Bare rock model Assumptions

•Amount of energy coming into the planet from sunlight is equal the amount of energy leaving the earth as IR.

Fin = Fout

•No atmosphere

(1-Isolar4

T4earth

Sun Light Earth Light

Earth

Tearth = 259 K = -14° C = 6°F

Energy Balance of a Bare Rock

How much solar energy reaches the Earth?

Sun is a nearly constant source of energy

Solar constant is the energy flux density of the solar emission at a distance (d)

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

• solar constant for Earth, So = 1367 W/m2

We know the solar constant S= 1367 W/m2

But not all solar energy is absorbed by the Earth. Some is reflected.

Earth albedoAlbedo is the fraction of sunlight which is reflected off a planet. The average albedo of the Earth is about 0.33.

For the Earth, α = 0.33 (33%)

(1)

Some Basic Information:

Area of a circle = r2

Area of a sphere = 4 r2

Let’s do some calculations

The intensity of incoming sunlight at the average distance from the sun to the Earth = 1350 W/m2

Reflected radiation = 30 % of incoming radiation = 1350 x 30 W/m2

100 = 400 W/m2

Therefore The energy absorbed by the Earth = 1350 – 400

= 950 W/m2

~ 1000 W/m2

The total absorbed solar radiation

= 1000 Wm-2 x Area of the circular shadow

Fin = 1000 Wm-2 X ( r2)

Where r = radius of the Earth

IR radiation emitted by the Earth = σ T4 W/m2

Total energy going out of earth as IR radiation = σ T4 X Area of the sphere

Fout = σ T4 x 4r2

Fout = σ T4 x 4r2

Fout = 5.67 x 10-8 x T4 x 4r2

Energy radiated from the Earth

Eout

Fin= 350 Wm-2 X ( r2)Fout = 5.67 x 10-8 x T4 x 4r2

Fin = Fout

1000 Wm-2 X ( r2)m2 = 5.67 x 10-8 Wm-2K-4 x T4 x 4r2 m2

1000 = 5.67 x 10-8 K-4 x T4 x 4

T4 = 1000 4 x 5.67 x 10-8 K-4

T = 257 K

If we know S and α we can calculate the temperature of the Earth. It is the temperature

we would expect if Earth behaves like a blackbody.

NOTE : This calculation can be done for any planet, provided

we know its solar (S )constant and albedo (α).

Simply the temperature of the Earth can be written as

T4 = S x ( 1- α) 4σ

Where s - Solar constantσ – Stephan constant - 5.67 x 10-8 W/m2K4

Α - albedo

(1-Isolar4

Earth

Atmosphere

Iup, atmosphere

Idown, atmosphere

Iup, ground

Boundary to Space

Tatm = 259 K

Tearth = 303 K = 86° F

A Planet with an Atmosphere