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 $R$. The temperature at which planet radiates the energy is

• A. $T_0\sqrt{\frac{R}{2d}}$
• B. $T_0\sqrt{\frac{2R}{d}}$
• C. $T_0\sqrt{\frac{R}{d}}$
• D. $T_0{\left(\frac{R}{d}\right)}^{1/4}$

Answer 1

Answer: (A)

$r=$ Radius of planet
$T_1=$ Temperature of planet at which energy is radiated
Power radiated by Sun, $P=4\pi R^2\sigma T_0^4$
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,
$\frac{R^2{T}_0^4\pi r^2\sigma }{d^2}=4\pi r^2\sigma T_1^4$
$R^2{T}_0^4=4d^2{T}_1^4$
$T_1=T_0\sqrt{\frac{R}{2d}}$