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A Level Physics 9702 · A Level · Topic 25 · ~4 min read

Astronomy and cosmology

Luminosity and radiant flux intensity, standard candles, Wien's displacement law and the Stefan–Boltzmann law to find stellar temperatures and radii, redshift, Hubble's law and the evidence for an expanding universe.

LuminosityRadiant flux intensity F = L/4πd²Standard candlesWien's lawStefan–Boltzmann lawStellar radiusRedshift Δλ/λ = v/cHubble's law
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Paper tag = where it is examinedDATA & FORMULAE given in the examC1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Luminosity L is the total power a star emits. At distance d, the radiant flux intensity is F = L/(4πd²): an inverse-square law.
  • Standard candles (Cepheid variables, type 1a supernovae) have known luminosity, so measuring F gives d.
  • Wien's law λmaxT = 2.9 × 10⁻³ m K gives T; Stefan–Boltzmann L = 4πσr²T⁴ then gives the radius r.
  • Redshift Δλ/λ ≈ v/c. Distant galaxies are receding, and Hubble's law v = H₀d shows the universe is expanding: evidence for the Big Bang.

By the end you can

  • Define luminosity and radiant flux intensity, and use F = L/(4πd²).
  • Understand standard candles and use them to find distances.
  • Use Wien's law and the Stefan–Boltzmann law to find surface temperatures and radii of stars.
  • Use Δλ/λ ≈ Δf/f ≈ v/c and Hubble's law, and explain the evidence for an expanding universe.

1Luminosity and distance

The inverse-square law

Luminosity — the total radiant power emitted by a star.

Radiant flux intensity — the radiant power passing normally through unit area.

Standard candle — an astronomical object of known luminosity.

F = L/(4πd²)
d from a standard candle: d = √(L/(4πF))
star, Ld2d3db = L/(4πd²)double d:b falls to ¼triple d: 1/9
The same power spreads over a sphere of area 4πd².
Worked example

The Sun's luminosity is 3.83 × 10²⁶ W and it is 1.50 × 10¹¹ m away. Find the radiant flux intensity at the Earth.

  1. F = 3.83 × 10²⁶/(4π × (1.50 × 10¹¹)²) C1
  2. F = 1.35 × 10³ W m⁻² A1
Worked example

A Cepheid of luminosity 1.0 × 10³⁰ W gives F = 2.0 × 10⁻¹² W m⁻² at the Earth. Find its distance.

  1. d = √(L/(4πF)) = √(1.0 × 10³⁰/(4π × 2.0 × 10⁻¹²)) C1
  2. d = 2.0 × 10²⁰ m A1

2Stellar temperatures and radii

Wien and Stefan–Boltzmann

λ / μmintensity0.511.522.5visible6000 K (blue), 5000 K, 4000 Khotter: peak moves to shorter λand total power rises as T⁴λmaxT = 2.9 × 10⁻³ m K
Hotter bodies peak at shorter wavelengths and emit much more power.
λmax T = 2.9 × 10⁻³ m K ; L = 4πσr²T⁴ DATA & FORMULAE
σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴
Worked example

A star's spectrum peaks at 350 nm and its luminosity is 4.0 × 10²⁸ W. Find its surface temperature and radius.

  1. T = 2.9 × 10⁻³/350 × 10⁻⁹ = 8300 K C1 A1
  2. r = √(L/(4πσT⁴)) = √(4.0 × 10²⁸/(4π × 5.67 × 10⁻⁸ × 8290⁴)) C1
  3. r = 3.5 × 10⁹ m A1
Common mistakeUsing the Celsius temperature or forgetting the fourth power. A small error in T is magnified four times in L.

3The expanding universe

Redshift and Hubble's law

Δλ/λ ≈ Δf/f ≈ v/c ; v = H₀d
for v much less than c
labgalaxyLines shifted to longer wavelength: the galaxy is moving away (redshift).
Absorption lines from a receding galaxy are shifted to longer wavelengths.
Worked example

A hydrogen line emitted at 656.3 nm is observed at 669.4 nm from a galaxy. Find its recession speed and its distance (H₀ = 2.3 × 10⁻¹⁸ s⁻¹).

  1. v = c Δλ/λ = 3.00 × 10⁸ × 13.1/656.3 = 6.0 × 10⁶ m s⁻¹ C1 A1
  2. d = v/H₀ = 2.6 × 10²⁴ m A1
  • Evidence for the Big Bang: all distant galaxies are redshifted, and v ∝ d, so the universe is expanding; running it backwards suggests it began from a very hot, dense state.
  • Age estimate: 1/H₀ ≈ 4.3 × 10¹⁷ s ≈ 14 billion years (ignoring changes in the expansion rate).
Common mistake"Galaxies move away from us because we are at the centre". Every observer sees the same recession: space itself is expanding.
Linking questionPhotons and spectra are in Quantum physics; the Doppler effect for sound is in Waves; and gravity is in Gravitational fields.

Test yourself — cover the answers first

Quick check

  1. Define luminosity.
  2. What is a standard candle?
  3. A star's temperature doubles at the same radius. By what factor does L change?
  4. Does redshift mean a source is approaching or receding?
  5. State Hubble's law.

Exam-style

  1. Two stars have the same luminosity; star A appears 9 times brighter than B. Compare their distances.
  2. Find λ_max for a star at 3500 K.
  3. Estimate the age of the universe in years if H₀ = 2.3 × 10⁻¹⁸ s⁻¹.
Show answers
  1. the total power radiated by a star
  2. an object of known luminosity
  3. 16
  4. receding
  5. recession speed is proportional to distance, v = H₀d
  6. F ∝ 1/d², so B is 3 times as far as A
  7. 8.3 × 10⁻⁷ m (red/infrared)
  8. 1/H₀ = 4.3 × 10¹⁷ s ≈ 1.4 × 10¹⁰ years
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by Cambridge University Press & Assessment. Syllabus codes 9709 and 9702 refer to the Cambridge International AS & A Level syllabuses.