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**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

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**Energy Balance of a Bare Rock**

Tearth = 259 K = -14° C = 6°F

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**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

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**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)

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**Some Basic Information:**

Area of a circle = r2 Area of a sphere = 4 r2

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**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

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**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

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

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**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 = 4 x 5.67 x 10-8 K-4 T = 257 K

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**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).

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**A Planet with an Atmosphere**

Tatm = 259 K Tearth = 303 K = 86° F

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