Question

Now we add a plexiglass atmosphere above the ground . The set-up of this environment has the following properties: -The atmosphere is completely transparent to the incoming solar radiation (it does not absorb any solar radiation) -The atmosphere’s temperature is Ta and it emits IR radiation according to the Stefan-Boltzmann law, both upward and downward by the same amount -The atmosphere completely absorbs the IR radiation coming from the ground -The ground temperature is Tg and the ground emits IR radiation according to the Stefan Boltzmann law -The ground completely absorbs all radiation it receives At the top of the atmosphere, solar radiation 238 w/m^2 goes down, while an upward IR radiation, originated from the atmosphere, goes out to space. From the energy balance at the top of the atmosphere, determine Ta in (K).

Answer 1

The set-up of this environment has the following properties which are given below as :

i. The atmosphere is completely transparent to the incoming solar radiation.

ii. The atmosphere’s temperature is T_a and it emits IR radiation according to the "Stefan-Boltzmann law".

iii. The atmosphere completely absorbs the IR radiation coming from the ground.

iv. The ground temperature is T_g and the ground emits IR radiation according to the "Stefan Boltzmann law".

v. The ground completely absorbs all radiation as it receives.

From the energy balance at the top of the atmosphere, determine T_a in (K).

Using a formula, we have

P = $\sigma$ A T_a⁴

(P / A) = $\sigma$ T_a⁴

F = $\sigma$ T_a⁴

where, $\sigma$ = Stefan Boltzmann constant = 5.67 x 10^-8 W/m².K⁴

F = energy flux per unit area = 238 W/m²

then, we get

(238 W/m²) = (5.67 x 10^-8 W/m².K⁴) T_a⁴

T_a⁴ = [(238 W/m²) / (5.67 x 10^-8 W/m².K⁴)]

T_a⁴ = 4.19753 x 10⁹ K⁴

T_a$\approx$ 254.5 K