Study Guide

Stellar radii (Wien displacement law, Stefan-Boltzmann law)

Physics· 12 min read

1. Wien's Displacement Law: Relating Temperature to Peak Wavelength★★☆☆☆⏱ 3 min

All stars behave approximately as ideal black bodies, so their emitted intensity spectrum follows a characteristic curve that peaks at a single wavelength. As the star gets hotter, this peak shifts to shorter, bluer wavelengths, a relationship formalised by Wien's Displacement Law.

📘 Definition

Wien's Displacement Law

λmaxT=b\lambda_{max} T = b

The product of the peak emission wavelength of a black body and its absolute surface temperature is a constant, independent of the size or composition of the body.

Example:

A 5800 K Sun-like star has a peak wavelength of ~500 nm, corresponding to visible green light.

📐 Worked Example

Calculate the surface temperature of a star whose emission spectrum peaks at 350 nm (ultraviolet). Use b = 2.9 × 10⁻³ m K.

  1. 1

    First convert peak wavelength to SI units:

  2. 2
    λmax=350×10−9 m\lambda_{max} = 350 \times 10^{-9} \text{ m}
  3. 3

    Rearrange Wien's law to solve for temperature:

  4. 4
    T=bλmaxT = \frac{b}{\lambda_{max}}
  5. 5

    Substitute values:

  6. 6
    T=2.9×10−3350×10−9≈8300 KT = \frac{2.9 \times 10^{-3}}{350 \times 10^{-9}} \approx 8300 \text{ K}
✓ Quick check

Test your understanding of Wien's law below:

  1. If a star's surface temperature doubles, what happens to its peak emission wavelength?

    • It doubles

    • It halves

    • It increases by 4x

    • It does not change

    Reveal answer
    It halves —

    Wien's law is an inverse proportionality, so doubling T reduces λ_max by a factor of 2.

2. Stefan-Boltzmann Law: Luminosity from Temperature and Radius★★★☆☆⏱ 4 min

While Wien's law gives surface temperature, the Stefan-Boltzmann Law describes the total power radiated per unit surface area of a black body. This quantity, multiplied by the total surface area of the star, gives its total luminosity.

📘 Definition

Stefan-Boltzmann Law

L=4πR2σT4L = 4 \pi R^2 \sigma T^4

The total luminosity of a spherical black body is proportional to the square of its radius and the fourth power of its absolute surface temperature.

Example:

A star with twice the Sun's radius and same temperature has 4 times the Sun's luminosity.

📐 Worked Example

Calculate the luminosity of a star with radius 7.0 × 10⁸ m (same as the Sun) and surface temperature 5800 K. Use σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.

  1. 1

    Write out the full Stefan-Boltzmann equation:

  2. 2
    L=4πR2σT4L = 4 \pi R^2 \sigma T^4
  3. 3

    Substitute all known values:

  4. 4
    L=4×π×(7.0×108)2×5.67×10−8×(5800)4L = 4 \times \pi \times (7.0 \times 10^8)^2 \times 5.67 \times 10^{-8} \times (5800)^4
  5. 5

    Calculate step by step to get final result:

  6. 6
    L≈3.9×1026 WL \approx 3.9 \times 10^{26} \text{ W}

Exam tip:

Note that luminosity scales with T⁴, so a small 10% increase in surface temperature increases total power output by ~46%.

3. Combining Both Laws to Calculate Stellar Radius★★★★☆⏱ 4 min

For most distant stars, we cannot measure radius directly, but we can observe peak wavelength (from spectral analysis) and luminosity (from apparent brightness and known distance). Combining the two laws lets us solve for R, the stellar radius.

🔬 Derivation
Goal:

Derive the formula for stellar radius R

Starting from:

Wien's law λ_max T = b and Stefan-Boltzmann law L = 4πR²σT⁴

  1. 1

    Rearrange Wien's law to express T in terms of λ_max:

  2. 2
    T=bλmaxT = \frac{b}{\lambda_{max}}
  3. 3

    Substitute this expression for T into the Stefan-Boltzmann equation:

  4. 4
    L=4πR2σ(bλmax)4L = 4 \pi R^2 \sigma \left( \frac{b}{\lambda_{max}} \right)^4
  5. 5

    Rearrange algebraically to isolate R:

  6. 6
    R=Lλmax44πσb4R = \sqrt{ \frac{L \lambda_{max}^4 }{4 \pi \sigma b^4} }
Result:

This final formula lets you calculate stellar radius using only three observable quantities: luminosity, peak wavelength, and the two universal constants.

📐 Worked Example

A star has luminosity 1.2 × 10²⁸ W, and peak emission at 420 nm. Calculate its radius. Use b = 2.9 × 10⁻³ m K, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.

  1. 1

    Convert peak wavelength to metres:

  2. 2
    λmax=420×10−9 m\lambda_{max} = 420 \times 10^{-9} \text{ m}
  3. 3

    Substitute all values into the derived radius equation:

  4. 4
    R=1.2×1028×(420×10−9)44×π×5.67×10−8×(2.9×10−3)4R = \sqrt{ \frac{1.2 \times 10^{28} \times (420 \times 10^{-9})^4 }{4 \times \pi \times 5.67 \times 10^{-8} \times (2.9 \times 10^{-3})^4 } }
  5. 5

    Evaluate numerator and denominator separately before taking the square root:

  6. 6
    R≈2.1×109 mR \approx 2.1 \times 10^9 \text{ m}

4. Common Pitfalls

Wrong move:

Forgetting to convert peak wavelength from nanometres to metres before calculation

Why:

The constants b and σ use SI units of metres, so mismatched units produce results 10⁹ times smaller/larger than correct values

Correct move:

Always convert all wavelength values to SI metres before substituting into any equation

Wrong move:

Treating the T⁴ term as T instead

Why:

The fourth power dependence of luminosity on temperature is easy to misread under exam pressure, leading to wildly incorrect luminosity values

Correct move:

Write the exponent 4 clearly next to T in your working, and double check before final calculation

Wrong move:

Using the area of a circle πR² instead of the total surface area of a sphere 4πR²

Why:

Students often confuse stellar cross-sectional area for total radiating surface area

Correct move:

Explicitly note that stars are 3D spheres, so total surface area is 4πR², not πR²

Wrong move:

Assuming you need to correct for stellar absorption lines in the spectrum

Why:

Real stellar spectra have absorption lines that shift the measured peak slightly, but exam questions simplify this

Correct move:

Exam questions will explicitly state to treat the star as an ideal black body, so no line correction is required

5. Quick Reference Cheatsheet

Law

Formula

Key Quantities

SI Units

Wien's Displacement Law

Peak wavelength, surface temperature

in m, T in K

Stefan-Boltzmann Law

Luminosity, radius, temperature

L in W, R in m, T in K

Combined Stellar Radius

Radius from observable data

R in m

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

  • 2024 · Paper 4

    Calculate star radius from spectrum data

  • 2023 · Paper 5

    Determine Wien constant from stellar data

  • 2022 · Paper 4

    Compare luminosity of two stars

What's Next

You have now mastered the two core laws used to calculate the physical properties of stars, a foundational skill for all further astronomy work. Next, you will apply these laws to classify stars on the Hertzsprung-Russell (HR) diagram, where you will map stellar temperature, luminosity and radius across the full lifecycle of different star types. You will also learn how to use parallax measurements to calculate the distance to nearby stars, a required input for finding luminosity from apparent brightness, which you used in this module's stellar radius calculations. These skills will prepare you for advanced topics such as stellar evolution and the life cycle of supergiants, white dwarfs and neutron stars.