Derive an expression for the fraction of energy received by the surface that comes from the atmosphere (this is the amount of energy that comes from the atmosphere divided by the sum of energy from the Sun and energy from the atmosphere). Using values in Table 4.1, calculate the fraction for Mercury, Earth, and Venus. Make the standard assumption that the atmosphere is transparent to visible photons but opaque to infrared photons.

Table 4.1

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Table 4.1 Data on the four inner planets in our Solar System Planet Mercury Venus Earth Mars Solar constant (W/m²) Albedo 10,000 2,650 1,360 580 0.1 0.7 0.3 0.15 Energy in (W/m²) 2,250 199 238 123 Observed surface temperature (K) 452 735 289 227 Inferred n 0.052 82 0.65 0.22