Irradiation of the earth's surface from the atmosphere exhibits strong spectral variation, mainly associated with the atmospheric window.

(a) For the range \(0 \leq \lambda<8 \mu \mathrm{m}\) the spectral emissivity of the sky under normal conditions can be approximated as \(\varepsilon_{\lambda, 1}=0.90\), while \(\varepsilon_{\lambda, 3}=0.85\) for \(\lambda>13 \mu \mathrm{m}\). Within the atmospheric window, \(8 \mu \mathrm{m} \leq \lambda \leq 13 \mu \mathrm{m}\), the spectral emissivity is approximately \(\varepsilon_{\lambda, 2}=0.05\), 0.8 , and 0.9 for clear, moderately cloudy, and cloudy skies. Calculate the effective sky temperature corresponding to an actual atmosphere temperature of \(280 \mathrm{~K}\) under the three sky conditions.

(b) Little water exists in the atmosphere under the cold, cloudless conditions of Antarctica where spectral emissivities have been measured to be approximately \(\varepsilon_{\lambda, 1}=0.75, \varepsilon_{\lambda, 2}=0.03\), and \(\varepsilon_{\lambda, 3}=0.75\) corresponding to an actual atmosphere temperature of \(220 \mathrm{~K}\). Determine the sky temperature under these extreme conditions. Explain how the coldest surface temperatures recorded on Earth have been found in Antarctica with values less than \(200 \mathrm{~K}\).