For this second earth the sun’s radiative output is 40% lower than earths, meaning the radiative input at the top of the atmosphere is 144 Wm^-2. This atmosphere absorbs 10% of incoming solar radiation and 90% of outgoing terrestrial radiation. What would be the second earth’s surface temperature and atmospheric temperature?
Atmosphere
E
0.1x
y
y
0.9E
x
Earth’s Surface
Let E be the radiation entering the top of the atmosphere from the sun, x be the total radiation emitted from the earth’s surface, and y be the total radiation emitted from the atmosphere.
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If we perform an energy balance for each level (earth’s surface and atmosphere), we get two equations and two unknowns as shown below
(1)
(2)
If we make y the subject of the formula for equation (1), we get:
(3)
If we substitute equation (2) into equation (3) above the result is:
(4)
If we substitute equation (4) into equation (2), we get:
From the given information, E = 144 Wm -2 .
Calculating the second earth’s surface and atmospheric temperature
From Stefan-Boltzmann law, we know that:
and
, where:
σ= 5.67 x 10 -8 Wm 2 K 4
Ts = the second earth’s surface temperature
Ta = the second earth’s atmospheric temperature
For the second earth whose sun’s radiative output is 40% lower than our current sun, the planet’s surface and atmospheric temperatures will be -257.35
C (257.35 K) and – 53.36
C (219.79 K), respectively.