Question

A planet having average surface temperature $T_{0}$ at an average distance $d$ from the sun. Assuming that the planet receives radiant energy from the sun only and it loses radiant energy only from the surface and neglecting all other atmospheric effects we conclude

• A. $T_{0}\propto d^{2}$
• B. $T_{0}\propto d^{-2}$
• C. $T_{0}\propto d^{1/2}$
• D. $T_{0}\propto d^{-1/2}$

Answer 1

Answer: (D)

$T_{0}\propto d^{-1/2}$

Answer 2

Energy received per second by the planet $=\frac{P}{4 \pi d^{2}}\left(\pi R^{2}\right)$ where, $P$ is power radiated by the sun and $R$ is the radius of the planet. Further, energy radiated per second by the planet according to Stefan's law is $\sigma\left(4 \pi R^{2}\right) T_{0}^{4}$. For thermal equilibrium to exist, we get $\frac{P}{4 \pi d^{2}}\left(\pi R^{2}\right)=\sigma\left(4 \pi R^{2}\right) T_{0}^{4}$ $\Rightarrow T_{0}^{4} \propto d^{-2}$ $\Rightarrow T_{0} \propto d^{-1 / 2}$