use the Stefan–Boltzmann law L = 4πσr 2 T

Stellar Radii

Learning Objective

By the end of this lesson you should be able to use the Stefan–Boltzmann law to calculate the radius of a star when its luminosity and surface temperature are known.

The Stefan–Boltzmann Law

The total power radiated by a black‑body surface of temperature $T$ is given by

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

where

• $L$ – luminosity (total power output) of the star (W)
• $r$ – radius of the star (m)
• $\sigma = 5.670\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$ – Stefan–Boltzmann constant
• $T$ – effective surface temperature (K)

Rearranging for Radius

Solving the Stefan–Boltzmann law for the radius gives

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

This expression shows that the radius depends on the square root of the luminosity and inversely on the square of the temperature.

Step‑by‑Step Procedure

1. Write down the known values of $L$ and $T$ for the star. Convert them to SI units if they are given in solar units.
2. Insert the value of the Stefan–Boltzmann constant $\sigma = 5.670\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$.
3. Calculate $T^{4}$.
4. Compute the denominator $4\pi\sigma T^{4}$.
5. Divide the luminosity $L$ by the denominator.
6. Take the square root of the result to obtain $r$.
7. If required, express $r$ in solar radii $R_{\odot}=6.96\times10^{8}\ \text{m}$.

Worked Example – The Sun

Given:

• Luminosity $L_{\odot}=3.828\times10^{26}\ \text{W}$
• Effective temperature $T_{\odot}=5778\ \text{K}$

Calculate the solar radius using the Stefan–Boltzmann law.

1. Compute $T_{\odot}^{4}$:

$$ T_{\odot}^{4} = (5778\ \text{K})^{4} = 1.11\times10^{15}\ \text{K}^{4} $$

2. Denominator:

$$ 4\pi\sigma T_{\odot}^{4} = 4\pi(5.670\times10^{-8})(1.11\times10^{15}) = 7.93\times10^{8}\ \text{W m}^{-2} $$

3. Ratio $L_{\odot}/(4\pi\sigma T_{\odot}^{4})$:

$$ \frac{3.828\times10^{26}}{7.93\times10^{8}} = 4.83\times10^{17}\ \text{m}^{2} $$

4. Square root:

$$ r_{\odot} = \sqrt{4.83\times10^{17}} = 6.96\times10^{8}\ \text{m} $$

This matches the accepted solar radius, confirming the method.

Typical Stellar Radii

Star (example)	Luminosity ($L/L_{\odot}$)	Effective Temperature (K)	Radius ($R/R_{\odot}$)
Sun	1.0	5778	1.0
Sirius A	25.4	9940	1.71
Betelgeuse (red supergiant)	1.2×10⁵	3500	887
Proxima Centauri (red dwarf)	0.0017	3042	0.14
Vega	40.1	9602	2.36

Common Pitfalls

• Forgetting the $T^{4}$ dependence – using $T$ instead of $T^{4}$ leads to radii that are off by orders of magnitude.
• Mixing units (e.g., using solar luminosities with SI constants). Convert all quantities to SI before inserting them into the formula.
• Neglecting the square‑root step when solving for $r$.
• Assuming every star behaves as a perfect black body; real stars have emissivity $<1$, which introduces a small correction factor.