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

> Physics · CIE A-Level 9702
> Source: https://www.owlsprep.com/study/cie-9702-u29-stellar-radii/

This module teaches you to use two core black body radiation laws to calculate the physical radius of distant stars, using only observable spectral data with no direct measurement required.

**Prerequisites:** [Understanding of black body radiation spectra](https://www.owlsprep.com/study/cie-9702-u29-black-body-radiation/); [Basic surface area of a sphere calculation](https://www.owlsprep.com/study/cie-9702-mathematical-skills-for-physics/)

## Learning objectives

- Recall and apply Wien's Displacement Law to relate peak wavelength of stellar emission to surface temperature
- Use the Stefan-Boltzmann Law to calculate total power output (luminosity) of a black body star
- Combine both laws to derive and calculate the radius of a distant star from observable quantities
- Identify common calculation errors in stellar radius determination

## Wien's Displacement Law: Relating Temperature to Peak Wavelength

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.

**Wien's Displacement Law** — 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.

*Notation:* \lambda_{max} T = b

*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. First convert peak wavelength to SI units:
2. $$\lambda_{max} = 350 \times 10^{-9} \text{ m}$$
3. Rearrange Wien's law to solve for temperature:
4. $$T = \frac{b}{\lambda_{max}}$$
5. Substitute values:
6. $$T = \frac{2.9 \times 10^{-3}}{350 \times 10^{-9}} \approx 8300 \text{ K}$$

**Check your understanding**

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

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

## Stefan-Boltzmann Law: Luminosity from Temperature and Radius

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.

**Stefan-Boltzmann Law** — 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.

*Notation:* L = 4 \pi R^2 \sigma T^4

*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. Write out the full Stefan-Boltzmann equation:
2. $$L = 4 \pi R^2 \sigma T^4$$
3. Substitute all known values:
4. $$L = 4 \times \pi \times (7.0 \times 10^8)^2 \times 5.67 \times 10^{-8} \times (5800)^4$$
5. Calculate step by step to get final result:
6. $$L \approx 3.9 \times 10^{26} \text{ W}$$

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

## Combining Both Laws to Calculate Stellar Radius

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:** Derive the formula for stellar radius R

*Starting from:* Wien's law λ_max T = b and Stefan-Boltzmann law L = 4πR²σT⁴

1. Rearrange Wien's law to express T in terms of λ_max:
2. $$T = \frac{b}{\lambda_{max}}$$
3. Substitute this expression for T into the Stefan-Boltzmann equation:
4. $$L = 4 \pi R^2 \sigma \left( \frac{b}{\lambda_{max}} \right)^4$$
5. Rearrange algebraically to isolate R:
6. $$R = \sqrt{ \frac{L \lambda_{max}^4 }{4 \pi \sigma b^4} }$$

*Conclusion:* 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. Convert peak wavelength to metres:
2. $$\lambda_{max} = 420 \times 10^{-9} \text{ m}$$
3. Substitute all values into the derived radius equation:
4. $$R = \sqrt{ \frac{1.2 \times 10^{28} \times (420 \times 10^{-9})^4 }{4 \times \pi \times 5.67 \times 10^{-8} \times (2.9 \times 10^{-3})^4 } }$$
5. Evaluate numerator and denominator separately before taking the square root:
6. $$R \approx 2.1 \times 10^9 \text{ m}$$

## Common pitfalls

- **Wrong:** Forgetting to convert peak wavelength from nanometres to metres before calculation
  - Why it fails: The constants b and σ use SI units of metres, so mismatched units produce results 10⁹ times smaller/larger than correct values
  - Correct: Always convert all wavelength values to SI metres before substituting into any equation
- **Wrong:** Treating the T⁴ term as T instead
  - Why it fails: The fourth power dependence of luminosity on temperature is easy to misread under exam pressure, leading to wildly incorrect luminosity values
  - Correct: Write the exponent 4 clearly next to T in your working, and double check before final calculation
- **Wrong:** Using the area of a circle πR² instead of the total surface area of a sphere 4πR²
  - Why it fails: Students often confuse stellar cross-sectional area for total radiating surface area
  - Correct: Explicitly note that stars are 3D spheres, so total surface area is 4πR², not πR²
- **Wrong:** Assuming you need to correct for stellar absorption lines in the spectrum
  - Why it fails: Real stellar spectra have absorption lines that shift the measured peak slightly, but exam questions simplify this
  - Correct: Exam questions will explicitly state to treat the star as an ideal black body, so no line correction is required

## Cheatsheet

| Law | Formula | Key Quantities | SI Units |
| --- | --- | --- | --- |
| Wien's Displacement Law | $\lambda_{max} T = b$ | Peak wavelength, surface temperature | $\lambda_{max}$ in m, T in K |
| Stefan-Boltzmann Law | $L = 4 \pi R^2 \sigma T^4$ | Luminosity, radius, temperature | L in W, R in m, T in K |
| Combined Stellar Radius | $R = \sqrt{\frac{L \lambda_{max}^4}{4 \pi \sigma b^4}}$ | Radius from observable data | R in m |

## 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.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-9702-u29-stellar-radii/