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.
(1) $S\propto T^4$
(2) $S\propto T^2$
(3) $S\propto {\theta }^2$
(4) $S\propto \theta$

• A. 1, 2 and 3 are correct
• B. 1 and 2 are correct
• C. 2 and 4 are correct
• D. 1 and 3 are correct

Answer 1

Answer: (D)

If $R$ is the radius of the sun and $T$ its temperature, then the energy emitted by the sun per $\sec$ through radiation in accordance with Stefan's law will be given by
$P=A\sigma T^4=4\pi R^2\sigma T^4$

Energy received per unit area per second on the earth of radius $r$
$S=\frac{P}{4\pi r^2}=\frac{4\pi R^2\sigma T^4}{4\pi r^2}$
$=\left(\sigma T^4\right){\left(\frac{R}{r}\right)}^2$
$=\frac{1}{4}\sigma T^4{\left(\frac{2R}{r}\right)}^2$
Angle subtended by the sun at the earth,
$\theta =\left(\frac{2R}{r}\right)$
Or $S=\frac{1}{4}\sigma T^4{\theta }^2$