The shape of the lesson
By the end of the lesson, learners can
- Spread thin over a sphere: Set the distance and the luminosity, and watch the received flux fall away.
- Luminosity, flux, standard candle: Three definitions and one equation carry the marks.
- 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
The luminosity L is the total power a star radiates. The radiant flux intensity F is the power we receive per unit area, and because the light spreads over a sphere of area 4πd², F = L / (4πd²). A standard candle is an object of known luminosity, so measuring F gives its distance d. Flux follows an inverse square law: move twice as far away and only a quarter of the flux arrives.
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, How bright, and how far.. A star pours out a fixed power, its luminosity. 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: Spread thin over a sphere | Set the distance and the luminosity, and watch the received flux fall away. Project tab 1 of the lab (Standard candles) and let a student drive it while the class predicts. | Whole class, then pairs |
| 17 to 30 min | Teach and check: Luminosity, flux, standard candle | Three definitions and one equation carry the marks. | Connect-Extend-Challenge |
| 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, 29 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 Luminosity, flux, standard candle, 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.
Connect-Extend-Challenge
After the new idea, each student writes three lines: how it connects to what they knew, how it extends their thinking, and what still challenges them. Pairs compare, then the challenges become the plenary.
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 |
|---|---|
| Keep luminosity (emitted, intrinsic, in W) apart from radiant flux intensity (received, in W m⁻²). | A nearby dim star can outshine a distant bright one. The method assumes the light is not absorbed on the way and spreads uniformly in all directions; and remember the area is 4πd², not πd² or 4πd. |
Standard candles and the inverse square law: topic worksheet
4 questions, 29 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 Sun has a luminosity of 3.83 × 10²⁶ W. The Earth orbits at a distance of 1.50 × 10¹¹ m.
(a) State what is meant by the luminosity of a star. [1]
(b) State what is meant by radiant flux intensity. [1]
(c) Calculate the radiant flux intensity received at the Earth. [3]
(d) State two assumptions made in using F = L / (4πd²). [2]
A type Ia supernova has a known luminosity of 1.0 × 10³⁶ W. In a distant galaxy, one is observed with a radiant flux intensity of 2.0 × 10⁻¹⁵ W m⁻².
(a) State what is meant by a standard candle. [1]
(b) Calculate the distance to the galaxy, in metres. [3]
(c) Convert this distance to megaparsecs (Mpc). [2]
(d) Suggest why standard candles are needed to find the distances to far galaxies. [1]
Two stars, A and B, have the same luminosity. Star A is at distance d and star B is at distance 3d.
(a) Calculate the ratio Fₐ / F₮ of the fluxes received from them. [2]
(b) State which star appears brighter, and why. [2]
(c) A third pair have equal luminosity, but one appears 100 times brighter than the other. Find the ratio of their distances. [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 total power (energy per second) radiated by the star ✓
- (b)The radiant power received per unit area (W m⁻²) ✓
- (c)F = L / (4πd²) = 3.83 × 10²⁶ / (4π × (1.50 × 10¹¹)²) ✓✓ = 1.36 × 10³ W m⁻² ✓
- (d)The star radiates uniformly in all directions ✓; no radiation is absorbed between the star and the observer ✓
- (a)An astronomical object of known luminosity (fixed by its type) ✓
- (b)d = √(L / 4πF) = √(1.0 × 10³⁶ / (4π × 2.0 × 10⁻¹⁵)) ✓✓ = 6.3 × 10²⁴ m ✓
- (c)6.3 × 10²⁴ / 3.09 × 10²² ✓ ≈ 2.0 × 10² Mpc (about 200 Mpc) ✓
- (d)They are too far for parallax; a known luminosity lets the measured flux give the distance ✓
- (a)F ∝ 1/d², so Fₐ/F₮ = (3d / d)² ✓ = 9 ✓
- (b)Star A ✓; it is closer, so its light is spread over a smaller sphere and the flux is greater ✓
- (c)100 = (d₮ / dₐ)² ✓, so d₮ / dₐ = 10 (the fainter one is 10 times further) ✓
- (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.
The luminosity of a star is best described as:
- Athe total power it radiates, in watts
- Bthe power we receive per square metre
- Cits surface temperature
- Dhow bright it looks from Earth
A star is moved three times further away. The radiant flux intensity we receive becomes:
- Aone ninth
- Bone third
- Cthree times larger
- Dunchanged
A standard candle is useful for finding distance because:
- Aits luminosity is known, so the measured flux gives the distance
- Bit is the closest star to Earth
- Cit has no redshift
- Dits flux is the same at every distance
The SI unit of radiant flux intensity F is:
- AW m⁻²
- BW
- CJ
- Dm
Quiz key
Luminosity L is the total power output of the star (W). It is intrinsic to the star; how bright it looks from Earth is the flux, which also depends on distance.
F = L / (4πd²), so F is proportional to 1/d². Tripling d divides the flux by 3² = 9.
If L is known (from the object's type, e.g. a Type Ia supernova) and F is measured, then d = √(L / 4πF).
Flux intensity is power per unit area, so W m⁻². Luminosity (total power) is in W.
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.