Question

The average temperature of the surface of the Earth is approximately 14◦C. What would the steady-state temperature of the Earth be if there were no greenhouse effect — i.e. if the emissivity of the earth was independent of wavelength? Give your answer in ◦C. Note that the radius of the Earth is 6.38 106 m, and the intensity of sunlight at the Earth’s location is 1366 W/m2. Hint, how much power from the sun would strike a plate with the same radius as the Earth? How does that power compare to the power that hits the Earth? Another hint — I didn’t give you an emissivity. Just write it as E, and if you remember that absorptivity and emissivity are the same, it will cancel out.

Answer 1

The Earth has absorptivity or emissivity of e. The radius is R. Therefore, if the intensity of sunlight at the Earth’s location is I, then the energy collected per unit time from Sun by Earth will be

$E_{c}=e\left (\pi R^{2}\times I \right )$

If the temperature of the Earth is T, then the energy radiated by Earth according to Stefan-Boltzmann law,

$E=e\left ( 4\pi R^{2} \right )\sigma T^{4}$

If there is no green house effect, then

$E=E_{c},\ e\left ( 4\pi R^{2} \right )\sigma T^{4}=e\left ( \pi R^{2}I \right )$

$\therefore T=\sqrt[4]{\frac{I}{4\sigma }}=\sqrt[4]{\frac{1366\ Wm^{-2}}{4\times 5.67\times 10^{-8}\ Wm^{^{-2}}K^{-4}}}=278.58\ K=5.58\ ^{o}C$