What would be the second earth’s surface temperature and atmospheric temperature?

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. 

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.