Question

The surface temperature of the Sun is $T_0$ and it is at average distance $d$ from a planet. The radius of the Sun is $R$. If $\theta$ is the angle subtended by the Sun at the planet, then temperature at which planet radiates energy is proportional to

• A. ${\theta }^{-1/2}$
• B. ${\theta }^2$
• C. ${\theta }^{1/2}$
• D. $\theta$

Answer 1

The correct option is C ${\theta }^{1/2}$

Let, $r=$ Radius of planet
$T_1=$ Temperature of planet at which energy is radiated

Power radiated by Sun, $P=\pi R^2\sigma T_0^4$

Energy received by planet

$=\frac{P}{4\pi d^2}\times \pi r^2=\frac{R^2{T}_0^4\pi r^2\sigma }{d^2}$

Energy radiated by planet

$=4\pi r^2\sigma T_1^4$

For thermal equilibrium,

$=\frac{R^2{T}_0^4\pi r^2\sigma }{d^2}=4\pi r^2\sigma T_1^4$

$R^2{T}_0^4=4d^2{T}_1^4$

$R^2{T}_{\circ }^4=4d^2{T}_1^4$

$T_1=T_{\circ }\sqrt{\frac{R}{2d}}$

As we know from the figure that
$2R=d\times \theta$

$d=\frac{2R}{\theta }$

$T_1=T_{\circ }\sqrt{\frac{R\times \theta }{2\times 2R}}$

$T_1=T_{\circ }\sqrt{\frac{\theta }{4}}$

$T_1\propto {\theta }^{1/2}$