Let us assume that sun radiates like a black body with surface temperature at $T s = 6000 K$ and earth absorbs radiations coming from sun only. If both earth and sun are considered as perfect spheres with distance between centre of earth and centre of sun to be $200$ times the radius of sun. Find the temperature (in Kelvin) of surface of earth in steady state (assume radiations incident on earth to be almost parallel)

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Solution

Power absorbed by earth = power emitted by earth.
$\Rightarrow \frac{e σ (4 π R_{S}^{2}) T_{S}^{4}}{4 π r^{2}} \times π R_{e}^{2} = e σ (2 π R_{e}^{2}) T_{e}^{4}$

$\Rightarrow T_{e} = T_{S} \sqrt{\frac{R_{S}}{2 R_{e}}} = T_{S} \sqrt{\frac{R_{S}}{2 \times 200 R_{S}}}$

$\Rightarrow T_{e} = \frac{T_{S}}{20} = 300 K$

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