# Why does the factor of $\frac{1}{4}$ appear in this equation?

This is an equation my professor wrote to calculate the equilibrium surface temperature of a planet:$$\frac{S}{4}(1-A) = \sigma T^4$$. (A is the albedo of the planet and S is the intensity of starlight at that distance).
Why is there a factor of $$\frac{1}{4}$$ on the left hand side?

It's simple geometry:

A planet absorbs energy from its star with its geometric cross-section $$\pi r^2$$.

The same planet re-radiates the energy via its complete surface $$4\pi r^2$$.

In equilibrium, in the incoming energy equals the re-radiated energy:

$$E_{in} = E_{out}\\ \pi r^2 \cdot S \cdot (1-A) = 4\pi r^2 \sigma T^4$$ Divide by the surface area of the planet and you arrive at $$\frac{S}{4}(1-A) = \sigma T^4$$

The $$\mathrm{\frac{S}{4}}$$ represents the area- and time-averaged incident solar flux and the whole term on the LHS represents solar flux emitted by the planet.

The factor of 1/4 comes from the fact that only a single hemisphere is lit at any moment in time (creates a factor of 1/2), and from integrating over angles of incident sunlight on the lit hemisphere (creating another factor of 1/2). (Wikipedia)