Green House Effect : Energy Balance Equation

 The amount of energy delivered to the Earth is equal to the energy lost from the Earth.

Otherwise, the Earth’s temperature would continually rise (or fall).

Incoming energy = outgoing energy Ein = Eout

As energy moves away from the sun, it is spread over a greater and greater area.

This is the Inverse Square Law

So = L / area of sphere

Where ,     L is the Sun’ luminosity.

                 So is  Solar Constant

So = L / (4 p rs-e ²) =   (3.9 x 10²⁶ W)/(4 x p  x (1.5 x 1011m)² ) = 1370 W/m²

So is the solar constant for Earth. It is determined by the distance between Earth (rs-e ) and sun L is the Sun’ luminosity.

Solar energy reaches the Earth,

Assuming solar radiation covers the area of a circle defined by the radius of the Earth (re )

Ein = So x (area of circle)

Ein = So (W/m²) x p re ² (m²)

But this is not quite correct as some energy is reflected away.

The actual amount of energy reaches the earth is,

Albedo (A) = % energy reflected away

Ein = So p re ² (1-A)

Ein re Albedo (A) = % energy reflected away

A= 0.3

Ein = So p re ² (1-A)

Ein = So p re ² (0.7)

The Energy emitted from the Earth,

Eout = F x (area of the Earth)

F = s T⁴

Area = 4 p re ²

Eout = (s T⁴) x (4 p re ²)

From Energy Balance condition:

Ein = Eout

So p re ² (1-A) = s T⁴ (4 p re ²)

T4 = So (1-A) /(4s)

For Earth:

So = 1370 W/m²;            A = 0.3;               s = 5.67 x 10^-8 W/m²K⁴

Now, finding temperature of Earth

T⁴ = (1370 W/m²)(1-0.3) /(4 *5.67 x 10^-8 W/m²K⁴)

T⁴ = 4.23 x 10⁹ (K⁴)

T = 255 K

So, expected temperature is  (255-273)= -18 ^oC

But, the actual temperature is warmer. The observed temperature (T_obs) is 15 ^oC

The difference between observed and expected temperatures (DT):

DT = T_obs - T_exp

DT = 15 - (-18)

DT = + 33 ^oC

This extra warmth is what we call the GREENHOUSE EFFECT. It is a result of warming of the Earth’s surface by the absorption of radiation by molecules in the atmosphere.