Question

A planet is at an average distance $d$ from the sun and its average surface temperature is $T$. Assume that the planet receives energy only from the sun, and loses energy only through radiation from its surface. Neglect atmospheric effects. If $T\propto d^{-n}$, the value of $n$

• A. $2$
• B. $1$
• C. $1/2$
• D. $1/4$

Answer 1

The correct option is C $1/2$

Let, $R=$ radius of planet
$P=$ power radiated by the sun
Energy received by planet :
$=\frac{P}{4\pi d^2}\times \pi R^2$
Energy radiated by planet :
$\propto 4\pi R^{2}e\sigma T^4$
For thermal equilibrium,
$4\pi R^{2}e\sigma T^4=\frac{P}{4\pi d^2}\times \pi R^2$
$T^4\propto \frac{1}{d^2}$
or $T\propto d^{-1/2}\therefore n=\frac{1}{2}$