Estimate stellar radii with the Stefan–Boltzmann law
Combine luminosity and surface temperature to estimate a star’s radius.
A Level · Cambridge International Physics 9702, 2025–2027 syllabus. Topic 25 was checked against Cambridge’s 2028–2030 syllabus; a full reconciliation of every remaining syllabus-year change is still outstanding. Online study supports the theory but cannot replace supervised practical work, full timed assessment or human subject-teacher review.
Before you begin
- Use powers of ten, inverse-square relationships, proportional reasoning and SI units; distinguish wavelength, frequency, speed, flux and luminosity.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which temperature scale belongs in L = 4πσr²T⁴?
What you will learn
- Use L = 4πσr²T⁴
- Estimate radius from luminosity and temperature
- Combine Wien’s law with the Stefan–Boltzmann law
Luminosity depends strongly on radius and temperature
The Stefan–Boltzmann law for a star is L = 4πσr²T⁴, where r is its radius, T is its surface temperature and σ is the Stefan–Boltzmann constant. Keep the temperature in kelvin and use SI units for L and r.
Rearrange to r = √(L/(4πσT⁴)). At fixed temperature, a more luminous star has a larger radius. At fixed luminosity, a hotter star needs a smaller radius.
Wien’s law supplies the temperature
If the peak wavelength is measured instead of temperature, estimate T using λmax T = b first. Then substitute that temperature into the Stefan–Boltzmann relation to find r.
Because temperature is raised to the fourth power, a small temperature change can have a large effect on the inferred luminosity for a given radius. Track powers carefully before taking the final square root.
Pause and explain
What is the correct rearrangement for radius?
Worked example
A star has L = 4.45 × 10²⁶ W and T = 5.00 × 10³ K. Use σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ to estimate r.
Show the worked solution
- Rearrange: r = √(L/(4πσT⁴)).
- T⁴ = (5.00 × 10³)⁴ = 6.25 × 10¹⁴ K⁴.
- r = √(4.45 × 10²⁶/(4π × 5.67 × 10⁻⁸ × 6.25 × 10¹⁴)) ≈ 1.00 × 10⁹ m.
Answer About 1.00 × 10⁹ m.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A star has L = 1.0 × 10²⁷ W and T = 1.0 × 10⁴ K. Estimate its radius using σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.
Give me a hint
Substitute the fourth power of the kelvin temperature before taking the square root.
Compare my reasoning
- T⁴ = 1.0 × 10¹⁶ K⁴.
- r² = 1.0 × 10²⁷/(4π × 5.67 × 10⁻⁸ × 1.0 × 10¹⁶) ≈ 1.40 × 10¹⁷ m².
- r ≈ 3.74 × 10⁸ m.
Approximately 3.7 × 10⁸ m.
Look for these in your work
- Uses T in kelvin
- Includes 4πσT⁴ in the denominator
- Takes the square root
Two stars have the same radius. Star B is twice as hot as star A. Find LB/LA.
Give me a hint
At fixed r, luminosity is proportional to T⁴.
Compare my reasoning
- LB/LA = (TB/TA)⁴.
- TB/TA = 2.
- LB/LA = 2⁴ = 16.
Star B is 16 times as luminous.
Look for these in your work
- Holds radius constant
- Uses the fourth-power dependence
- Calculates the ratio
A star’s peak wavelength is 580 nm and its luminosity is 4.45 × 10²⁶ W. Estimate its radius using b = 2.90 × 10⁻³ m K and σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.
Give me a hint
Find T from Wien’s law first, then use the stellar luminosity relation.
Compare my reasoning
- λmax = 580 nm = 5.80 × 10⁻⁷ m, so T = b/λmax = 5.00 × 10³ K.
- r = √(L/(4πσT⁴)).
- Substituting gives r ≈ 1.00 × 10⁹ m.
About 1.00 × 10⁹ m.
Look for these in your work
- Converts wavelength to metres
- Uses Wien’s law to find T
- Substitutes the temperature and finds r
Common mistakes
- Using T in Celsius rather than kelvin
- Forgetting the square root after isolating r²
- Using λmax directly in the Stefan–Boltzmann law without first finding T
What can you explain now?
Which temperature scale belongs in L = 4πσr²T⁴?
Compare with the explanation
Kelvin
The absolute temperature is used in the fourth-power law.
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Make it stick
Why must Wien’s law be used before the radius calculation when only a peak wavelength is supplied?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.