Question

The solar constant is the amount of heat energy received per second per unit area of a perfectly black surface placed at a mean distance of the Earth from the Sun, in the absence of Earths atmosphere, the surface being held perpendicular to the direction of Suns rays. Its value is $1388W/m^2$.If the solar constant for the earth is $s$. The surface temperature of the sun is $TK$, $D$ is the diameter of the Sun, $R$ is the mean distance of the Earth from the Sun. The sun subtends a small angle $\theta$ at the earth. Then correct options is/are :

• A. $s=\sigma T^4\left(\frac{D}{R}\right)$
• B. $s=\frac{\sigma T^4}{4}\left({\frac{D}{R}}^2\right)$
• C. $s=\frac{\sigma T^4}{4}{\theta }^2$
• D. $s=\frac{\sigma T^4}{4}{\left(\frac{R}{D}\right)}^2$

Answer 1

The correct option is B $s=\frac{\sigma T^4}{4}\left({\frac{D}{R}}^2\right)$

Total energy intercept = Solar Constant x Area

$E_{intercept}=k_s{A}_{earth}$

$\sigma A_{sun}{T}^4=k_s{A}_{earth}$

$k_s=\sigma T^4\frac{A_{sun}}{A_{earth}}=\sigma T^4\frac{4\pi {\left(\frac{D}{2}\right)}^2}{\pi R^2}=\sigma T^4{\left(\frac{D}{R}\right)}^2$

Hence, solar constant is $\sigma T^4{\left(\frac{D}{R}\right)}^2$