 106 m/s
Answer: galaxy B is moving away faster. 6 × 106 ÷ 1.2 × 106 = 5, so it is moving 5 times faster.
Common mistakes
Common mistakes
- Using the observed wavelength instead of Δλ on the top of the fraction. Subtract first.
- Dividing by the observed wavelength. You must divide by the reference wavelength, λ0.
- Using the wrong speed of light, such as 3 × 106 or 300. Use c = 3 × 108 m/s.
- Mixing units, such as Δλ in nm and λ0 in m. Convert so they match, or the units will not cancel.
- Stopping at Δλ ÷ λ0 and forgetting to multiply by c.
- Giving the observed wavelength as just Δλ. You must add it to λ0.
Exam-style question
A dark line in the spectrum of a galaxy has a reference wavelength of 480 nm. In the light from the galaxy the line is seen at 489.6 nm. The speed of light is 3 × 108 m/s.
(a) Calculate the change in wavelength. [1 mark]
(b) Calculate the velocity of the galaxy. [3 marks]
(c) A second galaxy is moving away at 1.2 × 107 m/s. Calculate the wavelength at which the same line is seen in the light from this galaxy. [3 marks]
Model answer
(a) Δλ = 489.6 − 480 = 9.6 nm (1)
(b) Δλ ÷ λ0 = 9.6 ÷ 480 = 0.02 (1); v = 0.02 × 3 × 108 (1); v = 6 × 106 m/s (1)
(c) v ÷ c = (1.2 × 107) ÷ (3 × 108) = 0.04 (1); Δλ = 0.04 × 480 = 19.2 nm (1); λ = 480 + 19.2 = 499.2 nm (1)
Exam tip
In part (c), the answer is the observed wavelength, not the change. Finish by adding Δλ to 480 nm, and write the unit.