1. 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

3. Energy Balance of a Bare Rock
Tearth = 259 K = -14° C = 6°F

4. 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

5. We know the solar constant S= 1367 W/m2
But not all solar energy is absorbed by the Earth. Some is reflected.
Earth albedo
Albedo 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)

6. Some Basic Information:
Area of a circle =  r2
Area of a sphere = 4  r2

7. Let’s do some calculations
The intensity of incoming sunlight at the average distance from the sun to the Earth = W/m2
Reflected radiation = 30 % of incoming radiation
= x W/m2
100
= 400 W/m2
Therefore
The energy absorbed by the Earth = 1350 – 400
= 950 W/m2
~ 1000 W/m2

8. 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

9. Energy radiated from 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 4r2
Fout = 5.67 x 10-8 x T4 x 4r2
Eout

10. Fin= 350 Wm-2 X ( r2) Fout = 5.67 x 10-8 x T4 x 4r2 Fin = Fout
1000 Wm-2 X ( r2)m2 = 5.67 x 10-8 Wm-2K-4 x T4 x 4r2 m2
1000 = 5.67 x 10-8 K-4 x T4 x 4
T4 =
4 x 5.67 x 10-8 K-4
T = 257 K

11. Simply the temperature of the Earth can be written as
T4 = S x ( 1- α) 4σ
Where s - Solar constant
σ – Stephan constant x 10-8 W/m2K4
Α - albedo
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).

12. A Planet with an Atmosphere
Tatm = 259 K
Tearth = 303 K = 86° F