Inverse Square Law of Radiation

Astrophysics | AQA A-Level Physics

Key Definitions

Luminosity ($L$): the total power output of radiation emitted by a star, measured in watts (W).

Brightness ($I$): the intensity of radiation received on Earth from a star, measured in watts per metre squared (W m$^{-2}$).

Luminosity vs brightness

Crucially, these are two different things. Luminosity tells us how much total power a star emits at its surface. Brightness tells us how much of that power we actually detect here on Earth.
The brightness of a star depends on two factors: its luminosity and its distance from Earth. A very luminous star far away can appear dimmer than a less luminous star nearby.

The inverse square law

Light from a star spreads out uniformly in all directions through a spherical shell. By the time it reaches Earth at distance $d$, the radiation has been spread over a sphere of surface area $4\pi d^2$.
The intensity (brightness) received at distance $d$ is therefore:

$$I = \frac{L}{4\pi d^2}$$

Where $I$ = apparent brightness (W m$^{-2}$), $L$ = luminosity (W), $d$ = distance to the star (m).
This equation assumes the source can be treated as a point, it radiates uniformly in all directions, and no radiation is absorbed or scattered between the star and the Earth.
The key part is the inverse square relationship: if you double the distance, the intensity drops by a factor of four. For stars with the same luminosity, the one with greater apparent brightness must be closer.

Common Mistake

When rearranging for distance, do not forget to square root. The equation gives $d^2 = \frac{L}{4\pi I}$, so $d = \sqrt{\frac{L}{4\pi I}}$. Missing the square root is a very common calculation error.