A spherical planet of radius $r$ , painted black, travels around the sun in a circular orbit at a distance $D$ from its centre $(r < < < D)$ . The sun, a sphere of radius $R$ , radiates as a blackbody at a temperature $T_{0} = 6000 K$ and subtends an angle 0.01 rad as seen from the planet. The equilibrium temperature $T$ of the planet is $x \times 100$ . Find $x$ .

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Detailed Solution

$\frac{2 R}{D} = 0.01 \frac{σ 4 π R^{2} T_{0}^{4}}{4 π D^{2}} π r^{2} = σ T^{4} 4 π r^{2} T^{4} = T_{0}^{4} \frac{R^{2}}{4 D^{2}} T = \frac{6000 \times 10^{- 1}}{2} T^{4} = \frac{T_{0}^{4}}{4} \times \frac{10^{- 4}}{4} = 300 K$

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