Key points
- Redshift: for speeds much less than c, Δλ/λ ≈ Δf/f ≈ v/c.
- Hubble's law: v = H₀d, so the gradient of a speed-distance graph is H₀ (about 70 km s⁻¹ Mpc⁻¹).
- The age of the universe is approximately 1/H₀, once H₀ is converted into s⁻¹.
- The cosmic microwave background, with a black-body peak at 2.7 K, is key evidence for the Big Bang.
- Dark matter explains galaxy rotation and dark energy explains the accelerating expansion.
What Hubble's law and redshift tell us
In the 1920s, Edwin Hubble and others found that light from distant galaxies is shifted towards the red end of the spectrum, and that the further away a galaxy is, the bigger the shift. Interpreted as a Doppler effect, this means the galaxies are moving away from us, and the more distant ones are moving away faster. That is Hubble's law, and it is the main observational evidence that the universe is expanding.
This guide covers the three connected ideas OCR A asks about: how redshift gives the speed of a galaxy, how Hubble's law links the speed to distance, and what the expansion tells us about the age and history of the universe.
Redshift and the Doppler equation
When a source of light moves away from you, the wavelengths of its light are stretched. Lines in the spectrum of a distant galaxy appear at longer wavelengths than the same lines measured in the laboratory. This is redshift.
Δλ/λ ≈ Δf/f ≈ v/c
- Δλ = observed wavelength − emitted (laboratory) wavelength / m
- λ = emitted wavelength / m
- Δf = change in frequency / Hz
- v = speed of the source along the line of sight / m s⁻¹
- c = speed of light = 3.00 × 10⁸ m s⁻¹
Valid only when v is much less than c. The ratio Δλ/λ is often written as the redshift z.
Astronomers know the wavelengths of lines for elements because they measure them in the laboratory. They find the same pattern in the light from a galaxy, then measure how far every line has shifted. The speed follows directly from Δλ/λ = v/c.
Strictly, the redshift of distant galaxies is caused by the expansion of space stretching the light as it travels, not by galaxies moving through space. At A-level you can use the Doppler equation as long as v is small compared with c, but you may be asked to say that the Universe is expanding.
Hubble's law and the Hubble constant
v = H₀d
- v = recession speed of the galaxy / km s⁻¹ (or m s⁻¹)
- H₀ = Hubble constant / km s⁻¹ Mpc⁻¹ (or s⁻¹)
- d = distance to the galaxy / Mpc (or m)
A speed-distance graph is a straight line through the origin with gradient H₀. Convert distance and speed to SI units if you need H₀ in s⁻¹.
The current value of the Hubble constant is about 70 km s⁻¹ Mpc⁻¹, which means that a galaxy 1 Mpc away is receding at about 70 km s⁻¹ and one 100 Mpc away at about 7000 km s⁻¹. Different measurement methods give values that differ by a few units and astronomers do not yet agree why. For exam questions, use the value given in the question.
Distance is the hard part. Speed comes from the spectrum, but distance needs a standard candle, an object of known luminosity such as a certain type of supernova. Its observed brightness (intensity) falls with the square of the distance, so comparing it with its known luminosity gives the distance.
Units of astronomical distance
| Unit | Meaning | Value in metres |
|---|---|---|
| Astronomical unit (AU) | Mean distance from the Earth to the Sun | 1.50 × 10¹¹ m |
| Light-year (ly) | Distance light travels in one year in a vacuum | 9.46 × 10¹⁵ m |
| Parsec (pc) | Distance at which 1 AU subtends an angle of 1 arcsecond | 3.09 × 10¹⁶ m (3.26 ly) |
The light-year follows from distance = speed × time: 3.00 × 10⁸ m s⁻¹ × (365.25 × 24 × 3600 s) = 9.47 × 10¹⁵ m, and the usual value is 9.46 × 10¹⁵ m. The parsec comes from parallax: as the Earth orbits the Sun, a nearby star appears to shift against distant stars. If the angle of parallax is p arcseconds, the distance to the star is 1/p parsecs. Large distances are measured in kiloparsecs (kpc), megaparsecs (Mpc) and so on, where 1 Mpc = 10⁶ pc = 3.09 × 10²² m.
The age of the universe: 1/H₀
If galaxies have always moved at their present speeds, then at time t ago they were all at the same place. A galaxy at distance d moving at speed v = H₀d has taken a time d/v = d/(H₀d) = 1/H₀ to get there. The distance cancels, so every galaxy gives the same answer.
t ≈ 1/H₀
- t = estimated age of the universe / s
- H₀ = Hubble constant in s⁻¹
H₀ must be in s⁻¹, not km s⁻¹ Mpc⁻¹, so you must convert both the kilometres and the megaparsecs into metres. Divide the answer in seconds by 3.15 × 10⁷ to get years.
The conversion step is where marks are lost, so here it is in full. Take H₀ = 70 km s⁻¹ Mpc⁻¹.
- Speed: 70 km s⁻¹ = 70 × 10³ m s⁻¹ = 7.0 × 10⁴ m s⁻¹.
- Distance: 1 Mpc = 10⁶ × 3.09 × 10¹⁶ m = 3.09 × 10²² m.
- H₀ = 7.0 × 10⁴ m s⁻¹ ÷ 3.09 × 10²² m = 2.27 × 10⁻¹⁸ s⁻¹.
- t = 1/H₀ = 1 ÷ 2.27 × 10⁻¹⁸ = 4.41 × 10¹⁷ s.
- In years: 4.41 × 10¹⁷ ÷ 3.15 × 10⁷ = 1.40 × 10¹⁰ years, about 14 billion years.
This is only an estimate. The expansion speed has not been constant (it slowed under gravity, then sped up under dark energy), so 1/H₀ is not exactly the age. The accepted age from detailed models is about 13.8 billion years, which is close to 1/H₀.
The cosmological principle and evidence for the Big Bang
The cosmological principle states that, on a large scale, the universe is homogeneous (the same everywhere, with matter evenly spread) and isotropic (it looks the same in every direction). It means there is no special place in the universe, which is why every observer sees all the other galaxies moving away. A useful picture is the dots on a balloon that is being blown up: every dot moves away from every other dot and none is the centre.
The Big Bang theory says that the universe began about 14 billion years ago, from an extremely hot, dense state, and has been expanding and cooling since. The evidence is:
- Hubble's law and redshift. Galaxies are moving apart, so in the past they were closer together.
- The cosmic microwave background (CMB). The universe is filled with microwave radiation that arrives from every direction, with a spectrum that matches a black body at 2.7 K. It is leftover radiation from when the universe cooled enough, a few hundred thousand years after the Big Bang, for atoms to form and light to travel freely. The expansion of the universe since then has stretched its wavelengths from visible and infrared to microwaves.
- The abundance of light elements. Models of the first minutes predict roughly three quarters hydrogen and one quarter helium by mass, which matches observations.
Dark matter and dark energy
Two observations show that visible matter is not the whole story.
- Dark matter. Stars at the edges of galaxies orbit much faster than the visible mass can explain: the gravitational force from visible matter alone would not hold them in orbit. Astronomers suggest that galaxies contain additional matter that does not emit or absorb light but does exert gravity. This is dark matter. We do not know what it is.
- Dark energy. Observations of distant supernovae show that the expansion of the universe is accelerating. Gravity should slow expansion, so something is driving it faster. This unknown cause is called dark energy.
Current models suggest that ordinary matter makes up only about 5% of the total mass-energy of the universe, dark matter about 27% and dark energy about 68%. These are estimates and they change as measurements improve. The key point is that the nature of 95% of the universe is not yet understood, and you should say clearly that it is an unsolved problem.
Remember the link to gravitational fields and Kepler's third law and orbits: galaxy rotation uses the same ideas, with centripetal force provided by gravity, as planets orbiting a star.
Worked examples
Speed from redshift
The hydrogen line that has a wavelength of 656 nm in the laboratory is observed at 663 nm in the light of a distant galaxy. Calculate the speed of recession of the galaxy.
- Δλ = 663 − 656 = 7 nm
- Δλ/λ = 7/656 = 1.07 × 10⁻²
- v = c × Δλ/λ = 3.00 × 10⁸ × 1.07 × 10⁻² = 3.2 × 10⁶ m s⁻¹
Answer 3.2 × 10⁶ m s⁻¹ (3.2 × 10³ km s⁻¹)
Distance from Hubble's law
Use the speed from the previous example, v = 3.2 × 10³ km s⁻¹, and H₀ = 70 km s⁻¹ Mpc⁻¹ to calculate the distance to the galaxy (a) in Mpc, (b) in metres, (c) in light-years.
- d = v/H₀ = 3200 / 70 = 45.7 Mpc
- In metres: 45.7 × 3.09 × 10²² = 1.41 × 10²⁴ m
- In light-years: 1 pc = 3.26 ly, so 45.7 × 10⁶ × 3.26 = 1.49 × 10⁸ ly
Answer (a) 46 Mpc (b) 1.4 × 10²⁴ m (c) 1.5 × 10⁸ light-years
Age of the universe from H₀
A student finds H₀ = 70 km s⁻¹ Mpc⁻¹. Estimate the age of the universe in years. (1 Mpc = 3.09 × 10²² m, 1 year = 3.15 × 10⁷ s)
- H₀ = 7.0 × 10⁴ m s⁻¹ / 3.09 × 10²² m = 2.27 × 10⁻¹⁸ s⁻¹
- t = 1/H₀ = 1 / 2.27 × 10⁻¹⁸ = 4.41 × 10¹⁷ s
- t in years = 4.41 × 10¹⁷ / 3.15 × 10⁷ = 1.40 × 10¹⁰ years
Answer 1.4 × 10¹⁰ years (about 14 billion years)
Parallax and units
A star has a parallax angle of 0.10 arcseconds. Calculate its distance in (a) parsecs, (b) metres, (c) light-years, and (d) state how long light from the star takes to reach the Earth.
- (a) d = 1/p = 1/0.10 = 10 pc
- (b) d = 10 × 3.09 × 10¹⁶ = 3.09 × 10¹⁷ m
- (c) d = 3.09 × 10¹⁷ / 9.46 × 10¹⁵ = 32.7 ly
- (d) Light travels 1 ly in 1 year, so it takes about 33 years
Answer (a) 10 pc (b) 3.1 × 10¹⁷ m (c) 33 ly (d) about 33 years
Common mistakes
How OCR A examines this topic
Expect calculations, graph questions and explanations:
- Using Δλ/λ ≈ v/c to find a speed, then v = H₀d to find a distance, often as a two-step question.
- Converting between AU, light-years, parsecs and metres, and converting H₀ to s⁻¹ to estimate the age of the universe.
- Finding H₀ from the gradient of a speed-distance graph, with a comment on uncertainty.
- Explaining the cosmological principle, the evidence for the Big Bang (especially the CMB), and the idea of dark matter and dark energy.
Synoptic links include the Hertzsprung-Russell diagram (Wien's law gives the CMB temperature and stellar spectra give the lines used for redshift), gravitational fields (galaxy rotation and dark matter), and practical skills such as uncertainties in a gradient. For the full set of topics see the Module 5 hub.
Practice questions
The calcium K line has a laboratory wavelength of 393 nm. In the spectrum of a galaxy it is measured at 411 nm. Calculate the speed of the galaxy.3 marks
Show the answer
Answer: 1.4 × 10⁷ m s⁻¹
Δλ = 411 − 393 = 18 nm. v = c × Δλ/λ = 3.00 × 10⁸ × 18/393 = 1.37 × 10⁷ m s⁻¹.
A galaxy is 250 Mpc from the Earth. Calculate this distance in metres and in light-years. (1 Mpc = 3.09 × 10²² m, 1 pc = 3.26 ly)3 marks
Show the answer
Answer: 7.7 × 10²⁴ m and 8.2 × 10⁸ ly
d = 250 × 3.09 × 10²² = 7.73 × 10²⁴ m. d = 250 × 10⁶ × 3.26 = 8.15 × 10⁸ ly.
Five galaxies have distances of 10, 20, 30, 40 and 50 Mpc and recession speeds of 720, 1380, 2250, 2760 and 3600 km s⁻¹. Plot speed against distance, draw a best-fit line through the origin and find H₀.4 marks
Show the answer
Answer: About 71 km s⁻¹ Mpc⁻¹ (accept 68 to 74)
The gradient of the best-fit line through the origin is H₀. A least-squares fit through the origin gives 71.4 km s⁻¹ Mpc⁻¹.
A value of H₀ is 2.2 × 10⁻¹⁸ s⁻¹. Calculate the age of the universe in years. (1 year = 3.15 × 10⁷ s)2 marks
Show the answer
Answer: 1.4 × 10¹⁰ years
t = 1/H₀ = 1 / 2.2 × 10⁻¹⁸ = 4.55 × 10¹⁷ s. In years: 4.55 × 10¹⁷ / 3.15 × 10⁷ = 1.44 × 10¹⁰ years.
Explain how the cosmic microwave background provides evidence for the Big Bang.4 marks
Show the answer
Answer: The CMB is radiation that arrives from all directions with a black-body spectrum corresponding to about 2.7 K. It was emitted when the universe was hot and dense, a few hundred thousand years after the Big Bang. As the universe expanded the wavelength was stretched (redshifted) into the microwave region. A universe that began hot and has expanded and cooled predicts this radiation.
Four points: radiation from all directions, black-body spectrum at 2.7 K, emitted when hot, stretched by expansion.
Next steps
To get more from this topic, read about the stars whose spectra give us redshift in the Hertzsprung-Russell diagram guide, and review the physics of orbits in gravitational fields. You can also practise this topic with step-by-step marking, which is free to start.
Frequently asked questions
What is Hubble's law?
Hubble's law says that the speed at which a galaxy is moving away from us is directly proportional to its distance, v = H₀d. The constant H₀ is about 70 km s⁻¹ Mpc⁻¹. It is evidence that the universe is expanding.
How do you calculate the age of the universe from the Hubble constant?
Convert H₀ into s⁻¹ by changing km to m and Mpc to m, then take the reciprocal. For 70 km s⁻¹ Mpc⁻¹, H₀ = 2.27 × 10⁻¹⁸ s⁻¹, so t = 1/H₀ = 4.4 × 10¹⁷ s, which is about 14 billion years.
What is the difference between a light-year, an astronomical unit and a parsec?
An astronomical unit is the mean Earth-Sun distance, 1.50 × 10¹¹ m. A light-year is the distance light travels in a year, 9.46 × 10¹⁵ m. A parsec is 3.09 × 10¹⁶ m, or 3.26 light-years, and is the distance at which 1 AU subtends an angle of one arcsecond.
What is the cosmological principle?
It states that on a large scale the universe is homogeneous and isotropic. Matter is spread evenly and looks the same in all directions, so there is no special centre and every observer sees galaxies moving away from them.
What are dark matter and dark energy?
Dark matter is unseen matter whose gravity explains why galaxies rotate faster than their visible mass allows. Dark energy is whatever is causing the expansion of the universe to accelerate. Neither has been identified, and together they make up most of the universe in current models.
Why is light from distant galaxies redshifted?
The universe is expanding, so the wavelength of the light is stretched as it travels. At A-level, you can treat it as a Doppler shift caused by the galaxy moving away, using Δλ/λ ≈ v/c when the speed is much less than the speed of light.
Related guides
Written for the OCR A Level Physics A (H556) specification. Questions are original and written in exam style; they are not taken from OCR papers. Spotted a mistake? Tell us.