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

Answer: (C)

$\frac{1}{2}$

Energy of sun near the planet is distributed in a sphere of radius d

Energy received by the planet$=\frac{4\pi r^2}{4\pi d^2}{E}_S=\frac{r^2}{d^2}{E}_S$

Energy radiated by the planet$=\sigma A{T}^4=\sigma \left(4\pi r^2\right)T^4$

For Thermal Equilibrium, Energy released=Energy Absorbed

$\frac{r^2}{d^2}{E}_S=\sigma \left(4\pi r^2\right)T^4$

$T=d^{-\frac{1}{2}}$

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