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

Question

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?

Answer 1

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 $$ \frac{S}{4}(1-A) = \sigma T^4 $$

Answer 2

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)