The shape of the lesson
By the end of the lesson, learners can
- Slide the peak toward the blue: Drag the temperature and watch the black-body curve: its peak slides to shorter wavelengths as the star heats up, exactly as Wien's law demands, while the whole curve climbs because the power per unit area grows as T⁴.
- Two laws, one radius: The marks come from the two laws and the way they combine.
- Standard candles: luminosity L (W); flux F = L / (4πd²); known L gives d = √(L / 4πF).
- Stellar radii: Wien λₘₐₓT = 2.90×10⁻³ m K; Stefan-Boltzmann L = 4πσr²T⁴.
- Cosmology: Δλ/λ ≈ v/c; Hubble v = H₀d; age ≈ 1/H₀.
One picture every idea returns to
A star radiates almost as a black body. Wien's displacement law says the peak wavelength is inversely proportional to temperature, λₘₐₓT = 2.90 × 10⁻³ m K, so colour gives T. The Stefan-Boltzmann law gives the total power, L = 4πσr²T⁴. Know L (from a standard candle) and T (from the colour), and you can solve for the radius r.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Read a star by its colour.. A star glows like a hot body, and the colour of that glow betrays its temperature. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 17 min | Teach and check: Slide the peak toward the blue | Drag the temperature and watch the black-body curve: its peak slides to shorter wavelengths as the star heats up, exactly as Wien's law demands, while the whole curve climbs because the power per unit area grows as T⁴. Project tab 2 of the lab (Stellar radii) and let a student drive it while the class predicts. | Whole class, then pairs |
| 17 to 30 min | Teach and check: Two laws, one radius | The marks come from the two laws and the way they combine. | Anticipation / Reaction Guide |
| 30 to 43 min | Check and plenary | The quick-check questions from the student lesson, answered on mini whiteboards one at a time; re-teach the one most get wrong, then a random call on the last. | Individual, then whole class |
| 43 to 45 min | Exit and homework | Set the worksheet below (4 questions, 30 marks) from the topic practice set. Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Two laws, one radius, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Anticipation / Reaction Guide
Before the lesson, students mark five statements true or false. After the teaching they mark them again and justify any change of mind in one sentence each. The statements come from the examiner traps below.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| In Wien's law λₘₐₓ is inversely proportional to T: a hotter star has a shorter peak wavelength, not a longer one. | Always use the absolute temperature in kelvin. In Stefan-Boltzmann, T is raised to the fourth power, so doubling T multiplies L by 16; and two stars at the same temperature but different luminosity must have different radii. |
Stellar radii: Wien and Stefan-Boltzmann: topic worksheet
4 questions, 30 marks. Written in the style of the papers; no past paper question is reproduced. These questions cover the whole topic, so sibling lessons share them.
The continuous spectrum of a star peaks at a wavelength of 480 nm.
(a) State Wien's displacement law. [1]
(b) Calculate the surface temperature of the star. [2]
(c) A second star appears red, with a peak at 700 nm. Calculate its surface temperature. [2]
(d) State which star is hotter, and how the colour tells you. [2]
A star has a luminosity of 4.0 × 10²⁸ W and a surface temperature of 1.0 × 10⁴ K.
(a) State the Stefan-Boltzmann law. [1]
(b) Calculate the power radiated per unit area of the star's surface. [2]
(c) Calculate the radius of the star. [3]
(d) Express this radius as a multiple of the Sun's radius. [2]
Two stars, X and Y, have the same radius. Star X has a surface temperature of 12000 K and star Y of 3000 K.
(a) Calculate the ratio of their peak wavelengths, λ₧/λₖ. [2]
(b) Calculate the ratio of their luminosities, L₧/Lₖ. [2]
(c) Comment on which star is more luminous and the role of temperature. [2]
Astronomers build a "distance ladder" using several methods together.
(a) State how each of standard candles, Wien's law and redshift contributes to studying a distant galaxy. [3]
(b) A Cepheid variable places a galaxy at 15 Mpc. Using H₀ = 70 km s⁻¹ Mpc⁻¹, calculate the expected recession speed. [2]
(c) A type Ia supernova in the same galaxy has a known luminosity and a measured flux. Explain how this independently confirms the distance. [2]
(d) Suggest why redshift, rather than standard candles, is used for the most distant galaxies. [2]
Answer key and marking notes
- (a)The peak wavelength is inversely proportional to absolute temperature: λₘₐₓT = 2.90 × 10⁻³ m K ✓
- (b)T = 2.90 × 10⁻³ / 480 × 10⁻⁹ ✓ = 6.0 × 10³ K (6040 K) ✓
- (c)T = 2.90 × 10⁻³ / 700 × 10⁻⁹ ✓ = 4.1 × 10³ K (4140 K) ✓
- (d)The 480 nm star is hotter ✓; a shorter (bluer) peak wavelength means a higher temperature ✓
- (a)The luminosity of a star is L = 4πσr²T⁴, where r is its radius and T its surface temperature ✓
- (b)σT⁴ = 5.67 × 10⁻⁸ × (1.0 × 10⁴)⁴ ✓ = 5.7 × 10⁸ W m⁻² ✓
- (c)r = √(L / 4πσT⁴) = √(4.0 × 10²⁸ / (4π × 5.7 × 10⁸)) ✓✓ = 2.4 × 10⁹ m ✓
- (d)r / R☉ = 2.4 × 10⁹ / 6.96 × 10⁸ ✓ ≈ 3.4 R☉ ✓
- (a)λ ∝ 1/T, so λ₧/λₖ = Tₖ/T₧ = 3000 / 12000 ✓ = 0.25 (X peaks at one quarter the wavelength) ✓
- (b)Same r, so L ∝ T⁴: L₧/Lₖ = (12000 / 3000)⁴ = 4⁴ ✓ = 256 ✓
- (c)X is far more luminous ✓; because L depends on T⁴, a fourfold temperature rise multiplies the output by 256 ✓
- (a)Standard candles give the distance from known L and measured F ✓; Wien's law gives the surface temperature from the peak wavelength ✓; redshift gives the recession speed (and, via Hubble, distance) ✓
- (b)v = H₀d = 70 × 15 ✓ = 1.05 × 10³ km s⁻¹ (1050 km s⁻¹) ✓
- (c)With L known and F measured, d = √(L / 4πF) ✓; this is a second, independent estimate that should agree with the Cepheid distance ✓
- (d)The most distant galaxies are too faint to resolve individual standard candles ✓; their redshift is still measurable, so Hubble's law gives the distance ✓
The lesson's own checks as a quiz
The quick-check and exam-style questions from the student lesson, printable with the key.
According to Wien's displacement law, a hotter star has a peak wavelength that is:
- Ashorter, so it looks bluer
- Blonger, so it looks redder
- Cunchanged
- Dalways in the ultraviolet
Two stars have the same radius, but star A is twice as hot as star B. The luminosity of A compared with B is:
- A16 times larger
- B2 times larger
- C4 times larger
- D8 times larger
To estimate a star's radius, you combine the Stefan-Boltzmann law with measurements of:
- Aits luminosity and its surface temperature
- Bits mass and its spin
- Cits redshift and its distance
- Dits age and its colour only
A red star and a blue star have the same luminosity. Which is the larger?
- Athe red star, because it is cooler so needs more area to radiate the same power
- Bthe blue star, because blue light has more energy
- Cthey must be the same size
- Dit cannot be decided at all
Quiz key
λₘₐₓT = constant, so λₘₐₓ is inversely proportional to T. Higher T means a shorter peak wavelength and a bluer colour.
L = 4πσr²T⁴. With r fixed, L is proportional to T⁴, so doubling T multiplies L by 2⁴ = 16.
r = √(L / 4πσT⁴). T comes from the colour (Wien) and L from a standard candle; together they give r.
The red star is cooler (lower T), so its power per unit area σT⁴ is much smaller. To reach the same total L it must have a larger surface area, hence a larger radius.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.