JEE Main & Advanced Physics Transmission of Heat Temperature of the Sun and Solar Constant

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

\[P=A\sigma {{T}^{4}}=4\pi {{R}^{2}}\sigma {{T}^{4}}\]

In reaching earth this energy will spread over a sphere of radius r (= average distance between sun and earth); so the intensity of solar radiation at the surface of earth (called solar constant S) will be given by

\[S=\frac{P}{4\pi {{r}^{2}}}=\frac{4\pi {{R}^{2}}\sigma {{T}^{4}}}{4\pi {{r}^{2}}}\]

i.e.\[T={{\left[ {{\left( \frac{r}{R} \right)}^{2}}\frac{S}{\sigma } \right]}^{1/4}}\] 

\[={{\left[ {{\left( \frac{1.5\times {{10}^{8}}}{7\times {{10}^{5}}} \right)}^{2}}\times \frac{1.4\times {{10}^{3}}}{5.67\times {{10}^{-8}}} \right]}^{1/4}}\tilde{-}5800\,K\]

As \[r=1.5\times {{10}^{8}}\,km,\] \[R=7\times {{10}^{5}}\,km,\]

\[S=2\frac{cal}{c{{m}^{2}}min}=1.4\frac{kW}{{{m}^{2}}}\] and \[\sigma =5.67\times {{10}^{-8}}\frac{W}{{{m}^{2}}{{K}^{4}}}\]

This result is in good agreement with the experimental value of temperature of sun, i.e., 6000 K.