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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 ...