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 the 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

Answer: (C)

$\frac{1}{2}$

Let P=power radiated by Sun

R=Radius of planet

E=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 equilibriums:

$\Rightarrow \frac{P}{4\pi d^2}\pi R^2=4\pi R^2\sigma T^4\Rightarrow T^4\alpha \frac{1}{d^2}\Rightarrow T\alpha \frac{1}{d^{1/2}}\Rightarrow T\alpha d^{-\frac{1}{2}}$

So n=$\frac{1}{2}$