A2 · Practice questions · Stellar radii

Colour gives the temperature, temperature gives the size.

Six original Cambridge-style questions on Wien's displacement law, the Stefan-Boltzmann law L = 4πσr²T⁴, and finding the radius of a star from its spectrum and luminosity.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

Keep these straight

Two laws, one radius.

Wien's displacement law: λ_maxT = 2.9 × 10⁻³ m K, so the peak wavelength of the spectrum gives the surface temperature.
Stefan-Boltzmann law: L = 4πσr²T⁴ with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, so L and T give the radius.
Order of work: spectrum to T, flux and distance to L, then r = √(L / 4πσT⁴).

Calculation

[2 marks]

The spectrum of a star peaks at a wavelength of 480 nm. Calculate the surface temperature of the star.

T = 2.9 × 10⁻³ / λ_max = 2.9 × 10⁻³ / 4.8 × 10⁻⁷. ✓
T = 6.0 × 10³ K. ✓

Analysis

[2 marks]

Star A appears red and star B appears blue-white. State which star has the higher surface temperature and explain how the colour shows this.

Star B, the blue-white one, is hotter. ✓
By Wien's law the peak wavelength is inversely proportional to the surface temperature, so a hotter star radiates most strongly at shorter (bluer) wavelengths while the cooler star peaks at longer (redder) wavelengths. ✓

Calculation

[3 marks]

A star has a surface temperature of 4200 K and a radius of 5.6 × 10⁸ m. Calculate its luminosity.

L = 4πσr²T⁴ = 4π × 5.67 × 10⁻⁸ × (5.6 × 10⁸)² × 4200⁴. ✓
r² = 3.14 × 10¹⁷ m² and T⁴ = 3.11 × 10¹⁴ K⁴. ✓
L = 6.9 × 10²⁶ W. ✓

Calculation

[3 marks]

A star has a luminosity of 1.5 × 10²⁸ W and its spectrum peaks at 290 nm. Calculate the surface temperature of the star and hence its radius.

T = 2.9 × 10⁻³ / 2.9 × 10⁻⁷ = 1.0 × 10⁴ K. ✓
r² = L / 4πσT⁴ = 1.5 × 10²⁸ / (4π × 5.67 × 10⁻⁸ × 1.0 × 10¹⁶) = 2.1 × 10¹⁸ m². ✓
r = 1.5 × 10⁹ m, about twice the Sun's radius. ✓

Analysis

[3 marks]

Two stars have the same luminosity. One has a surface temperature twice that of the other. Determine the ratio of their radii, and state which star is the larger.

With L fixed, r²T⁴ is constant, so r ∝ 1 / T². ✓
The hotter star has T twice as large, so its radius is 1 / 2² = 1/4 of the cooler star's radius; ratio cool : hot = 4 : 1. ✓
The cooler star is the larger: it must have a far greater surface area to radiate the same power at a lower temperature. ✓

Calculation

[4 marks]

A star is 8.0 × 10¹⁷ m from the Earth. Its radiant flux intensity at the Earth is 3.2 × 10⁻⁹ W m⁻² and its spectrum peaks at 1.0 × 10⁻⁶ m.

(a) Calculate the luminosity of the star. (b) Calculate its surface temperature. (c) Calculate its radius.

(a) L = 4πd²F = 4π × (8.0 × 10¹⁷)² × 3.2 × 10⁻⁹ = 2.6 × 10²⁷ W. ✓
(b) T = 2.9 × 10⁻³ / 1.0 × 10⁻⁶ = 2.9 × 10³ K. ✓
(c) r² = L / 4πσT⁴ = 2.57 × 10²⁷ / (4π × 5.67 × 10⁻⁸ × 7.07 × 10¹³) = 5.1 × 10²⁰ m². ✓
r = 2.3 × 10¹⁰ m, a red giant some thirty times the size of the Sun. ✓

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