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 is

• A. $2$
• B. $1$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{1}{2}$

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