Question

The solar constant for the earth is $S$. The surface temperature of the sun is $TK$. The sun subtends an angle $\theta$ at the earth.

• A. $S\propto T^4$
• B. $S\propto T^2$
• C. $S\propto {\theta }^2$
• D. $S\propto \theta$

Answer 1

Answer: (A), (C)

The correct options are
A $S\propto T^4$
C $S\propto {\theta }^2$

Let $R=$ radius of the sun, $d=$ distance of the earth from the sun.
Power radiated by the sun $=\left(4\pi R^2\right)\sigma T^4=P$.
Energy received per unit area per second normally on the earth
$=S=\frac{P}{4\pi d^2}=\frac{4\pi R^2\sigma T^4}{4\pi d^2}=\left(\sigma T^4\right){\left(\frac{R}{d}\right)}^2=\frac{1}{4}\sigma T^4{\left(\frac{2R}{d}\right)}^2$

Angle subtended by the sun at the earth $=\theta =\frac{2R}{d}$.
or, $S=\frac{\sigma }{4}{T}^4{\theta }^2$

Hence (A) and (C) are correct.