Stefan's Law

Astrophysics | AQA A-Level Physics

Key Definition

Stefan's law (Stefan-Boltzmann law): the total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body.

The general form

The total power $P$ radiated by a perfect black body depends on two factors: its absolute temperature and its surface area.
The Stefan-Boltzmann law can be written as:

$$P = \sigma A T^4$$

Where $P$ = total power emitted across all wavelengths (W), $\sigma$ = the Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$), $A$ = surface area of the body (m$^2$), $T$ = absolute temperature (K).

Applied to stars

Stars are approximately spherical, so their surface area is $A = 4\pi r^2$, where $r$ is the radius of the star.
Substituting this into Stefan's law gives the luminosity of a star:

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

Where $L$ = luminosity (W), $r$ = radius of the star (m), $\sigma$ = Stefan-Boltzmann constant, $T$ = surface temperature (K).
The key part is the $T^4$ dependence: even a small increase in temperature produces a very large increase in luminosity. Doubling the temperature increases the luminosity by a factor of $2^4 = 16$.
This equation is commonly used to calculate the radius of a star, given its luminosity and surface temperature:

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

Common Mistake

When calculating stellar radii, do not forget the square root. The equation gives $r^2 = \frac{L}{4\pi\sigma T^4}$, so you must take the square root at the end. Also make sure you express the radius in solar radii ($R_\odot = 6.96 \times 10^8$ m) if the question asks for a comparison with the Sun.