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

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.