use Wien’s displacement law and the Stefan–Boltzmann law to estimate the radius of a star

Stellar Radii

In this lesson we will see how two fundamental laws of thermal radiation – Wien’s displacement law and the Stefan–Boltzmann law – can be combined to give a simple method for estimating the radius of a star when its surface temperature and total luminosity are known.

Learning Objectives

• State Wien’s displacement law and the Stefan–Boltzmann law.
• Understand the physical meaning of each constant in the laws.
• Derive an expression for the radius of a star in terms of its luminosity and effective temperature.
• Apply the derived formula to real astronomical data (e.g., the Sun, Sirius).

Theoretical Background

Wien’s Displacement Law

For a black‑body radiator the wavelength at which the emitted power per unit wavelength is a maximum is inversely proportional to the temperature:

$$\lambda_{\max} = \frac{b}{T}$$

where

• $\lambda_{\max}$ is the peak wavelength (m),
• $T$ is the absolute temperature (K),
• $b = 2.898 \times 10^{-3}\ \text{m·K}$ is Wien’s constant.

Stefan–Boltzmann Law

The total power radiated per unit surface area of a black body is proportional to the fourth power of its temperature:

$$j^{\star} = \sigma T^{4}$$

where

• $j^{\star}$ is the radiant emittance (W·m⁻²),
• $\sigma = 5.670374419 \times 10^{-8}\ \text{W·m}^{-2}\text{·K}^{-4}$ is the Stefan–Boltzmann constant.

Deriving the Radius Formula

The total luminosity $L$ of a spherical star of radius $R$ is the product of its surface area $4\pi R^{2}$ and the radiant emittance $j^{\star}$:

$$L = 4\pi R^{2}\sigma T^{4}$$

Solving for $R$ gives:

$$R = \sqrt{\frac{L}{4\pi\sigma T^{4}}}$$

This expression shows that if we know a star’s luminosity (often expressed relative to the Sun’s luminosity $L_{\odot}$) and its effective temperature (obtained from the peak wavelength via Wien’s law), we can estimate its radius.

Step‑by‑Step Procedure

1. Measure or obtain the star’s peak wavelength $\lambda_{\max}$ from its spectrum.
2. Calculate the effective temperature using Wien’s law: $$T = \frac{b}{\lambda_{\max}}$$
3. Obtain the star’s luminosity $L$ (e.g., from distance and apparent brightness or from catalogues).
4. Insert $L$ and $T$ into the radius formula: $$R = \sqrt{\frac{L}{4\pi\sigma T^{4}}}$$
5. Convert $R$ to convenient units (e.g., solar radii $R_{\odot}$ where $R_{\odot}=6.96\times10^{8}\ \text{m}$).

Worked Example – The Sun

For the Sun we have:

• Peak wavelength $\lambda_{\max} \approx 5.0\times10^{-7}\ \text{m}$ (green light).
• Luminosity $L_{\odot}=3.828\times10^{26}\ \text{W}$.

Calculate the temperature:

$$T_{\odot} = \frac{2.898\times10^{-3}}{5.0\times10^{-7}} \approx 5.80\times10^{3}\ \text{K}$$

Now the radius:

$$R_{\odot} = \sqrt{\frac{3.828\times10^{26}}{4\pi(5.670374419\times10^{-8})(5.80\times10^{3})^{4}}} \approx 6.96\times10^{8}\ \text{m}$$

This matches the accepted solar radius, confirming the method.

Useful Constants

Constant	Symbol	Value	Units
Wien’s constant	$b$	2.898 × 10⁻³	m·K
Stefan–Boltzmann constant	$\sigma$	5.670374419 × 10⁻⁸	W·m⁻²·K⁻⁴
Solar luminosity	$L_{\odot}$	3.828 × 10²⁶	W
Solar radius	$R_{\odot}$	6.96 × 10⁸	m

Practice Questions

1. Betelgeuse has a measured peak wavelength of $1.0\times10^{-6}\ \text{m}$ and a luminosity of $1.2\times10^{5}\ L_{\odot}$. Estimate its radius in solar radii.
2. A newly discovered exoplanet‑host star shows a peak wavelength of $4.5\times10^{-7}\ \text{m}$ and an apparent magnitude that corresponds to a luminosity of $0.85\ L_{\odot}$. Determine its radius.
3. Explain why hotter stars tend to have smaller radii for a given luminosity compared with cooler stars.