Problem 4 Emissimty As we learned from the lecture, one of the 'complications' for simple Stefan-Boltzmann models of the radiative balance of a planet with an atmosphere is emissivity or absorp- tivity e that is, what if the atmosphere only absorbs some faction (so, 5 is between 0 and 1} of the outgoing infrared radiation? Recall from lecture that we can modify the Stefan-Boltzmann equation to account for imperfect emissivity (we'll assume that e is constant over all wavelengths, which in reality it seldom is), which changes our absorber from a blackbody to a 'graybody': 3mm = \"'1'" (1) Consider the following simple model of the Earth system in Figure 2, with a single atmosphere layer and emissivity of that layer equal to e. 4 (1 6.)ng 30 Figure 2: Schematic of a one-layer atmosphere with less than complete absorption Calculate the planetary temperature T9 for this simple Earth if e is equal to 0.80. Compare this to the value we get (from lecture is ne, no need to calculate this value again here} if we assume the planet has an atmosphere that behaves like a blackhody. How do these two temperature values compare to the true average Earth temperature? So, what do you think might this suggest about the Earth's atmosphere, with respect to whether it behaves like a blackbody or graybody'? Surface Temperature with Absorption (more complicated, signal layer atmosphere) 4 Sgray body I GOT Layer has emissivity 6 and temperature 601'; (1 0080/4 + (SOT: 6:32;} i 4 0T9 Problem Set #1 ! Constants and geophysical values: Assume the radius of the Sun to be 6.96 x 108 meters, and the 'radius' (by this, I mean the distance from the Sun to the Earth's orbit) of the Earth to be 1.5 x 1011 meters. The StefanBoltzmann constant is 5.67 X 10'8Wm'2K'4. The 'solar constant' (for our sun) can be treated as 1370Wm'2. The solar ux at the surface of the sun is 6.4 X 107Wm'2. You can assume the Earth system reects 30% of incident solar radiation